diff --git a/-tE3T4oBgHgl3EQfSwk7/content/tmp_files/2301.04435v1.pdf.txt b/-tE3T4oBgHgl3EQfSwk7/content/tmp_files/2301.04435v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..62e994d84082f4534d3f12dafd598b65d5de2bc6 --- /dev/null +++ b/-tE3T4oBgHgl3EQfSwk7/content/tmp_files/2301.04435v1.pdf.txt @@ -0,0 +1,2547 @@ +arXiv:2301.04435v1 [hep-th] 11 Jan 2023 +Holographic entanglement entropy in T T -deformed AdS3 +Miao Hea,b, Yuan Sunc +aSchool of Physics, Southeast University, Nanjing 211189, China +bShing-Tung Yau Center, Southeast University, Nanjing 210096, China +cCenter for Theoretical Physics and College of Physics, Jilin University, +Changchun 130012, People’s Republic of China +E-mail: hemiao@seu.edu.cn, sunyuan@jlu.edu.cn +Abstract +In this work, we study the holographic entanglement entropy in AdS3 gravity +with the certain mixed boundary condition, which turns out to correspond to T ¯T- +deformed 2D CFTs. +By employing the Chern-Simons formalism and Wilson line +technique, the exact holographic entanglement entropy in T ¯T-deformed BTZ black +hole is obtained. We also get the same formula by calculating the RT surface. The +holographic entanglement entropy agrees with the perturbation result derived from +both T ¯T-deformed CFTs and cutoff AdS3. +Moreover, our result also shows that +the deformed holographic entanglement entropy behaves like the zero temperature +CFT one for the large deformation parameter. Based on this result, the two intervals +entanglement entropy and phase transition between disconnected and connected phase +are also studied. + +Contents +1 +Introduction +1 +2 +Wilson lines and entanglement entropy in AdS3 +3 +2.1 +Wilson lines in AdS3 gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Equivalence to the geodesic equation +. . . . . . . . . . . . . . . . . . . . . . +6 +2.3 +Holographic entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.3.1 +Poincar´e AdS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.3.2 +BTZ black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +2.4 +Loops and thermal entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +3 +Holographic entanglement entropy in T ¯T - deformed AdS3 +12 +3.1 +T ¯T deformed AdS3 geometry +. . . . . . . . . . . . . . . . . . . . . . . . . . +12 +3.2 +T ¯T-deformed holographic entanglement entropy . . . . . . . . . . . . . . . . +14 +3.3 +Thermal entropy +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +3.4 +Two intervals entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . +19 +4 +Geodesic line method +22 +5 +Conclusion and discussion +24 +A Conventions +25 +B Wilson line defects +26 +1 +Introduction +The AdS/CFT correspondence gives a geometric interpretation to the conformal field theory. +This correspondence allows us to study quantum gravity from the conformal field theory, +and it achieves great success in 3D quantum gravity. +It is significant to generalize the +AdS/CFT correspondence by deforming the conformal field theory and investigating its +geometric interpretation. One of the deformed theories called T ¯T deformation was proposed +and its holographic descriptions were also explored [1–4]. It is interesting to establish the +holographic dictionary under T ¯T deformation. The holographic technique also provides us +with a gravitational method to study the T ¯T deformed CFT. +The T ¯T deformation is defined through the T ¯T flow equation +∂ST ¯T +∂µ += +� +d2xOT ¯T , +OT ¯T ≡ T ijTij + T 2, +1 + +where Tij is the stress tensor of the deformed theory. This flow equation generates a family +of integrable field theory if the original theory is integrable [1, 2]. The factorizable of T ¯T +operator leads to the Burgers equation for the deformed spectrum [5], so that the spectrum of +the deformed theory can be exactly solved. The partition function of the deformed theory +can be obtained from various methods, the result turns out that the deformed partition +function satisfies a differential equation or an integral transformation of the original one [6– +8]. The deformed partition function is still modular invariant [9]. According to the T ¯T +flow equation, the Lagrangian form and Hamiltonian form were also studied [10, 11]. There +are also some evidences shown that the T ¯T deformed theory is a non-local theory [12– +16]. In this irrelevant deformation, it is difficult to study the local properties, such as the +correlation function and entanglement entropy. These observables play the important role +in the quantum field theory. By using the perturbative method, the correlation functions +and entanglement entropy have also been obtained [21–31]. Some non-perturbative results +about the correlation function and entanglement were explored in [17–20]. However, there +is still an open question to calculate the correlation function and entanglement entropy in +T ¯T deformed theory. For a pedagogical review see [32]. +According to the AdS/CFT correspondence, the deformed theory can be investigated +by using the gravitational approach. There are two points of view to understand the T ¯T +deformed CFTs from gravity. The one is the T ¯T deformed CFTs dual to the AdS3 with +a finite radial cutoff [3, 4]. In this situation, the quasi-local energy of the cutoff region +matches the spectrum of the deformed theory. The T ¯T flow equation coincides with the +Hamilton-Jacobi equation governing the radial evolution of the classical gravity action in +AdS3. +Many holographic features of the T ¯T deformed CFT have been explored based +on the cutoff perspective [33–40]. +The other holographic perspective to understand the +T ¯T deformation is the AdS3 gravity with certain mixed boundary condition [41]. +The +boundary condition was derived from the flow equation and variational principle. It turned +out that the solution of the metric flow equation related to the higher order Fefferman- +Graham expansion, which leads to the mixed boundary condition. The mixed boundary +condition coincides with the induced metric on the finite radial cutoff. The AdS3 solutions +that satisfy the mixed boundary condition were also constructed through a field-dependent +coordinate transformation [41]. The dynamic coordinate transformation approach to T ¯T +was also found in field theoretic results [42, 43]. The deformed spectrum can also be obtained +from the deformed AdS3. The mixed boundary condition allows boundary graviton degree +of freedom, which turns out to be a T ¯T deformed theory [44–47]. The mixed boundary +condition provides us with another approach to studying the T ¯T deformation including the +entanglement entropy. +In this paper, we would like to investigate the entanglement entropy in T ¯T deformed CFT +from holography. For the cutoff perspective, the holographic entanglement was obtained +by calculating the length of cutoff geodesic line, and the results match perturbative CFT +results [22, 24]. The entanglement entropy in T ¯T deformation was also studied on both +the field theory side and holographic side in recent works [48–52]. We prefer to use the +mixed boundary condition perspective to study holographic entanglement entropy. Since +the deformed geometry is still AdS3, we will work in the SL(2, R)×SL(2, R) gauged Chern- +2 + +Simons formalism of AdS3 [53]. The Chern-Simons formalism has been used to study T ¯T +deformation in the literatures [44–46, 54–56]. In the gauge theory form, the holographic +entanglement entropy is encoded in the Wilson line of Chern-Simons [57]. Generally, the +Wilson lines depend on the path and representation of the gauge group. If we choose a +appropriate representation of sl(2, R), the trace over the representation can be formulated +into the path integral of a SL(2, R) × SL(2, R) invariant auxiliary theory. The on-shell +action of the auxiliary is equivalent to the length of geodesics in AdS3. In addition, the +Wilson line is a probe in gauge theory, just like a point particle in a curved background. +The Wilson lines give a back-reaction to the bulk geometry, and the resulting geometry +turns out to be a conical defect on the branch point, which exactly generates a n-sheet +manifold [57, 58]. Therefore, the Wilson line back reaction corresponds to the replica trick +along the ending points of the Wilson line on the boundary. These results told us that the +Wilson line is related to the entanglement entropy through +SEE = − log(WR(C)), +where the ending points of the Wilson line correspond to the interval on the boundary. +The thermal entropy also turned out corresponds to the Wilson loop. We use this tech- +nique for the deformed AdS3 geometry. +The single interval holographic entanglement +entropy is calculated exactly, which can reproduce the perturbative result obtained in +other literatures [22, 24, 51]. +We also consider the two intervals entanglement entropy +in T ¯T deformation, which implies a certain phase transition. Moreover, the holographic +entanglement entropy of T ¯T-deformed AdS3 in the non-perturbative region is also studied. +The results show that the entanglement entropy behaves like a zero temperature CFT one +for the large deformation parameter. +The paper is organized as follows: In section 2, we give an overview of the gravitational +Wilson line approach to obtain the holographic entanglement entropy. +In section 3, we +introduce the deformed AdS3 under T ¯T, which is parameterized by the deformed spectrum. +The holographic entanglement entropy is obtained using the Wilson line approach. We also +consider the two intervals entanglement entropy and its phase transition. The same result +is derived by calculating the RT surface in the deformed AdS3 in section 4. We summarize +our results and discussion in section 5. The appendix contains our conventions and Wilson +line defects. +2 +Wilson lines and entanglement entropy in AdS3 +This section is a review of using the Wilson lines technique to calculate the holographic +entanglement entropy, based on [57]. +By rewriting the AdS3 gravity in Chern-Simons +form, the Wilson line in an infinite-dimensional representation of the bulk gauge group +is related to the geodesics in the bulk. According to the Ryu-Takayanagi proposal [59, 60], +the holographic entanglement entropy or RT surface can be obtained through the Wilson +line approach. +3 + +2.1 +Wilson lines in AdS3 gravity +It is well-known that 3D general relativity has no local degrees of freedom, which is purely +topological and can be formulated as a Chern-Simons theory [53]. +In the case of AdS3 +gravity, the relevant Chern-Simons gauge group is SO(2, 2) ≃ SL(2, R) × SL(2, R), so +Einstein-Hilbert action can be written as +SEH[e, ω] = ICS[A] − ICS[ ¯A], +(2.1) +where the Chern-Simons action is +ICS[A] = k +4π +� +M +Tr +� +A ∧ dA + 2 +3A ∧ A ∧ A +� +, +k = 1 +4G. +(2.2) +The gauge fields A and ¯A are valued in sl(2, R), which are the linear combination of +gravitational vielbein and spin connection +A = (ωa + ea) La, +¯A = (ωa − ea) La. +(2.3) +The La are sl(2, R) generators, see Appendix A for our conventions. Variation of the action +leads to the equations of motion +F ≡ dA + A ∧ A = 0, +¯F ≡ d ¯A + ¯A ∧ ¯A = 0, +(2.4) +which are equivalent to the first order gravitational field equation and torsion free equation. +The AdS3 metric can also be recovered from the gauge fields via +gij = 1 +2Tr +� +(Ai − ¯Ai)(Aj − ¯Aj) +� +. +(2.5) +As a consequence, the AdS3 gravity is formulated into a Chern-Simons gauge theory. +By using the Chern-Simons form, we can introduce the gravitational Wilson lines in AdS3 +gravity +WR(C) = TrR +� +P exp +� +C +A +� +, +(2.6) +where R denotes a representation of sl(2, R), and C is a curve on M with two ending points +living on the boundary of M. If the path C is closed, it gives the Wilson loop which is +invariant under the gauge transformation +A → A′ = Λ−1(d + A)Λ. +(2.7) +We can use the Wilson lines to probe the bulk geometry, instead of a massive particle. The +massive particle moving in bulk is characterized by its mass m and spin s. These parameters +would contribute to the backreaction on the bulk geometry. The trajectory of the particle +can be understood as geodesics. When we turn to use the Wilson line to probe the bulk +geometry, we have to use the infinite-dimensional representations of sl(2, R), characterized +4 + +by (h, ¯h). So that the mass m and spin s of the particle can be encoded in the representation +of sl(2, R) through the relations m = h+ ¯h and s = h−¯h. For the representation of sl(2, R) +see Appendix A. +Note that infinite-dimensional representations of symmetry algebras can be regarded as +the Hilbert spaces of quantum mechanical systems in physics. The trace over all the states +in the representation R can be formulated into a path integral of an auxiliary quantum +mechanical system. Then the Wilson line can be written as +WR(C) = +� +DU exp [−S(U; A)C] . +(2.8) +where S(U; A)C is the action of the auxiliary quantum mechanical system that lives on +the Wilson line. The action should have a global symmetry group SL(2, R) × SL(2, R), so +that the Hilbert space of the system will be precisely the representation of sl(2, R) after +quantization. +For the free theory (without gauge fields), an appropriate system is described by a +particle moving on the group manifold [61], whose action reads +S(U, P)free = +� +C +ds +� +Tr +� +PU−1dU +ds +� ++ λ(s) +� +Tr +� +P 2� +− C +�� +, +(2.9) +where P lives in the Lie algebra sl(2, R) and U lives in Lie group SL(2, R). The trace in +this action means contraction with Cartan-Killing metric. The equations of motion for the +free theory are +U−1dU +ds + 2λP = 0, +(2.10) +dP +ds = 0, +(2.11) +TrP 2 = C. +(2.12) +This action has a SL(2, R) × SL(2, R) global symmetry, namely under the following global +gauge transformation +U(s) → LU(s)R, +P(s) → R−1P(s)R, +L, R ∈ SL(2, R), +(2.13) +the action (2.9) is invariant. +In [57], it turns out that the system coupled with the external gauge fields A and ¯A +should be +S(U, P; A)C = +� +C +ds +� +Tr +� +PU−1DsU +� ++ λ(s) +� +Tr +� +P 2� +− C +�� +, +(2.14) +where the covariant derivative is defined by +DsU = d +dsU + AsU − U ¯As, +As = Aµ +dxµ +ds . +(2.15) +5 + +The equations of motion become +U−1DsU + 2λP = 0, +(2.16) +d +dsP + +� ¯As, P +� += 0, +(2.17) +Tr P 2 = C. +(2.18) +After introducing the covariant derivative, the global symmetry (2.13) is enhanced to the +local gauge symmetry. The action (2.14) is invariant under local gauge transformation +Aµ → L(x) (Aµ + ∂µ) L−1(x), +¯Aµ → R−1(x) +� ¯Aµ + ∂µ +� +R(x), +(2.19) +U(s) → L(xµ(s))U(s)R(xµ(s)), +P(s) → R(xµ(s))P(s)R(xµ(s)). +(2.20) +We have to point out that the equations of motion do not change under these gauge +transformations. This feature is useful to construct the solutions of the equations of motion +from the free theory solutions. If the gauge fields A and ¯A are pure gauge, the solutions for +the equations (2.16)-(2.18) can be obtained from the free theory solution through the gauge +transformation (2.19) and (2.20). We will treat more details in section 2.3. +2.2 +Equivalence to the geodesic equation +This Wilson line probe should be equivalent to a massive particle moving in AdS3. Then we +will show that the usual geodesic equation with respect to the metric would appear in the +Wilson line path. We denote the Wilson line path in the bulk by xµ(s). Using the classical +equation of motion (2.16)-(2.18), the action (2.14) can be reduced into a second order one +S(U; A, ¯A)C = +√ +C +� +C +ds +� +Tr (U−1DsU)2. +(2.21) +In this form, the action is essentially a gauged sigma model, whose equation of motion reads +d +ds +�� +Au − ¯A +� +µ +dxµ +ds +� ++ +� ¯Aµ, Au +ν +� dxµ +ds +dxν +ds = 0, +(2.22) +where +Au +s = U−1 d +dsU + U−1AsU. +(2.23) +For the given gauge fields (A, ¯A), the equation of motion depends on the choice of path +xµ(s). From the perspective of the equation of motion, we learn that U(s) acts like a gauge +transformation on the connection A. There is a good choice for U(s), so that the particle +does not move in the auxiliary space, i.e. U(s) = 1. In this case, the equation of motion +reduces to +d +ds +� +ea +µ +dxµ +ds +� ++ ωa +µbeb +ν +dxµ +ds +dxν +ds = 0. +(2.24) +6 + +This is precisely the geodesic equation for the curve xµ(s) on a spacetime with vielbein +and spin connection which is equivalent to the more familiar Christoffel symbols forms. +Furthermore, the on-shell the action (2.14) for U(s) = 1 becomes +S(U; A, ¯A)C = +√ +2C +� +C +ds +� +gµν(x)dxµ +ds +dxν +ds , +(2.25) +which is manifestly the proper distance along the geodesic. +We have learned that the Wilson line in AdS3 gravity can be expressed as a path integral +of an auxiliary quantum mechanical system, whose action is (2.14). The on-shell action turns +out to be the proper distance along the geodesic. Thus in the classical limit, one can find +that the value of the Wilson line +WR(xi, xf) = exp(− +√ +2CL(xi, xf)), +(2.26) +where L(xi, xf) is the length of the bulk geodesic connecting these two endpoints on the +boundary. Holographically, it was proposed by Ryu and Takayanagi that the field-theoretical +entanglement entropies correspond to the length of the bulk geodesics ending on the bound- +ary [59, 60]. In terms of the Chern-Simons description of AdS3 gravity, we can calculate the +entanglement entropy from the Wilson line +SEE = − log(WR(C)). +(2.27) +In [57], it was also shown that the Wilson line backreaction on the geometry would create a +non-trivial holonomy, which can be interpreted as the conical singularity in the bulk. The +conical defects hence reproduce the field-theoretical entanglement entropy formula. In the +later of this paper, we would like to use the Wilson line technique to compute the holographic +entanglement entropy in Chern-Simons AdS3 gravity, including the T ¯T-deformed AdS3. +2.3 +Holographic entanglement entropy +In this section, we calculate WR(C) with C ending on the AdS3 boundary at two points +denoted by xi = x(si), xf = x(sf). Classically, we just need to calculate the on-shell action +of the auxiliary system +Son-shell = +� +C +ds Tr +� +PU−1DsU +� += −2C +� sf +si +dsλ(s), +(2.28) +which depends on the solution of the equations of motion. The solutions can be constructed +from the free theory solutions, i.e. (2.10)-(2.12), through the gauge transformation (2.19) +and (2.20). First of all, we should note the solutions to free theory, denoting them by U0(s) +and P0, are +U0(s) = u0 exp(−2α(s)P0), +with +dα(s) +ds += λ(s), +(2.29) +7 + +where P0 and u0 are constant. Next, we assume the bulk gauge fields are in pure gauge +A = L(x)dL−1(x), +¯A = R−1(x)dR(x). +(2.30) +In fact, most of the AdS3 solutions are in pure gauge, such as BTZ black hole and Ban˜ados +geometry. Then one can verify the following is the classical solution of (2.16)-(2.18) +U(s) = L(x(s))U0(s)R(x(s)), +P(s) = R−1(x(s))P0R(x(s)). +(2.31) +These solutions are directly obtained from the local gauge symmetry of the equations of +motion. As argued in [57], the boundary conditions for U(s) on the boundary ending points +can be chosen as +U(si) =L(x(si))u0 exp(−2α(si)P0)R(x(si)) = 1, +(2.32) +U(sf) =L(x(sf))u0 exp(−2α(sf)P0)R(x(sf)) = 1. +(2.33) +We then have to eliminate the initial value P0 and u0. Solving u0 from (2.32) and substituting +into (2.33), one can find +exp(−2∆αP0) =R(x(si))L(x(si))L−1(x(sf))R−1(x(sf)). +(2.34) +Taking the trace on both sides, we arrive at +cosh +� +−2∆α +√ +2C +� += 1 +2Tr +� +R(x(si))L(x(si))L−1(x(sf))R−1(x(sf)) +� +, +(2.35) +where we have used +Tr (exp(−2∆αP0)) = 2 cosh +� +−2∆α +√ +2C +� +. +(2.36) +Finally, according to (2.27), we obtain the holographic entanglement entropy formula +SEE = +√ +2C cosh−1 +�1 +2Tr +� +R(x(si))L(x(si))L−1(x(sf))R−1(x(sf)) +�� +. +(2.37) +We then use this formalism to check the holographic entanglement entropy in Poincare AdS3 +and BTZ black hole. +2.3.1 +Poincar´e AdS3 +For the case of Poincare AdS3, the line element reads +ds2 = dr2 +r2 + r2(dθ2 − dt2). +(2.38) +In terms of the Chern-Simons gauge connection, this geometry is described by +A =dr +r L0 + rL1(dθ + dt), +(2.39) +¯A = − dr +r L0 − rL−1(dθ − dt). +(2.40) +8 + +The gauge connections can be written in pure gauge form +A =LdL−1, +L = exp(− ln rL0) exp(−(θ + t)L1), +(2.41) +¯A =R−1dR, +R = exp((θ − t)L−1) exp(− ln rL0). +(2.42) +In order to calculate the entanglement entropy, we consider a time slice (t = 0) of this +geometry and impose the following boundary conditions for the ending points of the Wilson +line +r(si) = r(sf) = r0, +(2.43) +∆θ = θ(sf) − θ(si) = l, +(2.44) +which means we work on a constant radial boundary and the length of the interval is l. +Plugging (2.41) and (2.42) into (2.37), one obtain +SEE = +√ +2C cosh−1 +� +1 + r2 +0l2 +2 +� +. +(2.45) +Then taking the limit r0 ≫ 1, so that the result corresponds to the theory living on the +conformal boundary, we arrive at 1 +SEE = c +3 log +�l +ǫ +� +. +(2.46) +where the UV cutoff of the boundary field theory corresponds to the radial cutoff in the +bulk, and the central charge relarelatesthe expectation value of Casimir +ǫ = 1 +r0 +, +√ +2C = c +6. +(2.47) +The relation between the expectation value of Casimir and central charge can be derived by +calculating the Wilson line defect, for the details see Appendix B. This result is exactly the +entanglement entropy of CFT2. The same answer can also be obtained by solving the bulk +geodesic equation. However, in terms of the Wilson line form, we do not require the solution +of any differential equations and follow from purely algebraic operations. This technique +can be used for more complicated AdS3 geometry. +2.3.2 +BTZ black hole +For the BTZ black hole, the metric in Fefferman–Graham gauge is +ds2 = dr2 +r2 + r2 +� +dzd¯z + 1 +r2L0dz2 + 1 +r2 ¯L0d¯z2 + 1 +r4L0 ¯L0dzd¯z +� +, +(2.48) +1We have used the relation +cosh−1(x) ∼ log(2x) +for +x ≫ 1. +9 + +where L0 and ¯L0 are constants related to the mass and angular momentum of the black hole +L0 = M − J +2 +, +¯L0 = M + J +2 +. +(2.49) +The corresponding Chern-Simons gauge connections read +A =dr +r L0 + +� +rL1 − 1 +rL0L−1 +� +dz, +(2.50) +¯A = − dr +r L0 + +�1 +r +¯L0L1 − rL−1 +� +d¯z. +(2.51) +In this case, one can obtain +L (r, z, ¯z) = exp (− ln rL0) exp (−zL1 + L0zL−1) , +(2.52) +R (r, z, ¯z) = exp +� ¯L0¯zL1 − ¯zL−1 +� +exp (− ln rL0) . +(2.53) +In addition, such solutions can be parametrized as +A = b−1(d + a)b, +¯A = b(d + ¯a)b−1, +b = eln rL0, +(2.54) +Then a, ¯a are also flat connections, but do not depend on the radial coordinate +a = (L1 − L0L−1) dz, +(2.55) +¯a = +� ¯L0L1 − L−1 +� +d¯z. +(2.56) +Following the same steps in pure AdS3 and the boundary conditions for the ending points +of the Wilson line, we can get +Tr +� +R(r0, θ(si), 0)L(r0, θ(si), 0)L−1(r0, θ(sf), 0)R−1(r0, θ(sf), 0) +� += − 2 cosh +� +l +� +L0 +� +cosh +� +l +� ¯L0 +� ++ +� +L0 ¯L0 + r4 +0 +� +sinh +� +l√L0 +� +sinh +� +l +� ¯L0 +� +r2 +0 +√L0 +� ¯L0 +∼ +r2 +0 sinh +� +l√L0 +� +sinh +� +l +� ¯L0 +� +√L0 +� ¯L0 +, +(r0 ≫ 1) +(2.57) +This result leads to the entanglement entropy +SEE =c +6 log + + +r2 +0 sinh +� +l√L0 +� +sinh +� +l +� ¯L0 +� +√L0 +� ¯L0 + + . +(2.58) +If we consider the spinless black hole, i.e. L0 = ¯L0, the entanglement entropy reduces to +SEE =c +3 log +�β0 +πǫ sinh +�πl +β0 +�� +, +β0 = +π +√L0 +, +(2.59) +where β0 is the inverse temperature of the BTZ black hole [62–64]. This result coincides +with the entanglement entropy of a CFT in thermal state. +10 + +2.4 +Loops and thermal entropy +One can also consider the Wilson loops in AdS3. In this case, WR(C) turns out to be the +proper distance around the horizon, which corresponds to the black hole thermal entropy. +We will then check it in the BTZ black hole. Consider the Wilson loop along the S1 cycle +θ ∼ θ + 2π. In contrast to the open interval case, the closed path should be smooth and +hence impose the periodic boundary condition +U (sf) = U(si), +P (sf) = P(si). +(2.60) +According to (2.31), the boundary condition for P(s) implies +� +P0, R (si) R−1(sf) +� += 0, +(2.61) +Hence, the boundary condition for U(s) implies +exp (−2∆αP0) = u−1 +0 +� +L−1 (sf) L(si) +� +u0 +� +R(si)R−1 (sf) +� +. +(2.62) +In addition, note the relations +L−1 (sf) L(si) = exp +�� +dθaθ +� +, +(2.63) +R(si)R−1 (sf) = exp +� +− +� +dθ¯aθ +� +, +(2.64) +which are the holonomies of the connection, we can rewrite (2.62) as +exp (−2∆αP0) = u−1 +0 exp (2πaθ) u0 exp (−2π¯aθ) . +(2.65) +Here we just consider the case of BTZ black hole, so that one can perform the simple integral +over θ. +From (2.61), we learn that P0 and ¯aθ can be diagonalized simultaneously. If the initial +value of u0 is fixed, we can always choose a matrix V , such that aθ can also be diagonalized +by u0V +exp (−2∆αλP) = (u0V )−1 exp (2πaθ) u0V exp +� +−2π¯λθ +� += exp (2πλθ) exp +� +−2π¯λθ +� +, +(2.66) +where λP, λθ and ¯λθ are diagonalized matrix of P0, aθ and ¯aθ. Contracting (2.66) with L0, +we obtain the on-shell action for the loop +Sth = 2π +√ +2CTr +� +(λθ − ¯λθ)L0 +� +. +(2.67) +For the BTZ black hole, the diagonalized gauge connections are +λθ = 2 +� +L0L0, +¯λθ = −2 +� ¯L0L0. +(2.68) +Finally, the Wilson loop gives precisely the entropy of the BTZ black hole +Sth = 2π +�c +6L0 + 2π +�c +6 +¯L0. +(2.69) +11 + +3 +Holographic entanglement entropy in T ¯T - deformed +AdS3 +We turn to investigate the entanglement entropy of T ¯T deformed CFTs from the gravity +side. In [41], it is proposed that the holographic interpretation of T ¯T deformed CFTs is +still AdS3 gravity but with the mixed boundary condition. The AdS3 solutions associated +with the mixed boundary condition can be obtained from the Ba˜nados geometry through +a coordinate transformation. As the deformed geometry is still AdS3, we prefer to work in +Chern-Simons formulation. In this section, we introduce the T ¯T deformed AdS3 geometry. +The holographic entanglement entropy of T ¯T deformed CFTs can be obtained using the +Wilson line technique in the deformed AdS3. +3.1 +T ¯T deformed AdS3 geometry +We start from the general AdS3 solution with a flat conformal boundary, which is called the +Ba˜nados geometry [65]. In Fefferman-Graham gauge, the line element reads +ds2 = dr2 +r2 + r2 +� +dzd¯z + 1 +r2L(z)dz2 + 1 +r2 ¯L(¯z)d¯z2 + 1 +r4L(z) ¯L(¯z)dzd¯z +� +, +(3.1) +The parameters L(z) and ¯L(¯z) are arbitrary holomorphic and antiholomorphic functions, +which are related to the energy and angular momentum +L = E + J +2 +, +¯L = E − J +2 +. +(3.2) +The corresponding Chern-Simons gauge fields are +A =dr +r L0 + +� +rL1 − 1 +rL(z)L−1 +� +dz, +(3.3) +¯A = − dr +r L0 − +�1 +r +¯L(¯z)L1 − rL−1 +� +d¯z. +(3.4) +In this sense, the deformed Ba˜nados geometry can be constructed through a field-dependent +coordinate transformation [41], which reads +dz = +1 +1 − µ2Lµ ¯Lµ +(dw − µ ¯Lµd ¯w), +d¯z = +1 +1 − µ2Lµ ¯Lµ +(d ¯w − µLµdw), +(3.5) +where µ is the deformation parameter. +One should note that the parameters L and ¯L +in (3.1) would turn into Lµ and ¯Lµ under the coordinate transformation. Generally, the +parameters Lµ and ¯Lµ are different from the undeformed ones L and ¯L. The relations +between deformed parameters Lµ, ¯Lµ and undeformed parameters L, ¯L can be fixed by two +ways. The first one is that the deformation smoothly changes the spectrum but does not +change the local degeneracy of states. Therefore, in the bulk, this implies that the T ¯T +12 + +deformation does not change the horizon area of a black hole. +The second one is that +the deformed geometry can be transformed into the undeformed one without changing the +periodicity of the spatial coordinate. Indeed, the transformation is different from the inverse +of (3.5). These considerations lead to +Lµ(1 − µ ¯Lµ)2 +(1 − µ2Lµ ¯Lµ)2 = L, +¯Lµ(1 − µLµ)2 +(1 − µ2Lµ ¯Lµ)2 = ¯L. +(3.6) +One can turn to [41] for more details about fixing these relations. +By using the coordinate transformation (3.5), we obtain the deformed Chern-Simons +gauge fields +A =1 +rL0dr + +1 +1 − µ2Lµ ¯L µ +� +rL1 − 1 +rLµL−1 +� +(dw − µ ¯Lµd ¯w), +(3.7) +¯A = − 1 +rL0dr − +1 +1 − µ2Lµ ¯Lµ +�1 +r +¯LµL1 − rL−1 +� +(d ¯w − µLµdw). +(3.8) +Note that L(z) and ¯L(¯z) correspond to the charges of the solution in the Ba˜nados geometry. +However, in the deformed geometry, the parameters L(z) and ¯L(¯z) do not correspond to +the charges. Indeed, the deformed energy and angular momentum can be obtained from +both field theory and gravity side +Eµ = 1 +µ +� +1 − +� +1 − 2µ(L + ¯L) + µ2(L − ¯L)2 +� +, +Jµ = J. +(3.9) +Analogous to (3.2), we introduce the new parameters +Q = Eµ + Jµ +2 += 1 +2µ +� +1 + µ(L − ¯L) − +� +1 − 2µ(L + ¯L) + µ2(L − ¯L)2 +� +, +(3.10) +¯Q = Eµ − Jµ +2 += 1 +2µ +� +1 − µ(L − ¯L) − +� +1 − 2µ(L + ¯L) + µ2(L − ¯L)2 +� +. +(3.11) +We can regard Q and ¯Q as the generalized parameters of L and ¯L in the deformed geometry, +and Q and ¯Q reduce to L and ¯L in the limit µ → 0. We find it is more convenient to +parametrize the deformed gauge fields or metric in terms of these two independent charges. +In terms of these charges, the Chern-Simons gauge connection are formulated as +A =dr +r L0 + +1 − µQ +1 − µ(Q + ¯Q) +� +r(1 − µ ¯Q)L1 − 1 +rQL−1 +� +dw +− +µ ¯Q +1 − µ(Q + ¯Q) +� +r(1 − µ ¯Q)L1 − 1 +rQL−1 +� +d ¯w, +(3.12) +¯A = − dr +r L0 + +µQ +1 − µ(Q + ¯Q) +�1 +r +¯QL1 − r(1 − µQ)L−1 +� +dw +− +1 − µ ¯Q +1 − µ(Q + ¯Q) +�1 +r +¯QL1 − r(1 − µQ)L−1 +� +d ¯w, +(3.13) +13 + +In the following, we prefer to use the coordinates θ = (w + ¯w)/2, t = (w − ¯w)/2, where +t represents the time direction while θ denotes the spatial coordinate at the boundary with +the identification θ ∼ θ + 2π. We then have +Ar = 1 +rL0, +Aθ =r(1 − µ ¯Q)L1 − 1 +rQL−1, +At = K +� +r(1 − µ ¯Q)L1 − 1 +rQL−1 +� +, +(3.14) +¯Ar = −1 +rL0, +¯Aθ =1 +r +¯QL1 − r(1 − µQ)L−1, +¯At = ¯K +�1 +r +¯QL1 − r(1 − µQ)L−1 +� +, +(3.15) +where +K =1 + µ( ¯Q − Q) +1 − µ(Q + ¯Q), +¯K = −1 − µ( ¯Q − Q) +1 − µ(Q + ¯Q). +(3.16) +The radial gauge (2.54) still holds for the deformed gauge fields, so that the induced gauge +connections are +aθ =(1 − µ ¯Q)L1 − QL−1, +at = K +� +(1 − µ ¯Q)L1 − QL−1 +� +, +(3.17) +¯aθ = ¯QL1 − (1 − µQ)L−1, +¯at = ¯K +� +¯QL1 − (1 − µQ)L−1 +� +. +(3.18) +In addition, we can also write down the deformed +ds2 =dr2 +r2 + +1 +r2(1 − µ(Q + ¯Q))2× +� +Q(1 − µQ)(1 − µr2)dw + +� +µQ ¯Q + r2(1 − µQ)(1 − µ ¯Q) +� +d ¯w +� +× +� +¯Q(1 − µ ¯Q)(1 − µr2)d ¯w + +� +µQ ¯Q + r2(1 − µQ)(1 − µ ¯Q) +� +dw +� +. +(3.19) +We will use the deformed geometry to calculate the holographic entanglement entropy in +the T ¯T deformed CFTs. For simplicity, we just consider the constant charges Q and ¯Q, +namely we work in T ¯T deformed BTZ black hole. +3.2 +T ¯T -deformed holographic entanglement entropy +For the T ¯T-deformed AdS3, the metric still satisfies the Einstein equation or flat connection +condition in the Chern-Simons theory although it takes a complicated form. In the Poincar´e +AdS3, the Wilson line would produce a back-reaction in the bulk geometry. +The back- +reaction would then lead to a conical defect on the ending points of Wilson line, which +generates the n-sheet manifold on the boundary. +According to the replica trick on the +boundary field theory, the Wilson line exactly leads to the entanglement entropy. One can +turn to Appendix B for details. We can always transform the T ¯T-deformed AdS3 solution +into the Poincar´e form [66, 67]. However, the temperature (the period of Euclidean time) in +deformed AdS3 is different from the original one. The crucial point is that we have to identify +the deformed temperature and length of interval on the boundary under T ¯T deformation. +14 + +We will treat these considerations in more details and obtain the T ¯T deformed holographic +entanglement entropy in this section. +Now, we can use the Wilson line technique to calculate the holographic entanglement +entropy in T ¯T-deformed AdS3. First of all, we can give a glance at the Poincar´e AdS3, +which turns out correspond to the zero temperature entanglement entropy. In Fefferman- +Graham gauge, the Poincar´e AdS3 can be written as Ba˜nados geometry (3.1) with L and +¯L vanish. +In this case, the bulk geometry is the same as the undeformed one, so the +zero temperature entanglement entropy remains unchanged. This result coincides with the +perturbative calculation in field theory and cutoff perspective in the bulk [22, 24]. +We then consider the deformed BTZ black hole, in which the charges Q and ¯Q are +constants. For the deformed geometry, on a time slice, we obtain +L (r, θ, t = 0) = exp (− ln rL0) exp +� +− +� x +x0 +dxiai +� += exp (− ln rL0) exp +� +−(1 − µ ¯Q)θL1 + QθL−1 +� +, +(3.20) +R (r, θ, t = 0) = exp +�� x +x0 +dxi¯ai +� +exp (− ln rL0) += exp +� ¯QθL1 − (1 − µQ)θL−1 +� +exp (− ln rL0) . +(3.21) +As the deformed geometries are still AdS3 solution, we use the boundary condition for U(s) +U(si) = 1, +U(sf) = 1, +(3.22) +as well as the same boundary conditions for the ending points of the Wilson line +r(si) = r(sf) = r0, +(3.23) +∆θ = θ(sf) − θ(si) = l. +(3.24) +We should point out that the boundary condition for U is actually the unique choice because +of the Lorentz invariance at the boundary [57, 68]. As the T ¯T deformation does not break +Lorentz invariance, we can use the same boundary condition (3.22) for U. It seems that l +is just the length of the interval in the deformed boundary. But it equals to the deformed +length of interval, because the length is defined in the (w, ¯w) coordinates. +Using the gauge transformation (2.31), one can get the solution U(s) for the Wilson line +coupled to the deformed gauge fields. The boundary condition for U(s) and ending points +15 + +boundary condition for the Wilson line imply +Tr +� +(R(si)L(si)) (R (sf) L (sf))−1 � +=2 cosh +� +l +� +¯Q (1 − µQ) +� +cosh +� +l +� +Q(1 − µ ¯Q) +� ++ +r2 +0 +� ¯Q(1 − µQ) +� +Q(1 − µ ¯Q) sinh +� +l +� ¯Q(1 − µQ) +� +sinh +� +l +� +Q(1 − µ ¯Q) +� +Q ¯Q ++ +Q ¯Q sinh +� +l +� ¯Q(1 − µQ) +� +sinh +� +l +� +Q(1 − µ ¯Q) +� +r2 +0 +� ¯Q(1 − µQ) +� +Q(1 − µ ¯Q) +∼ +r2 +0 +� ¯Q(1 − µQ) +� +Q(1 − µ ¯Q) sinh +� +l +� ¯Q(1 − µQ) +� +sinh +� +l +� +Q(1 − µ ¯Q) +� +Q ¯Q +. +(3.25) +In the last step, we consider the r0 ≫ 1 limit. It is straightforward to get the holographic +entanglement entropy for T ¯T deformation +SEE = +√ +2C cosh−1 + + +r2 +0 +� ¯Q(1 − µQ) +� +Q(1 − µ ¯Q) sinh +� +l +� ¯Q(1 − µQ) +� +sinh +� +l +� +Q(1 − µ ¯Q) +� +2Q ¯Q + + +∼c +6 log + + +r2 +0 +� ¯Q(1 − µQ) +� +Q(1 − µ ¯Q) sinh +� +l +� ¯Q(1 − µQ) +� +sinh +� +l +� +Q(1 − µ ¯Q) +� +Q ¯Q + + . +(3.26) +If the original geometry is non-rotating BTZ black hole, namely Q = ¯Q, the deformed +entanglement entropy becomes +SEE =c +3 log + + +r0 +� +Q(1 − µQ) sinh +� +l +� +Q(1 − µQ) +� +Q + + . +(3.27) +For the deformed BTZ black hole, the temperature can be obtained by analysing the period +of Euclidean time, which is discussed in the next section (4.10). We quote the result here +β = 1 +T = π(1 − 2µQ) +� +Q(1 − µQ) +. +(3.28) +This temperature can also be derived using the first law of thermodynamics, and we will +show it in section 3.3. For the limit µ → 0, the temperature reduce to the BTZ black hole +temperature. The length of interval l is already the deformed one, which can be seen from +the coordinate transformation (3.5) on a time slice. In terms of the deformed temperature, +we can express the entanglement entropy as +SEE = c +3 log +�� +β2 + 4µπ2 + β +2πǫ +sinh +� +πl +� +β2 + 4µπ2 +�� +. +(3.29) +16 + +This is actually the T ¯T deformed entanglement entropy obtained from the holographic ap- +proach. For µ = 0, the deformed entanglement entropy reduce to the familiar entanglement +entropy of CFT at finite temperature. For the small µ, we can obtain the perturbative +result +SEE = c +3 log +� β +πǫ sinh +�πl +β +�� ++ µc +3 +�π2 +β2 − 2π3l +β3 coth +�πl +β +�� ++ O(µ2). +(3.30) +In the “low temperature” limit β ≫ l, up to the first order, the entanglement entropy +becomes +SEE-low =c +3 log +� β +πǫ sinh +�πl +β +�� ++ µc +3 +�π2 +β2 +� ++ O(µ2). +(3.31) +In the “high temperature” limit β ≪ l, the first order corrected entanglement entropy is +SEE-high =c +3 log +� β +πǫ sinh +�πl +β +�� +− 2µc +3 +π3l +β3 coth +�πl +β +� ++ O(µ2). +(3.32) +The “high temperature” result coincides with the result obtained from both boundary field +side and AdS3 with cutoff perspective [22, 24]2. We apply the Wlison line approach to the +T ¯T-deformed AdS3 and obtain the holographic entanglement entropy formula, which agree +with the perturbation results. However, the “low temperature” result is different from the +cutoff AdS3 perspective. +We are more interested in the non-perturbative result. +In order to make sure the +entanglement entropy is real, we have +− β2 +4π2 < µ, +(3.34) +which means the holographic description maybe lose when µ out of this region. For µ > 0 +the entanglement entropy is always real. In the following discussion, we just consider the +µ > 0 case, which also corresponds to the cutoff perspective. For a fixed temperature, we +can consider the entanglement entropy for large deformation parameter +SEE = c +3 log +� l +2ǫ +� ++ βc +6π +1 +√µ + +�cl2 +72 − β2c +24π2 +� 1 +µ + O +� 1 +µ +� +. +(3.35) +The leading order coincides with the entanglement entropy of the zero temperature CFT +with the length of interval l/2. This result implies the T ¯T deformation behaves like the +2Note that our convention is different from Ref. [22]. In [22], the deformation parameter is related to +the radial cutoff r2 +c = +6 +µπc, while we have r2 +c = 1 +µ in this paper. Therefore, if one replaces µ by µπc +6 , the +equation (3.32) becomes +SEE-high = c +3 log +� β +πǫ sinh +�πl +β +�� +− µπ4c2l +9β3 +coth +�πl +β +� +. +(3.33) +which is exactly the result in [22]. +17 + +free theory at the large µ limit. The similar feature was also found in [69, 70], in which +the authors shown that at the level of the equations of motion the left- and right-chiral +sectors of T ¯T deformed free theories are decoupled when the deformation parameter is +sent to infinity. Moreover, the Casini-Huerta entropic c-function [71] for the T ¯T deformed +entanglement entropy is +C(l, µ) = ldSEE +dl += +πcl +3 +� +β2 + 4π2µ +coth +� +πl +� +β2 + 4π2µ +� +, +(3.36) +which is always positive, and does not depend on the ultraviolet regulator. We also find +that +∂C(l, µ) +∂l += πc +3 + + + + +coth +� +πl +√ +β2+4π2µ +� +� +β2 + 4π2µ +− +πlcsch2 +� +πl +√ +β2+4π2µ +� +β2 + 4π2µ + + + + ≥ 0, +(3.37) +which implies the entropic c-function is non–decreasing along the renormalization group +flow towards the ultraviolet. The similar result was also found in single trace T ¯T deforma- +tion [72]. +3.3 +Thermal entropy +The thermal entropy of the deformed BTZ black hole can also be calculated from the Wilson +loop. As discussed in section 2.4, the thermal entropy can be obtained by diagonalizing the +induced gauge connections aθ and ¯aθ in (3.17) and (3.18). For the deformed BTZ black +hole, the diagonalized gauge connections read +λθ = 2 +� +Q(1 − µ ¯Q)L0 = 2 +√ +LL0, +(3.38) +¯λθ = −2 +� +¯Q(1 − µQ)L0 = −2 +� +¯LL0. +(3.39) +Finally, according to (2.67), we obtain the thermal entropy +S = 2π +�c +6L + 2π +�c +6 +¯L, +(3.40) +which is the same as the BTZ black hole entropy. This result means the black hole entropy +does not change under the T ¯T deformation. On the field theory side, the degeneracy of +states do not change under the T ¯T flow. +For the deformed theory, the thermal entropy should be expressed in terms of the +deformed energy. In case of Q = ¯Q, the entropy can be written as +S = 4π +�c +6Q(1 − µQ) = 2π +�c +6Eµ(2 − µEµ), +(3.41) +18 + +which agrees with the result in [3]. The thermal entropy can help us to define the tempera- +ture in the T ¯T-deformed theory. In fact, according to the first law of thermodynamics, the +temperature can be determined by +T = ∂Eµ +∂S = +� +6 +c +� +Q(1 − µQ) +π(1 − 2µQ) ∼ +� +Q(1 − µQ) +π(1 − 2µQ) , +(3.42) +where we have used the convention k = c/6 = 1 in the definiton of temperature. This is +actually the temperature we have used in (3.28). +3.4 +Two intervals entanglement entropy +We proceed to consider the entanglement entropy of the system consists of two disjoint +intervals. For the single interval case, we have shown that the entanglement entropy is the +Wilson line or length of geodesic in AdS3 with ending points on the spatial infinity boundary +for both Brown-Henneaux boundary condition and mixed boundary condition. According +to Ryu-Takayanagi’s proposal [59, 60], we have two choices for how to draw the geodesics +that end on the ending points of two intervals, which are shown in Figure 1. For each choice, +the two intervals entanglement entropy decouples into a sum of single interval cases. The +Figure 1: +The two minimal surfaces for the two intervals boundary region. We consider the +two intervals have the same length l separated by x. The left is the disconnected case, and +the right is the connected case. +two intervals holographic entanglement entropy should be the minimal one of them +SEE-2 = min{Sdis, Scon}. +(3.43) +This implies that there are two phases of the entanglement entropy. It turns out that there +actually exist a phase transition between the connected and disconnected phase [73]. +We first brief review the zero temperature entanglement entropy of two disjoint intervals. +We assume the two intervals have the same length l separated by x, described in Figure 1. +Then the difference between two phases is +∆S = Sdis − Scon = c +3 log +� +l2 +x(2l + x) +� +. +(3.44) +19 + +COF6CI60One can find the phase transition critical point is determined by the cross-ratio +η = +l2 +(l + x)2 = 1 +2 +or +x +l = +√ +2 − 1. +(3.45) +For the finite temperature case, the similar phase transition was shown in [74, 75]. However, +there is no quantity like cross-ratio to illustrate the critical point. +Now we would like to investigate the similar feature for the T ¯T deformed entanglement +entropy. For the different choices of Wilson lines or RT surfaces, we have +Sdis =c +3 log + + +π2µ + 1 +2β +�� +β2 + 4π2µ + β +� +π2ǫ2 +sinh2 +� +πl +� +β2 + 4µπ2 +� + , +(3.46) +Scon =c +3 log + + +π2µ + 1 +2β +�� +β2 + 4π2µ + β +� +π2ǫ2 +sinh +� +πx +� +β2 + 4µπ2 +� +sinh +� +π(2l + x) +� +β2 + 4µπ2 +� + . +(3.47) +The two intervals entanglement entropy is the minimal one of them. In order to determine +which is the minimal one and under what conditions the phase transition happens, we +consider the difference between two RT surfaces +∆S =Sdis − Scon = c +3 log + + + + +sinh2 +� +πl +√ +β2+4µπ2 +� +sinh +� +πx +√ +β2+4µπ2 +� +sinh +� +π(2l+x) +√ +β2+4µπ2 +� + + + + . +(3.48) +This quantity is also related to the mutual information between two disjoint subsystems. +From (3.48), we learn that ∆S behaves like the undeformed one but with different tem- +perature. We first consider the low temperature and high temperature limit. For the low +temperature limit β ≫ 1, we have +∆S = c +3 log +� +l2 +x(2l + x) +� ++ O +� +1/β2� +. +(3.49) +The leading order is exactly the zero temperature case. +The phase transition occur at +x/l = +√ +2−1 and does not depend on the deformation parameter. For the high temperature +limit β ≪ 1, we have +∆S = c +3 log + + +cosh +� +l +√µ +� +− 1 +cosh +� +l+x +√µ +� +− cosh +� +l +√µ +� + + + O +� +β2� +. +(3.50) +In this case, the critical point depends on the deformation parameter. +We find it is convenient to introduce the following parameters +˜l = x +l , +˜x = x +β , +˜µ = µ +β2. +(3.51) +20 + +In terms of the new parameters, the ∆S reduces to +∆S = c +3 log + + + + +sinh2 +� +π˜x +˜l√ +1+4˜µπ2 +� +sinh +� +π˜x +√ +1+4˜µπ2 +� +sinh +� +π(2+˜l)˜x +˜l√ +1+4˜µπ2 +� + + + + , +(3.52) +in which the temperature is implicit. We plot the critical lines ∆S = 0 in (˜l, ˜x) plane for +different deformation parameters in Figure 2. Then we consider some special limit about +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.00 +0.05 +0.10 +0.15 +0.20 +l +∼ +x +∼ +Critical lines: ΔS +=0 +μ∼=-0.02 +μ∼=-0.01 +μ∼=0 +μ∼=0.01 +μ∼=0.02 +μ∼=0.03 +μ∼=0.4 +Figure 2: +Plot the critical lines ∆S = 0 in ˜l − ˜x plane for different deformation parameters. +The critical lines separate the connected phase (left side) and disconnected phase (right +side). +The green line corresponds to the undeformed case. +The dashed line denotes the +zero temperature critical line ˜l = +√ +2 − 1. The critical lines tend to the zero temperature case +with the increase of deformation parameter. +the critical lines. For ˜x ≪ 1, we have +∆S = c +3 +� +log +� +1 +˜l2 + 2˜l +� +− π2(˜l + 1)2˜x2 +3˜l2 (1 + 4˜µπ2) +� ++ O +� +˜x3� +. +(3.53) +The leading order is just the zero temperature case and also does not depend on the +deformation parameter. This result can be seen from Figure 2 that the critical lines coincide +with the zero temperature one for small ˜x. +It is interesting to investigate the µ dependence of phase transition. For the small ˜µ, there +is actually exist a phase transition, which has been discussed in [24] using the perturbative +method. We can also see from Figure 2 the critical line is around the undeformed case for +21 + +both ˜µ < 0 and ˜µ > 0. For the ˜µ ≫ 1 region, we have +∆S = c +3 log +� +1 +˜l2 + 2˜l +� +− c(˜l + 1)2˜x2 +36˜l2˜µ ++ O(1/˜µ2). +(3.54) +The leading order is the just the zero temperature case. One can also see from Figure 2 +that the critical lines would become the zero temperature one as the increase of deformation +parameters. This result implies the T ¯T deformed theory becomes a decoupled free theory +for large µ limit [69, 70]. +These results show that there still exist the phase transition for two intervals entangle- +ment entropy under T ¯T deformation. The transition point is depends on the deformation +parameter. The T ¯T deformation does not introduce new phases. For large deformation +parameter, the the critical point is the same as zero temperature CFT case, it would be +interesting to study this feature from the field theoretic results. +4 +Geodesic line method +In this section we re-compute the holographic entanglement entropy in BTZ background +with mix boundary condition using RT formula, i.e., identifying the holographic entan- +glement entropy as the geodesic distance. The results turn out to be consistent with the +computation via Wilson line method. +The metric of BTZ black hole with mass M and angular momentum J takes the form +(2.48). +3 For simplicity we consider the case where the black hole being static J = 0. It +follows from (3.6) that the deformed parameters Lµ, ¯Lµ are constant and satisfy +Lµ = ¯Lµ = 1 − µM ± √1 − 2µM +Mµ2 +, +(4.1) +where only the solution with “-” is well defined in µ → 0 limit. We start from the following +metric +ds2 =dr2 +r2 + r2� +dzd¯z + 1 +r2(Lµdz2 + ¯Lµd¯z2) + 1 +r4Lµ ¯Lµdzd¯z +� +, +(4.2) +in which we have replaced the L, ¯L by Lµ, ¯Lµ in the BTZ black hole solution, so that we +can obtain the deformed BTZ only by using the coordinate transformation. Let z = x + iy, +and define +r = +� +Lµeρ, +x = +¯x +� +4Lµ +, +y = +¯y +� +4Lµ +, +(4.3) +then the metric becomes the global AdS3 +ds2 =dρ2 + cosh2 ρd¯x2 + sinh2 ρd¯y2, +(4.4) +3We follow the convention in [41], and set 4πG = 1, l = 1 and R = 2π (periodicity of spatial dimension) +in their paper. We also use r which is related with the radial coordinate ρ in [41] as r2 = 1/ρ. The cutoff +in [41] locates at ρ = ρc = µ, then in r-coordinate, r0 = rc = 1/√µ. +22 + +where ¯y is treated as the Euclidean time and ¯x the spatial coordinate. The requirement +of no conical singularity in ρ − ¯y plane implies the identification ¯y ∼ ¯y + 2π, where the +periodicity is related with the temperature for BTZ black hole. It is convenient to work in +embedding coordinate +Y 0 = cosh ρ cosh ¯x, +Y 3 = cosh ρ sinh ¯x, +Y 1 = sinh ρ sin ¯y, +Y 2 = sinh ρ cos ¯y. +(4.5) +In this coordinate system the BTZ black hole is a hypersurface −(Y 0)2 + (Y 3)2 + (Y 1)2 + +(Y 2)2 = −1 in the background ds2 = −d(Y 0)2 + d(Y 1)2 + d(Y 2)2 + d(Y 3)2. The geodesic +distant d between two points Y a +1 , Y b +2 is simply computed by +cosh d = −Y1 · Y2 = Y 0 +1 Y 0 +2 − Y 1 +1 Y 1 +2 − Y 2 +1 Y 2 +2 − Y 3 +1 Y 3 +2 . +(4.6) +The deformed metric corresponding to T ¯T deformation can be obtained by transforma- +tion of +dz = +1 +1 − µ2Lµ ¯Lµ +(dw − µ ¯Lµd ¯w), +d¯z = +1 +1 − µ2Lµ ¯Lµ +(d ¯w − µLµdw). +(4.7) +In the present case, (4.7) can be solved straightforwardly as +z = +1 +1 − µ2Lµ ¯Lµ +(w − µ ¯Lµ ¯w), +¯z = +1 +1 − µ2Lµ ¯Lµ +( ¯w − µLµw). +(4.8) +And its inverse +w = z + µ ¯Lµ¯z, +¯w = µLµz + ¯z, +(4.9) +where w = θ + it, ¯w = θ − it. From the periodicity of ¯y discussed above, we can work out +the periodic of t, which is +t ∼ t + 2π(1 − µLµ) +� +4Lµ += t + β, +β = π(1 − 2µQ) +� +Q(1 − µQ) +, +(4.10) +where the β is the inverse temperature of deformed black hole, as well as the inverse +temperature of the T ¯T deformed CFT. +To compute the HEE of a single interval, we consider two endding points on the boundary +locate at (r1, t1, θ1) = ( +� +Lµeρ0, 0, 0) and (r2, t2, θ2) = ( +� +Lµeρ0, 0, l) respectively. +Then +w1 = ¯w1 = 0, w2 = ¯w2 = l +z1 = ¯z1 = 0, +z2 = ¯z2 = +l +1 + µLµ +. +(4.11) +In terms of embedding coordinates +Y 0 +1 = cosh ρ0, +Y 3 +1 = 0, +Y 1 +1 = +0, +Y 2 +1 = sinh ρ0, +(4.12) +and +Y 0 +2 = cosh ρ0 cosh +� +4Lµz2, +Y 3 +2 = cosh ρ sinh +� +4Lµz2, +Y 1 +2 = 0, +Y 2 +2 = sinh ρ0. +(4.13) +23 + +Finally using (4.6), the geodesic distance between the points is +cosh d = cosh2 ρ0 cosh +� +4Lµz2 − sinh2 ρ0 += +Q +2r2 +0(1 − µQ) sinh2 l +� +Q(1 − µQ) + cosh2 l +� +Q(1 − µQ) ++ r2 +0(1 − µQ) +2Q +sinh2 l +� +Q(1 − µQ), +(4.14) +where we made the replacement +� +Lµz2 = l +� +Q(1 − µQ). It follows that the HEE is +SEE = 1 +4G cosh−1 +� +Q +2r2 +0(1 − µQ) sinh2 l +� +Q(1 − µQ) + cosh2 l +� +Q(1 − µQ) ++ r2 +0(1 − µQ) +2Q +sinh2 l +� +Q(1 − µQ) +� +. +(4.15) +For the r0 → ∞ limit, note the definition of temperature (4.10) and relation 1/4G = c/6, +we arrive at +SEE = c +3 log +�� +β2 + 4µπ2 + β +2πǫ +sinh +� +πl +� +β2 + 4µπ2 +�� +, +ǫ = 1 +r0 +. +(4.16) +This is coincide with (3.29) in the case of non-rotating BTZ black hole. We obtain the same +holographic entanglement entropy formula by calculating the RT surface in the deformed +BTZ black hole. +5 +Conclusion and discussion +The T ¯T deformed CFT was proposed dual to the AdS3 with a certain mixed boundary +condition. The AdS3 with mixed boundary condition or the T ¯T-deformed AdS3 geometry +can be obtained from the Ban˜ados geometry using the dynamical change of coordinates. +In this paper, we studied the holographic entanglement entropy in the T ¯T-deformed AdS3 +under this situation. In terms of Chern-Simons form, we derived the exact holographic +entanglement entropy formula using the Wilson line technique. For the zero temperature +case, the entanglement entropy turned out unchanged under the T ¯T deformation. For the +finite temperature case, we calculated the Wilson line with ending points on the boundary +of deformed AdS3. After identifying the deformed temperature and length of interval on +the boundary, we found the Wilson line lead to holographic entanglement entropy formula, +which is closely related to the entanglement entropy in T ¯T-deformed CFTs. +The same +formula was also obtained by calculating the RT surface in the T ¯T-deformed BTZ black +hole. The deformed entanglement entropy formula can reproduce the known perturbative +results, which were obtained from both field theory and cutoff AdS3. We also showed that +the entropic c-function is always positive and non–decreasing along the renormalization +24 + +group flow towards the ultraviolet. For the non-perturbative region, our results show that +the entanglement entropy behaves like entanglement entropy of CFT at zero temperature. +Moreover, we also considered the two intervals entanglement entropy and found there still +exist a certain phase transition between disconnected and connected phase. It turned out +that the critical point for the phase transition depends on the deformation parameters. The +critical point is sensitive to the deformation parameter for the high temperature region. But +the critical point becomes independent of deformation parameter for the low temperature +region. For a fixed temperature, the critical point tends to the zero temperature case at +large deformation parameter, which is shown in Figure 2. +Finally, we want to point out that the holographic entanglement entropy formula was +derived from the holographic study and the formula agrees with the pertubative result. +However, we still need an exact calculation from T ¯T-deformed CFTs. In addition, since we +found the entanglement entropy behaves like a free CFT, it would be interesting to study +the T ¯T deformation for large deformation parameter following [69, 70]. +Acknowledgements +We are grateful to Song He for suggesting this topic. We would like to thank Yunfeng Jiang, +Zhangcheng Liu, Hao Ouyang, Qiang Wen and Long Zhao for helpful discussions. This work +is supported by the National Natural Science Foundation of China (No.12105113). +A +Conventions +In this paper, we choose the following standard Lie algebra generators of sl(2, R) +L−1 = +� 0 +1 +0 +0 +� +, +L0 = +� 1 +2 +0 +0 +−1 +2 +� +, +L1 = +� +0 +0 +−1 +0 +� +, +(A.1) +whose commutators simplify to +[La, Lb] = (a − b)La+b, +a, b ∈ {0, ±1}. +(A.2) +The non-zero components of non-degenerate bilinear form are given by +Tr(L0L0) = 1 +2, +Tr(L−1L1) = Tr(L1L−1) = −1. +(A.3) +We use the following representation of the sl(2, R) Lie algebra, i.e. the highest-weight +representation. The highest-weight state |h⟩ satisfies +L1|h⟩ = 0, +L0|h⟩ = h|h⟩. +(A.4) +There is an infinite tower of descendant states found by acting with the raising operator +|h, n⟩ = (L−1)n|h⟩. +(A.5) +25 + +These states form an irreducible, unitary, and infinite-dimensional representation of sl(2, R). +The quadratic Casimir operator of the algebra is +C = 2L2 +0 − (L1L−1 + L−1L1), +(A.6) +which commutes with all the elements of the algebra. The expectation value of Casimir +operator on highest-weight state is +C = ⟨h|C|h⟩ = 2h2 − 2h. +(A.7) +B +Wilson line defects +The Wilson line as a probe in the bulk will produce a back-reaction in the bulk. To solve +for this back-reaction, we consider the total action +S = SCS[A] − SCS[ ¯A] + B + S(U; A, ¯A)C. +(B.1) +where B denotes the boundary term, the last term is the auxiliary action associated with +the Wilson line. For different boundary conditions, there will be different boundary terms. +In case of the T ¯T deformation, the boundary term turns out to be +B = k +4π +� +∂M +d2x1 +µ +�� +1 − 2µ +� +Tr(AθAθ) + Tr( ¯Aθ ¯Aθ) +� ++ µ2 � +Tr(AθAθ) − Tr( ¯Aθ ¯Aθ) +�2 − 1 +� +. +(B.2) +This boundary term leads to the T ¯T deformed spectrum and can also help to reduce the +gravitational action to T ¯T deformed Alekseev-Shatashvili action on the boundary [45]. The +boundary term does not contribute to the equation of motion, but the Wilson line term will +contribute as a source for the equations of motion +k +2πFµν = +� +dsdxρ +ds εµνρδ(3)(x − x(s))UPU−1, +(B.3) +k +2π +¯Fµν = − +� +dsdxρ +ds εµνρδ(3)(x − x(s))P. +(B.4) +We can choose the Wilson line trajectory as a bulk geodesic, the corresponding Wilson line +variables is +r(s) = s, +U(s) = 1, +P(s) = +√ +2CL0. +(B.5) +Contracting (B.3) and (B.4) with the tangent vector to the curve, we find the non-vanishing +components of field strength F, ¯F are tangent to the curve +Fµν +dxµ +ds = 0, +(B.6) +¯Fµν +dxµ +ds = 0. +(B.7) +26 + +Since we can always transform the AdS3 solution into the Poincar´e coordinate [66, 67], we +just consider the Poincar´e AdS3. The solution is asymptotic AdS3 in Poincar´e coordinate +A =L(asource + d)L−1, +L = e− ln rL0e−zL1, +(B.8) +¯A =R−1(asource + d)R, +R = e−¯zL−1e− ln rL0, +(B.9) +where the coupling to the source is taken into account by +asource = +� +C +2 +1 +k +�dz +z − d¯z +¯z +� +L0. +(B.10) +With the help of the identities ∂ 1 +¯z = ¯∂ 1 +z = πδ(2)(z, ¯z), one can verify these connections satisfy +the sourced equations of motion. The connections are flat except for where the Wilson line +sources them. We can obtain the specific form of the gauge field +A =L0 +dr +r + rL1dz + +� +C +2 +1 +k +�dz +z − d¯z +¯z +� +(L0 − rzL1), +(B.11) +¯A = − L0 +dr +r − rL−1d¯z + +� +C +2 +1 +k +�dz +z − d¯z +¯z +� +(L0 − r¯zL−1). +(B.12) +This solution produces the metric +ds2 = dr2 +r2 + +r2 � +− +√ +2 +√ +Ck (zd¯z − ¯zdz)2 + C (zd¯z − ¯zdz)2 − 2k2z¯zdzd¯z +� +2k2z¯z +. +(B.13) +Consider the map from plane to cylinder (τ, ϑ) +z = eτ+iϑ, +¯z = eτ−iϑ, +(B.14) +the metric becomes +ds2 =dr2 +r2 − r2e2τ + + +dτ 2 + +dϑ2 �√ +2C − k +�2 +k2 + + + . +(B.15) +One can see this is precisely the metric for AdS3 with a conical singularity surrounding the +Wilson line. The boundary geometry with Wilson line back-reaction becomes the n-sheet +cylinder if we set the defect angle to be 2π(1 − 1 +n). Then we can find the relation +√ +2C +k += (n − 1) + O((n − 1)2). +(B.16) +Since the Wilson line action generates the n-sheet manifold, the partition function for n- +sheet manifold can be written as +Zn = log WR(C) = − +√ +2CL(xi, xj), +(B.17) +27 + +therefore the entanglement entropy can be obtained +SEE = lim +n→1 +1 +1 − n log WR(C) = kL(xi, xj), +(B.18) +which is coincide with the RT formula. +The stress tensor corresponds to Poincar´e AdS3 vanishes, namely L = 0 in (3.1). For +the BTZ black hole, the stress tensor is a constant. According to the transformation law +of the stress-tensor, we can transform the stress tensor to a constant by using a conformal +map. After rescaling the radial coordinate, the BTZ black hole becomes Poincar´e AdS3 +geometry with different period of the time direction. 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D 87, no.12, 126012 (2013) [arXiv:1212.4764 [hep-th]]. +33 + diff --git a/-tE3T4oBgHgl3EQfSwk7/content/tmp_files/load_file.txt b/-tE3T4oBgHgl3EQfSwk7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dad2fe49e35fd520ee3f82400957b6f67c1faeae --- /dev/null +++ b/-tE3T4oBgHgl3EQfSwk7/content/tmp_files/load_file.txt @@ -0,0 +1,1469 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf,len=1468 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='04435v1 [hep-th] 11 Jan 2023 Holographic entanglement entropy in T T -deformed AdS3 Miao Hea,b, Yuan Sunc aSchool of Physics, Southeast University, Nanjing 211189, China bShing-Tung Yau Center, Southeast University, Nanjing 210096, China cCenter for Theoretical Physics and College of Physics, Jilin University, Changchun 130012, People’s Republic of China E-mail: hemiao@seu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='cn, sunyuan@jlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='cn Abstract In this work, we study the holographic entanglement entropy in AdS3 gravity with the certain mixed boundary condition, which turns out to correspond to T ¯T- deformed 2D CFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' By employing the Chern-Simons formalism and Wilson line technique, the exact holographic entanglement entropy in T ¯T-deformed BTZ black hole is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We also get the same formula by calculating the RT surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The holographic entanglement entropy agrees with the perturbation result derived from both T ¯T-deformed CFTs and cutoff AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Moreover, our result also shows that the deformed holographic entanglement entropy behaves like the zero temperature CFT one for the large deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Based on this result, the two intervals entanglement entropy and phase transition between disconnected and connected phase are also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Contents 1 Introduction 1 2 Wilson lines and entanglement entropy in AdS3 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1 Wilson lines in AdS3 gravity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='2 T ¯T-deformed holographic entanglement entropy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3 Thermal entropy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='4 Two intervals entanglement entropy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 19 4 Geodesic line method 22 5 Conclusion and discussion 24 A Conventions 25 B Wilson line defects 26 1 Introduction The AdS/CFT correspondence gives a geometric interpretation to the conformal field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This correspondence allows us to study quantum gravity from the conformal field theory, and it achieves great success in 3D quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' It is significant to generalize the AdS/CFT correspondence by deforming the conformal field theory and investigating its geometric interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' One of the deformed theories called T ¯T deformation was proposed and its holographic descriptions were also explored [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' It is interesting to establish the holographic dictionary under T ¯T deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The holographic technique also provides us with a gravitational method to study the T ¯T deformed CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The T ¯T deformation is defined through the T ¯T flow equation ∂ST ¯T ∂µ = � d2xOT ¯T , OT ¯T ≡ T ijTij + T 2, 1 where Tij is the stress tensor of the deformed theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This flow equation generates a family of integrable field theory if the original theory is integrable [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The factorizable of T ¯T operator leads to the Burgers equation for the deformed spectrum [5], so that the spectrum of the deformed theory can be exactly solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The partition function of the deformed theory can be obtained from various methods, the result turns out that the deformed partition function satisfies a differential equation or an integral transformation of the original one [6– 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The deformed partition function is still modular invariant [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' According to the T ¯T flow equation, the Lagrangian form and Hamiltonian form were also studied [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' There are also some evidences shown that the T ¯T deformed theory is a non-local theory [12– 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In this irrelevant deformation, it is difficult to study the local properties, such as the correlation function and entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' These observables play the important role in the quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' By using the perturbative method, the correlation functions and entanglement entropy have also been obtained [21–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Some non-perturbative results about the correlation function and entanglement were explored in [17–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' However, there is still an open question to calculate the correlation function and entanglement entropy in T ¯T deformed theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For a pedagogical review see [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' According to the AdS/CFT correspondence, the deformed theory can be investigated by using the gravitational approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' There are two points of view to understand the T ¯T deformed CFTs from gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The one is the T ¯T deformed CFTs dual to the AdS3 with a finite radial cutoff [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In this situation, the quasi-local energy of the cutoff region matches the spectrum of the deformed theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The T ¯T flow equation coincides with the Hamilton-Jacobi equation governing the radial evolution of the classical gravity action in AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Many holographic features of the T ¯T deformed CFT have been explored based on the cutoff perspective [33–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The other holographic perspective to understand the T ¯T deformation is the AdS3 gravity with certain mixed boundary condition [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The boundary condition was derived from the flow equation and variational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' It turned out that the solution of the metric flow equation related to the higher order Fefferman- Graham expansion, which leads to the mixed boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The mixed boundary condition coincides with the induced metric on the finite radial cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The AdS3 solutions that satisfy the mixed boundary condition were also constructed through a field-dependent coordinate transformation [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The dynamic coordinate transformation approach to T ¯T was also found in field theoretic results [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The deformed spectrum can also be obtained from the deformed AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The mixed boundary condition allows boundary graviton degree of freedom, which turns out to be a T ¯T deformed theory [44–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The mixed boundary condition provides us with another approach to studying the T ¯T deformation including the entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In this paper, we would like to investigate the entanglement entropy in T ¯T deformed CFT from holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the cutoff perspective, the holographic entanglement was obtained by calculating the length of cutoff geodesic line, and the results match perturbative CFT results [22, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The entanglement entropy in T ¯T deformation was also studied on both the field theory side and holographic side in recent works [48–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We prefer to use the mixed boundary condition perspective to study holographic entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Since the deformed geometry is still AdS3, we will work in the SL(2, R)×SL(2, R) gauged Chern- 2 Simons formalism of AdS3 [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The Chern-Simons formalism has been used to study T ¯T deformation in the literatures [44–46, 54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In the gauge theory form, the holographic entanglement entropy is encoded in the Wilson line of Chern-Simons [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Generally, the Wilson lines depend on the path and representation of the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' If we choose a appropriate representation of sl(2, R), the trace over the representation can be formulated into the path integral of a SL(2, R) × SL(2, R) invariant auxiliary theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The on-shell action of the auxiliary is equivalent to the length of geodesics in AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In addition, the Wilson line is a probe in gauge theory, just like a point particle in a curved background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The Wilson lines give a back-reaction to the bulk geometry, and the resulting geometry turns out to be a conical defect on the branch point, which exactly generates a n-sheet manifold [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Therefore, the Wilson line back reaction corresponds to the replica trick along the ending points of the Wilson line on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' These results told us that the Wilson line is related to the entanglement entropy through SEE = − log(WR(C)), where the ending points of the Wilson line correspond to the interval on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The thermal entropy also turned out corresponds to the Wilson loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We use this tech- nique for the deformed AdS3 geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The single interval holographic entanglement entropy is calculated exactly, which can reproduce the perturbative result obtained in other literatures [22, 24, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We also consider the two intervals entanglement entropy in T ¯T deformation, which implies a certain phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Moreover, the holographic entanglement entropy of T ¯T-deformed AdS3 in the non-perturbative region is also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The results show that the entanglement entropy behaves like a zero temperature CFT one for the large deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The paper is organized as follows: In section 2, we give an overview of the gravitational Wilson line approach to obtain the holographic entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In section 3, we introduce the deformed AdS3 under T ¯T, which is parameterized by the deformed spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The holographic entanglement entropy is obtained using the Wilson line approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We also consider the two intervals entanglement entropy and its phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The same result is derived by calculating the RT surface in the deformed AdS3 in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We summarize our results and discussion in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The appendix contains our conventions and Wilson line defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 2 Wilson lines and entanglement entropy in AdS3 This section is a review of using the Wilson lines technique to calculate the holographic entanglement entropy, based on [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' By rewriting the AdS3 gravity in Chern-Simons form, the Wilson line in an infinite-dimensional representation of the bulk gauge group is related to the geodesics in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' According to the Ryu-Takayanagi proposal [59, 60], the holographic entanglement entropy or RT surface can be obtained through the Wilson line approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1 Wilson lines in AdS3 gravity It is well-known that 3D general relativity has no local degrees of freedom, which is purely topological and can be formulated as a Chern-Simons theory [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In the case of AdS3 gravity, the relevant Chern-Simons gauge group is SO(2, 2) ≃ SL(2, R) × SL(2, R), so Einstein-Hilbert action can be written as SEH[e, ω] = ICS[A] − ICS[ ¯A], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1) where the Chern-Simons action is ICS[A] = k 4π � M Tr � A ∧ dA + 2 3A ∧ A ∧ A � , k = 1 4G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='2) The gauge fields A and ¯A are valued in sl(2, R), which are the linear combination of gravitational vielbein and spin connection A = (ωa + ea) La, ¯A = (ωa − ea) La.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3) The La are sl(2, R) generators, see Appendix A for our conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Variation of the action leads to the equations of motion F ≡ dA + A ∧ A = 0, ¯F ≡ d ¯A + ¯A ∧ ¯A = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='4) which are equivalent to the first order gravitational field equation and torsion free equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The AdS3 metric can also be recovered from the gauge fields via gij = 1 2Tr � (Ai − ¯Ai)(Aj − ¯Aj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='5) As a consequence, the AdS3 gravity is formulated into a Chern-Simons gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' By using the Chern-Simons form, we can introduce the gravitational Wilson lines in AdS3 gravity WR(C) = TrR � P exp � C A � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='6) where R denotes a representation of sl(2, R), and C is a curve on M with two ending points living on the boundary of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' If the path C is closed, it gives the Wilson loop which is invariant under the gauge transformation A → A′ = Λ−1(d + A)Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='7) We can use the Wilson lines to probe the bulk geometry, instead of a massive particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The massive particle moving in bulk is characterized by its mass m and spin s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' These parameters would contribute to the backreaction on the bulk geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The trajectory of the particle can be understood as geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' When we turn to use the Wilson line to probe the bulk geometry, we have to use the infinite-dimensional representations of sl(2, R), characterized 4 by (h, ¯h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' So that the mass m and spin s of the particle can be encoded in the representation of sl(2, R) through the relations m = h+ ¯h and s = h−¯h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the representation of sl(2, R) see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Note that infinite-dimensional representations of symmetry algebras can be regarded as the Hilbert spaces of quantum mechanical systems in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The trace over all the states in the representation R can be formulated into a path integral of an auxiliary quantum mechanical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Then the Wilson line can be written as WR(C) = � DU exp [−S(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' A)C] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='8) where S(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' A)C is the action of the auxiliary quantum mechanical system that lives on the Wilson line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The action should have a global symmetry group SL(2, R) × SL(2, R), so that the Hilbert space of the system will be precisely the representation of sl(2, R) after quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the free theory (without gauge fields), an appropriate system is described by a particle moving on the group manifold [61], whose action reads S(U, P)free = � C ds � Tr � PU−1dU ds � + λ(s) � Tr � P 2� − C �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='9) where P lives in the Lie algebra sl(2, R) and U lives in Lie group SL(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The trace in this action means contraction with Cartan-Killing metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The equations of motion for the free theory are U−1dU ds + 2λP = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='10) dP ds = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='11) TrP 2 = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='12) This action has a SL(2, R) × SL(2, R) global symmetry, namely under the following global gauge transformation U(s) → LU(s)R, P(s) → R−1P(s)R, L, R ∈ SL(2, R), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='13) the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='9) is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In [57], it turns out that the system coupled with the external gauge fields A and ¯A should be S(U, P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' A)C = � C ds � Tr � PU−1DsU � + λ(s) � Tr � P 2� − C �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='14) where the covariant derivative is defined by DsU = d dsU + AsU − U ¯As, As = Aµ dxµ ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='15) 5 The equations of motion become U−1DsU + 2λP = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='16) d dsP + � ¯As, P � = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='17) Tr P 2 = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='18) After introducing the covariant derivative, the global symmetry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='13) is enhanced to the local gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='14) is invariant under local gauge transformation Aµ → L(x) (Aµ + ∂µ) L−1(x), ¯Aµ → R−1(x) � ¯Aµ + ∂µ � R(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='19) U(s) → L(xµ(s))U(s)R(xµ(s)), P(s) → R(xµ(s))P(s)R(xµ(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='20) We have to point out that the equations of motion do not change under these gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This feature is useful to construct the solutions of the equations of motion from the free theory solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' If the gauge fields A and ¯A are pure gauge, the solutions for the equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='18) can be obtained from the free theory solution through the gauge transformation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='19) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We will treat more details in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='2 Equivalence to the geodesic equation This Wilson line probe should be equivalent to a massive particle moving in AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Then we will show that the usual geodesic equation with respect to the metric would appear in the Wilson line path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We denote the Wilson line path in the bulk by xµ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Using the classical equation of motion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='18), the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='14) can be reduced into a second order one S(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' A, ¯A)C = √ C � C ds � Tr (U−1DsU)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='21) In this form, the action is essentially a gauged sigma model, whose equation of motion reads d ds �� Au − ¯A � µ dxµ ds � + � ¯Aµ, Au ν � dxµ ds dxν ds = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='22) where Au s = U−1 d dsU + U−1AsU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='23) For the given gauge fields (A, ¯A), the equation of motion depends on the choice of path xµ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' From the perspective of the equation of motion, we learn that U(s) acts like a gauge transformation on the connection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' There is a good choice for U(s), so that the particle does not move in the auxiliary space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' U(s) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In this case, the equation of motion reduces to d ds � ea µ dxµ ds � + ωa µbeb ν dxµ ds dxν ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='24) 6 This is precisely the geodesic equation for the curve xµ(s) on a spacetime with vielbein and spin connection which is equivalent to the more familiar Christoffel symbols forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Furthermore, the on-shell the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='14) for U(s) = 1 becomes S(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' A, ¯A)C = √ 2C � C ds � gµν(x)dxµ ds dxν ds , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='25) which is manifestly the proper distance along the geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We have learned that the Wilson line in AdS3 gravity can be expressed as a path integral of an auxiliary quantum mechanical system, whose action is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The on-shell action turns out to be the proper distance along the geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Thus in the classical limit, one can find that the value of the Wilson line WR(xi, xf) = exp(− √ 2CL(xi, xf)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='26) where L(xi, xf) is the length of the bulk geodesic connecting these two endpoints on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Holographically, it was proposed by Ryu and Takayanagi that the field-theoretical entanglement entropies correspond to the length of the bulk geodesics ending on the bound- ary [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In terms of the Chern-Simons description of AdS3 gravity, we can calculate the entanglement entropy from the Wilson line SEE = − log(WR(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='27) In [57], it was also shown that the Wilson line backreaction on the geometry would create a non-trivial holonomy, which can be interpreted as the conical singularity in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The conical defects hence reproduce the field-theoretical entanglement entropy formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In the later of this paper, we would like to use the Wilson line technique to compute the holographic entanglement entropy in Chern-Simons AdS3 gravity, including the T ¯T-deformed AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3 Holographic entanglement entropy In this section, we calculate WR(C) with C ending on the AdS3 boundary at two points denoted by xi = x(si), xf = x(sf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Classically, we just need to calculate the on-shell action of the auxiliary system Son-shell = � C ds Tr � PU−1DsU � = −2C � sf si dsλ(s), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='28) which depends on the solution of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The solutions can be constructed from the free theory solutions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='10)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='12), through the gauge transformation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='19) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' First of all, we should note the solutions to free theory, denoting them by U0(s) and P0, are U0(s) = u0 exp(−2α(s)P0), with dα(s) ds = λ(s), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='29) 7 where P0 and u0 are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Next, we assume the bulk gauge fields are in pure gauge A = L(x)dL−1(x), ¯A = R−1(x)dR(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='30) In fact, most of the AdS3 solutions are in pure gauge, such as BTZ black hole and Ban˜ados geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Then one can verify the following is the classical solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='16)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='18) U(s) = L(x(s))U0(s)R(x(s)), P(s) = R−1(x(s))P0R(x(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='31) These solutions are directly obtained from the local gauge symmetry of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' As argued in [57], the boundary conditions for U(s) on the boundary ending points can be chosen as U(si) =L(x(si))u0 exp(−2α(si)P0)R(x(si)) = 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='32) U(sf) =L(x(sf))u0 exp(−2α(sf)P0)R(x(sf)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='33) We then have to eliminate the initial value P0 and u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Solving u0 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='32) and substituting into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='33), one can find exp(−2∆αP0) =R(x(si))L(x(si))L−1(x(sf))R−1(x(sf)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='34) Taking the trace on both sides, we arrive at cosh � −2∆α √ 2C � = 1 2Tr � R(x(si))L(x(si))L−1(x(sf))R−1(x(sf)) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='35) where we have used Tr (exp(−2∆αP0)) = 2 cosh � −2∆α √ 2C � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='36) Finally, according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='27), we obtain the holographic entanglement entropy formula SEE = √ 2C cosh−1 �1 2Tr � R(x(si))L(x(si))L−1(x(sf))R−1(x(sf)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='37) We then use this formalism to check the holographic entanglement entropy in Poincare AdS3 and BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1 Poincar´e AdS3 For the case of Poincare AdS3, the line element reads ds2 = dr2 r2 + r2(dθ2 − dt2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='38) In terms of the Chern-Simons gauge connection, this geometry is described by A =dr r L0 + rL1(dθ + dt), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='39) ¯A = − dr r L0 − rL−1(dθ − dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='40) 8 The gauge connections can be written in pure gauge form A =LdL−1, L = exp(− ln rL0) exp(−(θ + t)L1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='41) ¯A =R−1dR, R = exp((θ − t)L−1) exp(− ln rL0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='42) In order to calculate the entanglement entropy, we consider a time slice (t = 0) of this geometry and impose the following boundary conditions for the ending points of the Wilson line r(si) = r(sf) = r0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='43) ∆θ = θ(sf) − θ(si) = l, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='44) which means we work on a constant radial boundary and the length of the interval is l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Plugging (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='41) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='42) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='37), one obtain SEE = √ 2C cosh−1 � 1 + r2 0l2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='45) Then taking the limit r0 ≫ 1, so that the result corresponds to the theory living on the conformal boundary, we arrive at 1 SEE = c 3 log �l ǫ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='46) where the UV cutoff of the boundary field theory corresponds to the radial cutoff in the bulk, and the central charge relarelatesthe expectation value of Casimir ǫ = 1 r0 , √ 2C = c 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='47) The relation between the expectation value of Casimir and central charge can be derived by calculating the Wilson line defect, for the details see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This result is exactly the entanglement entropy of CFT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The same answer can also be obtained by solving the bulk geodesic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' However, in terms of the Wilson line form, we do not require the solution of any differential equations and follow from purely algebraic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This technique can be used for more complicated AdS3 geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='2 BTZ black hole For the BTZ black hole, the metric in Fefferman–Graham gauge is ds2 = dr2 r2 + r2 � dzd¯z + 1 r2L0dz2 + 1 r2 ¯L0d¯z2 + 1 r4L0 ¯L0dzd¯z � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='48) 1We have used the relation cosh−1(x) ∼ log(2x) for x ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 9 where L0 and ¯L0 are constants related to the mass and angular momentum of the black hole L0 = M − J 2 , ¯L0 = M + J 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='49) The corresponding Chern-Simons gauge connections read A =dr r L0 + � rL1 − 1 rL0L−1 � dz, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='50) ¯A = − dr r L0 + �1 r ¯L0L1 − rL−1 � d¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='51) In this case, one can obtain L (r, z, ¯z) = exp (− ln rL0) exp (−zL1 + L0zL−1) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='52) R (r, z, ¯z) = exp � ¯L0¯zL1 − ¯zL−1 � exp (− ln rL0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='53) In addition, such solutions can be parametrized as A = b−1(d + a)b, ¯A = b(d + ¯a)b−1, b = eln rL0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='54) Then a, ¯a are also flat connections, but do not depend on the radial coordinate a = (L1 − L0L−1) dz, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='55) ¯a = � ¯L0L1 − L−1 � d¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='56) Following the same steps in pure AdS3 and the boundary conditions for the ending points of the Wilson line, we can get Tr � R(r0, θ(si), 0)L(r0, θ(si), 0)L−1(r0, θ(sf), 0)R−1(r0, θ(sf), 0) � = − 2 cosh � l � L0 � cosh � l � ¯L0 � + � L0 ¯L0 + r4 0 � sinh � l√L0 � sinh � l � ¯L0 � r2 0 √L0 � ¯L0 ∼ r2 0 sinh � l√L0 � sinh � l � ¯L0 � √L0 � ¯L0 , (r0 ≫ 1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='57) This result leads to the entanglement entropy SEE =c 6 log \uf8eb \uf8ed r2 0 sinh � l√L0 � sinh � l � ¯L0 � √L0 � ¯L0 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='58) If we consider the spinless black hole, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' L0 = ¯L0, the entanglement entropy reduces to SEE =c 3 log �β0 πǫ sinh �πl β0 �� , β0 = π √L0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='59) where β0 is the inverse temperature of the BTZ black hole [62–64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This result coincides with the entanglement entropy of a CFT in thermal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='4 Loops and thermal entropy One can also consider the Wilson loops in AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In this case, WR(C) turns out to be the proper distance around the horizon, which corresponds to the black hole thermal entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We will then check it in the BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Consider the Wilson loop along the S1 cycle θ ∼ θ + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In contrast to the open interval case, the closed path should be smooth and hence impose the periodic boundary condition U (sf) = U(si), P (sf) = P(si).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='60) According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='31), the boundary condition for P(s) implies � P0, R (si) R−1(sf) � = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='61) Hence, the boundary condition for U(s) implies exp (−2∆αP0) = u−1 0 � L−1 (sf) L(si) � u0 � R(si)R−1 (sf) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='62) In addition, note the relations L−1 (sf) L(si) = exp �� dθaθ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='63) R(si)R−1 (sf) = exp � − � dθ¯aθ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='64) which are the holonomies of the connection, we can rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='62) as exp (−2∆αP0) = u−1 0 exp (2πaθ) u0 exp (−2π¯aθ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='65) Here we just consider the case of BTZ black hole, so that one can perform the simple integral over θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='61), we learn that P0 and ¯aθ can be diagonalized simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' If the initial value of u0 is fixed, we can always choose a matrix V , such that aθ can also be diagonalized by u0V exp (−2∆αλP) = (u0V )−1 exp (2πaθ) u0V exp � −2π¯λθ � = exp (2πλθ) exp � −2π¯λθ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='66) where λP, λθ and ¯λθ are diagonalized matrix of P0, aθ and ¯aθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Contracting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='66) with L0, we obtain the on-shell action for the loop Sth = 2π √ 2CTr � (λθ − ¯λθ)L0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='67) For the BTZ black hole, the diagonalized gauge connections are λθ = 2 � L0L0, ¯λθ = −2 � ¯L0L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='68) Finally, the Wilson loop gives precisely the entropy of the BTZ black hole Sth = 2π �c 6L0 + 2π �c 6 ¯L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='69) 11 3 Holographic entanglement entropy in T ¯T - deformed AdS3 We turn to investigate the entanglement entropy of T ¯T deformed CFTs from the gravity side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In [41], it is proposed that the holographic interpretation of T ¯T deformed CFTs is still AdS3 gravity but with the mixed boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The AdS3 solutions associated with the mixed boundary condition can be obtained from the Ba˜nados geometry through a coordinate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' As the deformed geometry is still AdS3, we prefer to work in Chern-Simons formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In this section, we introduce the T ¯T deformed AdS3 geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The holographic entanglement entropy of T ¯T deformed CFTs can be obtained using the Wilson line technique in the deformed AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1 T ¯T deformed AdS3 geometry We start from the general AdS3 solution with a flat conformal boundary, which is called the Ba˜nados geometry [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In Fefferman-Graham gauge, the line element reads ds2 = dr2 r2 + r2 � dzd¯z + 1 r2L(z)dz2 + 1 r2 ¯L(¯z)d¯z2 + 1 r4L(z) ¯L(¯z)dzd¯z � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1) The parameters L(z) and ¯L(¯z) are arbitrary holomorphic and antiholomorphic functions, which are related to the energy and angular momentum L = E + J 2 , ¯L = E − J 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='2) The corresponding Chern-Simons gauge fields are A =dr r L0 + � rL1 − 1 rL(z)L−1 � dz, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3) ¯A = − dr r L0 − �1 r ¯L(¯z)L1 − rL−1 � d¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='4) In this sense, the deformed Ba˜nados geometry can be constructed through a field-dependent coordinate transformation [41], which reads dz = 1 1 − µ2Lµ ¯Lµ (dw − µ ¯Lµd ¯w), d¯z = 1 1 − µ2Lµ ¯Lµ (d ¯w − µLµdw), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='5) where µ is the deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' One should note that the parameters L and ¯L in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1) would turn into Lµ and ¯Lµ under the coordinate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Generally, the parameters Lµ and ¯Lµ are different from the undeformed ones L and ¯L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The relations between deformed parameters Lµ, ¯Lµ and undeformed parameters L, ¯L can be fixed by two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The first one is that the deformation smoothly changes the spectrum but does not change the local degeneracy of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Therefore, in the bulk, this implies that the T ¯T 12 deformation does not change the horizon area of a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The second one is that the deformed geometry can be transformed into the undeformed one without changing the periodicity of the spatial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Indeed, the transformation is different from the inverse of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' These considerations lead to Lµ(1 − µ ¯Lµ)2 (1 − µ2Lµ ¯Lµ)2 = L, ¯Lµ(1 − µLµ)2 (1 − µ2Lµ ¯Lµ)2 = ¯L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='6) One can turn to [41] for more details about fixing these relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' By using the coordinate transformation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='5), we obtain the deformed Chern-Simons gauge fields A =1 rL0dr + 1 1 − µ2Lµ ¯L µ � rL1 − 1 rLµL−1 � (dw − µ ¯Lµd ¯w), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='7) ¯A = − 1 rL0dr − 1 1 − µ2Lµ ¯Lµ �1 r ¯LµL1 − rL−1 � (d ¯w − µLµdw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='8) Note that L(z) and ¯L(¯z) correspond to the charges of the solution in the Ba˜nados geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' However, in the deformed geometry, the parameters L(z) and ¯L(¯z) do not correspond to the charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Indeed, the deformed energy and angular momentum can be obtained from both field theory and gravity side Eµ = 1 µ � 1 − � 1 − 2µ(L + ¯L) + µ2(L − ¯L)2 � , Jµ = J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='9) Analogous to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='2), we introduce the new parameters Q = Eµ + Jµ 2 = 1 2µ � 1 + µ(L − ¯L) − � 1 − 2µ(L + ¯L) + µ2(L − ¯L)2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='10) ¯Q = Eµ − Jµ 2 = 1 2µ � 1 − µ(L − ¯L) − � 1 − 2µ(L + ¯L) + µ2(L − ¯L)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='11) We can regard Q and ¯Q as the generalized parameters of L and ¯L in the deformed geometry, and Q and ¯Q reduce to L and ¯L in the limit µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We find it is more convenient to parametrize the deformed gauge fields or metric in terms of these two independent charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In terms of these charges, the Chern-Simons gauge connection are formulated as A =dr r L0 + 1 − µQ 1 − µ(Q + ¯Q) � r(1 − µ ¯Q)L1 − 1 rQL−1 � dw − µ ¯Q 1 − µ(Q + ¯Q) � r(1 − µ ¯Q)L1 − 1 rQL−1 � d ¯w, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='12) ¯A = − dr r L0 + µQ 1 − µ(Q + ¯Q) �1 r ¯QL1 − r(1 − µQ)L−1 � dw − 1 − µ ¯Q 1 − µ(Q + ¯Q) �1 r ¯QL1 − r(1 − µQ)L−1 � d ¯w, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='13) 13 In the following, we prefer to use the coordinates θ = (w + ¯w)/2, t = (w − ¯w)/2, where t represents the time direction while θ denotes the spatial coordinate at the boundary with the identification θ ∼ θ + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We then have Ar = 1 rL0, Aθ =r(1 − µ ¯Q)L1 − 1 rQL−1, At = K � r(1 − µ ¯Q)L1 − 1 rQL−1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='14) ¯Ar = −1 rL0, ¯Aθ =1 r ¯QL1 − r(1 − µQ)L−1, ¯At = ¯K �1 r ¯QL1 − r(1 − µQ)L−1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='15) where K =1 + µ( ¯Q − Q) 1 − µ(Q + ¯Q), ¯K = −1 − µ( ¯Q − Q) 1 − µ(Q + ¯Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='16) The radial gauge (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='54) still holds for the deformed gauge fields, so that the induced gauge connections are aθ =(1 − µ ¯Q)L1 − QL−1, at = K � (1 − µ ¯Q)L1 − QL−1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='17) ¯aθ = ¯QL1 − (1 − µQ)L−1, ¯at = ¯K � ¯QL1 − (1 − µQ)L−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='18) In addition, we can also write down the deformed ds2 =dr2 r2 + 1 r2(1 − µ(Q + ¯Q))2× � Q(1 − µQ)(1 − µr2)dw + � µQ ¯Q + r2(1 − µQ)(1 − µ ¯Q) � d ¯w � × � ¯Q(1 − µ ¯Q)(1 − µr2)d ¯w + � µQ ¯Q + r2(1 − µQ)(1 − µ ¯Q) � dw � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='19) We will use the deformed geometry to calculate the holographic entanglement entropy in the T ¯T deformed CFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For simplicity, we just consider the constant charges Q and ¯Q, namely we work in T ¯T deformed BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='2 T ¯T -deformed holographic entanglement entropy For the T ¯T-deformed AdS3, the metric still satisfies the Einstein equation or flat connection condition in the Chern-Simons theory although it takes a complicated form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In the Poincar´e AdS3, the Wilson line would produce a back-reaction in the bulk geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The back- reaction would then lead to a conical defect on the ending points of Wilson line, which generates the n-sheet manifold on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' According to the replica trick on the boundary field theory, the Wilson line exactly leads to the entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' One can turn to Appendix B for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We can always transform the T ¯T-deformed AdS3 solution into the Poincar´e form [66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' However, the temperature (the period of Euclidean time) in deformed AdS3 is different from the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The crucial point is that we have to identify the deformed temperature and length of interval on the boundary under T ¯T deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 14 We will treat these considerations in more details and obtain the T ¯T deformed holographic entanglement entropy in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Now, we can use the Wilson line technique to calculate the holographic entanglement entropy in T ¯T-deformed AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' First of all, we can give a glance at the Poincar´e AdS3, which turns out correspond to the zero temperature entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In Fefferman- Graham gauge, the Poincar´e AdS3 can be written as Ba˜nados geometry (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1) with L and ¯L vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In this case, the bulk geometry is the same as the undeformed one, so the zero temperature entanglement entropy remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This result coincides with the perturbative calculation in field theory and cutoff perspective in the bulk [22, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We then consider the deformed BTZ black hole, in which the charges Q and ¯Q are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the deformed geometry, on a time slice, we obtain L (r, θ, t = 0) = exp (− ln rL0) exp � − � x x0 dxiai � = exp (− ln rL0) exp � −(1 − µ ¯Q)θL1 + QθL−1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='20) R (r, θ, t = 0) = exp �� x x0 dxi¯ai � exp (− ln rL0) = exp � ¯QθL1 − (1 − µQ)θL−1 � exp (− ln rL0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='21) As the deformed geometries are still AdS3 solution, we use the boundary condition for U(s) U(si) = 1, U(sf) = 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='22) as well as the same boundary conditions for the ending points of the Wilson line r(si) = r(sf) = r0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='23) ∆θ = θ(sf) − θ(si) = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='24) We should point out that the boundary condition for U is actually the unique choice because of the Lorentz invariance at the boundary [57, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' As the T ¯T deformation does not break Lorentz invariance, we can use the same boundary condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='22) for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' It seems that l is just the length of the interval in the deformed boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' But it equals to the deformed length of interval, because the length is defined in the (w, ¯w) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Using the gauge transformation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='31), one can get the solution U(s) for the Wilson line coupled to the deformed gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The boundary condition for U(s) and ending points ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='boundary condition for the Wilson line imply ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='(R(si)L(si)) (R (sf) L (sf))−1 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='=2 cosh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='¯Q (1 − µQ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='cosh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='Q(1 − µ ¯Q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ¯Q(1 − µQ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='Q(1 − µ ¯Q) sinh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ¯Q(1 − µQ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='sinh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='Q(1 − µ ¯Q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='Q ¯Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='Q ¯Q sinh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ¯Q(1 − µQ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='sinh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='Q(1 − µ ¯Q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ¯Q(1 − µQ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='Q(1 − µ ¯Q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='∼ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ¯Q(1 − µQ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='Q(1 − µ ¯Q) sinh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ¯Q(1 − µQ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='sinh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='Q(1 − µ ¯Q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='Q ¯Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='25) In the last step, we consider the r0 ≫ 1 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' It is straightforward to get the holographic entanglement entropy for T ¯T deformation SEE = √ 2C cosh−1 \uf8eb \uf8ed r2 0 � ¯Q(1 − µQ) � Q(1 − µ ¯Q) sinh � l � ¯Q(1 − µQ) � sinh � l � Q(1 − µ ¯Q) � 2Q ¯Q \uf8f6 \uf8f8 ∼c 6 log \uf8eb \uf8ed r2 0 � ¯Q(1 − µQ) � Q(1 − µ ¯Q) sinh � l � ¯Q(1 − µQ) � sinh � l � Q(1 − µ ¯Q) � Q ¯Q \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='26) If the original geometry is non-rotating BTZ black hole, namely Q = ¯Q, the deformed entanglement entropy becomes SEE =c 3 log \uf8eb \uf8ed r0 � Q(1 − µQ) sinh � l � Q(1 − µQ) � Q \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='27) For the deformed BTZ black hole, the temperature can be obtained by analysing the period of Euclidean time, which is discussed in the next section (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We quote the result here β = 1 T = π(1 − 2µQ) � Q(1 − µQ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='28) This temperature can also be derived using the first law of thermodynamics, and we will show it in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the limit µ → 0, the temperature reduce to the BTZ black hole temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The length of interval l is already the deformed one, which can be seen from the coordinate transformation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='5) on a time slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In terms of the deformed temperature, we can express the entanglement entropy as SEE = c 3 log �� β2 + 4µπ2 + β 2πǫ sinh � πl � β2 + 4µπ2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='29) 16 This is actually the T ¯T deformed entanglement entropy obtained from the holographic ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For µ = 0, the deformed entanglement entropy reduce to the familiar entanglement entropy of CFT at finite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the small µ, we can obtain the perturbative result SEE = c 3 log � β πǫ sinh �πl β �� + µc 3 �π2 β2 − 2π3l β3 coth �πl β �� + O(µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='30) In the “low temperature” limit β ≫ l, up to the first order, the entanglement entropy becomes SEE-low =c 3 log � β πǫ sinh �πl β �� + µc 3 �π2 β2 � + O(µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='31) In the “high temperature” limit β ≪ l, the first order corrected entanglement entropy is SEE-high =c 3 log � β πǫ sinh �πl β �� − 2µc 3 π3l β3 coth �πl β � + O(µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='32) The “high temperature” result coincides with the result obtained from both boundary field side and AdS3 with cutoff perspective [22, 24]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We apply the Wlison line approach to the T ¯T-deformed AdS3 and obtain the holographic entanglement entropy formula, which agree with the perturbation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' However, the “low temperature” result is different from the cutoff AdS3 perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We are more interested in the non-perturbative result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In order to make sure the entanglement entropy is real, we have − β2 4π2 < µ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='34) which means the holographic description maybe lose when µ out of this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For µ > 0 the entanglement entropy is always real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In the following discussion, we just consider the µ > 0 case, which also corresponds to the cutoff perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For a fixed temperature, we can consider the entanglement entropy for large deformation parameter SEE = c 3 log � l 2ǫ � + βc 6π 1 √µ + �cl2 72 − β2c 24π2 � 1 µ + O � 1 µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='35) The leading order coincides with the entanglement entropy of the zero temperature CFT with the length of interval l/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This result implies the T ¯T deformation behaves like the 2Note that our convention is different from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In [22], the deformation parameter is related to the radial cutoff r2 c = 6 µπc, while we have r2 c = 1 µ in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Therefore, if one replaces µ by µπc 6 , the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='32) becomes SEE-high = c 3 log � β πǫ sinh �πl β �� − µπ4c2l 9β3 coth �πl β � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='33) which is exactly the result in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 17 free theory at the large µ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The similar feature was also found in [69, 70], in which the authors shown that at the level of the equations of motion the left- and right-chiral sectors of T ¯T deformed free theories are decoupled when the deformation parameter is sent to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Moreover, the Casini-Huerta entropic c-function [71] for the T ¯T deformed entanglement entropy is C(l, µ) = ldSEE dl = πcl 3 � β2 + 4π2µ coth � πl � β2 + 4π2µ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='36) which is always positive, and does not depend on the ultraviolet regulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We also find that ∂C(l, µ) ∂l = πc 3 \uf8eb \uf8ec \uf8ec \uf8ed coth � πl √ β2+4π2µ � � β2 + 4π2µ − πlcsch2 � πl √ β2+4π2µ � β2 + 4π2µ \uf8f6 \uf8f7 \uf8f7 \uf8f8 ≥ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='37) which implies the entropic c-function is non–decreasing along the renormalization group flow towards the ultraviolet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The similar result was also found in single trace T ¯T deforma- tion [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3 Thermal entropy The thermal entropy of the deformed BTZ black hole can also be calculated from the Wilson loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' As discussed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='4, the thermal entropy can be obtained by diagonalizing the induced gauge connections aθ and ¯aθ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='17) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the deformed BTZ black hole, the diagonalized gauge connections read λθ = 2 � Q(1 − µ ¯Q)L0 = 2 √ LL0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='38) ¯λθ = −2 � ¯Q(1 − µQ)L0 = −2 � ¯LL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='39) Finally, according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='67), we obtain the thermal entropy S = 2π �c 6L + 2π �c 6 ¯L, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='40) which is the same as the BTZ black hole entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This result means the black hole entropy does not change under the T ¯T deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' On the field theory side, the degeneracy of states do not change under the T ¯T flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the deformed theory, the thermal entropy should be expressed in terms of the deformed energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In case of Q = ¯Q, the entropy can be written as S = 4π �c 6Q(1 − µQ) = 2π �c 6Eµ(2 − µEµ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='41) 18 which agrees with the result in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The thermal entropy can help us to define the tempera- ture in the T ¯T-deformed theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In fact, according to the first law of thermodynamics, the temperature can be determined by T = ∂Eµ ∂S = � 6 c � Q(1 − µQ) π(1 − 2µQ) ∼ � Q(1 − µQ) π(1 − 2µQ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='42) where we have used the convention k = c/6 = 1 in the definiton of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This is actually the temperature we have used in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='4 Two intervals entanglement entropy We proceed to consider the entanglement entropy of the system consists of two disjoint intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the single interval case, we have shown that the entanglement entropy is the Wilson line or length of geodesic in AdS3 with ending points on the spatial infinity boundary for both Brown-Henneaux boundary condition and mixed boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' According to Ryu-Takayanagi’s proposal [59, 60], we have two choices for how to draw the geodesics that end on the ending points of two intervals, which are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For each choice, the two intervals entanglement entropy decouples into a sum of single interval cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The Figure 1: The two minimal surfaces for the two intervals boundary region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We consider the two intervals have the same length l separated by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The left is the disconnected case, and the right is the connected case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' two intervals holographic entanglement entropy should be the minimal one of them SEE-2 = min{Sdis, Scon}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='43) This implies that there are two phases of the entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' It turns out that there actually exist a phase transition between the connected and disconnected phase [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We first brief review the zero temperature entanglement entropy of two disjoint intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We assume the two intervals have the same length l separated by x, described in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Then the difference between two phases is ∆S = Sdis − Scon = c 3 log � l2 x(2l + x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='44) 19 COF6CI60One can find the phase transition critical point is determined by the cross-ratio η = l2 (l + x)2 = 1 2 or x l = √ 2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='45) For the finite temperature case, the similar phase transition was shown in [74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' However, there is no quantity like cross-ratio to illustrate the critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Now we would like to investigate the similar feature for the T ¯T deformed entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the different choices of Wilson lines or RT surfaces, we have Sdis =c 3 log \uf8eb \uf8ed π2µ + 1 2β �� β2 + 4π2µ + β � π2ǫ2 sinh2 � πl � β2 + 4µπ2 �\uf8f6 \uf8f8 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='46) Scon =c 3 log \uf8eb \uf8ed π2µ + 1 2β �� β2 + 4π2µ + β � π2ǫ2 sinh � πx � β2 + 4µπ2 � sinh � π(2l + x) � β2 + 4µπ2 �\uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='47) The two intervals entanglement entropy is the minimal one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In order to determine which is the minimal one and under what conditions the phase transition happens, we consider the difference between two RT surfaces ∆S =Sdis − Scon = c 3 log \uf8eb \uf8ec \uf8ec \uf8ed sinh2 � πl √ β2+4µπ2 � sinh � πx √ β2+4µπ2 � sinh � π(2l+x) √ β2+4µπ2 � \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='48) This quantity is also related to the mutual information between two disjoint subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='48), we learn that ∆S behaves like the undeformed one but with different tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We first consider the low temperature and high temperature limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the low temperature limit β ≫ 1, we have ∆S = c 3 log � l2 x(2l + x) � + O � 1/β2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='49) The leading order is exactly the zero temperature case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The phase transition occur at x/l = √ 2−1 and does not depend on the deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the high temperature limit β ≪ 1, we have ∆S = c 3 log \uf8eb \uf8ed cosh � l √µ � − 1 cosh � l+x √µ � − cosh � l √µ � \uf8f6 \uf8f8 + O � β2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='50) In this case, the critical point depends on the deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We find it is convenient to introduce the following parameters ˜l = x l , ˜x = x β , ˜µ = µ β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='51) 20 In terms of the new parameters, the ∆S reduces to ∆S = c 3 log \uf8eb \uf8ec \uf8ec \uf8ed sinh2 � π˜x ˜l√ 1+4˜µπ2 � sinh � π˜x √ 1+4˜µπ2 � sinh � π(2+˜l)˜x ˜l√ 1+4˜µπ2 � \uf8f6 \uf8f7 \uf8f7 \uf8f8 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='52) in which the temperature is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We plot the critical lines ∆S = 0 in (˜l, ˜x) plane for different deformation parameters in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Then we consider some special limit about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='20 l ∼ x ∼ Critical lines: ΔS =0 μ∼=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='02 μ∼=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='01 μ∼=0 μ∼=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='01 μ∼=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='02 μ∼=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='03 μ∼=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='4 Figure 2: Plot the critical lines ∆S = 0 in ˜l − ˜x plane for different deformation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The critical lines separate the connected phase (left side) and disconnected phase (right side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The green line corresponds to the undeformed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The dashed line denotes the zero temperature critical line ˜l = √ 2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The critical lines tend to the zero temperature case with the increase of deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' the critical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For ˜x ≪ 1, we have ∆S = c 3 � log � 1 ˜l2 + 2˜l � − π2(˜l + 1)2˜x2 3˜l2 (1 + 4˜µπ2) � + O � ˜x3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='53) The leading order is just the zero temperature case and also does not depend on the deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This result can be seen from Figure 2 that the critical lines coincide with the zero temperature one for small ˜x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' It is interesting to investigate the µ dependence of phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the small ˜µ, there is actually exist a phase transition, which has been discussed in [24] using the perturbative method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We can also see from Figure 2 the critical line is around the undeformed case for 21 both ˜µ < 0 and ˜µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the ˜µ ≫ 1 region, we have ∆S = c 3 log � 1 ˜l2 + 2˜l � − c(˜l + 1)2˜x2 36˜l2˜µ + O(1/˜µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='54) The leading order is the just the zero temperature case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' One can also see from Figure 2 that the critical lines would become the zero temperature one as the increase of deformation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This result implies the T ¯T deformed theory becomes a decoupled free theory for large µ limit [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' These results show that there still exist the phase transition for two intervals entangle- ment entropy under T ¯T deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The transition point is depends on the deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The T ¯T deformation does not introduce new phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For large deformation parameter, the the critical point is the same as zero temperature CFT case, it would be interesting to study this feature from the field theoretic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 4 Geodesic line method In this section we re-compute the holographic entanglement entropy in BTZ background with mix boundary condition using RT formula, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=', identifying the holographic entan- glement entropy as the geodesic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The results turn out to be consistent with the computation via Wilson line method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The metric of BTZ black hole with mass M and angular momentum J takes the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 3 For simplicity we consider the case where the black hole being static J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='6) that the deformed parameters Lµ, ¯Lµ are constant and satisfy Lµ = ¯Lµ = 1 − µM ± √1 − 2µM Mµ2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1) where only the solution with “-” is well defined in µ → 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We start from the following metric ds2 =dr2 r2 + r2� dzd¯z + 1 r2(Lµdz2 + ¯Lµd¯z2) + 1 r4Lµ ¯Lµdzd¯z � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='2) in which we have replaced the L, ¯L by Lµ, ¯Lµ in the BTZ black hole solution, so that we can obtain the deformed BTZ only by using the coordinate transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Let z = x + iy, and define r = � Lµeρ, x = ¯x � 4Lµ , y = ¯y � 4Lµ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3) then the metric becomes the global AdS3 ds2 =dρ2 + cosh2 ρd¯x2 + sinh2 ρd¯y2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='4) 3We follow the convention in [41], and set 4πG = 1, l = 1 and R = 2π (periodicity of spatial dimension) in their paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We also use r which is related with the radial coordinate ρ in [41] as r2 = 1/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The cutoff in [41] locates at ρ = ρc = µ, then in r-coordinate, r0 = rc = 1/√µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 22 where ¯y is treated as the Euclidean time and ¯x the spatial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The requirement of no conical singularity in ρ − ¯y plane implies the identification ¯y ∼ ¯y + 2π, where the periodicity is related with the temperature for BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' It is convenient to work in embedding coordinate Y 0 = cosh ρ cosh ¯x, Y 3 = cosh ρ sinh ¯x, Y 1 = sinh ρ sin ¯y, Y 2 = sinh ρ cos ¯y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='5) In this coordinate system the BTZ black hole is a hypersurface −(Y 0)2 + (Y 3)2 + (Y 1)2 + (Y 2)2 = −1 in the background ds2 = −d(Y 0)2 + d(Y 1)2 + d(Y 2)2 + d(Y 3)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The geodesic distant d between two points Y a 1 , Y b 2 is simply computed by cosh d = −Y1 · Y2 = Y 0 1 Y 0 2 − Y 1 1 Y 1 2 − Y 2 1 Y 2 2 − Y 3 1 Y 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='6) The deformed metric corresponding to T ¯T deformation can be obtained by transforma- tion of dz = 1 1 − µ2Lµ ¯Lµ (dw − µ ¯Lµd ¯w), d¯z = 1 1 − µ2Lµ ¯Lµ (d ¯w − µLµdw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='7) In the present case, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='7) can be solved straightforwardly as z = 1 1 − µ2Lµ ¯Lµ (w − µ ¯Lµ ¯w), ¯z = 1 1 − µ2Lµ ¯Lµ ( ¯w − µLµw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='8) And its inverse w = z + µ ¯Lµ¯z, ¯w = µLµz + ¯z, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='9) where w = θ + it, ¯w = θ − it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' From the periodicity of ¯y discussed above, we can work out the periodic of t, which is t ∼ t + 2π(1 − µLµ) � 4Lµ = t + β, β = π(1 − 2µQ) � Q(1 − µQ) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='10) where the β is the inverse temperature of deformed black hole, as well as the inverse temperature of the T ¯T deformed CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' To compute the HEE of a single interval, we consider two endding points on the boundary locate at (r1, t1, θ1) = ( � Lµeρ0, 0, 0) and (r2, t2, θ2) = ( � Lµeρ0, 0, l) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Then w1 = ¯w1 = 0, w2 = ¯w2 = l z1 = ¯z1 = 0, z2 = ¯z2 = l 1 + µLµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='11) In terms of embedding coordinates Y 0 1 = cosh ρ0, Y 3 1 = 0, Y 1 1 = 0, Y 2 1 = sinh ρ0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='12) and Y 0 2 = cosh ρ0 cosh � 4Lµz2, Y 3 2 = cosh ρ sinh � 4Lµz2, Y 1 2 = 0, Y 2 2 = sinh ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='13) 23 Finally using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='6), the geodesic distance between the points is cosh d = cosh2 ρ0 cosh � 4Lµz2 − sinh2 ρ0 = Q 2r2 0(1 − µQ) sinh2 l � Q(1 − µQ) + cosh2 l � Q(1 − µQ) + r2 0(1 − µQ) 2Q sinh2 l � Q(1 − µQ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='14) where we made the replacement � Lµz2 = l � Q(1 − µQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' It follows that the HEE is SEE = 1 4G cosh−1 � Q 2r2 0(1 − µQ) sinh2 l � Q(1 − µQ) + cosh2 l � Q(1 − µQ) + r2 0(1 − µQ) 2Q sinh2 l � Q(1 − µQ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='15) For the r0 → ∞ limit, note the definition of temperature (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='10) and relation 1/4G = c/6, we arrive at SEE = c 3 log �� β2 + 4µπ2 + β 2πǫ sinh � πl � β2 + 4µπ2 �� , ǫ = 1 r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='16) This is coincide with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='29) in the case of non-rotating BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We obtain the same holographic entanglement entropy formula by calculating the RT surface in the deformed BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' 5 Conclusion and discussion The T ¯T deformed CFT was proposed dual to the AdS3 with a certain mixed boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The AdS3 with mixed boundary condition or the T ¯T-deformed AdS3 geometry can be obtained from the Ban˜ados geometry using the dynamical change of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In this paper, we studied the holographic entanglement entropy in the T ¯T-deformed AdS3 under this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In terms of Chern-Simons form, we derived the exact holographic entanglement entropy formula using the Wilson line technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the zero temperature case, the entanglement entropy turned out unchanged under the T ¯T deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the finite temperature case, we calculated the Wilson line with ending points on the boundary of deformed AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' After identifying the deformed temperature and length of interval on the boundary, we found the Wilson line lead to holographic entanglement entropy formula, which is closely related to the entanglement entropy in T ¯T-deformed CFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The same formula was also obtained by calculating the RT surface in the T ¯T-deformed BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The deformed entanglement entropy formula can reproduce the known perturbative results, which were obtained from both field theory and cutoff AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We also showed that the entropic c-function is always positive and non–decreasing along the renormalization 24 group flow towards the ultraviolet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the non-perturbative region, our results show that the entanglement entropy behaves like entanglement entropy of CFT at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Moreover, we also considered the two intervals entanglement entropy and found there still exist a certain phase transition between disconnected and connected phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' It turned out that the critical point for the phase transition depends on the deformation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The critical point is sensitive to the deformation parameter for the high temperature region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' But the critical point becomes independent of deformation parameter for the low temperature region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For a fixed temperature, the critical point tends to the zero temperature case at large deformation parameter, which is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Finally, we want to point out that the holographic entanglement entropy formula was derived from the holographic study and the formula agrees with the pertubative result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' However, we still need an exact calculation from T ¯T-deformed CFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In addition, since we found the entanglement entropy behaves like a free CFT, it would be interesting to study the T ¯T deformation for large deformation parameter following [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Acknowledgements We are grateful to Song He for suggesting this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We would like to thank Yunfeng Jiang, Zhangcheng Liu, Hao Ouyang, Qiang Wen and Long Zhao for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' This work is supported by the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='12105113).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' A Conventions In this paper, we choose the following standard Lie algebra generators of sl(2, R) L−1 = � 0 1 0 0 � , L0 = � 1 2 0 0 −1 2 � , L1 = � 0 0 −1 0 � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1) whose commutators simplify to [La, Lb] = (a − b)La+b, a, b ∈ {0, ±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='2) The non-zero components of non-degenerate bilinear form are given by Tr(L0L0) = 1 2, Tr(L−1L1) = Tr(L1L−1) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3) We use the following representation of the sl(2, R) Lie algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' the highest-weight representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The highest-weight state |h⟩ satisfies L1|h⟩ = 0, L0|h⟩ = h|h⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='4) There is an infinite tower of descendant states found by acting with the raising operator |h, n⟩ = (L−1)n|h⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='5) 25 These states form an irreducible, unitary, and infinite-dimensional representation of sl(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The quadratic Casimir operator of the algebra is C = 2L2 0 − (L1L−1 + L−1L1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='6) which commutes with all the elements of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The expectation value of Casimir operator on highest-weight state is C = ⟨h|C|h⟩ = 2h2 − 2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='7) B Wilson line defects The Wilson line as a probe in the bulk will produce a back-reaction in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' To solve for this back-reaction, we consider the total action S = SCS[A] − SCS[ ¯A] + B + S(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' A, ¯A)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1) where B denotes the boundary term, the last term is the auxiliary action associated with the Wilson line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For different boundary conditions, there will be different boundary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' In case of the T ¯T deformation, the boundary term turns out to be B = k 4π � ∂M d2x1 µ �� 1 − 2µ � Tr(AθAθ) + Tr( ¯Aθ ¯Aθ) � + µ2 � Tr(AθAθ) − Tr( ¯Aθ ¯Aθ) �2 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='2) This boundary term leads to the T ¯T deformed spectrum and can also help to reduce the gravitational action to T ¯T deformed Alekseev-Shatashvili action on the boundary [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The boundary term does not contribute to the equation of motion, but the Wilson line term will contribute as a source for the equations of motion k 2πFµν = � dsdxρ ds εµνρδ(3)(x − x(s))UPU−1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3) k 2π ¯Fµν = − � dsdxρ ds εµνρδ(3)(x − x(s))P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='4) We can choose the Wilson line trajectory as a bulk geodesic, the corresponding Wilson line variables is r(s) = s, U(s) = 1, P(s) = √ 2CL0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='5) Contracting (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='3) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='4) with the tangent vector to the curve, we find the non-vanishing components of field strength F, ¯F are tangent to the curve Fµν dxµ ds = 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='6) ¯Fµν dxµ ds = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='7) 26 Since we can always transform the AdS3 solution into the Poincar´e coordinate [66, 67], we just consider the Poincar´e AdS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The solution is asymptotic AdS3 in Poincar´e coordinate A =L(asource + d)L−1, L = e− ln rL0e−zL1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='8) ¯A =R−1(asource + d)R, R = e−¯zL−1e− ln rL0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='9) where the coupling to the source is taken into account by asource = � C 2 1 k �dz z − d¯z ¯z � L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='10) With the help of the identities ∂ 1 ¯z = ¯∂ 1 z = πδ(2)(z, ¯z), one can verify these connections satisfy the sourced equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The connections are flat except for where the Wilson line sources them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' We can obtain the specific form of the gauge field A =L0 dr r + rL1dz + � C 2 1 k �dz z − d¯z ¯z � (L0 − rzL1), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='11) ¯A = − L0 dr r − rL−1d¯z + � C 2 1 k �dz z − d¯z ¯z � (L0 − r¯zL−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='12) This solution produces the metric ds2 = dr2 r2 + r2 � − √ 2 √ Ck (zd¯z − ¯zdz)2 + C (zd¯z − ¯zdz)2 − 2k2z¯zdzd¯z � 2k2z¯z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='13) Consider the map from plane to cylinder (τ, ϑ) z = eτ+iϑ, ¯z = eτ−iϑ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='14) the metric becomes ds2 =dr2 r2 − r2e2τ \uf8eb \uf8ec \uf8eddτ 2 + dϑ2 �√ 2C − k �2 k2 \uf8f6 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='15) One can see this is precisely the metric for AdS3 with a conical singularity surrounding the Wilson line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The boundary geometry with Wilson line back-reaction becomes the n-sheet cylinder if we set the defect angle to be 2π(1 − 1 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' Then we can find the relation √ 2C k = (n − 1) + O((n − 1)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='16) Since the Wilson line action generates the n-sheet manifold, the partition function for n- sheet manifold can be written as Zn = log WR(C) = − √ 2CL(xi, xj), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='17) 27 therefore the entanglement entropy can be obtained SEE = lim n→1 1 1 − n log WR(C) = kL(xi, xj), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='18) which is coincide with the RT formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The stress tensor corresponds to Poincar´e AdS3 vanishes, namely L = 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the BTZ black hole, the stress tensor is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' According to the transformation law of the stress-tensor, we can transform the stress tensor to a constant by using a conformal map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' After rescaling the radial coordinate, the BTZ black hole becomes Poincar´e AdS3 geometry with different period of the time direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' For the deformed BTZ black hole, we can perform the following coordinate transformation to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='19) w = (1 − µQ)ξ + Q¯ξ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='19) ¯w = (1 − µ ¯Q)¯ξ + ¯Qξ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='20) r = (1 − µQ)(1 − µ ¯Q)˜r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='21) so that the metric becomes the same as BTZ black hole ds2 = d˜r2 ˜r2 + ˜r2 � dξd¯ξ + 1 ˜r2 � Ldξ2 + ¯Ld¯ξ2� + L ¯L ˜r4 dξd¯ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content='22) One should note that the temperature (the period of Euclidean time) is different from the original BTZ black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfSwk7/content/2301.04435v1.pdf'} +page_content=' The above consideration for the holographic entanglement entropy still holds for BTZ black hole and deformed BTZ black hole.' metadata={'source': 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+size 5505069 diff --git a/19E2T4oBgHgl3EQfiwfE/content/tmp_files/2301.03962v1.pdf.txt b/19E2T4oBgHgl3EQfiwfE/content/tmp_files/2301.03962v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..20b83bb4f195ea4fc1a03f03240a19f761337a70 --- /dev/null +++ b/19E2T4oBgHgl3EQfiwfE/content/tmp_files/2301.03962v1.pdf.txt @@ -0,0 +1,4497 @@ +A Unified Theory of Diversity in Ensemble Learning +Danny Wood +danny.wood@manchester.ac.uk† +Tingting Mu +tingting.mu@manchester.ac.uk† +Andrew M. Webb +andrew@awebb.info† +Henry W. J. Reeve +henry.reeve@bristol.ac.uk∗ +Mikel Lujan +mikel.lujan@manchester.ac.uk† +Gavin Brown +gavin.brown@manchester.ac.uk† +† Department of Computer Science, University of Manchester, UK +∗ Department of Mathematics, University of Bristol, UK +Abstract +We present a theory of ensemble diversity, explaining the nature and effect of +diversity for a wide range of supervised learning scenarios. +This challenge, of +understanding ensemble diversity, has been referred to as the “holy grail” of ensemble +learning, an open question for over 30 years. Our framework reveals that diversity +is in fact a hidden dimension in the bias-variance decomposition of an ensemble. +In particular, we prove a family of exact bias-variance-diversity decompositions, +for both classification and regression losses, e.g., squared, and cross-entropy. The +framework provides a methodology to automatically identify the combiner rule +enabling such a decomposition, specific to the loss. The formulation of diversity +is therefore dependent on just two design choices: +the loss, and the combiner. +For certain choices (e.g., 0-1 loss with majority voting) the effect of diversity is +necessarily dependent on the target label. Experiments illustrate how we can use our +framework to understand the diversity-encouraging mechanisms of popular ensemble +methods: Bagging, Boosting, and Random Forests. +Keywords: ensemble, diversity, bias-variance decomposition, Bregman divergence +1 +Introduction +Ensemble methods have enabled state-of-the-art results for decades: +from early industrial +computer vision (Viola & Jones, 2001) to the deep learning revolution (Krizhevsky et al., 2012), +and inter-disciplinary applications (Cao et al., 2020). An accepted mantra is that ensembles work +best when the individuals have a “diversity” of predictions—often induced by classical methods +such as Bagging (Breiman, 1996), but diversity-encouraging heuristics are rife in the literature +(Brown et al., 2005). Given this, we trust that the combination will “average out” the errors of +the individuals. One reason for the popularity of such methods is clear: the very idea of ensembles +is an appealing anthropomorphism, invoking analogies to human committees, and “wisdom of +the crowds”. Unfortunately, such analogies have limitations. More formal approaches have been +pursued, in particular for quantifying diversity. It is obvious that we do not want all predictions +to be identical; and, it is equally obvious we do not want them to be different just for the sake +of it, sacrificing overall performance. +We want something in-between these two—the so-called +error/diversity tradeoff. However, here we encounter the problem of formally defining “diversity” +1 +arXiv:2301.03962v1 [cs.LG] 10 Jan 2023 + +and its relation to ensemble error. In general, there is no agreement on how to quantify diversity, +except in the limited case of regression with an arithmetic mean ensemble (Krogh & Vedelsby, +1994; Ueda & Nakano, 1996). For classification and other scenarios, there are dozens of proposed +diversity measures (Kuncheva, 2014). A comprehensive theory of ensemble diversity has been an +open problem for over 30 years. +Motivation: +Our primary motivation is to fill this ‘gap’ in current ensemble theory, providing +a solid foundation to understand and study ensemble diversity. +However, there are also many +practical reasons to pursue this. Diverse ensembles can be more computationally efficient than +single large models, with the same generalisation performance (Kondratyuk et al., 2020). Diverse +ensembles are robust against adversarial attacks (Biggio et al., 2011), and can counteract covariate +shift (Sinha et al., 2020). Advantages are also found in important application areas (Cao et al., +2020) and well beyond supervised learning (Carreira-Perpinán & Raziperchikolaei, 2016). +It is +important to note that these use-cases do not follow a common approach: they either adopt some +measurement of diversity picked from historical literature, or propose their own novel metric. There +is, therefore, good reason to pursue a unified theory, where diversity is derived from first principles. +This challenge has proven non-trivial: surveys of progress can be found in Dietterich (2000); Brown +et al. (2005); Zhou (2012), and Kuncheva (2014). Diversity is nowadays referred to as a heuristic +with no precise definition, and, it has been said: +“There is no doubt that understanding diversity is the holy grail in the field of +ensemble learning” (Zhou, 2012, Sec 5.1, page 100). +Summary of our Results: +In contrast to previous efforts which define novel diversity measures, +we take loss functions and decompose them, exposing terms that naturally account for diversity. We +show that diversity is a hidden dimension in the bias-variance decomposition of the ensemble loss. +In particular, we prove exact bias-variance-diversity decompositions, applying for a broad range of +losses, taking a common form: +(expected loss) = (average bias) + (average variance) − (diversity), +where diversity is a measure of member disagreement, independent of the target. +This is an +alternative to known results (Ueda & Nakano, 1996) in the special case of squared loss, but +generalises the formal notion of diversity to many other losses, e.g., the cross-entropy, and +the Poisson regression loss. +A notable exception is the 0-1 loss—where we prove that such a +decomposition cannot hold, for any combiner rule. In spite of this, we are still able to quantify +diversity and measure its effects, with the caveat that the effects are dependent on the target +variable. +Overall, we argue that diversity is best understood as a measure of model fit, in precisely the +same sense as bias and variance, but accounting for statistical dependencies between ensemble +members. +With single models, we have a bias/variance trade-off. +With an ensemble we have +a bias/variance/diversity trade-off—varying both with individual model capacity, and similarities +between model predictions. +2 + +2 +Problem Statement: What is Ensemble Diversity? +One of the earliest works on ensemble methods (at least in the machine learning community) was +Hansen & Salamon (1990). +This work trained multiple neural networks, each with a different +training data subset, and combined them by majority vote. Many subsequent algorithms followed +this “parallel" strategy, notably Bagging (Breiman, 1996), and Random Forests (Breiman, 2001). +The well-known boosting family of algorithms (Schapire et al., 1998) exploit a similar principle, but +construct models sequentially, providing each with data based on the errors of previous models. +These approaches, parallel and sequential (see Figure 1), are the most common schemes to construct +ensembles (Kuncheva, 2014). +Sample +1 +Sample +2 +Sample +M +Model +1 +Model +2 +Model +M +Sample +1 +Sample +2 +Sample +M +Model +1 +Model +2 +Model +M +Data + Labels +Data + Labels +Ensemble +Ensemble +Figure 1: Parallel vs sequential ensemble construction. Both can be seen as creating “diverse" +models in some sense—either implicitly (independently re-sampling the training data), or explicitly +(re-sampling according to the errors of earlier models). +So why do these strategies work? Both can be understood heuristically in terms of “diversity", +in the sense coined by Opitz & Shavlik (1996), referring to differences in generalisation behavior +among a group of models. In a review, Dietterich (2000) explains: +“An accurate classifier is one that has an error rate of better than random guessing +on new x values. Two classifiers are diverse if they make different errors on new +data points.” (Dietterich, 2000) +In this sense, both approaches foster diversity—either implicitly by randomly perturbing the data +for each model, or explicitly by constructing each data set to address the errors of other models +(Brown et al., 2005). +The implicit approach to generating ensemble diversity has been widely +adopted in deep learning; in their recent book, Goodfellow et al. (2016) note that the sources of +randomness in the initialisation and training of deep networks “are often enough to cause different +members of the ensemble to make partially independent errors," so each ensemble member can see +all training data, while still being “diverse". +Given the success of ensembles, there have been many attempts to explain why they work, in terms +of the diversity. Goodfellow et al. (2016, p. 249) wrote “The reason that model averaging works is +that different models will usually not make all the same errors on the test set”. While this is true, +statements like this are not the formal treatment we desire. +3 + +What are we looking for? +A theory of ensemble diversity would ideally have three key +ingredients: +1. a definition of diversity as a measure of disagreement between the ensemble members, +independently of the target variable; +2. this measure should have a clear relation to the overall ensemble error; and, +3. the theory should have a clear relation to previously established results, and expand our +understanding in other learning scenarios. +The first point ensures that diversity can be discussed solely as a property of the ensemble, a +phenomenon in its own right. The second point ensures we can interpret what effect the diversity +has on our ultimate objective: reducing the ensemble error. The third point relates to the only +known scenario where this can be considered a “solved” problem: regression using squared loss, +with an arithmetic mean ensemble. We now review this. +Known results for regression ensembles: +Krogh & Vedelsby (1994) showed that, for an +arithmetic mean combiner, using squared loss, the ensemble loss is guaranteed to be less than or +equal to the average individual loss. Given a target y ∈ R, a member prediction qi(x), and an +ensemble ¯q(x) = +1 +M +�M +i=1 qi(x), we have, +�¯q(x) − y +�2 = +1 +M +M +� +i=1 +�qi(x) − y +�2 − 1 +M +M +� +i=1 +�qi(x) − ¯q(x) +�2. +(1) +The left hand side is the ensemble loss for a single test point (x, y). The first term on the right is +the average individual loss. The second is known as the ambiguity—measuring the disagreement +of individuals, as a spread around the ensemble prediction. Since this term is non-negative, it +guarantees the ensemble loss will be less than or equal to the average loss. This result is often +erroneously cited as the reason why all ensembles work. However, the expression above applies if +and only if we use the squared loss with an arithmetic mean combiner. If we use squared loss with +a different combiner, the result no longer holds. +A deeper understanding came from Ueda & Nakano (1996)—though under the same loss/combiner +assumptions. They extended the bias-variance theory of Geman et al. (1992) to show that the +expected squared loss of the ensemble decomposes into three terms: +ED +� +(q(x) − y)2� += bias(q) + 1 +M variance + +� +1 − 1 +M +� +covar. +(2) +This is the ensemble bias, plus +1 +M times the average variance, and the third term involves the +covariance of model pairs, averaged over all +�M +2 +� pairs of members. Thus, we have a three-way +bias/variance/covariance trade-off, where the covariance term completely captures the notion of +diversity. It is the trade-off that determines the overall expected ensemble loss, where a strongly +negative covariance indicates a diverse ensemble. +As mentioned, the expressions above do not apply beyond squared loss with the arithmetic mean +combiner. +A significant community effort has been directed to find corresponding notions of +diversity for classification problems. We review this next. +4 + +Known results for classifier ensembles: +For classification problems, we might have estimates +of the class probability distribution, or just labels Understanding diversity in these scenarios has +proven more challenging. +An early result by Tumer & Ghosh (1996) demonstrated that the +correlation between pairs of ensemble averaged class probabilities had a simple relationship to the +overall ensemble classification error, at least in a region close to decision boundaries. The analysis +was extended to weighted combinations by Fumera & Roli (2005), under similar assumptions. +Brown (2009) and Zhou & Li (2010) proposed information theoretic analyses, showing that diversity +manifests as both low- and high-order interactions between ensemble members. Buschjäger et al. +(2020) used a Taylor approximation on twice-differentiable losses, showing an exact decomposition +when higher derivatives are zero, e.g. squared loss, but not cross-entropy. Similarly, Ortega et al. +(2022) decomposed upper bounds on losses, again only obtaining an equality for squared loss. +Kuncheva & Whitaker (2003) took another approach, examining diversity measures for their +empirical relationship to the ensemble error. +One strategy they proposed was to choose a +discrepancy metric δ(qi, qj) ∈ R between the predictions of two models at point x, and defining +“diversity” by averaging over all pairs of ensemble members: +diversity(q1, .., qM) = +1 +M(M − 1) +M +� +i=1 +� +j̸=i +δ(qi, qj). +(3) +The diversity measure is then evaluated for its correlation to the overall ensemble performance, +and seen as more successful if it has high correlation, illustrated in Figure 2. +diversity measure A +ensemble improvement +relative to baseline +(0-1 loss) +diversity measure B +Figure 2: Accuracy/diversity for two (hypothetical) diversity measures. Measure B (right) is more +“successful”, as it has stronger correlation to performance improvement. +Several measures (including non-pairwise measures) were explored for both class labels and class +probability distributions, with no single measure proving more successful than any other. Almost +20 years on, novel diversity heuristics and measures are still being proposed, e.g., Jan & Verma +(2019); Rame & Cord (2021); Wu et al. (2021). +Our approach to the problem: +In our work we build on the strong foundation of bias-variance +theory (Heskes, 1998; Pfau, 2013; Wood et al., 2022). This results in exact bias-variance-diversity +decompositions for several ensemble losses, and a deeper understanding of the importance of the +ensemble combination rule. +5 + +3 +A Very Short Introduction to Bias-Variance Decompositions +Consider a training set D = {(xi, yi)}n +i=1 drawn from a random variable D ∼ P(X, Y )n. From +this, we learn a model q(x; D), predicting the conditional mean EY |X=x[Y ]. We adopt the following +notation for expectation over the true data distribution, +EXY +� +· · · +� +def= +� +PX(x) +� +PY |X(y | x) +� +· · · +� +dy dx, +and over training sets: +ED +� +· · · +� +def= +� +PD(D) +� +· · · +� +dD, +where PX and PD are the usual probability density functions. For regression problems, we write +the squared risk of a particular (trained) model, q(x; D), as +R(q) +def= EXY +� +(q(X; D) − Y )2� +. +(4) +Geman et al. (1992) considered the expected risk, i.e., risk in expectation over training sets D drawn +from D, showing it decomposes into three terms: +ED +� +R(q) +� +� +�� +� +expected +risk(q) += EX +� +σ2 +Y |X +� �� � +noise ++ +� +ED [q(X; D)] − EY |X[Y ] +�2 +� +�� +� +bias(q) ++ ED +�� +q(X; D) − ED [q(X; D)] +�2� +� +�� +� +variance(q) +� +. +(5) +The first term, σ2 +Y |X = EY |X[ +� +Y − EY |X[Y ] +�2], is the irreducible noise in the problem. The second +term is the bias—the loss of the expected predictor against the conditional mean EY |X[Y ]. The final +term is the variance, expressing the variation in q due to different training sets. These concepts +are often explained with a dartboard diagram, as in Figure 3. +Figure 3: The classic dartboard analogy for explaining bias and variance. +The bullseye (yellow circle) is the target for a single test point, and each blue dot is a prediction +from a model trained with a different training set. A model with high bias, low variance (e.g., linear +regression) will be insensitive to small training data changes, but have an expected value far from +the target. A model with low bias, high variance (e.g., a regression tree) will have an expected +value that is close to the target, but will be very sensitive to training data changes, meaning any +given model is likely to overfit. Note that D does not have to refer to training data; there is a +decomposition with respect to any initial condition (e.g., initial weights for neural networks). Note +also that we may use the term bias (correspondingly variance) to indicate the value at point x, or +6 + +in expectation over the distribution of X—which of the two is intended will be made clear from +context. +As model capacity increases, bias tends to decrease, and variance tends to increase: the bias-variance +trade-off. +This is a simplified view, which does not always capture more complex underlying +behavior e.g., Belkin et al. (2018). The terms in Equation (5) can be estimated from data (details +in Appendix B) illustrated in Figure 4. +0 +5 +10 +15 +20 +25 +30 +maximum depth +0.0 +0.2 +0.4 +0.6 +0.8 +expected risk +bias + noise +variance +Figure 4: Building regression trees of increasing depth (California Housing data). +Bias-variance decompositions apply for more than just squared loss. +Geman et al. (1992) +is a widely-appreciated result. However, similar decompositions hold for other losses, e.g., the KL- +divergence of class probability estimates (Heskes, 1998), which we illustrate with a “dartboard" for +k = 3 classes in Figure 5. +Figure 5: Bias/variance for the KL divergence. Yellow circle is the target for a test point x. Blue +star is the normalised geometric mean of the model distribution. +The bias/variance terms take different functional forms for each loss. For example, Geman et al’s +original bias term is often referred to as ‘squared bias’, but the square turns out to be an artefact of +using this particular loss, and is not present in other cases. Furthermore, the terms obey a different +geometry, defined by the loss function of interest—this manifests in the ‘expected model’ being +replaced by other forms, e.g., a normalised geometric mean for the KL-divergence. +7 + +4 +A Unified Theory of Ensemble Diversity +Ensemble “diversity” is a popular, but variously defined idea. Bias and variance, on the other hand, +are clear-cut and precisely defined. It therefore makes sense to build stronger bridges between the +two. Our approach is exactly this, revealing diversity as a hidden dimension in the bias-variance +decomposition of an ensemble. We argue that diversity should be considered in exactly the same +manner as bias/variance—simply another aspect of the model fitting process. We first describe +how the ideas apply to the squared loss, then generalise it to other losses. +4.1 +Ensemble Diversity via the ‘Double Decomposition’ Trick +Our approach to understanding diversity is ‘unified’ in the sense that the same methodology can +be applied to numerous different losses. We refer to this as the ‘double decomposition’ trick, which +we will now describe. +The first step is to recognise that the ambiguity decomposition (Krogh & Vedelsby, 1994) is a +special case of the bias-variance decomposition (Geman et al., 1992). We can see this most clearly +by stating the bias-variance decomposition at a single test point (x, y), i.e., omitting the noise term. +All results still apply with noise, but are easier to explain in the noise-free scenario. For brevity, +we omit the dependence on x and D, taking it as understood that the model q is dependent on +both. This gives us a simpler form: +ED +� +(q − y)2� += (ED [q] − y)2 +� +�� +� +bias(q) ++ ED +� +(q − ED [q])2� +� +�� +� +variance(q) +. +(6) +We now just replace each occurrence of the expectation ED [· · ·], with a uniformly weighted +arithmetic mean +1 +M +�M +i=1[· · · ] over a set of M models, +1 +M +M +� +i=1 +(qi − y)2 = +� 1 +M +M +� +i=1 +qi − y +�2 ++ 1 +M +M +� +i=1 +� +qi − 1 +M +M +� +i=1 +qi +�2 +. +(7) +Noting that the ensemble combiner is q = +1 +M +�M +i=1 qi, we simply rearrange the terms of Equation +(7), and obtain the ambiguity decomposition. +� +q − y +�2 += 1 +M +M +� +i=1 +(qi − y)2 +� +�� +� +average loss +− 1 +M +M +� +i=1 +� +qi − q +�2 +� +�� +� +ambiguity +. +(8) +Thus, the ambiguity decomposition is a special case of the bias-variance decomposition, replacing +expectations ED with an arithmetic mean +1 +M +�M +i=1. +We define the double decomposition trick as the successive application of the ambiguity and +bias/variance decompositions. This exposes a natural term quantifying the diversity, specific to +the loss. This methodology is the key contribution of our work, illustrated in Figure 6, to be re- +used throughout the paper with a range of different losses. For squared loss, we apply Equation (8), +then Equation (6), and obtain Theorem 1. Since the derivation is trivial following the proposed +trick, we omit proof detail. +8 + +ED +� +ensemble loss +� +�� +� +� +Apply Equation (8) +ED[ average loss +− +ambiguity ] +Apply Equation (6) +bias ++ +variance +− +diversity +Figure 6: The ‘double decomposition’ trick, shown here for squared loss, but applicable to any loss +which admits a bias-variance decomposition. +Theorem 1 (Bias-Variance-Diversity Decomposition for Squared Loss) For an ensemble +of models q1, . . . , qM, where q = +1 +M +�M +i=1 qi, the expected loss of q decomposes as, +ED +� +(q − y)2 � += +1 +M +M +� +i=1 +(ED [qi] − y)2 +� +�� +� +bias ++ +1 +M +M +� +i=1 +ED +� +(qi − ED [qi])2� +� +�� +� +variance +− ED +� +1 +M +M +� +i=1 +(qi − q)2 +� +� +�� +� +diversity +. +(9) +This has decomposed the expected loss into: the average bias, the average variance, and the +expectation of the ensemble ambiguity. It is this expected ambiguity term that we consider as the +ensemble diversity, which we highlight has the opposite sign to the other terms. In Figure 7 we +estimate the terms1 for Bagged trees on the California housing data. +0 +10 +20 +30 +ensemble size +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +squared error +expected risk +0 +10 +20 +30 +ensemble size +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +average bias + noise +average variance +diversity +Figure 7: Decomposing the expected ensemble loss (Bagging depth 8 regression trees). +The expected risk decreases with M. Looking at the loss components: the bias and variance are +constant—this is as we might anticipate, since the form/capacity of the individuals is constant, it +is only the number of them, M, that we change. In contrast, the diversity increases with M— +subtracting from the expected risk—and the improvement is determined entirely by diversity. This +is of course different if we vary something other than M. +Figure 8 fixes M = 10, but varies +1Note that D can be a joint random variable over data/conditions for each individual—a short discussion on this +is in Appendix C.4. +9 + +tree depth—all three components now change, and we see overall performance is determined by a +bias-variance-diversity trade-off. +Figure 8: Bagging M = 10 trees, varying maximum depth. +Further experiments (with deep/shallow trees and Random Forests) are in Appendix A. For squared +loss, the decomposition can be seen as an alternative to the bias-variance-covariance decomposition +(Ueda & Nakano, 1996), seen in Equation (2). We will compare and contrast these later (Section +5.5.2) once the general case is established. +4.2 +Generalising Bias-Variance-Diversity to Other Losses +As mentioned in Section 3, bias-variance decompositions in a form similar to Geman et al. (1992), +are known for several other losses (Heskes, 1998; Buja et al., 2005; Pfau, 2013; Wood et al., 2022). +We can define a general form, covering all these cases, as follows. +Definition 1 (Generalised Bias-Variance Decomposition) For +a +loss +function +L, +a +generalised bias-variance decomposition is defined, +ED [L(y, q)] +� +�� +� +expected loss += L(y, q∗) +� +�� +� +bias ++ ED [ V(q∗, q) ] +� +�� +� +variance +, +(10) +where V(·, ·) is a non-negative dissimilarity function, and q∗ +def= arg minz ED +� +V(z, q) +� +is the +“centroid" of the model distribution. +A +corresponding +generalised +ambiguity +decomposition +is +induced +by +the +bias-variance +decomposition, and can be stated as follows. +Definition 2 (Generalised Ambiguity Decomposition) For +a +finite +set +of +models +{q1, . . . , qM}, the ambiguity decomposition, under loss function L, is defined, +L(y, q) = 1 +M +M +� +i=1 +L(y, qi) +� +�� +� +average loss +− 1 +M +M +� +i=1 +V(q, qi) +� +�� +� +ambiguity +, +(11) +where q +def= arg minz 1 +M +�M +i=1 V(z, qi). +10 + +These are templates, generalising other decompositions. We can recover Geman et al. (1992) with +L(y, q) = V(y, q) = (y−q)2, where the centroid is q∗ = argminz ED +�(z − q)2� = ED [q]. Or, if L and +V are the KL-divergence of class distributions, we recover Heskes (1998). In this case, the centroid +is different—turning out to be a normalised geometric mean, i.e. q∗ = argminz ED [V(z, q)] = +Z−1 exp(ED [ln q]). The centroid is determined by the form of the dissimilarity, V. These examples +both have V = L. However, in general, V and L do not have to be of the same form—as is the case +for margin losses, discussed in Section 7. +We now present our main result in Theorem 2, a generalised bias-variance-diversity decomposition. +It is derived via the strategy in Figure 6, using Equations (11) and (10). +Theorem 2 (Generalised Bias-Variance-Diversity Decomposition) Consider +a +set +of +models {q1, . . . , qM}, evaluated by a loss function L. +Assuming a bias-variance decomposition +holds in the form of Definition 1, the following generalised bias-variance-diversity decomposition +also holds. +ED [L(y, q)] = 1 +M +M +� +i=1 +L(y, q∗ +i ) +� +�� +� +average bias ++ 1 +M +M +� +i=1 +ED [V(q∗ +i , qi)] +� +�� +� +average variance +− ED +� +1 +M +M +� +i=1 +V(q, qi) +� +� +�� +� +diversity +, +(12) +where q∗ def= arg minz ED +� +V(z, q) +� +and the combiner is q +def= arg minz 1 +M +�M +i=1 V(z, qi). +There is now a bias/variance/diversity trade-off. As individual models increase in capacity, +their average bias decreases. Without regularisation, their average variance would increase. These +determine only part of the ensemble behavior. The final part is the diversity. A critical point +here is diversity always subtracts from the expected risk. This is not to say that greater diversity +always reduces expected risk—it only reduces it given a fixed bias and variance. Ultimately, the +three-way trade-off of bias/variance/diversity is what determines the overall ensemble performance. +It is worth highlighting that diversity is defined similarly to bias/variance, involving an expectation +over D, as opposed to being a property of a single training run. We also note, this applies for both +dependent and independent training schemes—discussion in Appendix C.4. +The ensemble combiner q is a centroid. We refer to this as the centroid combiner rule, the +minimiser of the average dissimilarity, q +def= arg minz 1 +M +�M +i=1 V(z, qi). Krogh & Vedelsby (1994) +assumed the combiner was an arithmetic mean, q = +1 +M +�M +i=1 qi. Audhkhasi et al. (2013) and Jiang +et al. (2017) proposed generalisations of the ambiguity decomposition for classification, though +both still assumed must be an arithmetic mean. In contrast, here the combiner is defined in terms +of the dissimilarity V, itself a consequence of L. +4.3 +Summary +We presented a framework to understand diversity, applicable for any loss where a bias-variance +decomposition holds in the form of Definition 1. This revealed diversity as a measure of model +fit, in precisely the same sense as bias/variance, with a concrete application for squared loss. In +Section 5 we present an application with Bregman divergences, covering many losses as special +cases. In Section 6 we show necessary modifications to enable an understanding for 0-1 loss, and +discuss limitations in Section 7. +11 + +5 +Diversity for Bregman Divergences +In this section, we will apply the framework developed in Section 4 to the class of Bregman +divergences (Bregman, 1967), which covers many popular losses as special cases. +5.1 +The Basics of Bregman Divergences +A Bregman divergence Bφ (p, q) is defined in terms of a generator function, φ. Let φ : S → R be a +strictly convex function defined on a convex set S ⊆ Rk, such that φ is differentiable on ri(S)—the +relative interior of S. The Bregman divergence Bφ : S × ri(S) → R+ is defined, +Bφ (p, q) +def= φ (p) − φ (q) − ⟨∇φ (q) , (p − q)⟩, +(13) +where ⟨·, ·⟩ denotes an inner product, and ∇φ(q) denotes the gradient vector of φ at q. Different +choices of φ lead to different losses. With φ (q) = q2, the gradient vector ∇φ(q) is a scalar derivative +dφ(q)/dq = 2q, and we recover a squared loss, Bφ (p, q) = (p − q)2. +q = 0.3 +p = 1.0 +1.0 +0.0 +�(p) +B�(p, q) = 0.49 +�(q) +Bφ (p, q) += +φ (p) − φ (q) − ⟨∇φ (q) , (p − q)⟩ += +p2 − q2 − ⟨2q, (p − q)⟩ += +p2 − q2 − 2pq + 2q2 += +p2 + q2 − 2pq += +(p − q)2 +Figure 9: Bregman divergence illustrated for the generator φ (q) = q2. For this example, we have +a divergence Bφ (p, q) = (p − q)2 = (1.0 − 0.3)2 = 0.49. +Alternatively, we can take a vector q ∈ Rk−1, for a k-class problem. Note, this is not a probability +vector summing to one. It is, however, the minimal description of the distribution, as the kth class +probability is 1 − � +c q(c). With a particular generator (see final row of Table 1) we recover the the +KL-divergence between the distributions in Rk. +Loss function +Bφ (p, q) +Generator φ (q) +Domain S +Squared loss +(p − q)2 +q2 +q ∈ R +Itakura-Saito +p +q − ln p +q − 1 +− ln q +q ∈ R+ +Poisson loss +p ln p +q − (p − q) +q ln q − q +q ∈ R+ +KL-divergence +� p(c) ln p(c) +q(c) + +� +1 − � p(c)� +ln 1−� +p(c) +1−� +q(c) +� q(c) ln q(c) + +� +1 − � q(c)� +ln +� +1 − � q(c)� +q ∈ [0, 1]k−1 +s.t. � +c q(c) ≤ 1 +Table 1: Common loss functions and their Bregman generators. +12 + +5.2 +Relating Bias, Variance, and Ambiguity for Bregman Divergences +As discussed in the previous section, our framework requires the existence of a bias-variance +decomposition for the loss at hand. In this case of Bregman divergences, Pfau (2013) proved a +bias-variance decomposition, which is written for a single point (x, y) as: +ED [Bφ (y, q)] = Bφ (y, q∗) + ED [Bφ (q∗, q)] , +(14) +where the value of q∗ can be obtained in closed-form as +q∗ +def= arg min +z +ED +� +Bφ (z, q) +� += [∇φ]−1 � +ED [∇φ (q)] +� +. +(15) +This definition of q∗ is known in the information geometry literature, as a left Bregman centroid +(Nielsen & Nock, 2009). If φ is the generator for a squared loss, then q∗ = ED [q], the expected +model. With other losses, the form of q∗ changes—we detail several examples in Appendix C.1. +Considering Equation (14), the corresponding ambiguity decomposition can be written as follows, +again simply replacing expectations by finite averages and rearranging terms. +Theorem 3 (Bregman Ambiguity Decomposition) For a target label y ∈ S and a set of +predictions q1, . . . , qM ∈ ri(S), +Bφ (y, q) = 1 +M +M +� +i=1 +Bφ (y, qi) − 1 +M +M +� +i=1 +Bφ (q, qi) +(16) +where q = [∇φ]−1� 1 +M +� +i ∇φ(qi) +�. +For the ambiguity decomposition to apply, the ensemble combiner q is constrained to be of the +same form as q∗. When combining ensemble members, we will refer to this form as the Bregman +centroid combiner. Before we present the formulation of diversity in this case, we offer a brief +discussion on this combiner, clarifying the relation to existing methods. +Definition 3 (Bregman Centroid Combiner) The Bregman centroid combiner is the left +Bregman centroid for a set of M predictions. +This is the minimizer of the average divergence +from all points in the set. The centroid combiner q for {q1, . . . qM} is +q +def= +arg min +z +1 +M +M +� +i=1 +Bφ (z, qi) += +[∇φ]−1 � 1 +M +M +� +i=1 +∇φ(qi) +� +. +(17) +The Bregman centroid combiner is a generalised f-mean, with f = ∇φ, also known as a Kolmogorov +mean or quasi-arithmetic mean. For φ(q) = q2, this is the arithmetic mean of the points, i.e., the +centre of mass. +However, other φ generators define a non-linear transformation, meaning the +centroid and the centre of mass are different in general. The centroid combiner reproduces several +known combiners as special cases, shown in Table 2. Using this principle comes with a distinct +advantage—through the ambiguity decomposition, we are guaranteed that the ensemble loss will be +less than or equal to the average loss, generalising the well-known squared loss case. An equivalent +definition was considered by Gupta et al. (2022) under the assumption of i.i.d. models. Our analysis +both complements and extends this by removing the assumption, and more fully characterising the +properties of ensembles using this combination rule. +13 + +Loss function +Centroid Combiner +Name +Squared loss +1 +M +�M +i=1 qi +Arithmetic mean +Poisson regression loss +�M +i=1 q +1 +M +i +Geometric mean +KL-divergence +Z−1 �M +i=1 +� +q(c) +i +� 1 +M +Normalised geometric mean +Itakura-Saito loss +1 +�� +1 +M +�M +i=1 +1 +qi +� +Harmonic mean +Table 2: Centroid combiners (i.e., left Bregman centroid of the ensemble) for various losses. +The centroid combiner can be understood as an ensemble averaging operation, but in a new +coordinate system, where the mapping between coordinate systems is defined by the gradient of the +Bregman generator with respect to its argument, ∇φ(q). This is illustrated in Figure 10 for the +KL-divergence. +q +∂ +∂qi φ(qi) +η ≈ 2.16 +q ≈ 0.896 +q1 = 0.7 +η1 ≈ 0.8473 +q2 = 0.97 +η2 ≈ 3.476 +Figure 10: Ensemble averaging in the geometry defined by the KL-divergence. +We use notation q for the primal coordinate system, and η for the dual coordinate system. In this +simple illustration we are predicting a single probability p ∈ (0, 1). The primal-dual mapping is +the gradient ηi = +∂ +∂qi φ(qi) = ln +qi +1−qi , plotted as the blue curve. Two points in the primal {q1 = +0.7, q2 = 0.97} are mapped to the dual {η1 ≈ 0.8473, η2 ≈ 3.476}, then combined via arithmetic +mean (η ≈ 2.16), and finally mapped back by the inverse operation q = exp(η)/(1+exp(η)) ≈ 0.896. +The centroid combiner is therefore an arithmetic mean ensemble in the dual coordinate system, +which is equivalent to the centroid of the models in the primal coordinate system, using the Bregman +divergence as the measure of dissimilarity. In the case of KL divergence, this is equivalent to a +normalised geometric mean in the primal coordinate system. The primal coordinate system for KL +is a probability simplex—Figure 11 shows this for the 3-class case, i.e., a Bregman divergence on +constrained vectors in R2. +14 + +class 3 +class 1 +class 2 +Figure 11: Combining M = 4 predictions in the probability simplex. +The centroid combiner is shown as the blue star, minimising average KL-divergence from all +predictions. The arithmetic mean is also shown (pink star). Note that points are connected in +the simplex not by straight lines, but by dual geodesics defined via the generator φ (Nielsen & +Nock, 2009). A different φ (and therefore a different loss) would result in a different primal-dual +mapping, and thus a different ensemble combiner rule. +5.3 +A Bias-Variance-Diversity Decomposition for Bregman Divergences +We can now apply the double decomposition trick, as in Section 4. Doing so shows that an expected +Bregman divergence (when using the centroid combiner) decomposes as follows. +Theorem 4 (Bregman Bias-Variance-Diversity decomposition) +For an ensemble q1(X; D), . . . , qM(X; D), let q∗ +i be the left Bregman centroid of qi (i.e., q∗ +i +def= +[∇φ]−1 (ED [∇φ(qi)])) and define q +def= [∇φ]−1 � +1 +M +�M +i=1 ∇φ (qi) +� +. Then we have the decomposition, +ED [EXY [Bφ (Y, q)]] = +EXY +� +Bφ +� +Y, Y +� +� +�� +� +noise ++ 1 +M +M +� +i=1 +Bφ +� +Y, q∗ +i +� +� +�� +� +average bias ++ 1 +M +M +� +i=1 +ED [Bφ (q∗ +i , qi)] +� +�� +� +average variance +− ED +� +1 +M +M +� +i=1 +Bφ (q, qi) +� +� +�� +� +diversity +� +, +where Y = EY|X [Y]. +Examples for different losses (i.e., different Bregman generators) are shown in Table 3. One point +that this makes clear is that the mathematical formulation of diversity is specific to the loss function. +15 + +Expected Ensemble Loss +Average Bias +Average Variance +Diversity +Squared +EY +� +ED +� +(q − Y )2�� +1 +M +M +� +i=1 +(q∗ +i − Y )2 +1 +M +M +� +i=1 +ED +� +(qi − q∗ +i )2� +ED +� 1 +M +M +� +i=1 +� +qi − q +�2 � +KL-divergence (Bernoulli) +EY +� +ED +� +Y ln Y +q + (1−Y ) ln 1−Y +1−q +�� +1 +M +M +� +i=1 +Y ln Y +q∗ +i + (1 − Y ) ln 1−Y +1−q∗ +i +1 +M +M +� +i=1 +ED +� +q∗ +i ln +q∗ +i +qi + (1 − q∗ +i ) ln +1−q∗ +i +1−qi +� +ED +� +1 +M +M +� +i=1 +q ln q +qi + (1 − q) ln 1−q +1−qi +� +KL-divergence (Multinoulli) +EY [ED [K(Y || q)]] +1 +M +M +� +i=1 +K(Y || q∗ +i ) +1 +M +M +� +i=1 +ED [K(q∗ +i || qi)] +ED +� +1 +M +M +� +i=1 +K(q || qi) +� +Itakura-Saito +EY +� +ED +� Y +q − ln Y +q − 1�� +1 +M +M +� +i=1 +Y +q∗ +i +− ln Y +q∗ +i +− 1 +1 +M +M +� +i=1 +ED +� +q∗ +i +qi − ln q∗ +i +qi − 1 +� +ED +� +1 +M +M +� +i=1 +� +q +qi − ln q +qi − 1 +�� +Poisson +EY +� +ED +� +Y ln Y +q − (Y − q)�� +1 +M +M +� +i=1 +� +Y ln Y +q∗ +i +− (Y − q∗ +i ) +� +1 +M +M +� +i=1 +� +ED [qi] − q∗ +i +� +ED +� +1 +M +M +� +i=1 +qi − +M +� +i=1 +q1/M +i +� +Table 3: Bias, variance and diversity terms under different Bregman divergences. In all cases, the expectation over p(x) is omitted and +the expressions given are for a single x. Refer to Table 5 for definitions of the left Bregman centroid q∗. +16 + +5.4 +Empirical Demonstration for the Cross-Entropy Loss +In this section we show some illustrative experiments with the cross-entropy loss. +5.4.1 +Estimating Bias, Variance, and Diversity +Theorem 4 is particularly powerful in that it applies across a range of losses. In Table 3 we saw the +KL-divergence, but we can also extend this with little effort to a decomposition for the ubiquitous +cross-entropy loss. +Theorem 5 Let y be a one-hot class vector of length k, and q ∈ Rk be a model’s prediction of the +class distribution. Define a set of such models {qi}M +i=1, and their combination q as their normalised +geometric mean. The following decomposition holds. +−ED [y · ln q] +� +�� +� +expected +cross-entropy += − 1 +M +M +� +i=1 +y · ln q∗ +i +� +�� +� +average bias ++ 1 +M +M +� +i=1 +ED [K(q∗ +i || qi)] +� +�� +� +average variance +− ED +� +1 +M +M +� +i=1 +K(q || qi) +� +� +�� +� +diversity +, +(18) +Here, the centroid combiner q is a normalised geometric mean (see Appendix C.3 for more detail). +With neural network ensembles, this is equivalent to averaging the network logits, followed by a +softmax—a well-established practice, e.g., Hinton et al. (2015). In Figure 12 we compare Bagging +of single-layer MLPs on MNIST, using small networks of 20 nodes, versus larger networks with +100 nodes. The expected risks decompose into bias/variance/diversity components, and we observe +similar patterns as we saw with the squared loss. Further experiments can be found in Appendix A. +0 +5 +10 +15 +20 +ensemble size +0.1 +0.2 +0.3 +Expected Risk +Expected Risk +Small Network +Larger Network +0 +5 +10 +15 +20 +ensemble size +0.00 +0.05 +0.10 +0.15 +0.20 +Small Network +average bias + noise +average variance +diversity +0 +5 +10 +15 +20 +ensemble size +Larger Network +Figure 12: Decomposing expected ensemble cross-entropy for Bagging small/larger MLPs. +Overall, the ensemble of larger networks have performed better, and the reason for this can be +explained by examining the expected loss components. The larger networks are can be observed to +have both lower average variance2 and lower average bias. Since we are varying ensemble size, the +average bias/variance terms remain constant, and the only factor that changes with M (in both +ensembles) is the diversity, which converges closer to the variance in the case of larger networks. +2The lower variance is counter to the ML folklore that increasing model capacity should also increase variance. It +is, however, consistent with recent observations (Yang et al., 2020). +17 + +5.4.2 +Examining the Correlation of Diversity and Classification Error +Figure 2 showed a toy “error/diversity” scatter plot, in the style popularised by Kuncheva & +Whitaker (2003). In such a plot, each point is one ensemble, showing its performance improvement +(0-1 loss), against a diversity measure. A higher correlation is seen to be a more successful diversity +measure, as it explains the performance improvement. Our framework defines diversity specific to +each loss function. For cross-entropy, the diversity is written in Equation (18), defined separately +below. +Definition 4 Diversity, when using cross entropy loss, is defined as follows. +diversity(q1, ..., .qM) = ED +� +1 +M +M +� +i=1 +K(q || qi) +� +(19) +Figure 13 shows a scatter plot corresponding to the experiment in Figure 12 on MNIST with +Bagged neural networks, combined via a normalised geometric mean. Further experiments are in +Appendix A. +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +diversity +0.005 +0.010 +0.015 +0.020 +0.025 +average individual error +- ensemble error +M = 2 +M = 3 +M = 4 +M = 5 +M = 20 +M = 2 +M = 3 +M = 4 +M = 5 +M = 20 +Ensemble of Smaller Networks +Ensemble of Larger Networks +Figure 13: Diversity plot for Bagged MLPs. +The x-axis is the diversity, estimated on a +validation set. The y-axis is the difference of +error rates, on a final test set—the average +individual error, minus the ensemble error, i.e., +the gain of the ensemble, over the average +individual. +We +see +a +strong +correlation +in +both +configurations. +The smaller networks (blue +triangles, Pearson’s r2 = 0.998) have greater +diversity than the larger networks (orange +circles, +r2 += +0.992). +The plot must be +read in the context that overall, the larger +networks +significantly +outperformed +the +smaller networks—it simply shows that the +performance came from more powerful base +models, as opposed to their diversity. +One might wonder why there is such a strong +correlation in both cases. +If we remember +the alternative view of the same experiment, +Figure +12, +we +see +that +bias/variance +are +constant, and it is only the diversity that changes. When any change is observed in the overall +ensemble cross-entropy, we know it is caused by a change in diversity. Therefore, if we can assume +strong correlation between the ensemble cross-entropy and the 0-1 loss, then there will be a similar +strong correlation between diversity and 0-1 loss. +We can do the same for decision tree classifiers, even though they output only class labels, as +estimated class probabilities can be obtained through various schemes. We use a simple method, +counting label frequency in leaf nodes, and thus can evaluate the cross-entropy diversity. +In +Figure 14 we compare Bagging decision trees (unlimited depth) and Random Forests on MNIST. +In both cases the trees are combined by obtaining probabilities and combining via normalised +18 + +geometric mean. We use the same procedure as before: diversity estimated on validation data, and +the 0-1 loss measured on a final test set. +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +diversity +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +average individual error + + ensemble error +M = 2 +M = 2 +M = 3 +M = 3 +M = 4 +M = 4 +M = 5 +M = 5 +M = 20 +M = 20 +bagging +random forests +Figure 14: Ensembles of tree classifiers. +Again we see strong correlation of diversity and +performance gain. +Bagging has a correlation +r2 ≈ 0.996, whilst Random Forests has r2 ≈ +0.999. +For a fixed M, +we can compare +corresponding points, where the only difference +is the additional split-point randomisation of +the forest. At M = 20, RF provides a reduction +in generalisation error of ≈ 14.5%, versus only +≈ 9% for Bagging. It interesting to note this +is solely due to increased diversity generated by +random feature splits. +We might now wonder, with this diversity +measure, will we always see a strong correlation +between diversity and reduction in 0-1 loss? +The answer is no, for a very good reason that +highlights a critical point in our understanding +of diversity. +In Figure 15 we fix at M = 10 bagged trees, and +vary their depth. The expected loss reduces— +however, now it is not solely due to diversity. +Now, the bias and variance also change rapidly, +and the correlation of 0-1 loss/diversity is much lower. +2 +4 +6 +8 +10 +max depth +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +expected risk +2 +4 +6 +8 +10 +max depth +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +average bias + noise +average variance +diversity +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +diversity +0.07 +0.08 +0.09 +0.10 +0.11 +0.12 +average individual error + + ensemble error +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Figure 15: Bagging M = 10 trees, varying depth, and correlation is now r2 ≈ 0.59. +When varying any other parameter than M, one should not expect to see a strong correlation of +performance improvement and diversity. This is because, if we vary any parameter that alters +individual capacities, then the average bias/variance also changes, and diversity is not the only +factor in play. The overall performance is decided by a 3-way trade-off, just as there is a 2-way +trade-off of bias/variance in single models. It would be interesting to explore this with ensembles +of very deep neural networks, where the bias/variance trade-off seems to not act as classical theory +predicts (Belkin et al., 2018). +19 + +5.5 +Further Properties of the Decomposition +In this section, we further explore properties of the decomposition we have proposed. First, we +consider the properties of homogeneous vs heterogeneous ensembles, then two scenarios of interest— +the relation to the bias-variance-covariance decomposition (Ueda & Nakano, 1996), and the common +practice of averaging class probability estimates. +5.5.1 +Homogeneous vs Heterogeneous Ensembles +An ensemble is said to be heterogeneous if the individual members are from different model families, +or homogeneous if they are the same. The expected divergence of the ensemble from the target has +a bias-variance decomposition: +ED [Bφ (y, q)] +� +�� +� +expected ensemble loss += Bφ (y, q∗) +� +�� +� +ensemble bias ++ ED [Bφ (q∗, q)] +� +�� +� +ensemble variance +. +(20) +These ensemble bias/variance terms can be related that of the individual models, {qi}M +i=1. +Theorem 6 The ensemble bias and ensemble variance can be re-written as: +Bφ (y, q∗) +� +�� +� +ensemble bias += +1 +M +M +� +i=1 +Bφ (y, q∗ +i ) +� +�� +� +average bias +− ∆, +(21) +ED [Bφ (q∗, q)] +� +�� +� +ensemble variance += ∆ + +1 +M +M +� +i=1 +ED [Bφ (q∗ +i , qi)] +� +�� +� +average variance +− ED +� +1 +M +M +� +i=1 +Bφ (q, qi) +� +� +�� +� +diversity +, +(22) +where the common term is ∆ = +1 +M +�M +i=1 Bφ (q∗, q∗ +i ), referred to as the model “disparity”, accounting +for diversity in the model families. +Equation (21) can be proven by applying Theorem 3 to a set of centroid models, {q∗ +i }M +i=1. +Equation (22) can be proven similarly, applying Theorem 4 but substituting y = q∗. +If the models comprising the ensemble are all from the same family (i.e., ‘homogeneous’, as is +common with Bagging/Random Forests) then q∗ +i = q∗ +j = q∗, meaning ∆ = 0, and we can draw +some conclusions: +• the ensemble bias is equal to the average bias, i.e., there is no reduction in bias. +• the ensemble variance is guaranteed to be less than or equal to the average variance: the +amount by which the ensemble variance is reduced is exactly the diversity. +• the diversity is upper-bounded by the average variance. +Alternatively, if ∆ > 0 (as we may expect to occur with boosting) then the ensemble is +heterogeneous. In this case it can be noted that the ensemble bias is always reduced. However, we +can make no such simple statement about the ensemble variance, since it has both the addition of +the disparity, and the subtraction of the diversity. +20 + +5.5.2 +Relation to the Bias-Variance-Covariance Decomposition +For the case of squared loss, our decomposition in Equation (1) can be contrasted with the bias- +variance-covariance decomposition of Ueda & Nakano (1996), which states: +ED +� +(q − y)2 � += +(23) +(ED [q] − y)2 +� +�� +� +bias(q) ++ 1 +M +1 +M +M +� +i=1 +ED +� +(qi − ED [qi])2� +� +�� +� +1/M × variance ++ 1 +M2 +� +i,j +ED [(qi − ED [qi])(qj − ED [qj]))] +� +�� +� +(1−1/M) × covariance +. +This decomposition relies on a simple property for the variance of linear combinations of random +variables: +V ar(aX1 + bX2) = a2V ar(X1) + b2V ar(X2) + 2abCov(X1, X2), +(24) +where X1, X2 represent two model outputs and aX1 + bX2 represents the ensemble combination. +When the combination rule is non-linear, this property (and hence this as a route to understand +diversity) no longer applies. +The covariance can be either positive or negative. Our diversity term, however, is always non- +negative, growing with more disagreement around the ensemble decision. +The covariance is +a fundamentally pairwise computation—it is likely that this form inspired the many published +pairwise diversity measures (Kuncheva, 2014). Diversity in the Bregman case is written in a non- +pairwise manner—the expected average deviation around the ensemble prediction. We conjecture +that this term cannot be expressed as solely pairwise operations, implying that pairwise measures +may be fundamentally limited. +Further differences are found in examining the bias components of each decomposition. Ours is the +average individual bias, whereas Ueda & Nakano’s is the ensemble bias: +bias += +1 +M +M +� +i=1 +(ED [qi] − y)2, +(25) +bias(q) += +(ED [q] − y)2. +(26) +Ueda & Nakano observed a re-writing of their term: (ED [q] − y)2 = ( 1 +M +�M +i=1[ED [qi] − y])2, and +described this as “the square of the average biases”. +This follows from language in statistical +estimation theory, where E[ˆθ] − θ is the bias of ˆθ as an estimate of a population value, θ. However, +we remind the reader that the square is an artefact of using the squared loss function. This square +is not present in generalised forms of the bias-variance decomposition e.g., Pfau (2013). Thus, +(ED [q] − y)2 should be referred to as simply the “bias of the ensemble”. The difference between +bias and bias(q) is however an interesting quantity. With some simple algebra, we can re-write the +ensemble bias term as follows: +bias(q) = bias − +1 +M2 +� +i,j +(ED [qi] − ED [qj])2. +(27) +This shows that their term is in fact made up of two components: the average of the individual +biases, and the disparity term introduced above. If the models are homogeneous, i.e., of the same +family, then this term will be zero. +21 + +5.5.3 +Cross Entropy Loss: Averaging estimates of class probabilities +When combining class probability estimates, a very popular strategy is to take their arithmetic +mean, e.g., Lakshminarayanan et al. (2017), but, if we use the cross-entropy, this is not the centroid +combiner. We might wonder what effect this has. The proposition below demonstrates that the +cross-entropy loss of the ensemble is still guaranteed to be less than the average loss of its members, +but the ambiguity becomes dependent on the target. +Proposition 7 Assume a target probability vector, y ∈ Rk, and a set of models {qi}M +i=1 combined +by averaging, i.e. q† = +1 +M +�M +i=1 qi, then the cross-entropy of q and y is +−y · ln q† +� +�� +� +ensemble cross-entropy += +− 1 +M +M +� +i=1 +y · ln qi +� +�� +� +average cross-entropy +− +k +� +c=1 +y(c) ln +1 +M +�M +j=1 q(c) +j +��M +i=1 q(c) +i +� 1 +M +� +�� +� +ambiguity (target-dependent) +, +(28) +where the second term is non-negative, thus the ensemble loss is guaranteed less than or equal to +the average individual loss. +This property can be observed without the framework we have presented thus far, by taking the +difference between −y · ln q† and − 1 +M +�M +i=1 y · ln qi. Proposition 7 was observed independently +by Ivaşcu et al. (2021), for the case of two classes. However, only with our framework can we +identify that the normalised geometric mean is the necessary combiner to make the final term +target-independent. If we take the expectation of this with respect to D, we have the following +result, proved in Appendix C.5. +Proposition 8 (Diversity for Averaged Probabilities is target-dependent) +Let q† = +1 +M +�M +i=1 qi, with qi ∈ [0, 1]k. The expected cross-entropy admits the decomposition: +ED +� +−y · ln q†) +� += +− 1 +M +M +� +i=1 +y · ln q∗ +i +� +�� +� +average bias ++ +1 +M +M +� +i=1 +ED [K(q∗ +i || qi)] +� +�� +� +average variance +− ED +� +��� +k +� +c=1 +y(c) ln +1 +M +�M +j=1 q(c) +j +��M +i=1 q(c) +i +� 1 +M +� +��� +� +�� +� +dependency +. +The final term is now dependent on the target y. +For this reason we avoid using the name +“diversity”, and instead refer to it as a “dependency” term. +We emphasise that we make no +claims about the empirical superiority of one combiner versus another. We simply observe that the +centroid combiner is the only case where diversity is independent of y. +Gupta et al. (2022) also studied properties of ensemble bias/variance with an arithmetic mean +combiner, showing that (under an i.i.d. +model assumption), the ensemble variance is always +reduced. At the same time, they raised a concern, that this may potentially increase the ensemble +bias (above the average bias), dependent on the label distribution. Our proposition adds insight: +the overall expected loss will always be less than the average. Thus, even if there is an increase in +ensemble bias, it is always more than compensated by the reduction in ensemble variance, leading +to lower overall expected loss. +22 + +6 +Diversity for the 0-1 Loss +In this section we show the necessary modifications to our framework, that will enable a stronger +understanding of diversity in the case of 0-1 loss, with particular focus on majority voting ensembles. +6.1 +The Nature of Bias/Variance for the 0-1 Loss is very different +The 0-1 loss is defined L0/1 : S ×S → {0, 1} over a finite set S such that L0/1(y, q) = 1, when q ̸= y, +or 0 otherwise. That is, a loss of 1 when q is incorrect, and 0 otherwise. With this definition, the +quantities we might intuitively understand as bias and variance, do not sum to the classification +error, discussed at length by Domingos (2000), i.e., +ED +� +L0/1(y, q) +� +̸= L0/1(y, q∗) + ED +� +L0/1(q∗, q) +� +. +(29) +where q∗ = argminz∈S ED +� +L0/1(z, q) +� +is the modal value of the distribution. Many authors tried +to find alternative decompositions, with much debate on what axioms the terms should obey, e.g., +Kohavi et al. (1996); James & Hastie (1997); Heskes (1998). Given this literature, one might be +tempted to say we just need to keep searching for the “right” definitions of bias and variance, that +will then sum to the expected loss. In fact it turns out that a definition of bias/variance fitting the +abstract form in Definition 1 does not exist. We present this in the Theorem below, with proof in +Appendix D—to the best of our knowledge, this is the first formal proof of this fact. +Theorem 9 (Non-existence of a Bias-Variance Decomposition for 0-1 loss) The 0-1 loss +cannot be decomposed as a bias term plus a variance term, where the bias is the loss of some +deterministic model q∗ derived from the distribution of q and the variance term is independent of +the label y. I.e., there is no V and q∗ such that in general +ED +� +L0/1(y, q) +� +� +�� +� +expected loss += L0/1(y, q∗) +� +�� +� +bias ++ V[q] +� �� � +variance +. +(30) +We note that this is an even more general form than the bias-variance decompositions stated earlier, +in that the q∗ is not required to be a centroid. +Corollary 10 (Non-existence of an ambiguity decomposition for 0-1 loss) Given a set of +models q1, .., qM which predict labels drawn from a finite set S, there exists no rule for constructing +q and function V, independent of y, such that +L0/1(y, q) = 1 +M +M +� +i=1 +L0/1(y, qi) − V({qi}M +i=1) +(31) +This has significant implications. In many works (Wu et al., 2021; Kuncheva & Whitaker, 2003) +authors have attempted to define a measure of diversity that correlates with the improvement in +ensemble accuracy (e.g., Q-statistics). The above shows that if we quantify the improvement as +the difference between the ensemble loss and the average individual loss, this quantity cannot be +expressed independently of the target variable, for any combiner rule. This is of course disappointing; +however, there is still a way forward. +23 + +6.2 +Understanding the Effects of Bias, Variance, and Diversity +James & Hastie (1997) present an insightful viewpoint: that we should not be interested in the +bias/variance of 0-1 loss for their own sake, but for their effect on the expected loss. +If we had a model q that was constant with variations in D, the loss will also be constant. But if +the model varies at all, it will cause a change in the loss. Therefore the effect of this model variance +is visible in how the loss changes. Compare the expected loss of q (which varies with D), and the +loss of q∗ (which is constant with D). The variance-effect is +variance-effect = EY +� +ED +� +L0/1(Y, q) +�� +− EY +� +L0/1(Y, q∗) +� +, +(32) +i.e., their difference. The bias-effect is defined similarly, and accounts for the fact that there may +be noise in the target. The bias-effect3 is the expected change in the loss, when using the centroid +model q∗, compared to using the Bayes prediction Y ∗, +bias-effect = EY +� +L0/1(Y, q∗) − L0/1(Y, Y ∗) +� +. +(33) +Combining these, they derive a decomposition as follows. +Theorem 11 (Bias/Variance Effect decomposition, James & Hastie (1997)) +ED +� +EY +� +L0/1(Y, q) +�� += +(34) +EY +� +L0/1(Y, Y ∗) +� +� +�� +� +noise ++ EY +� +L0/1(Y, q∗) − L0/1(Y, Y ∗) +� +� +�� +� +bias-effect ++ EY +� +ED +� +L0/1(Y, q) − L0/1(Y, q∗) +�� +� +�� +� +variance-effect +. +where Y ∗ = argminz∈S EY +� +L0/1(Y, z) +� +and q∗ = argminz∈S ED +� +L0/1(z, q) +� +. +For some losses, the effects of bias/variance coincide with the quantities themselves, and the above +reduces to the familiar bias-variance decomposition. However, for the 0-1 loss, the effects and the +terms themselves are different. +We note again that for any bias-variance decomposition (including James & Hastie), we can state +a corresponding ambiguity decomposition. As before, we replace expectations by finite averages, +and evaluate at a single target y. +Proposition 12 (Ambiguity-Effect Decomposition) For an ensemble q1, . . . , qM with any +combiner rule q, and correct label y ∈ S +L0/1(y, q) = 1 +M +M +� +i=1 +L0/1(y, qi) +� +�� +� +average loss +− 1 +M +M +� +i=1 +� +L0/1(y, qi) − L0/1(y, q) +� +� +�� +� +ambiguity-effect +. +(35) +The proof of this is trivial, but the result is powerful. It generalises the ambiguity decomposition to +the 0-1 loss, while acknowledging Corollary 10, that the difference between the ensemble loss and +the average loss cannot be expressed independently of the target. The ambiguity effect measures the +average change in the loss, when using individual members instead of the ensemble. We now use +the double-decomposition trick to derive the effect of diversity in 0-1 loss. Using the results above, +we apply the double decomposition as illustrated in Figure 16. +3James & Hastie (1997) actually use the term “systematic effect", while we refer to it as the “bias-effect" to more +accurately reflect its role in our framework, and relation to other decompositions. +24 + +ED +� +ensemble loss +� +�� +� +� +Apply Equation (35) +ED[ average loss +− +ambiguity-effect ] +Apply Equation (34) +bias-effect ++ +variance-effect +− +diversity-effect +Figure 16: The double decomposition trick using James & Hastie (1997). +Theorem 13 (Bias-Variance-Diversity effect decomposition) Given a loss function L : S × +S → R+ and an ensemble of models q1, . . . , qM, where q is the majority vote. +EY +� +ED +� +L0/1(Y, q) +�� += +EY +� +L0/1(Y, Y ∗) +� +� +�� +� +noise ++ +1 +M +M +� +i=1 +EY +� +L0/1(Y, q∗ +i ) − L0/1(Y, Y ∗) +� +� +�� +� +average bias-effect ++ 1 +M +M +� +i=1 +ED +� +EY +� +L0/1(Y, qi) − L(Y, q∗ +i ) +�� +� +�� +� +average variance-effect +− ED +� +EY +� +1 +M +M +� +i=1 +� +L0/1(Y, qi) − L(Y, q) +��� +� +�� +� +diversity-effect +. +We obtained three terms: the effects of average bias, average variance, and diversity. The diversity- +effect is dependent on the target Y , an unavoidable property of the 0-1 loss. +6.3 +Estimating the Effects of Bias/Variance/Diversity +As before, we can estimate these terms. We illustrate this by comparing Bagging and Random +Forests on MNIST, combining predictions by majority voting. +5 +10 +15 +20 +Ensemble Size +0.00 +0.05 +0.10 +0.15 +0-1 Loss +Ensemble 0-1 Loss +Bagging +Random Forests +3 +6 +9 +12 +15 +18 +Ensemble size +Bagging +average bias-effect +average variance-effect +diversity-effect +3 +6 +9 +12 +15 +18 +Ensemble size +Random Forest +Figure 17: Bias/Variance/Diversity effect for Bagging vs Random Forests. +25 + +We see familiar behaviors—as we increase M, the bias-effect and variance-effect remain constant, +but the diversity-effect increases. The variance-effect for Random Forests is significantly higher +than for Bagging, though this is compensated for with the diversity-effect for large ensembles. +Theorem 13 can be extended relatively simply for weighted voting. The extension and proof are +provided in Appendix D.2, while here we utilise the result to analyse boosting. +We plot the +components (Figure 18) this time boosting low-variance decision stumps. +0 +100 +200 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Bagging +ensemble error +0 +100 +200 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +AdaBoost +0 +100 +200 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +LogitBoost +0 +100 +200 +Ensemble size +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +average bias-effect +average variance-effect +diversity-effect +0 +100 +200 +Ensemble size +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +100 +200 +Ensemble size +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Figure 18: Mease data, ensembling decision stumps. +We note some interesting differences between the parallel model (Bagging) and the sequential +models (LogitBoost and AdaBoost). +• As before, the bias/variance are constant for Bagging. However for boosting, the terms vary +with ensemble size, M. +This is caused by the non-homogeneous nature of the ensemble +members, specialising to different parts of the data. +• In boosting, the diversity-effect can be greater than the variance-effect. This is due to the +fact that ensemble members are designed to be complimentary: with disagreements actively +encouraged, as opposed to being a property of random sampling. +More generally, with Bagging the decrease in error comes from increasing diversity, but the story +for boosting models is more complex, with the overall performance of the model being determined +by a complex trade-off between the three components. +26 + +6.4 +How Can Diversity-Effect be Negative? +It is possible for diversity-effect to take negative values. +In this case, given the signs in the +expression, it would be harmful for the expected risk. We now examine under what circumstances +this situation might arise. +A Theoretical Model. +Assuming a binary classifier, define ϵ = ED +� +EY +� +L0/1(y, q(x; D)) +�� +as +the probability of error (over the distribution of D) at a test point. If we assume M classifiers make +errors independently then, for odd M, the diversity-effect can be written: +DE = ϵ − +M−1 +2 +� +i=0 +� +M +i +� +ϵM−i(1 − ϵ)i, +(36) +where the second term is the majority voting error for independent models (Hansen & Salamon, +1990). We plot this, varying ϵ, in Figure 19. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ϵ +−0.4 +−0.2 +0.0 +0.2 +0.4 +diversity-effect +M=5 +M=11 +M=51 +M=101 +M=501 +Figure 19: Diversity-effect for independent models. +The diversity-effect is positive when the probability of error is less than 0.5, i.e., the models do better +than random guessing. Conversely, when the models are worse than random guessing (ϵ ≥ 0.5), +the diversity effect is negative. This gives some comfort, as negative effects from diversity should +only be seen in pathological scenarios. However, it should be remembered that the assumptions of +this theoretical model are rather strong. +A Real Case, for 10 Classes. +We can see a remarkably similar pattern with real data, Figure 20, +where we consider unconstrained Bagged Decision Trees on MNIST. Each point on the figure is +the ensemble evaluated at a single test point. The test points below the solid line have a negative +diversity effect. +27 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +average individual error +−0.2 +−0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +diversity-effect +Average individual error vs diversity-effect +for Bagged Decision Trees +examples in test set +Figure 20: Scatter plot of average individual error against diversity effect. Bagging trees on MNIST, +M = 20. +Similar observations can be made regarding random guessing and negative diversity. A random +guesser on this k = 10 class problem would have ϵ = 0.9. It can be observed in Figure 20 that the +points above this all have negative diversity effect. However, it can be noted that there are many +test points where diversity-effect is negative, but the individual error is not worse than random +guessing, i.e., less than 0.9. It can be shown that, for independent classifiers, in order to have a +negative diversity effect, it mus be the case that ϵ > 0.5, i.e., the ensemble members makes errors +more than half the time. Note that this is independent of the number of classes. This is formalised +with Theorem 14 below—proof in Appendix D.2. +Theorem 14 Assume an ensemble of models making independent errors on a k-class problem, with +each model predicting the correct class with probability p. If p > 0.5, then the diversity-effect is +guaranteed to be non-negative. +Overall, in theoretical and empirical scenarios, we see the same phenomenon. Ensembles with strong +individual performance tend to benefit from diversity. And, it is only for pathologically bad models +(worse than random guessing) the diversity is detrimental to the overall ensemble performance. +The issue of positive/negative diversity turns out be closely related to the phenomenon of “good” +and “bad” diversity (Brown & Kuncheva, 2010). They showed that, restricting the label to y ∈ +{−1, +1}, and q as a majority vote, the following holds: +L0/1(y, q) = +1 +M +M +� +i=1 +L0/1(y, qi) − yq 1 +M +M +� +i=1 +L0/1(q, qi). +(37) +If yq > 0, the diversity subtracts from the average error; when the opposite is true, yq < 0, it adds +to the error. The former case is referred to as “good” diversity, and the latter is “bad” diversity. +Comparing this with Theorem 12, we see the diversity-effect is in fact the expected value of the +“good”/“bad” diversity term. The idea of good/bad diversity was generalised to the multi-class +case by Didaci et al. (2013), and the same relation applies. +28 + +6.5 +Summary +We examined the properties of ensemble diversity under the 0-1 loss. We started with a proof +showing the non-existence of a 0-1 bias-variance decomposition with target-independent variance. +A consequence of this is, there cannot be a corresponding ambiguity decomposition. Therefore, +there is no decomposition of the 0-1 loss such that the diversity can be expressed independently of +the target. +In spite of this target-dependence, we can formulate diversity in terms of its effect on the error, +using the bias-variance ‘effect’ decompositions of James & Hastie (1997). Thus, we have the same +conclusion as in the previous section: the role of diversity can be formulated as a hidden degree of +freedom in a decomposition of the ensemble error. +7 +Discussion: Limitations & Future Work +Our framework applies for any loss admitting a bias-variance decomposition, giving a clear +methodology to understand the form and nature of diversity. +The framework is not without +limitations. Here we outline one such limitation, and discuss some potential future work. +In many learning scenarios, the loss we minimise is not always the loss in which we are ultimately +interested—so-called surrogate losses. The most obvious here is the cross-entropy, where we are +ultimately interested in the 0-1 loss. Margin losses like the logistic or exponential loss are another +important example, since boosting algorithms like AdaBoost/LogitBoost can be seen as minimising +such losses. Wood et al. (2022) analysed bias-variance decompositions for margin losses. Using +these results, we can indeed obtain bias-variance-diversity decompositions applying for boosting +models—some with target-independent diversity (e.g. +LogitBoost), and some target-dependent +(e.g. AdaBoost). However, Mease & Wyner (2008) showed strong evidence that the additive model +form in AdaBoost/LogitBoost results in a disconnect between the surrogate margin loss and the 0-1 +loss. In particular, the surrogate loss can go up (sometimes exponentially fast) whilst the 0-1 loss +on a hold-out sample is going down. This implies that any analysis of the surrogate (including loss +decompositions) does not necessarily give meaningful insights on the ultimate quantity of interest, +the 0-1 loss. Furthermore, with boosting, the individual models are more naturally interpreted as +learning to correct the errors of previous ensemble members rather than perform well in their own +right, making interpretation of the average bias term problematic. +Regarding future work, a natural line of research might be to enforce diversity in some sense, i.e., +using our diversity measures as a regulariser in the construction of an ensemble itself. Negative +Correlation (NC) Learning Liu & Yao (1999) uses the squared loss ambiguity decomposition, +Equation (1), to encourage diversity for regression ensembles, analysed by Brown et al. (2006). +The Bregman ambiguity decomposition, Equation (16), implies that the NC algorithm is a special +case of a wider family of diversity encouraging losses—the case for cross-entropy was explored in +Webb et al. (2021). However, given the full framework proposed in this paper, it is clear that many +other opportunities exist, including introducing heterogeneity into the ensemble, or in implicit +ensemble creation techniques like MC-Dropout. +29 + +8 +Conclusion +We have presented a unified theory of ensemble diversity. A key insight is that it is not the task (e.g. +classification/regression) that matters, but the loss function. We demonstrated that a natural basis +for the concept of diversity can be found as a hidden dimension in the bias-variance decomposition +of the ensemble loss. Diversity emerges naturally when considered from this point of view—as one +part of a bias-variance-diversity decomposition, specific to the chosen loss function, all taking a +common form: +expected risk = (average bias) + (average variance) − (diversity). +The gives a clear relationship between the ensemble performance and diversity, measured as +a target-independent quantity. +The only other scenario where was previously available is for +squared loss with an arithmetic mean combiner (Ueda & Nakano, 1996). Our framework is an +alternative in this case, but also generalises the notion of diversity to a wide range of other losses. +The framework provides a methodology to automatically identify the combiner rule that enables +such a decomposition, which we define as the centroid combiner rule. This generalises the idea of +ensemble “averaging” to many other scenarios. The case of 0-1 loss is particularly interesting—we +prove that, for any combiner, a target-independent diversity term cannot exist. Following James & +Hastie (1997), we introduced a diversity-effect term which, though target-dependent, allows us to +understand the role that diversity plays. +Notable properties of the framework are: +(i) It offers a unified view of diversity for a wide range of losses. For any loss L, if a +bias-variance decomposition holds, then we can apply the double decomposition trick and +obtain a definition of diversity. As such, our framework can be used to understand diversity +for, e.g., squared loss, cross-entropy loss, the Poisson regression loss, and several margin- +based losses. +(ii) It shows that diversity is a measure of model fit, just like bias/variance. Diversity +is a measure of model fit—a hidden dimension in the expected loss of an ensemble, +accounting for statistical dependencies between the individual models. +Just as bias and +variance change with model characteristics, the same applies to diversity. +This gives a +three-way bias/variance/diversity trade-off. It may be possible (though outside the scope +of this paper) to use diversity as a regularisation target. +We therefore have a broad and precise formulation of diversity, with clear conditions for when it +(and its effects) can, and cannot, be expressed independently of the target. This challenge has been +referred to as the “holy grail” of ensemble learning (Zhou, 2012, Sec 5.1), an open question for over +30 years. We believe this work provides a solid foundation from which to explore new directions in +ensemble learning. +Acknowledgements +Funding in direct support of this research: EPSRC EP/N035127/1 (LAMBDA project). GB would +like to thank LK and FR for a career’s worth of inspiration. +30 + +A +Additional Experimental Results +Further results for Squared Loss +We extend results for squared loss, shown in Figure 7. In Table 4 we see results from three ensembles +(each M = 30 regression trees), compared to a single tree. We use a Bagging with constrained +depth trees (max depth 8) and compare against unlimited depth trees, and a Random Forest. +Single tree +(depth 8) +Bagging +(depth 8) +Bagging +(unconstrained) +Random Forest +0.47 +0.35 +0.30 +0.28 +Table 4: California housing data: MSE of a single tree versus ensembles of 30 trees. +We observe that the Random Forest is the best choice here, followed up closely by the unconstrained +Bagging. +Figure 21 explains their performance by decomposing risk into bias, variance, and +diversity—also showing how the components change as we grow the ensemble. +0 +10 +20 +30 +ensemble size +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Bagging (max depth 8) +0 +10 +20 +30 +ensemble size +Bagging (unconstrained) +average bias + noise +average variance +diversity +0 +10 +20 +30 +ensemble size +Random Forest +Figure 21: Decomposing the expected risk of three decision tree ensembles. +We observe the same behaviour as in Figure 7. The diversity increases with M, and is upper- +bounded by the value of the average variance. +A higher average variance effectively raises the +“ceiling” to which diversity can rise. The average variance is higher as we move from depth-limited +trees to unlimited depth, and higher again with the random split-points in the Random Forest (here +we use the square root of the number of features). The higher average variance is compensated +for by the diversity, causing Random Forest to be the best option. It is notable that for large +ensembles, the expected risk of the ensemble is almost entirely due to the value of the average +bias (≈ 0.28 in the case of unconstrained trees), with diversity having essentially cancelled out the +average variance of the individual models. This behaviour is not just a quirk of this data set, in +fact it holds as long as the individuals are all from the same model family, i.e., the ensemble is +homogeneous—the general case is discussed in Section 5.5. +31 + +Further results for cross-entropy +In Figure 22, we present additional Accuracy/diversity plots for neural network ensembles for +different data sets. In each case, the squared Person’s correlation coefficient is shown in the legend. +The following configuration was used in all MLP experiments: +• learning rate: 0.1 (Stochastic gradient descent) +• num epochs: 50 (MNIST), 200 (other data sets) +• hidden layer size (20 small/100 larger) +• number of trials: 100 +where each trial uses a 90% sub-sample of the full training data, as outlined in Figure 23. +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +diversity +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +0.014 +0.016 +average individual error +- ensemble error +M = 2 +M = 20 +M = 2 +M = 20 +phoneme +Smaller Networks, R2 = 0.97 +Larger Networks, R2 = 0.95 +0.000 +0.025 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +diversity +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +0.014 +0.016 +average individual error +- ensemble error +M = 2 +M = 2 +M = 20 +M = 20 +spambase +Smaller Networks, R2 = 0.99 +Larger Networks, R2 = 0.96 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +diversity +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +average individual error +- ensemble error +M = 2 +M = 2 +M = 20 +M = 20 +landsat +Smaller Networks, R2 = 0.99 +Larger Networks, R2 = 0.98 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +diversity +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +average individual error +- ensemble error +M = 2 +M = 2 +M = 20 +M = 20 +south german credit +Smaller Networks, R2 = 0.93 +Larger Networks, R2 = 0.71 +Figure 22: Error/diversity relationship observed across four data sets, comparing ensembles of small +networks (20 hidden node, blue dots) versus large (100 hidden node, orange dots). +32 + +B +Methodology for Estimating Bias, Variance, and Diversity +Here, we present our methodology for estimating the bias, variance and diversity terms from data. +Algorithm 1 shows the procedure for experiments where we estimate diversity of ensembles of +different sizes, such as in the experiments for Bagging and Random Forests. Notably, an ensemble +of size M + 1 is created by using the members of the ensemble of size M, rather creating a new +ensemble of size M + 1 from scratch. We also present a visualisation of the sub-sampling scheme +used for Bagging in Figure 23. +Algorithm parameters: model, n_trials, ensemble_size, train_data, test_data +Output: test_preds: an array of model predictions of size +n_trials × ensemble_size × n_test_data +for k ∈ {1, . . . , n_trials} do +for j ∈ {1, . . . , len(test_data)} do +trial_data ← 90% of train_data, sampled without replacement; +for i ∈ {1, . . . ,ensemble_size} do +member_data ← bootstrap of trial_data; +ith ensemble member ← copy of model trained on member_data; +test_preds[k, i, j] ← prediction of test_data of the ith ensemble member in the kth +trial, jth test data point; +end +end +end +Algorithm 1: Algorithm for collecting data to estimate diversity of bootstrap ensemble while +varying ensemble size +Training Data +Test Data +bootstraps +Trials +Dataset +90% +Sub-sample +90% +Sub-sample +90% +Sub-sample +90% +Sub-sample +Ensembles +Ensemble Members +bootstraps +bootstraps +bootstraps +Figure 23: Visualisation of the sub-sampling scheme used for Bagging ensembles. +The result of Algorithm 1 is that we get an array of size (D, M, N), where D is the number of +trials, M is the ensemble size and N is the number of test points. +For Bregman divergences, the average bias and average variance are calculated by estimating the +central model of each ensemble member by replacing the expectation with an average. Writing Dj +33 + +as the full training data for the jth trial +q∗,est +i +(x) = [∇φ]−1 +� +� 1 +D +D +� +j=1 +∇φ(qi(x; Dj)) +� +� ≈ q∗ +i (x) +With this estimate of q∗(x) the average bias and average variance are computed as +average bias ≈ 1 +M +1 +N +M +� +i=1 +N +� +j=1 +Bφ +� +yj, q∗,est +i +(xj) +� +, +(38) +average variance ≈ 1 +D +1 +M +1 +N +D +� +k=1 +M +� +i=1 +N +� +j=1 +Bφ +� +q∗,est +i +(xj), qi(x; Dk) +� +. +(39) +Diversity is calculated similarly, with q defined as the centroid combiner of the M ensemble members +in a given trial: +diversity ≈ 1 +D +1 +M +1 +N +D +� +k=1 +M +� +i=1 +N +� +j=1 +Bφ (q(xj, Dk), qi(x; Dk)) . +(40) +34 + +C +Proofs and further explanations for Section 5 +C.1 +Bregman Ambiguity and Bregman Diversity +The following sections will make use of Pfau (2013)’s bias-variance decomposition, which for +referencing purposes we restate here. +Theorem 15 (Bregman Bias-Variance Decomposition Pfau (2013)) Given +a +loss +Bφ (Y, q(X; D)), the expectation of the risk with respect to D can be written, +ED +� +EXY +�� +Bφ(Y, q(X; D)) +�� += +EX +� +EY|X +� +Bφ +� +Y, Y +�� +� +�� +� +noise ++ Bφ +� +Y, q∗(X) +� +� +�� +� +bias ++ ED [Bφ (q∗(X), q(X; D))] +� +�� +� +variance +� +, +where Y = EY|X[Y], the conditional mean of the vector Y, and +q∗(x) +def= +arg min +z +ED +� +Bφ (z, q(x; D)) +� += +[∇φ]−1 � +ED [∇φ (q(x; D))] +� +(41) +The centroid q∗ takes different forms dependent on the generator used. Examples are below. +Loss +Gradient +η = ∇φ(q) +Inverse Grad. +q = [∇φ]−1 (η) +Left Bregman Centroid +q∗ = [∇φ]−1 � +ED [∇φ (qD)] +� +Squared +2q +1 +2η +ED[qD] +Itakura-Saito +−1 +q +− 1 +η +1 +�� +ED [1/qD] +� +Poisson loss +ln q +exp(η) +exp +�ED [ln qD] +� +KL-divergence +ln +q(c) +1−�k−1 +c′=1 q(c′) +exp(η(c)) +1+�k−1 +c′=1 exp(η(c′)) +1 +Z exp +� +ED [ln qD] +� +Table 5: Common losses (see Table 1) and their Bregman centroids. In the case of KL-divergence, +Z is a normalizer to ensure a valid distribution. +Whilst we assert that the ambiguity decomposition is a special case of this (which emerges trivially +if we assume zero noise, and replace expectations by summations), it can be proven independently, +as shown below. +Theorem 3 (Bregman Ambiguity Decomposition) For a target label y ∈ S and a set of +predictions q1, . . . , qM ∈ ri(S), +Bφ (y, q) = 1 +M +M +� +i=1 +Bφ (y, qi) − 1 +M +M +� +i=1 +Bφ (q, qi) +(16) +where q = [∇φ]−1� 1 +M +� +i ∇φ(qi) +�. +35 + +Proof 1 Take the average loss over the M models, and subtract the loss of the ensemble: +1 +M +M +� +i=1 +Bφ (y, qi) − Bφ (y, q) += 1 +M +M +� +i=1 +� +φ(y) − φ(qi) − ⟨∇φ(qi), y − qi⟩ +� +− +� +φ(y) − φ(q) − ⟨∇φ(q), y − q⟩ +� += 1 +M +M +� +i=1 +� +φ(y) − φ(qi) − ⟨∇φ(qi), y − qi⟩ − φ(y) + φ(q) + ⟨∇φ(q), y − q⟩ +� += 1 +M +M +� +i=1 +� +φ(q) − φ(qi) − ⟨∇φ(qi), y − qi⟩ + ⟨∇φ(q), y − q⟩ +� +Now expand the inner products and use the definition of q = [∇φ]−1� 1 +M +� +i ∇φ(qi) +�, to note that +∇φ(q) = +1 +M +� +i ∇φ(qi). += 1 +M +M +� +i=1 +� +φ(q) − φ(qi) − y · ∇φ(qi) + qi · ∇φ(qi) + y · 1 +M +M +� +i=1 +∇φ(qi) − q · 1 +M +M +� +i=1 +∇φ(qi) +� += 1 +M +M +� +i=1 +� +φ(q) − φ(qi) + qi · ∇φ(qi) − q · 1 +M +M +� +i=1 +∇φ(qi) +� += 1 +M +M +� +i=1 +� +φ(q) − φ(qi) − ∇φ(qi) · [q − qi] +� += 1 +M +M +� +i=1 +Bφ (q, qi) +which after rearranging completes the proof. +Theorem 4 (Bregman Bias-Variance-Diversity decomposition) +For an ensemble q1(X; D), . . . , qM(X; D), let q∗ +i be the left Bregman centroid of qi (i.e., q∗ +i +def= +[∇φ]−1 (ED [∇φ(qi)])) and define q +def= [∇φ]−1 � +1 +M +�M +i=1 ∇φ (qi) +� +. Then we have the decomposition, +ED [EXY [Bφ (Y, q)]] = +EXY +� +Bφ +� +Y, Y +� +� +�� +� +noise ++ 1 +M +M +� +i=1 +Bφ +� +Y, q∗ +i +� +� +�� +� +average bias ++ 1 +M +M +� +i=1 +ED [Bφ (q∗ +i , qi)] +� +�� +� +average variance +− ED +� +1 +M +M +� +i=1 +Bφ (q, qi) +� +� +�� +� +diversity +� +, +where Y = EY|X [Y]. +Proof 2 Starting with the ambiguity decomposition, we have +ED [EXY [Bφ (Y, q)]] = ED +� +EXY +� +1 +M +M +� +i=1 +Bφ (Y, qi) +�� +− ED +� +EXY +� +1 +M +M +� +i=1 +Bφ (q, qi) +�� +(42) +Applying Theorem 15 to the first term on the RHS, we have +ED +� +EXY +� 1 +M +M +� +i=1 +Bφ(Y, qi) +�� += +EX +� +EY|X +� +Bφ +� +Y, Y +�� ++ +1 +M +M +� +i=1 +Bφ +� +Y, q∗ +i +� ++ +1 +M +M +� +i=1 +ED [Bφ (q∗ +i , qi)] +� +(43) +Plugging Equation (43) into (42) completes the proof. +36 + +C.2 +The Importance of Parameter Encoding in the KL-divergence +In order to apply our decomposition to the cross-entropy, we use the fact that when the target is +one-hot encoded, the cross-entropy coincides with the KL-divergence. There are two ways in which +we can express the KL-divergence as a Bregman divergence, either using the full-length probability +vectors, p ∈ Rk giving the generator +φfull(p) = +k +� +c=1 +p(c) ln p(c) +(44) +or using the minimally parameterised vectors, �p ∈ Rk−1, where the last entry is omitted: +φmin(�p) = +k−1 +� +c=1 +�p(c) ln �p(c) + (1 − +k−1 +� +c′=1 +�p(c′)) ln(1 − +k−1 +� +c′=1 +�p(c′)) +(45) +Given two probability vectors in the appropriate form, either formulation gives a Bregman +divergence is equivalent to the KL-divergence Nielsen & Nock (2009), i.e., +Bφfull (p, q) = Bφmin (�p, �q) = K(p || q) +However, we prefer the second as it exhibits desirable properties. In particular, it is necessary +to use the second to ensure that the Bregman centroid is always a valid distribution on the +probability simplex. +In this case, the centroid combiner is the normalised geometric mean, as +we now demonstrate. +To show this we consider M minimally parameterised vectors, qi ∈ Rk−1 (note that we have +dropped tilde above q for simplicity). Our claim is that the centroid combiner is the normalised +geometric mean q = [∇φ]−1 � +1 +M +�M +i=1 ∇φ(qi) +� +, is of the form +q(c) = [∇φmin]−1 +� +1 +M +M +� +i=1 +∇φmin(qi) +� += +�M +i=1 q(c) +i +1 +M +�k +c′=1 +�M +i=1 �q(c′) 1 +M +, +(46) +where �q denotes the extension of the k − 1 length vector into a full k length probability vector. +Plugging in the gradients from Table 5, we start with +q(c) = +exp +� +1 +M +�M +i=1 ln +q(c) +i +1−�k−1 +c′=1 q(c′) +i +� +1 + �k−1 +c′=1 exp +� +1 +M +�M +i=1 ln +q(c′) +i +1−�k−1 +c′′=1 q(c′′) +i +�. +Note that the numerator here can be rearranged: +exp +� +1 +M +M +� +i=1 +ln +q(c) +i +1 − �k−1 +c′=1 q(c′) +i +� += +M +� +i=1 +� +1 − +k−1 +� +c′=1 +q(c′) +i +�− 1 +M +M +� +i=1 +� +q(c) +i +� 1 +M , +37 + +and the denominator can be written +1 + +k−1 +� +c′=1 +exp +� +1 +M +M +� +i=1 +ln +q(c′) +m +1 − �k−1 +c′′=1 q(c′′) +m +� += 1 + +k−1 +� +c′=1 +M +� +i=1 +q(c′) +i +1 +M +� +1 − +k−1 +� +c′′=1 +q(c′′) +i +�− 1 +M += 1 + +M +� +i′=1 +� +1 − +k−1 +� +c′′=1 +q(c′′) +i′ +�− 1 +M k−1 +� +c′=1 +M +� +i=1 +q(c′) +i +1 +M += +M +� +i′=1 +� +1 − +k−1 +� +c′′=1 +q(c′′) +i′ +�− 1 +M +� +� +� +M +� +i=1 +� +1 − +k−1 +� +c′′=1 +q(c′′) +i +� 1 +M ++ +k−1 +� +c′=1 +M +� +i=1 +q(c′) +i +1 +M +� +� +� . +Putting the numerator and denominator back into the second expression of Equation (46) and +using the definition of �q, we find the first terms in both products cancel and we are left with the +required result. +Full length k Probability Vector +If we do not use the minimally parameterized vectors, we +would have the Bregman generator φ(p) = �k +c=1 p(c) ln p(c). This gives the geometric mean, rather +than the normalised version. To see this, we first note that +(∇φ(p))(c) = 1 + ln p(c) = η(c) +� +[∇φ]−1 (η) +�(c) = exp +� +η(c) − 1 +� +, +and therefore the centroid combiner is +� +[∇φ]−1� 1 +M +M +� +m=1 +∇φ(qm) +��(c) += exp +� +1 +M +M +� +i=1 +1 + ln q(c) +i +− 1 +� += exp +� +1 +M +M +� +i=1 +ln q(c) +i +� += +M +� +i=1 +q(c) +i +1 +M . +Note that this means that q is not necessarily a valid probability vector. In fact, it is a valid +probability vector only if q1 = . . . = qM. +C.3 +Decomposing the Cross-Entropy +Theorem 5 Let y be a one-hot class vector of length k, and q ∈ Rk be a model’s prediction of the +class distribution. Define a set of such models {qi}M +i=1, and their combination q as their normalised +geometric mean. The following decomposition holds. +−ED [y · ln q] +� +�� +� +expected +cross-entropy += − 1 +M +M +� +i=1 +y · ln q∗ +i +� +�� +� +average bias ++ 1 +M +M +� +i=1 +ED [K(q∗ +i || qi)] +� +�� +� +average variance +− ED +� +1 +M +M +� +i=1 +K(q || qi) +� +� +�� +� +diversity +, +(18) +Proof 3 From Theorem 4, using the generator φmin, we have +ED [Bφmin (y, q)] = 1 +M +M +� +i=1 +Bφmin (y, q∗ +i ) + 1 +M +M +� +i=1 +ED [Bφmin (q∗ +i , qi)] + 1 +M +M +� +i=1 +Bφmin (q, qi) +38 + +Using the equivalence with KL-divergences, we have +ED [K(y || q)] = 1 +M +M +� +i=1 +K(y || q∗ +i ) + 1 +M +M +� +i=1 +ED [K(q∗ +i || qi)] + 1 +M +M +� +i=1 +K(q || qi) +Since y is one-hot y · ln y = 0 and therefore for any vector q, we can write K(y || q)) = −y · ln q. +Applying this to the expected risk and average bias terms in the above gives +ED [y · ln q] = − 1 +M +M +� +i=1 +y · ln q∗ +i + 1 +M +M +� +i=1 +ED [K(q∗ +i || qi)] − ED +� +1 +M +M +� +i=1 +K(q || qi) +� +. +C.4 +Sources of Stochasticity in the Bias-Variance-Diversity Decomposition +In this appendix, we clarify exactly what is meant when we write ED [·] when dealing ensembles and +the bias-variance-diversity decomposition. In particular, we show how D is constructed to account +for the stochasticity in individual ensemble members and the interactions between them. +In the bias-variance decomposition of a single model, we consider q as dependent on a random +variable D during its training. This might represent the random sample of training data, or initial +weights in a neural net, or any other source of stochastic behavior during learning. We can write +the expected loss with respect to D, at a point as +ED [L(y, q(x; D)] . +When considering an ensemble, the situation becomes more complicated, as there are multiple +sources of stochasticity. For instance, if we use Bagging, we first consider the overall training set to +be a sample of n i.i.d. observations from P(X, Y ), then each ensemble member receives a random +bootstrap from that same data sample. Hence, each ensemble member is influenced by a different— +though not necessarily independent—random variable (determining the bootstrap sample they each +receive). We write the random variable representing the training data of the ith ensemble member +as Di, and therefore can write the ith mode output as q(x; Di). Continuing the example, we can +write the expected loss of this model as +EDi[L(y, q(x; Di)]. +Furthermore, we can define D = (D1, D2, . . . , DM), such that D is a random vector containing all +M random variables of the individual ensemble members. Now, due to the law of total expectation +we can write this using D instead of Di, i.e. +ED[L(y, q(x; Di)] = EDi[L(y, q(x; Di)]. +even when there dependencies between the individual Di. We see therefore that the expectation +over D reduces to Di for individual models, and the decomposition applies even with D being +vector with dependencies. +39 + +C.5 +Further Properties of the Bias-Variance-Diversity Decomposition +This Appendix contains proofs for results in Section 5.5. +Proposition 7 Assume a target probability vector, y ∈ Rk, and a set of models {qi}M +i=1 combined +by averaging, i.e. q† = +1 +M +�M +i=1 qi, then the cross-entropy of q and y is +−y · ln q† +� +�� +� +ensemble cross-entropy += +− 1 +M +M +� +i=1 +y · ln qi +� +�� +� +average cross-entropy +− +k +� +c=1 +y(c) ln +1 +M +�M +j=1 q(c) +j +��M +i=1 q(c) +i +� 1 +M +� +�� +� +ambiguity (target-dependent) +, +(28) +where the second term is non-negative, thus the ensemble loss is guaranteed less than or equal to +the average individual loss. +Proof 4 Take the average cross-entropy, and subtract the ensemble cross entropy: +− 1 +M +M +� +i=1 +y · ln qi − +� +− y · ln q†� += +k +� +c=1 +y(c) ln q†(c) − 1 +M +M +� +i=1 +k +� +c=1 +y(c) ln q(c) +i += +k +� +c=1 +y(c) ln q†(c) − +k +� +c=1 +y(c) ln +� � +i +q(c) +i +�1/M += +k +� +c=1 +y(c) ln +� +� +� +� +q†(c) +�M +i=1 +� +q(c) +i +� 1 +M +� +� +� +� +Using the definition of q† and rearranging completes the derivation. From the arithmetic-geometric +mean inequality, q†(c) ≥ �M +i=1 +� +q(c) +i +� 1 +M , implying that the term inside the logarithm is greater or +equal to 1, and the overall term is non-negative. +Proposition 8 (Diversity for Averaged Probabilities is target-dependent) +Let q† = +1 +M +�M +i=1 qi, with qi ∈ [0, 1]k. The expected cross-entropy admits the decomposition: +ED +� +−y · ln q†) +� += +− 1 +M +M +� +i=1 +y · ln q∗ +i +� +�� +� +average bias ++ +1 +M +M +� +i=1 +ED [K(q∗ +i || qi)] +� +�� +� +average variance +− ED +� +��� +k +� +c=1 +y(c) ln +1 +M +�M +j=1 q(c) +j +��M +i=1 q(c) +i +� 1 +M +� +��� +� +�� +� +dependency +. +Proof 5 Starting with Equation (28) and taking the expectation over D, we have +−ED +� +y · ln q†� += ED +� +− 1 +M +M +� +i=1 +y · ln qi +� +− ED +� +��� +k +� +c=1 +y(c) ln +1 +M +�M +j=1 q(c) +j +��M +i=1 q(c) +i +� 1 +M +� +��� . +Now, applying the bias variance decomposition to each ensemble member we get the result. +40 + +D +Proofs and Additional Material for Section 6 +D.1 +Proofs for Section 6 +In this section, we prove a theorem stated in Section 6 claiming that one cannot construct a bias- +variance decomposition for the 0-1 loss, where the bias term is the 0-1 loss of some deterministic +prediction. +Theorem 9 (Non-existence of a Bias-Variance Decomposition for 0-1 loss) The 0-1 loss +cannot be decomposed as a bias term plus a variance term, where the bias is the loss of some +deterministic model q∗ derived from the distribution of q and the variance term is independent of +the label y. I.e., there is no V and q∗ such that in general +ED +� +L0/1(y, q) +� +� +�� +� +expected loss += L0/1(y, q∗) +� +�� +� +bias ++ V[q] +� �� � +variance +. +(30) +Proof 6 Rearranging Equation (10), for a decomposition to hold, the 0-1 loss needs to satisfy +V[q] = ED +� +L0/1(y, q) +� +− L0/1(y, q∗) +(47) +and therefore we require the right hand side to be independent of y. We show that this cannot be +the case by constructing an example such that there is no valid choice of q∗ that makes the right +hand side independent of y. +The two class case: +First, consider the case when there are two classes, i.e., S = {1, 2}. +For a fixed x, let P(q(x; D) = 1) = 0.6. Since L0/1 : S × S → {0, 1} we necessarily have that +q∗ ∈ S = {1, 2} in order for L0/1(y, q∗) to be defined. We now show that for both possible q∗, the +RHS above is dependent on y. When q∗ = 1, we have +ED +� +L0/1(y, q) +� +− L0/1(y, q∗) = +� +0.4 − 0 = 0.4 +if y = 1 +0.6 − 1 = −0.4 +if y = 2 +Alternatively, when q∗ = 2, +ED +� +L0/1(y, q) +� +− L0/1(y, q∗) = +� +0.4 − 1 = −0.6 +if y = 1 +0.6 − 0 = 0.6 +if y = 2 +For both possible values of q∗, the value of ED +� +L0/1(y, q) +� +− L0/1(y, q∗) is dependent on y. +The multiclass case: +We set P(q(x; D) = 1) = 0.6, and P(q(x; D) = 2) = 0.4, and zero +probability mass for all other classes. From the two-class case we know that when q∗ ∈ {1, 2}, there +is a dependency on y. This persists for q∗ ∈ {3, . . . , k}, where we have +ED +� +L0/1(y, q) +� +− L0/1(y, q∗) = +� +� +� +� +� +� +� +� +� +� +� +� +� +0.4 − 1 = −0.6 +if y = 1 +0.6 − 1 = −0.4 +if y = 2 +1 − 0 = 1 +if y = q∗ +1 − 1 = 0 +otherwise. +For all possible values of q∗, the expression ED +� +L0/1(y, q) +� +− L0/1(y, q∗) is dependent on the value +of y, and therefore the bias-variance decomposition cannot exist. +41 + +Theorem 13 (Bias-Variance-Diversity effect decomposition) Given a loss function L : S × +S → R+ and an ensemble of models q1, . . . , qM, where q is the majority vote. +EY +� +ED +� +L0/1(Y, q) +�� += +EY +� +L0/1(Y, Y ∗) +� +� +�� +� +noise ++ +1 +M +M +� +i=1 +EY +� +L0/1(Y, q∗ +i ) − L0/1(Y, Y ∗) +� +� +�� +� +average bias-effect ++ 1 +M +M +� +i=1 +ED +� +EY +� +L0/1(Y, qi) − L(Y, q∗ +i ) +�� +� +�� +� +average variance-effect +− ED +� +EY +� +1 +M +M +� +i=1 +� +L0/1(Y, qi) − L(Y, q) +��� +� +�� +� +diversity-effect +. +Proof 7 Note that several terms on the right cancel, reducing to the left-hand side. +D.2 +Diversity-Effect Decomposition for Weighted Ensembles +In this section, we derive and justify the bias-variance-diversity-effect decomposition for weighted +majority vote. Given an ensemble of classification models q1, . . . , qM with f(x; D) ∈ {1, . . . , k}, +and weights for those models α1(D), . . . , αM(D), we consider the ambiguity-effect decomposition +for weighted plurality vote. The weighted majority vote is +q(x; D) = argmin +c∈{1,...,k} +M +� +i=1 +αi(D) +�M +j=1 αj(D)L0/1(c, qi(x; D)), +similarly, the central vote of an ensemble member is given by +q∗ +i (x; D) = argmin +c∈{1,...,k} +ED +� +αi(D) +ED [αi(D)]L0/1(c, qi(x; D)) +� +, +with ties broken randomly (the tie break procedure can be thought of as part of the random variable +D, since D implicitly contains all sources of stochasticity related to the model). +Theorem 16 (Ambiguity-Effect Decomposition for Weighted Majority Vote) With +this, we can define a weighted effect decomposition as +L0/1(y, q(x; D)) = +M +� +i=1 +ai(D)L0/1(y, qi(x; D) +� +�� +� +weighted average loss +− +� +������ +M +� +i=1 +ai(D)L0/1(y, qi(x; D)) − L0/1(y, q(x; D)) +� +�� +� +ambiguity-effect +� +������ +, +where ai = +αi +�M +j=1 αj . +The validity of this theorem can be verified simply by cancelling terms on the right-hand side. The +theorem tells us that the loss of an ensemble can be decomposed into a positive term, giving the +(weighted) average loss of the ensemble members, and an ambiguity-effect term, which quantifies +how much better (or worse) the performance of the ensemble is over the average ensemble member. +Using the same principle, we can also construct a bias-variance decomposition which takes into +account a weighting α(D). +42 + +Theorem 17 (Bias-variance-effect Decomposition for Weighted Majority Vote) +ED +� +αL0/1(y, f) +� += ED [α] L0/1(y, f∗) + +� +ED +� +αL0/1(y, f) +� +− ED [α] L0/1(y, f∗) +� +Again, the proof of the result is immediate from considering which terms on the right cancel. +However, it is worth considering how the decomposition works and what the terms mean. Consider +the decomposition when we replace α with a normalised version +α +ED[α], we get the following: +ED +� +α +ED [α]L(y, f) +� += L0/1(y, f∗) + +� +ED +� +α +ED [α]L0/1(y, f) +� +− L0/1(y, f∗) +� +. +This is exactly bias-variance-effect decomposition that we have seen previously, but re-weighting the +contributions of the different data sets. In fact, the two are equivalent, with the weights defining a +new probability density function. Taking PD(D) as the probability density function over data sets, +the decomposition above is exactly the bias-variance-effect decomposition with the new probability +density function QD(D) = PD(D) +α(D) +ED[α(D)]. +We can also easily reintroduce label noise and expose a noise term, and turning the bias into a +bias-effect: +EY +� +ED +� +α +ED [α]L(Y, f) +�� += EY +� +L0/1(Y, Y ∗) +� +� +�� +� +noise ++ EY +� +L0/1(Y, f∗) − L(Y, Y ∗) +� +� +�� +� +weighted bias-effect +� +ED +� +EY +� +α +ED [α]L0/1(Y, f) − L0/1(Y, f∗) +��� +� +�� +� +weighted variance-effect +. +We can now apply the double decomposition trick, getting the following bias-variance-diversity +effect decomposition for weighted majority vote. +Theorem 18 (Bias-Variance-Diversity-Effect for Weighted Voting) Given M +classifiers +q1, . . . , qM, where the ensemble is a weighted majority vote, i.e., q = argminz∈S +�M +i=1 αiL0/1(z, qi) +for weights α1, . . . , αM ∈ R+, the ensemble loss admits the following decomposition, where the +normalised weight is ai = αi/�M +j=1 αj. +EY +� +ED +� +L0/1(Y, q) +�� += EY +� +L0/1(Y, Y ∗) +� +� +�� +� +noise ++ +M +� +i=1 +ED +� +ai EY +� +L0/1(Y, q∗ +i ) − L0/1(Y, Y ∗) +�� +� +�� +� +weighted average bias-effect ++ +M +� +i=1 +ED +� +ai EY +� +L0/1(Y, qi) − L0/1(Y, q∗ +i ) +�� +� +�� +� +weighted average variance-effect +− ED +� +EY +� M +� +i=1 +ai L0/1(Y, qi) − L0/1(Y, q) +�� +� +�� +� +diversity-effect +� +, +where q∗ +i = argminz∈S ED +� +αiL0/1(z, qi) +� +, noting that αi and qi are both dependent on D. +43 + +As before, the veracity of this decomposition can be seen by cancelling like terms on the right-hand +side. AdaBoost produces a set of binary classifiers hi(x; D) ∈ {−1, +1} and corresponding weights +αi(D) ∈ R, so setting qi = hi allows immediate application of the decomposition. +LogitBoost +does not produce classifier/weight pairs, but instead a set of regression models each gi(x; D) ∈ R. +We can apply the decomposition by separating these into sign/magnitude components, giving a +classification and weight: qi = sign(gi(x; D)) and αi(D) = |gi(x; D)|. +Theorem 14 Assume an ensemble of models making independent errors on a k-class problem, with +each model predicting the correct class with probability p. If p > 0.5, then the diversity-effect is +guaranteed to be non-negative. +Proof 8 From the multi-class classification problem, we construct a binary classification problem +and define the models qbin +i +such that qbin +i += “y” when qi = y and qbin +i += “not y” when qi ̸= y. We +define qbin as being the majority vote of qbin +i +. “y” winning the majority vote is sufficient but not +necessary for y to win the plurality vote in the multi-class setting, so we get P(q = y) ≥ P(qbin = +“y”). For the binary classification problem, we use Condorcet’s Jury Theorem (see, e.g., Berend +& Sapir (2005)) to get that the probability of the ensemble being correct is greater than or equal to +any given individual being correct and therefore +P(q = y) ≥ P(qbin = “y”) ≥ P(qbin +i += “y”) = P(qi = y), +with the second inequality being strict when M ≥ 3 and P(qi = y) < 1. Plugging this into the +diversity-effect definition, for i.i.d. models we have +DE += +ED +� +1 +M +M +� +i=1 +L(y, qi) − L(y, q) +� += +1 − P(qi = y) − 1 + P(q = y) += +P(q = y) − P(qi = y) +≥ +0. +References +K. Audhkhasi, A. Sethy, B. Ramabhadran, and S.S. 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Springer, 2010. +47 + diff --git a/19E2T4oBgHgl3EQfiwfE/content/tmp_files/load_file.txt b/19E2T4oBgHgl3EQfiwfE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..39ae56ef85bf5046dcb4fbcc76e4538bf474b081 --- /dev/null +++ b/19E2T4oBgHgl3EQfiwfE/content/tmp_files/load_file.txt @@ -0,0 +1,1739 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf,len=1738 +page_content='A Unified Theory of Diversity in Ensemble Learning Danny Wood danny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='wood@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='uk† Tingting Mu tingting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='mu@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='uk† Andrew M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Webb andrew@awebb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='info† Henry W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Reeve henry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='reeve@bristol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='uk∗ Mikel Lujan mikel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='lujan@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='uk† Gavin Brown gavin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='brown@manchester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='uk† † Department of Computer Science, University of Manchester, UK ∗ Department of Mathematics, University of Bristol, UK Abstract We present a theory of ensemble diversity, explaining the nature and effect of diversity for a wide range of supervised learning scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This challenge, of understanding ensemble diversity, has been referred to as the “holy grail” of ensemble learning, an open question for over 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Our framework reveals that diversity is in fact a hidden dimension in the bias-variance decomposition of an ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In particular, we prove a family of exact bias-variance-diversity decompositions, for both classification and regression losses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', squared, and cross-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The framework provides a methodology to automatically identify the combiner rule enabling such a decomposition, specific to the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The formulation of diversity is therefore dependent on just two design choices: the loss, and the combiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For certain choices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 0-1 loss with majority voting) the effect of diversity is necessarily dependent on the target label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Experiments illustrate how we can use our framework to understand the diversity-encouraging mechanisms of popular ensemble methods: Bagging, Boosting, and Random Forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Keywords: ensemble, diversity, bias-variance decomposition, Bregman divergence 1 Introduction Ensemble methods have enabled state-of-the-art results for decades: from early industrial computer vision (Viola & Jones, 2001) to the deep learning revolution (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2012), and inter-disciplinary applications (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' An accepted mantra is that ensembles work best when the individuals have a “diversity” of predictions—often induced by classical methods such as Bagging (Breiman, 1996), but diversity-encouraging heuristics are rife in the literature (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Given this, we trust that the combination will “average out” the errors of the individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' One reason for the popularity of such methods is clear: the very idea of ensembles is an appealing anthropomorphism, invoking analogies to human committees, and “wisdom of the crowds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Unfortunately, such analogies have limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' More formal approaches have been pursued, in particular for quantifying diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It is obvious that we do not want all predictions to be identical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' and, it is equally obvious we do not want them to be different just for the sake of it, sacrificing overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We want something in-between these two—the so-called error/diversity tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, here we encounter the problem of formally defining “diversity” 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='03962v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='LG] 10 Jan 2023 and its relation to ensemble error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In general, there is no agreement on how to quantify diversity, except in the limited case of regression with an arithmetic mean ensemble (Krogh & Vedelsby, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Ueda & Nakano, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For classification and other scenarios, there are dozens of proposed diversity measures (Kuncheva, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A comprehensive theory of ensemble diversity has been an open problem for over 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Motivation: Our primary motivation is to fill this ‘gap’ in current ensemble theory, providing a solid foundation to understand and study ensemble diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, there are also many practical reasons to pursue this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Diverse ensembles can be more computationally efficient than single large models, with the same generalisation performance (Kondratyuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Diverse ensembles are robust against adversarial attacks (Biggio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2011), and can counteract covariate shift (Sinha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Advantages are also found in important application areas (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2020) and well beyond supervised learning (Carreira-Perpinán & Raziperchikolaei, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It is important to note that these use-cases do not follow a common approach: they either adopt some measurement of diversity picked from historical literature, or propose their own novel metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' There is, therefore, good reason to pursue a unified theory, where diversity is derived from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This challenge has proven non-trivial: surveys of progress can be found in Dietterich (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Zhou (2012), and Kuncheva (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Diversity is nowadays referred to as a heuristic with no precise definition, and, it has been said: “There is no doubt that understanding diversity is the holy grail in the field of ensemble learning” (Zhou, 2012, Sec 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1, page 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Summary of our Results: In contrast to previous efforts which define novel diversity measures, we take loss functions and decompose them, exposing terms that naturally account for diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We show that diversity is a hidden dimension in the bias-variance decomposition of the ensemble loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In particular, we prove exact bias-variance-diversity decompositions, applying for a broad range of losses, taking a common form: (expected loss) = (average bias) + (average variance) − (diversity), where diversity is a measure of member disagreement, independent of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This is an alternative to known results (Ueda & Nakano, 1996) in the special case of squared loss, but generalises the formal notion of diversity to many other losses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', the cross-entropy, and the Poisson regression loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A notable exception is the 0-1 loss—where we prove that such a decomposition cannot hold, for any combiner rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In spite of this, we are still able to quantify diversity and measure its effects, with the caveat that the effects are dependent on the target variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Overall, we argue that diversity is best understood as a measure of model fit, in precisely the same sense as bias and variance, but accounting for statistical dependencies between ensemble members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' With single models, we have a bias/variance trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' With an ensemble we have a bias/variance/diversity trade-off—varying both with individual model capacity, and similarities between model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 2 2 Problem Statement: What is Ensemble Diversity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' One of the earliest works on ensemble methods (at least in the machine learning community) was Hansen & Salamon (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This work trained multiple neural networks, each with a different training data subset, and combined them by majority vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Many subsequent algorithms followed this “parallel" strategy, notably Bagging (Breiman, 1996), and Random Forests (Breiman, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The well-known boosting family of algorithms (Schapire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 1998) exploit a similar principle, but construct models sequentially, providing each with data based on the errors of previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' These approaches, parallel and sequential (see Figure 1), are the most common schemes to construct ensembles (Kuncheva, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Sample 1 Sample 2 Sample M Model 1 Model 2 Model M Sample 1 Sample 2 Sample M Model 1 Model 2 Model M Data + Labels Data + Labels Ensemble Ensemble Figure 1: Parallel vs sequential ensemble construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Both can be seen as creating “diverse" models in some sense—either implicitly (independently re-sampling the training data), or explicitly (re-sampling according to the errors of earlier models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' So why do these strategies work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Both can be understood heuristically in terms of “diversity", in the sense coined by Opitz & Shavlik (1996), referring to differences in generalisation behavior among a group of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In a review, Dietterich (2000) explains: “An accurate classifier is one that has an error rate of better than random guessing on new x values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Two classifiers are diverse if they make different errors on new data points.” (Dietterich, 2000) In this sense, both approaches foster diversity—either implicitly by randomly perturbing the data for each model, or explicitly by constructing each data set to address the errors of other models (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The implicit approach to generating ensemble diversity has been widely adopted in deep learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' in their recent book, Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2016) note that the sources of randomness in the initialisation and training of deep networks “are often enough to cause different members of the ensemble to make partially independent errors," so each ensemble member can see all training data, while still being “diverse".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Given the success of ensembles, there have been many attempts to explain why they work, in terms of the diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2016, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 249) wrote “The reason that model averaging works is that different models will usually not make all the same errors on the test set”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' While this is true, statements like this are not the formal treatment we desire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 3 What are we looking for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A theory of ensemble diversity would ideally have three key ingredients: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' a definition of diversity as a measure of disagreement between the ensemble members, independently of the target variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' this measure should have a clear relation to the overall ensemble error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' and, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' the theory should have a clear relation to previously established results, and expand our understanding in other learning scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The first point ensures that diversity can be discussed solely as a property of the ensemble, a phenomenon in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The second point ensures we can interpret what effect the diversity has on our ultimate objective: reducing the ensemble error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The third point relates to the only known scenario where this can be considered a “solved” problem: regression using squared loss, with an arithmetic mean ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We now review this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Known results for regression ensembles: Krogh & Vedelsby (1994) showed that, for an arithmetic mean combiner, using squared loss, the ensemble loss is guaranteed to be less than or equal to the average individual loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Given a target y ∈ R, a member prediction qi(x), and an ensemble ¯q(x) = 1 M �M i=1 qi(x), we have, �¯q(x) − y �2 = 1 M M � i=1 �qi(x) − y �2 − 1 M M � i=1 �qi(x) − ¯q(x) �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (1) The left hand side is the ensemble loss for a single test point (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The first term on the right is the average individual loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The second is known as the ambiguity—measuring the disagreement of individuals, as a spread around the ensemble prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Since this term is non-negative, it guarantees the ensemble loss will be less than or equal to the average loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This result is often erroneously cited as the reason why all ensembles work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, the expression above applies if and only if we use the squared loss with an arithmetic mean combiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' If we use squared loss with a different combiner, the result no longer holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A deeper understanding came from Ueda & Nakano (1996)—though under the same loss/combiner assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' They extended the bias-variance theory of Geman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (1992) to show that the expected squared loss of the ensemble decomposes into three terms: ED � (q(x) − y)2� = bias(q) + 1 M variance + � 1 − 1 M � covar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2) This is the ensemble bias, plus 1 M times the average variance, and the third term involves the covariance of model pairs, averaged over all �M 2 � pairs of members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Thus, we have a three-way bias/variance/covariance trade-off, where the covariance term completely captures the notion of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It is the trade-off that determines the overall expected ensemble loss, where a strongly negative covariance indicates a diverse ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' As mentioned, the expressions above do not apply beyond squared loss with the arithmetic mean combiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A significant community effort has been directed to find corresponding notions of diversity for classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We review this next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 4 Known results for classifier ensembles: For classification problems, we might have estimates of the class probability distribution, or just labels Understanding diversity in these scenarios has proven more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' An early result by Tumer & Ghosh (1996) demonstrated that the correlation between pairs of ensemble averaged class probabilities had a simple relationship to the overall ensemble classification error, at least in a region close to decision boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The analysis was extended to weighted combinations by Fumera & Roli (2005), under similar assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Brown (2009) and Zhou & Li (2010) proposed information theoretic analyses, showing that diversity manifests as both low- and high-order interactions between ensemble members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Buschjäger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2020) used a Taylor approximation on twice-differentiable losses, showing an exact decomposition when higher derivatives are zero, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' squared loss, but not cross-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Similarly, Ortega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2022) decomposed upper bounds on losses, again only obtaining an equality for squared loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Kuncheva & Whitaker (2003) took another approach, examining diversity measures for their empirical relationship to the ensemble error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' One strategy they proposed was to choose a discrepancy metric δ(qi, qj) ∈ R between the predictions of two models at point x, and defining “diversity” by averaging over all pairs of ensemble members: diversity(q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='., qM) = 1 M(M − 1) M � i=1 � j̸=i δ(qi, qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (3) The diversity measure is then evaluated for its correlation to the overall ensemble performance, and seen as more successful if it has high correlation, illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' diversity measure A ensemble improvement relative to baseline (0-1 loss) diversity measure B Figure 2: Accuracy/diversity for two (hypothetical) diversity measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Measure B (right) is more “successful”, as it has stronger correlation to performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Several measures (including non-pairwise measures) were explored for both class labels and class probability distributions, with no single measure proving more successful than any other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Almost 20 years on, novel diversity heuristics and measures are still being proposed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', Jan & Verma (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Rame & Cord (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Our approach to the problem: In our work we build on the strong foundation of bias-variance theory (Heskes, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Pfau, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This results in exact bias-variance-diversity decompositions for several ensemble losses, and a deeper understanding of the importance of the ensemble combination rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 5 3 A Very Short Introduction to Bias-Variance Decompositions Consider a training set D = {(xi, yi)}n i=1 drawn from a random variable D ∼ P(X, Y )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' From this, we learn a model q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D), predicting the conditional mean EY |X=x[Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We adopt the following notation for expectation over the true data distribution, EXY � · · � def= � PX(x) � PY |X(y | x) � · · � dy dx, and over training sets: ED � · · � def= � PD(D) � · · � dD, where PX and PD are the usual probability density functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For regression problems, we write the squared risk of a particular (trained) model, q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D), as R(q) def= EXY � (q(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D) − Y )2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (4) Geman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (1992) considered the expected risk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', risk in expectation over training sets D drawn from D, showing it decomposes into three terms: ED � R(q) � � �� � expected risk(q) = EX � σ2 Y |X � �� � noise + � ED [q(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)] − EY |X[Y ] �2 � �� � bias(q) + ED �� q(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D) − ED [q(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)] �2� � �� � variance(q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (5) The first term, σ2 Y |X = EY |X[ � Y − EY |X[Y ] �2], is the irreducible noise in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The second term is the bias—the loss of the expected predictor against the conditional mean EY |X[Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The final term is the variance, expressing the variation in q due to different training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' These concepts are often explained with a dartboard diagram, as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Figure 3: The classic dartboard analogy for explaining bias and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The bullseye (yellow circle) is the target for a single test point, and each blue dot is a prediction from a model trained with a different training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A model with high bias, low variance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', linear regression) will be insensitive to small training data changes, but have an expected value far from the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A model with low bias, high variance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', a regression tree) will have an expected value that is close to the target, but will be very sensitive to training data changes, meaning any given model is likely to overfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Note that D does not have to refer to training data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' there is a decomposition with respect to any initial condition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', initial weights for neural networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Note also that we may use the term bias (correspondingly variance) to indicate the value at point x, or 6 in expectation over the distribution of X—which of the two is intended will be made clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' As model capacity increases, bias tends to decrease, and variance tends to increase: the bias-variance trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This is a simplified view, which does not always capture more complex underlying behavior e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', Belkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The terms in Equation (5) can be estimated from data (details in Appendix B) illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 0 5 10 15 20 25 30 maximum depth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='8 expected risk bias + noise variance Figure 4: Building regression trees of increasing depth (California Housing data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Bias-variance decompositions apply for more than just squared loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Geman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (1992) is a widely-appreciated result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, similar decompositions hold for other losses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', the KL- divergence of class probability estimates (Heskes, 1998), which we illustrate with a “dartboard" for k = 3 classes in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Figure 5: Bias/variance for the KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Yellow circle is the target for a test point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Blue star is the normalised geometric mean of the model distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The bias/variance terms take different functional forms for each loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For example, Geman et al’s original bias term is often referred to as ‘squared bias’, but the square turns out to be an artefact of using this particular loss, and is not present in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Furthermore, the terms obey a different geometry, defined by the loss function of interest—this manifests in the ‘expected model’ being replaced by other forms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', a normalised geometric mean for the KL-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 7 4 A Unified Theory of Ensemble Diversity Ensemble “diversity” is a popular, but variously defined idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Bias and variance, on the other hand, are clear-cut and precisely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It therefore makes sense to build stronger bridges between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Our approach is exactly this, revealing diversity as a hidden dimension in the bias-variance decomposition of an ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We argue that diversity should be considered in exactly the same manner as bias/variance—simply another aspect of the model fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We first describe how the ideas apply to the squared loss, then generalise it to other losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 Ensemble Diversity via the ‘Double Decomposition’ Trick Our approach to understanding diversity is ‘unified’ in the sense that the same methodology can be applied to numerous different losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We refer to this as the ‘double decomposition’ trick, which we will now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The first step is to recognise that the ambiguity decomposition (Krogh & Vedelsby, 1994) is a special case of the bias-variance decomposition (Geman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We can see this most clearly by stating the bias-variance decomposition at a single test point (x, y), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', omitting the noise term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' All results still apply with noise, but are easier to explain in the noise-free scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For brevity, we omit the dependence on x and D, taking it as understood that the model q is dependent on both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This gives us a simpler form: ED � (q − y)2� = (ED [q] − y)2 � �� � bias(q) + ED � (q − ED [q])2� � �� � variance(q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (6) We now just replace each occurrence of the expectation ED [· · ·], with a uniformly weighted arithmetic mean 1 M �M i=1[· · · ] over a set of M models, 1 M M � i=1 (qi − y)2 = � 1 M M � i=1 qi − y �2 + 1 M M � i=1 � qi − 1 M M � i=1 qi �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (7) Noting that the ensemble combiner is q = 1 M �M i=1 qi, we simply rearrange the terms of Equation (7), and obtain the ambiguity decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' � q − y �2 = 1 M M � i=1 (qi − y)2 � �� � average loss − 1 M M � i=1 � qi − q �2 � �� � ambiguity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (8) Thus, the ambiguity decomposition is a special case of the bias-variance decomposition, replacing expectations ED with an arithmetic mean 1 M �M i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We define the double decomposition trick as the successive application of the ambiguity and bias/variance decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This exposes a natural term quantifying the diversity, specific to the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This methodology is the key contribution of our work, illustrated in Figure 6, to be re- used throughout the paper with a range of different losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For squared loss, we apply Equation (8), then Equation (6), and obtain Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Since the derivation is trivial following the proposed trick, we omit proof detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 8 ED � ensemble loss � �� � � Apply Equation (8) ED[ average loss − ambiguity ] Apply Equation (6) bias + variance − diversity Figure 6: The ‘double decomposition’ trick, shown here for squared loss, but applicable to any loss which admits a bias-variance decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 1 (Bias-Variance-Diversity Decomposition for Squared Loss) For an ensemble of models q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , qM, where q = 1 M �M i=1 qi, the expected loss of q decomposes as, ED � (q − y)2 � = 1 M M � i=1 (ED [qi] − y)2 � �� � bias + 1 M M � i=1 ED � (qi − ED [qi])2� � �� � variance − ED � 1 M M � i=1 (qi − q)2 � � �� � diversity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (9) This has decomposed the expected loss into: the average bias, the average variance, and the expectation of the ensemble ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It is this expected ambiguity term that we consider as the ensemble diversity, which we highlight has the opposite sign to the other terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In Figure 7 we estimate the terms1 for Bagged trees on the California housing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 0 10 20 30 ensemble size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 squared error expected risk 0 10 20 30 ensemble size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 average bias + noise average variance diversity Figure 7: Decomposing the expected ensemble loss (Bagging depth 8 regression trees).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The expected risk decreases with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Looking at the loss components: the bias and variance are constant—this is as we might anticipate, since the form/capacity of the individuals is constant, it is only the number of them, M, that we change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In contrast, the diversity increases with M— subtracting from the expected risk—and the improvement is determined entirely by diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This is of course different if we vary something other than M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Figure 8 fixes M = 10, but varies 1Note that D can be a joint random variable over data/conditions for each individual—a short discussion on this is in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 9 tree depth—all three components now change, and we see overall performance is determined by a bias-variance-diversity trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Figure 8: Bagging M = 10 trees, varying maximum depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Further experiments (with deep/shallow trees and Random Forests) are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For squared loss, the decomposition can be seen as an alternative to the bias-variance-covariance decomposition (Ueda & Nakano, 1996), seen in Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We will compare and contrast these later (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2) once the general case is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 Generalising Bias-Variance-Diversity to Other Losses As mentioned in Section 3, bias-variance decompositions in a form similar to Geman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (1992), are known for several other losses (Heskes, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Buja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Pfau, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We can define a general form, covering all these cases, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Definition 1 (Generalised Bias-Variance Decomposition) For a loss function L, a generalised bias-variance decomposition is defined, ED [L(y, q)] � �� � expected loss = L(y, q∗) � �� � bias + ED [ V(q∗, q) ] � �� � variance , (10) where V(·, ·) is a non-negative dissimilarity function, and q∗ def= arg minz ED � V(z, q) � is the “centroid" of the model distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A corresponding generalised ambiguity decomposition is induced by the bias-variance decomposition, and can be stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Definition 2 (Generalised Ambiguity Decomposition) For a finite set of models {q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , qM}, the ambiguity decomposition, under loss function L, is defined, L(y, q) = 1 M M � i=1 L(y, qi) � �� � average loss − 1 M M � i=1 V(q, qi) � �� � ambiguity , (11) where q def= arg minz 1 M �M i=1 V(z, qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 10 These are templates, generalising other decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We can recover Geman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (1992) with L(y, q) = V(y, q) = (y−q)2, where the centroid is q∗ = argminz ED �(z − q)2� = ED [q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Or, if L and V are the KL-divergence of class distributions, we recover Heskes (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In this case, the centroid is different—turning out to be a normalised geometric mean, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q∗ = argminz ED [V(z, q)] = Z−1 exp(ED [ln q]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The centroid is determined by the form of the dissimilarity, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' These examples both have V = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, in general, V and L do not have to be of the same form—as is the case for margin losses, discussed in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We now present our main result in Theorem 2, a generalised bias-variance-diversity decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It is derived via the strategy in Figure 6, using Equations (11) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 2 (Generalised Bias-Variance-Diversity Decomposition) Consider a set of models {q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , qM}, evaluated by a loss function L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Assuming a bias-variance decomposition holds in the form of Definition 1, the following generalised bias-variance-diversity decomposition also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' ED [L(y, q)] = 1 M M � i=1 L(y, q∗ i ) � �� � average bias + 1 M M � i=1 ED [V(q∗ i , qi)] � �� � average variance − ED � 1 M M � i=1 V(q, qi) � � �� � diversity , (12) where q∗ def= arg minz ED � V(z, q) � and the combiner is q def= arg minz 1 M �M i=1 V(z, qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' There is now a bias/variance/diversity trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' As individual models increase in capacity, their average bias decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Without regularisation, their average variance would increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' These determine only part of the ensemble behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The final part is the diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A critical point here is diversity always subtracts from the expected risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This is not to say that greater diversity always reduces expected risk—it only reduces it given a fixed bias and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Ultimately, the three-way trade-off of bias/variance/diversity is what determines the overall ensemble performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It is worth highlighting that diversity is defined similarly to bias/variance, involving an expectation over D, as opposed to being a property of a single training run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We also note, this applies for both dependent and independent training schemes—discussion in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The ensemble combiner q is a centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We refer to this as the centroid combiner rule, the minimiser of the average dissimilarity, q def= arg minz 1 M �M i=1 V(z, qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Krogh & Vedelsby (1994) assumed the combiner was an arithmetic mean, q = 1 M �M i=1 qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Audhkhasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2013) and Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2017) proposed generalisations of the ambiguity decomposition for classification, though both still assumed must be an arithmetic mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In contrast, here the combiner is defined in terms of the dissimilarity V, itself a consequence of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 Summary We presented a framework to understand diversity, applicable for any loss where a bias-variance decomposition holds in the form of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This revealed diversity as a measure of model fit, in precisely the same sense as bias/variance, with a concrete application for squared loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In Section 5 we present an application with Bregman divergences, covering many losses as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In Section 6 we show necessary modifications to enable an understanding for 0-1 loss, and discuss limitations in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 11 5 Diversity for Bregman Divergences In this section, we will apply the framework developed in Section 4 to the class of Bregman divergences (Bregman, 1967), which covers many popular losses as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 The Basics of Bregman Divergences A Bregman divergence Bφ (p, q) is defined in terms of a generator function, φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Let φ : S → R be a strictly convex function defined on a convex set S ⊆ Rk, such that φ is differentiable on ri(S)—the relative interior of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The Bregman divergence Bφ : S × ri(S) → R+ is defined, Bφ (p, q) def= φ (p) − φ (q) − ⟨∇φ (q) , (p − q)⟩, (13) where ⟨·, ·⟩ denotes an inner product, and ∇φ(q) denotes the gradient vector of φ at q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Different choices of φ lead to different losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' With φ (q) = q2, the gradient vector ∇φ(q) is a scalar derivative dφ(q)/dq = 2q, and we recover a squared loss, Bφ (p, q) = (p − q)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 �(p) B�(p, q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='49 �(q) Bφ (p, q) = φ (p) − φ (q) − ⟨∇φ (q) , (p − q)⟩ = p2 − q2 − ⟨2q, (p − q)⟩ = p2 − q2 − 2pq + 2q2 = p2 + q2 − 2pq = (p − q)2 Figure 9: Bregman divergence illustrated for the generator φ (q) = q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For this example, we have a divergence Bφ (p, q) = (p − q)2 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Alternatively, we can take a vector q ∈ Rk−1, for a k-class problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Note, this is not a probability vector summing to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It is, however, the minimal description of the distribution, as the kth class probability is 1 − � c q(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' With a particular generator (see final row of Table 1) we recover the the KL-divergence between the distributions in Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Loss function Bφ (p, q) Generator φ (q) Domain S Squared loss (p − q)2 q2 q ∈ R Itakura-Saito p q − ln p q − 1 − ln q q ∈ R+ Poisson loss p ln p q − (p − q) q ln q − q q ∈ R+ KL-divergence � p(c) ln p(c) q(c) + � 1 − � p(c)� ln 1−� p(c) 1−� q(c) � q(c) ln q(c) + � 1 − � q(c)� ln � 1 − � q(c)� q ∈ [0, 1]k−1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' � c q(c) ≤ 1 Table 1: Common loss functions and their Bregman generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 Relating Bias, Variance, and Ambiguity for Bregman Divergences As discussed in the previous section, our framework requires the existence of a bias-variance decomposition for the loss at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In this case of Bregman divergences, Pfau (2013) proved a bias-variance decomposition, which is written for a single point (x, y) as: ED [Bφ (y, q)] = Bφ (y, q∗) + ED [Bφ (q∗, q)] , (14) where the value of q∗ can be obtained in closed-form as q∗ def= arg min z ED � Bφ (z, q) � = [∇φ]−1 � ED [∇φ (q)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (15) This definition of q∗ is known in the information geometry literature, as a left Bregman centroid (Nielsen & Nock, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' If φ is the generator for a squared loss, then q∗ = ED [q], the expected model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' With other losses, the form of q∗ changes—we detail several examples in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Considering Equation (14), the corresponding ambiguity decomposition can be written as follows, again simply replacing expectations by finite averages and rearranging terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 3 (Bregman Ambiguity Decomposition) For a target label y ∈ S and a set of predictions q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , qM ∈ ri(S), Bφ (y, q) = 1 M M � i=1 Bφ (y, qi) − 1 M M � i=1 Bφ (q, qi) (16) where q = [∇φ]−1� 1 M � i ∇φ(qi) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For the ambiguity decomposition to apply, the ensemble combiner q is constrained to be of the same form as q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' When combining ensemble members, we will refer to this form as the Bregman centroid combiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Before we present the formulation of diversity in this case, we offer a brief discussion on this combiner, clarifying the relation to existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Definition 3 (Bregman Centroid Combiner) The Bregman centroid combiner is the left Bregman centroid for a set of M predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This is the minimizer of the average divergence from all points in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The centroid combiner q for {q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' qM} is q def= arg min z 1 M M � i=1 Bφ (z, qi) = [∇φ]−1 � 1 M M � i=1 ∇φ(qi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (17) The Bregman centroid combiner is a generalised f-mean, with f = ∇φ, also known as a Kolmogorov mean or quasi-arithmetic mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For φ(q) = q2, this is the arithmetic mean of the points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', the centre of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, other φ generators define a non-linear transformation, meaning the centroid and the centre of mass are different in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The centroid combiner reproduces several known combiners as special cases, shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Using this principle comes with a distinct advantage—through the ambiguity decomposition, we are guaranteed that the ensemble loss will be less than or equal to the average loss, generalising the well-known squared loss case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' An equivalent definition was considered by Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2022) under the assumption of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Our analysis both complements and extends this by removing the assumption, and more fully characterising the properties of ensembles using this combination rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 13 Loss function Centroid Combiner Name Squared loss 1 M �M i=1 qi Arithmetic mean Poisson regression loss �M i=1 q 1 M i Geometric mean KL-divergence Z−1 �M i=1 � q(c) i � 1 M Normalised geometric mean Itakura-Saito loss 1 �� 1 M �M i=1 1 qi � Harmonic mean Table 2: Centroid combiners (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', left Bregman centroid of the ensemble) for various losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The centroid combiner can be understood as an ensemble averaging operation, but in a new coordinate system, where the mapping between coordinate systems is defined by the gradient of the Bregman generator with respect to its argument, ∇φ(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This is illustrated in Figure 10 for the KL-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q ∂ ∂qi φ(qi) η ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='16 q ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='896 q1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='7 η1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='8473 q2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='97 η2 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='476 Figure 10: Ensemble averaging in the geometry defined by the KL-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We use notation q for the primal coordinate system, and η for the dual coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In this simple illustration we are predicting a single probability p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The primal-dual mapping is the gradient ηi = ∂ ∂qi φ(qi) = ln qi 1−qi , plotted as the blue curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Two points in the primal {q1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='7, q2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='97} are mapped to the dual {η1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='8473, η2 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='476}, then combined via arithmetic mean (η ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='16), and finally mapped back by the inverse operation q = exp(η)/(1+exp(η)) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The centroid combiner is therefore an arithmetic mean ensemble in the dual coordinate system, which is equivalent to the centroid of the models in the primal coordinate system, using the Bregman divergence as the measure of dissimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In the case of KL divergence, this is equivalent to a normalised geometric mean in the primal coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The primal coordinate system for KL is a probability simplex—Figure 11 shows this for the 3-class case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', a Bregman divergence on constrained vectors in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 14 class 3 class 1 class 2 Figure 11: Combining M = 4 predictions in the probability simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The centroid combiner is shown as the blue star, minimising average KL-divergence from all predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The arithmetic mean is also shown (pink star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Note that points are connected in the simplex not by straight lines, but by dual geodesics defined via the generator φ (Nielsen & Nock, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A different φ (and therefore a different loss) would result in a different primal-dual mapping, and thus a different ensemble combiner rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 A Bias-Variance-Diversity Decomposition for Bregman Divergences We can now apply the double decomposition trick, as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Doing so shows that an expected Bregman divergence (when using the centroid combiner) decomposes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 4 (Bregman Bias-Variance-Diversity decomposition) For an ensemble q1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , qM(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D), let q∗ i be the left Bregman centroid of qi (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', q∗ i def= [∇φ]−1 (ED [∇φ(qi)])) and define q def= [∇φ]−1 � 1 M �M i=1 ∇φ (qi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Then we have the decomposition, ED [EXY [Bφ (Y, q)]] = EXY � Bφ � Y, Y � � �� � noise + 1 M M � i=1 Bφ � Y, q∗ i � � �� � average bias + 1 M M � i=1 ED [Bφ (q∗ i , qi)] � �� � average variance − ED � 1 M M � i=1 Bφ (q, qi) � � �� � diversity � , where Y = EY|X [Y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Examples for different losses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', different Bregman generators) are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' One point that this makes clear is that the mathematical formulation of diversity is specific to the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='Expected Ensemble Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='Average Bias ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='Average Variance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='Diversity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='Squared ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='EY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='(q − Y )2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='(q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i − Y )2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='(qi − q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i )2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='qi − q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='�2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='KL-divergence (Bernoulli) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='EY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='Y ln Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i + (1 − Y ) ln 1−Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1−q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='qi + (1 − q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ) ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1−q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1−qi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q ln q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='qi + (1 − q) ln 1−q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1−qi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='KL-divergence (Multinoulli) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='EY [ED [K(Y || q)]] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='K(Y || q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED [K(q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i || qi)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='K(q || qi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='Itakura-Saito ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='EY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q − ln Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q − 1�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='− ln Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='qi − ln q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='qi − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='qi − ln q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='qi − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='Poisson ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='EY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='Y ln Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q − (Y − q)�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='Y ln Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='− (Y − q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED [qi] − q∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='ED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='qi − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='q1/M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='Table 3: Bias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' variance and diversity terms under different Bregman divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In all cases, the expectation over p(x) is omitted and the expressions given are for a single x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Refer to Table 5 for definitions of the left Bregman centroid q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 Empirical Demonstration for the Cross-Entropy Loss In this section we show some illustrative experiments with the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 Estimating Bias, Variance, and Diversity Theorem 4 is particularly powerful in that it applies across a range of losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In Table 3 we saw the KL-divergence, but we can also extend this with little effort to a decomposition for the ubiquitous cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 5 Let y be a one-hot class vector of length k, and q ∈ Rk be a model’s prediction of the class distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Define a set of such models {qi}M i=1, and their combination q as their normalised geometric mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The following decomposition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' −ED [y · ln q] � �� � expected cross-entropy = − 1 M M � i=1 y · ln q∗ i � �� � average bias + 1 M M � i=1 ED [K(q∗ i || qi)] � �� � average variance − ED � 1 M M � i=1 K(q || qi) � � �� � diversity , (18) Here, the centroid combiner q is a normalised geometric mean (see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 for more detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' With neural network ensembles, this is equivalent to averaging the network logits, followed by a softmax—a well-established practice, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In Figure 12 we compare Bagging of single-layer MLPs on MNIST, using small networks of 20 nodes, versus larger networks with 100 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The expected risks decompose into bias/variance/diversity components, and we observe similar patterns as we saw with the squared loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Further experiments can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 0 5 10 15 20 ensemble size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 Expected Risk Expected Risk Small Network Larger Network 0 5 10 15 20 ensemble size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='20 Small Network average bias + noise average variance diversity 0 5 10 15 20 ensemble size Larger Network Figure 12: Decomposing expected ensemble cross-entropy for Bagging small/larger MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Overall, the ensemble of larger networks have performed better, and the reason for this can be explained by examining the expected loss components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The larger networks are can be observed to have both lower average variance2 and lower average bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Since we are varying ensemble size, the average bias/variance terms remain constant, and the only factor that changes with M (in both ensembles) is the diversity, which converges closer to the variance in the case of larger networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 2The lower variance is counter to the ML folklore that increasing model capacity should also increase variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It is, however, consistent with recent observations (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 Examining the Correlation of Diversity and Classification Error Figure 2 showed a toy “error/diversity” scatter plot, in the style popularised by Kuncheva & Whitaker (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In such a plot, each point is one ensemble, showing its performance improvement (0-1 loss), against a diversity measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A higher correlation is seen to be a more successful diversity measure, as it explains the performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Our framework defines diversity specific to each loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For cross-entropy, the diversity is written in Equation (18), defined separately below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Definition 4 Diversity, when using cross entropy loss, is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' diversity(q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='qM) = ED � 1 M M � i=1 K(q || qi) � (19) Figure 13 shows a scatter plot corresponding to the experiment in Figure 12 on MNIST with Bagged neural networks, combined via a normalised geometric mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Further experiments are in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='25 diversity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='025 average individual error ensemble error M = 2 M = 3 M = 4 M = 5 M = 20 M = 2 M = 3 M = 4 M = 5 M = 20 Ensemble of Smaller Networks Ensemble of Larger Networks Figure 13: Diversity plot for Bagged MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The x-axis is the diversity, estimated on a validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The y-axis is the difference of error rates, on a final test set—the average individual error, minus the ensemble error, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', the gain of the ensemble, over the average individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We see a strong correlation in both configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The smaller networks (blue triangles, Pearson’s r2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='998) have greater diversity than the larger networks (orange circles, r2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The plot must be read in the context that overall, the larger networks significantly outperformed the smaller networks—it simply shows that the performance came from more powerful base models, as opposed to their diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' One might wonder why there is such a strong correlation in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' If we remember the alternative view of the same experiment, Figure 12, we see that bias/variance are constant, and it is only the diversity that changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' When any change is observed in the overall ensemble cross-entropy, we know it is caused by a change in diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Therefore, if we can assume strong correlation between the ensemble cross-entropy and the 0-1 loss, then there will be a similar strong correlation between diversity and 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We can do the same for decision tree classifiers, even though they output only class labels, as estimated class probabilities can be obtained through various schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We use a simple method, counting label frequency in leaf nodes, and thus can evaluate the cross-entropy diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In Figure 14 we compare Bagging decision trees (unlimited depth) and Random Forests on MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In both cases the trees are combined by obtaining probabilities and combining via normalised 18 geometric mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We use the same procedure as before: diversity estimated on validation data, and the 0-1 loss measured on a final test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 diversity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='14 average individual error ensemble error M = 2 M = 2 M = 3 M = 3 M = 4 M = 4 M = 5 M = 5 M = 20 M = 20 bagging random forests Figure 14: Ensembles of tree classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Again we see strong correlation of diversity and performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Bagging has a correlation r2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='996, whilst Random Forests has r2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For a fixed M, we can compare corresponding points, where the only difference is the additional split-point randomisation of the forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' At M = 20, RF provides a reduction in generalisation error of ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5%, versus only ≈ 9% for Bagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It interesting to note this is solely due to increased diversity generated by random feature splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We might now wonder, with this diversity measure, will we always see a strong correlation between diversity and reduction in 0-1 loss?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The answer is no, for a very good reason that highlights a critical point in our understanding of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In Figure 15 we fix at M = 10 bagged trees, and vary their depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The expected loss reduces— however, now it is not solely due to diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Now, the bias and variance also change rapidly, and the correlation of 0-1 loss/diversity is much lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 2 4 6 8 10 max depth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='00 average bias + noise average variance diversity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='50 diversity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='12 average individual error ensemble error 1 2 3 4 5 6 7 8 9 10 Figure 15: Bagging M = 10 trees, varying depth, and correlation is now r2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' When varying any other parameter than M, one should not expect to see a strong correlation of performance improvement and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This is because, if we vary any parameter that alters individual capacities, then the average bias/variance also changes, and diversity is not the only factor in play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The overall performance is decided by a 3-way trade-off, just as there is a 2-way trade-off of bias/variance in single models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It would be interesting to explore this with ensembles of very deep neural networks, where the bias/variance trade-off seems to not act as classical theory predicts (Belkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 Further Properties of the Decomposition In this section, we further explore properties of the decomposition we have proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' First, we consider the properties of homogeneous vs heterogeneous ensembles, then two scenarios of interest— the relation to the bias-variance-covariance decomposition (Ueda & Nakano, 1996), and the common practice of averaging class probability estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 Homogeneous vs Heterogeneous Ensembles An ensemble is said to be heterogeneous if the individual members are from different model families, or homogeneous if they are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The expected divergence of the ensemble from the target has a bias-variance decomposition: ED [Bφ (y, q)] � �� � expected ensemble loss = Bφ (y, q∗) � �� � ensemble bias + ED [Bφ (q∗, q)] � �� � ensemble variance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (20) These ensemble bias/variance terms can be related that of the individual models, {qi}M i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 6 The ensemble bias and ensemble variance can be re-written as: Bφ (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q∗) � �� � ensemble bias = 1 M M � i=1 Bφ (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q∗ i ) � �� � average bias − ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (21) ED [Bφ (q∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q)] � �� � ensemble variance = ∆ + 1 M M � i=1 ED [Bφ (q∗ i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' qi)] � �� � average variance − ED � 1 M M � i=1 Bφ (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' qi) � � �� � diversity ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (22) where the common term is ∆ = 1 M �M i=1 Bφ (q∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q∗ i ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' referred to as the model “disparity”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' accounting for diversity in the model families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Equation (21) can be proven by applying Theorem 3 to a set of centroid models, {q∗ i }M i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Equation (22) can be proven similarly, applying Theorem 4 but substituting y = q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' If the models comprising the ensemble are all from the same family (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', ‘homogeneous’, as is common with Bagging/Random Forests) then q∗ i = q∗ j = q∗, meaning ∆ = 0, and we can draw some conclusions: the ensemble bias is equal to the average bias, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', there is no reduction in bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' the ensemble variance is guaranteed to be less than or equal to the average variance: the amount by which the ensemble variance is reduced is exactly the diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' the diversity is upper-bounded by the average variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Alternatively, if ∆ > 0 (as we may expect to occur with boosting) then the ensemble is heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In this case it can be noted that the ensemble bias is always reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, we can make no such simple statement about the ensemble variance, since it has both the addition of the disparity, and the subtraction of the diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 Relation to the Bias-Variance-Covariance Decomposition For the case of squared loss, our decomposition in Equation (1) can be contrasted with the bias- variance-covariance decomposition of Ueda & Nakano (1996), which states: ED � (q − y)2 � = (23) (ED [q] − y)2 � �� � bias(q) + 1 M 1 M M � i=1 ED � (qi − ED [qi])2� � �� � 1/M × variance + 1 M2 � i,j ED [(qi − ED [qi])(qj − ED [qj]))] � �� � (1−1/M) × covariance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This decomposition relies on a simple property for the variance of linear combinations of random variables: V ar(aX1 + bX2) = a2V ar(X1) + b2V ar(X2) + 2abCov(X1, X2), (24) where X1, X2 represent two model outputs and aX1 + bX2 represents the ensemble combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' When the combination rule is non-linear, this property (and hence this as a route to understand diversity) no longer applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The covariance can be either positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Our diversity term, however, is always non- negative, growing with more disagreement around the ensemble decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The covariance is a fundamentally pairwise computation—it is likely that this form inspired the many published pairwise diversity measures (Kuncheva, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Diversity in the Bregman case is written in a non- pairwise manner—the expected average deviation around the ensemble prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We conjecture that this term cannot be expressed as solely pairwise operations, implying that pairwise measures may be fundamentally limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Further differences are found in examining the bias components of each decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Ours is the average individual bias, whereas Ueda & Nakano’s is the ensemble bias: bias = 1 M M � i=1 (ED [qi] − y)2, (25) bias(q) = (ED [q] − y)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (26) Ueda & Nakano observed a re-writing of their term: (ED [q] − y)2 = ( 1 M �M i=1[ED [qi] − y])2, and described this as “the square of the average biases”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This follows from language in statistical estimation theory, where E[ˆθ] − θ is the bias of ˆθ as an estimate of a population value, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, we remind the reader that the square is an artefact of using the squared loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This square is not present in generalised forms of the bias-variance decomposition e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', Pfau (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Thus, (ED [q] − y)2 should be referred to as simply the “bias of the ensemble”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The difference between bias and bias(q) is however an interesting quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' With some simple algebra, we can re-write the ensemble bias term as follows: bias(q) = bias − 1 M2 � i,j (ED [qi] − ED [qj])2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (27) This shows that their term is in fact made up of two components: the average of the individual biases, and the disparity term introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' If the models are homogeneous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', of the same family, then this term will be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 Cross Entropy Loss: Averaging estimates of class probabilities When combining class probability estimates, a very popular strategy is to take their arithmetic mean, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', Lakshminarayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2017), but, if we use the cross-entropy, this is not the centroid combiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We might wonder what effect this has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The proposition below demonstrates that the cross-entropy loss of the ensemble is still guaranteed to be less than the average loss of its members, but the ambiguity becomes dependent on the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Proposition 7 Assume a target probability vector, y ∈ Rk, and a set of models {qi}M i=1 combined by averaging, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q† = 1 M �M i=1 qi, then the cross-entropy of q and y is −y · ln q† � �� � ensemble cross-entropy = − 1 M M � i=1 y · ln qi � �� � average cross-entropy − k � c=1 y(c) ln 1 M �M j=1 q(c) j ��M i=1 q(c) i � 1 M � �� � ambiguity (target-dependent) , (28) where the second term is non-negative, thus the ensemble loss is guaranteed less than or equal to the average individual loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This property can be observed without the framework we have presented thus far, by taking the difference between −y · ln q† and − 1 M �M i=1 y · ln qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Proposition 7 was observed independently by Ivaşcu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2021), for the case of two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, only with our framework can we identify that the normalised geometric mean is the necessary combiner to make the final term target-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' If we take the expectation of this with respect to D, we have the following result, proved in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Proposition 8 (Diversity for Averaged Probabilities is target-dependent) Let q† = 1 M �M i=1 qi, with qi ∈ [0, 1]k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The expected cross-entropy admits the decomposition: ED � −y · ln q†) � = − 1 M M � i=1 y · ln q∗ i � �� � average bias + 1 M M � i=1 ED [K(q∗ i || qi)] � �� � average variance − ED � ��� k � c=1 y(c) ln 1 M �M j=1 q(c) j ��M i=1 q(c) i � 1 M � ��� � �� � dependency .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The final term is now dependent on the target y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For this reason we avoid using the name “diversity”, and instead refer to it as a “dependency” term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We emphasise that we make no claims about the empirical superiority of one combiner versus another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We simply observe that the centroid combiner is the only case where diversity is independent of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2022) also studied properties of ensemble bias/variance with an arithmetic mean combiner, showing that (under an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' model assumption), the ensemble variance is always reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' At the same time, they raised a concern, that this may potentially increase the ensemble bias (above the average bias), dependent on the label distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Our proposition adds insight: the overall expected loss will always be less than the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Thus, even if there is an increase in ensemble bias, it is always more than compensated by the reduction in ensemble variance, leading to lower overall expected loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 22 6 Diversity for the 0-1 Loss In this section we show the necessary modifications to our framework, that will enable a stronger understanding of diversity in the case of 0-1 loss, with particular focus on majority voting ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 The Nature of Bias/Variance for the 0-1 Loss is very different The 0-1 loss is defined L0/1 : S ×S → {0, 1} over a finite set S such that L0/1(y, q) = 1, when q ̸= y, or 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' That is, a loss of 1 when q is incorrect, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' With this definition, the quantities we might intuitively understand as bias and variance, do not sum to the classification error, discussed at length by Domingos (2000), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', ED � L0/1(y, q) � ̸= L0/1(y, q∗) + ED � L0/1(q∗, q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (29) where q∗ = argminz∈S ED � L0/1(z, q) � is the modal value of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Many authors tried to find alternative decompositions, with much debate on what axioms the terms should obey, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', Kohavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' James & Hastie (1997);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Heskes (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Given this literature, one might be tempted to say we just need to keep searching for the “right” definitions of bias and variance, that will then sum to the expected loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In fact it turns out that a definition of bias/variance fitting the abstract form in Definition 1 does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We present this in the Theorem below, with proof in Appendix D—to the best of our knowledge, this is the first formal proof of this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 9 (Non-existence of a Bias-Variance Decomposition for 0-1 loss) The 0-1 loss cannot be decomposed as a bias term plus a variance term, where the bias is the loss of some deterministic model q∗ derived from the distribution of q and the variance term is independent of the label y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', there is no V and q∗ such that in general ED � L0/1(y, q) � � �� � expected loss = L0/1(y, q∗) � �� � bias + V[q] � �� � variance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (30) We note that this is an even more general form than the bias-variance decompositions stated earlier, in that the q∗ is not required to be a centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Corollary 10 (Non-existence of an ambiguity decomposition for 0-1 loss) Given a set of models q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='., qM which predict labels drawn from a finite set S, there exists no rule for constructing q and function V, independent of y, such that L0/1(y, q) = 1 M M � i=1 L0/1(y, qi) − V({qi}M i=1) (31) This has significant implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In many works (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Kuncheva & Whitaker, 2003) authors have attempted to define a measure of diversity that correlates with the improvement in ensemble accuracy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', Q-statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The above shows that if we quantify the improvement as the difference between the ensemble loss and the average individual loss, this quantity cannot be expressed independently of the target variable, for any combiner rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This is of course disappointing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' however, there is still a way forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 23 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 Understanding the Effects of Bias, Variance, and Diversity James & Hastie (1997) present an insightful viewpoint: that we should not be interested in the bias/variance of 0-1 loss for their own sake, but for their effect on the expected loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' If we had a model q that was constant with variations in D, the loss will also be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' But if the model varies at all, it will cause a change in the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Therefore the effect of this model variance is visible in how the loss changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Compare the expected loss of q (which varies with D), and the loss of q∗ (which is constant with D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The variance-effect is variance-effect = EY � ED � L0/1(Y, q) �� − EY � L0/1(Y, q∗) � , (32) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', their difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The bias-effect is defined similarly, and accounts for the fact that there may be noise in the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The bias-effect3 is the expected change in the loss, when using the centroid model q∗, compared to using the Bayes prediction Y ∗, bias-effect = EY � L0/1(Y, q∗) − L0/1(Y, Y ∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (33) Combining these, they derive a decomposition as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 11 (Bias/Variance Effect decomposition, James & Hastie (1997)) ED � EY � L0/1(Y, q) �� = (34) EY � L0/1(Y, Y ∗) � � �� � noise + EY � L0/1(Y, q∗) − L0/1(Y, Y ∗) � � �� � bias-effect + EY � ED � L0/1(Y, q) − L0/1(Y, q∗) �� � �� � variance-effect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' where Y ∗ = argminz∈S EY � L0/1(Y, z) � and q∗ = argminz∈S ED � L0/1(z, q) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For some losses, the effects of bias/variance coincide with the quantities themselves, and the above reduces to the familiar bias-variance decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, for the 0-1 loss, the effects and the terms themselves are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We note again that for any bias-variance decomposition (including James & Hastie), we can state a corresponding ambiguity decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' As before, we replace expectations by finite averages, and evaluate at a single target y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Proposition 12 (Ambiguity-Effect Decomposition) For an ensemble q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , qM with any combiner rule q, and correct label y ∈ S L0/1(y, q) = 1 M M � i=1 L0/1(y, qi) � �� � average loss − 1 M M � i=1 � L0/1(y, qi) − L0/1(y, q) � � �� � ambiguity-effect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (35) The proof of this is trivial, but the result is powerful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It generalises the ambiguity decomposition to the 0-1 loss, while acknowledging Corollary 10, that the difference between the ensemble loss and the average loss cannot be expressed independently of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The ambiguity effect measures the average change in the loss, when using individual members instead of the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We now use the double-decomposition trick to derive the effect of diversity in 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Using the results above, we apply the double decomposition as illustrated in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 3James & Hastie (1997) actually use the term “systematic effect", while we refer to it as the “bias-effect" to more accurately reflect its role in our framework, and relation to other decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 24 ED � ensemble loss � �� � � Apply Equation (35) ED[ average loss − ambiguity-effect ] Apply Equation (34) bias-effect + variance-effect − diversity-effect Figure 16: The double decomposition trick using James & Hastie (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 13 (Bias-Variance-Diversity effect decomposition) Given a loss function L : S × S → R+ and an ensemble of models q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , qM, where q is the majority vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' EY � ED � L0/1(Y, q) �� = EY � L0/1(Y, Y ∗) � � �� � noise + 1 M M � i=1 EY � L0/1(Y, q∗ i ) − L0/1(Y, Y ∗) � � �� � average bias-effect + 1 M M � i=1 ED � EY � L0/1(Y, qi) − L(Y, q∗ i ) �� � �� � average variance-effect − ED � EY � 1 M M � i=1 � L0/1(Y, qi) − L(Y, q) ��� � �� � diversity-effect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We obtained three terms: the effects of average bias, average variance, and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The diversity- effect is dependent on the target Y , an unavoidable property of the 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 Estimating the Effects of Bias/Variance/Diversity As before, we can estimate these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We illustrate this by comparing Bagging and Random Forests on MNIST, combining predictions by majority voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 5 10 15 20 Ensemble Size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='15 0-1 Loss Ensemble 0-1 Loss Bagging Random Forests 3 6 9 12 15 18 Ensemble size Bagging average bias-effect average variance-effect diversity-effect 3 6 9 12 15 18 Ensemble size Random Forest Figure 17: Bias/Variance/Diversity effect for Bagging vs Random Forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 25 We see familiar behaviors—as we increase M, the bias-effect and variance-effect remain constant, but the diversity-effect increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The variance-effect for Random Forests is significantly higher than for Bagging, though this is compensated for with the diversity-effect for large ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 13 can be extended relatively simply for weighted voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The extension and proof are provided in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2, while here we utilise the result to analyse boosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We plot the components (Figure 18) this time boosting low-variance decision stumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 0 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 Bagging ensemble error 0 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 AdaBoost 0 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 LogitBoost 0 100 200 Ensemble size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 average bias-effect average variance-effect diversity-effect 0 100 200 Ensemble size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 0 100 200 Ensemble size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 Figure 18: Mease data, ensembling decision stumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We note some interesting differences between the parallel model (Bagging) and the sequential models (LogitBoost and AdaBoost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' As before, the bias/variance are constant for Bagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However for boosting, the terms vary with ensemble size, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This is caused by the non-homogeneous nature of the ensemble members, specialising to different parts of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In boosting, the diversity-effect can be greater than the variance-effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This is due to the fact that ensemble members are designed to be complimentary: with disagreements actively encouraged, as opposed to being a property of random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' More generally, with Bagging the decrease in error comes from increasing diversity, but the story for boosting models is more complex, with the overall performance of the model being determined by a complex trade-off between the three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 26 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 How Can Diversity-Effect be Negative?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It is possible for diversity-effect to take negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In this case, given the signs in the expression, it would be harmful for the expected risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We now examine under what circumstances this situation might arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A Theoretical Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Assuming a binary classifier, define ϵ = ED � EY � L0/1(y, q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)) �� as the probability of error (over the distribution of D) at a test point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' If we assume M classifiers make errors independently then, for odd M, the diversity-effect can be written: DE = ϵ − M−1 2 � i=0 � M i � ϵM−i(1 − ϵ)i, (36) where the second term is the majority voting error for independent models (Hansen & Salamon, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We plot this, varying ϵ, in Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 ϵ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 diversity-effect M=5 M=11 M=51 M=101 M=501 Figure 19: Diversity-effect for independent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The diversity-effect is positive when the probability of error is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', the models do better than random guessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Conversely, when the models are worse than random guessing (ϵ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5), the diversity effect is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This gives some comfort, as negative effects from diversity should only be seen in pathological scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, it should be remembered that the assumptions of this theoretical model are rather strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A Real Case, for 10 Classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We can see a remarkably similar pattern with real data, Figure 20, where we consider unconstrained Bagged Decision Trees on MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Each point on the figure is the ensemble evaluated at a single test point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The test points below the solid line have a negative diversity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 average individual error −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 diversity-effect Average individual error vs diversity-effect for Bagged Decision Trees examples in test set Figure 20: Scatter plot of average individual error against diversity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Bagging trees on MNIST, M = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Similar observations can be made regarding random guessing and negative diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A random guesser on this k = 10 class problem would have ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It can be observed in Figure 20 that the points above this all have negative diversity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, it can be noted that there are many test points where diversity-effect is negative, but the individual error is not worse than random guessing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It can be shown that, for independent classifiers, in order to have a negative diversity effect, it mus be the case that ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', the ensemble members makes errors more than half the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Note that this is independent of the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This is formalised with Theorem 14 below—proof in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 14 Assume an ensemble of models making independent errors on a k-class problem, with each model predicting the correct class with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' If p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5, then the diversity-effect is guaranteed to be non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Overall, in theoretical and empirical scenarios, we see the same phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Ensembles with strong individual performance tend to benefit from diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' And, it is only for pathologically bad models (worse than random guessing) the diversity is detrimental to the overall ensemble performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The issue of positive/negative diversity turns out be closely related to the phenomenon of “good” and “bad” diversity (Brown & Kuncheva, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' They showed that, restricting the label to y ∈ {−1, +1}, and q as a majority vote, the following holds: L0/1(y, q) = 1 M M � i=1 L0/1(y, qi) − yq 1 M M � i=1 L0/1(q, qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (37) If yq > 0, the diversity subtracts from the average error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' when the opposite is true, yq < 0, it adds to the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The former case is referred to as “good” diversity, and the latter is “bad” diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Comparing this with Theorem 12, we see the diversity-effect is in fact the expected value of the “good”/“bad” diversity term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The idea of good/bad diversity was generalised to the multi-class case by Didaci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2013), and the same relation applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 28 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 Summary We examined the properties of ensemble diversity under the 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We started with a proof showing the non-existence of a 0-1 bias-variance decomposition with target-independent variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A consequence of this is, there cannot be a corresponding ambiguity decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Therefore, there is no decomposition of the 0-1 loss such that the diversity can be expressed independently of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In spite of this target-dependence, we can formulate diversity in terms of its effect on the error, using the bias-variance ‘effect’ decompositions of James & Hastie (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Thus, we have the same conclusion as in the previous section: the role of diversity can be formulated as a hidden degree of freedom in a decomposition of the ensemble error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 7 Discussion: Limitations & Future Work Our framework applies for any loss admitting a bias-variance decomposition, giving a clear methodology to understand the form and nature of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The framework is not without limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Here we outline one such limitation, and discuss some potential future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In many learning scenarios, the loss we minimise is not always the loss in which we are ultimately interested—so-called surrogate losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The most obvious here is the cross-entropy, where we are ultimately interested in the 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Margin losses like the logistic or exponential loss are another important example, since boosting algorithms like AdaBoost/LogitBoost can be seen as minimising such losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Wood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2022) analysed bias-variance decompositions for margin losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Using these results, we can indeed obtain bias-variance-diversity decompositions applying for boosting models—some with target-independent diversity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' LogitBoost), and some target-dependent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' AdaBoost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, Mease & Wyner (2008) showed strong evidence that the additive model form in AdaBoost/LogitBoost results in a disconnect between the surrogate margin loss and the 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In particular, the surrogate loss can go up (sometimes exponentially fast) whilst the 0-1 loss on a hold-out sample is going down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This implies that any analysis of the surrogate (including loss decompositions) does not necessarily give meaningful insights on the ultimate quantity of interest, the 0-1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Furthermore, with boosting, the individual models are more naturally interpreted as learning to correct the errors of previous ensemble members rather than perform well in their own right, making interpretation of the average bias term problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Regarding future work, a natural line of research might be to enforce diversity in some sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', using our diversity measures as a regulariser in the construction of an ensemble itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Negative Correlation (NC) Learning Liu & Yao (1999) uses the squared loss ambiguity decomposition, Equation (1), to encourage diversity for regression ensembles, analysed by Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The Bregman ambiguity decomposition, Equation (16), implies that the NC algorithm is a special case of a wider family of diversity encouraging losses—the case for cross-entropy was explored in Webb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, given the full framework proposed in this paper, it is clear that many other opportunities exist, including introducing heterogeneity into the ensemble, or in implicit ensemble creation techniques like MC-Dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 29 8 Conclusion We have presented a unified theory of ensemble diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A key insight is that it is not the task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' classification/regression) that matters, but the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We demonstrated that a natural basis for the concept of diversity can be found as a hidden dimension in the bias-variance decomposition of the ensemble loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Diversity emerges naturally when considered from this point of view—as one part of a bias-variance-diversity decomposition, specific to the chosen loss function, all taking a common form: expected risk = (average bias) + (average variance) − (diversity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The gives a clear relationship between the ensemble performance and diversity, measured as a target-independent quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The only other scenario where was previously available is for squared loss with an arithmetic mean combiner (Ueda & Nakano, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Our framework is an alternative in this case, but also generalises the notion of diversity to a wide range of other losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The framework provides a methodology to automatically identify the combiner rule that enables such a decomposition, which we define as the centroid combiner rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This generalises the idea of ensemble “averaging” to many other scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The case of 0-1 loss is particularly interesting—we prove that, for any combiner, a target-independent diversity term cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Following James & Hastie (1997), we introduced a diversity-effect term which, though target-dependent, allows us to understand the role that diversity plays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Notable properties of the framework are: (i) It offers a unified view of diversity for a wide range of losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For any loss L, if a bias-variance decomposition holds, then we can apply the double decomposition trick and obtain a definition of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' As such, our framework can be used to understand diversity for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', squared loss, cross-entropy loss, the Poisson regression loss, and several margin- based losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (ii) It shows that diversity is a measure of model fit, just like bias/variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Diversity is a measure of model fit—a hidden dimension in the expected loss of an ensemble, accounting for statistical dependencies between the individual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Just as bias and variance change with model characteristics, the same applies to diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This gives a three-way bias/variance/diversity trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It may be possible (though outside the scope of this paper) to use diversity as a regularisation target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We therefore have a broad and precise formulation of diversity, with clear conditions for when it (and its effects) can, and cannot, be expressed independently of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This challenge has been referred to as the “holy grail” of ensemble learning (Zhou, 2012, Sec 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1), an open question for over 30 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We believe this work provides a solid foundation from which to explore new directions in ensemble learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Acknowledgements Funding in direct support of this research: EPSRC EP/N035127/1 (LAMBDA project).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' GB would like to thank LK and FR for a career’s worth of inspiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 30 A Additional Experimental Results Further results for Squared Loss We extend results for squared loss, shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In Table 4 we see results from three ensembles (each M = 30 regression trees), compared to a single tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We use a Bagging with constrained depth trees (max depth 8) and compare against unlimited depth trees, and a Random Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Single tree (depth 8) Bagging (depth 8) Bagging (unconstrained) Random Forest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='28 Table 4: California housing data: MSE of a single tree versus ensembles of 30 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We observe that the Random Forest is the best choice here, followed up closely by the unconstrained Bagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Figure 21 explains their performance by decomposing risk into bias, variance, and diversity—also showing how the components change as we grow the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 0 10 20 30 ensemble size 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 Bagging (max depth 8) 0 10 20 30 ensemble size Bagging (unconstrained) average bias + noise average variance diversity 0 10 20 30 ensemble size Random Forest Figure 21: Decomposing the expected risk of three decision tree ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We observe the same behaviour as in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The diversity increases with M, and is upper- bounded by the value of the average variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' A higher average variance effectively raises the “ceiling” to which diversity can rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The average variance is higher as we move from depth-limited trees to unlimited depth, and higher again with the random split-points in the Random Forest (here we use the square root of the number of features).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The higher average variance is compensated for by the diversity, causing Random Forest to be the best option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' It is notable that for large ensembles, the expected risk of the ensemble is almost entirely due to the value of the average bias (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='28 in the case of unconstrained trees), with diversity having essentially cancelled out the average variance of the individual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This behaviour is not just a quirk of this data set, in fact it holds as long as the individuals are all from the same model family, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', the ensemble is homogeneous—the general case is discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 31 Further results for cross-entropy In Figure 22, we present additional Accuracy/diversity plots for neural network ensembles for different data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In each case, the squared Person’s correlation coefficient is shown in the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The following configuration was used in all MLP experiments: learning rate: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 (Stochastic gradient descent) num epochs: 50 (MNIST), 200 (other data sets) hidden layer size (20 small/100 larger) number of trials: 100 where each trial uses a 90% sub-sample of the full training data, as outlined in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='06 diversity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='016 average individual error ensemble error M = 2 M = 20 M = 2 M = 20 phoneme Smaller Networks, R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='97 Larger Networks, R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='175 diversity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='016 average individual error ensemble error M = 2 M = 2 M = 20 M = 20 spambase Smaller Networks, R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='99 Larger Networks, R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='30 diversity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='030 average individual error ensemble error M = 2 M = 2 M = 20 M = 20 landsat Smaller Networks, R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='99 Larger Networks, R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='6 diversity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='035 average individual error ensemble error M = 2 M = 2 M = 20 M = 20 south german credit Smaller Networks, R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='93 Larger Networks, R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='71 Figure 22: Error/diversity relationship observed across four data sets, comparing ensembles of small networks (20 hidden node, blue dots) versus large (100 hidden node, orange dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 32 B Methodology for Estimating Bias, Variance, and Diversity Here, we present our methodology for estimating the bias, variance and diversity terms from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Algorithm 1 shows the procedure for experiments where we estimate diversity of ensembles of different sizes, such as in the experiments for Bagging and Random Forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Notably, an ensemble of size M + 1 is created by using the members of the ensemble of size M, rather creating a new ensemble of size M + 1 from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We also present a visualisation of the sub-sampling scheme used for Bagging in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Algorithm parameters: model, n_trials, ensemble_size, train_data, test_data Output: test_preds: an array of model predictions of size n_trials × ensemble_size × n_test_data for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , n_trials} do for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , len(test_data)} do trial_data ← 90% of train_data, sampled without replacement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' ,ensemble_size} do member_data ← bootstrap of trial_data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' ith ensemble member ← copy of model trained on member_data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' test_preds[k, i, j] ← prediction of test_data of the ith ensemble member in the kth trial, jth test data point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' end end end Algorithm 1: Algorithm for collecting data to estimate diversity of bootstrap ensemble while varying ensemble size Training Data Test Data bootstraps Trials Dataset 90% Sub-sample 90% Sub-sample 90% Sub-sample 90% Sub-sample Ensembles Ensemble Members bootstraps bootstraps bootstraps Figure 23: Visualisation of the sub-sampling scheme used for Bagging ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The result of Algorithm 1 is that we get an array of size (D, M, N), where D is the number of trials, M is the ensemble size and N is the number of test points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For Bregman divergences, the average bias and average variance are calculated by estimating the central model of each ensemble member by replacing the expectation with an average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Writing Dj 33 as the full training data for the jth trial q∗,est i (x) = [∇φ]−1 � � 1 D D � j=1 ∇φ(qi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Dj)) � � ≈ q∗ i (x) With this estimate of q∗(x) the average bias and average variance are computed as average bias ≈ 1 M 1 N M � i=1 N � j=1 Bφ � yj, q∗,est i (xj) � , (38) average variance ≈ 1 D 1 M 1 N D � k=1 M � i=1 N � j=1 Bφ � q∗,est i (xj), qi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Dk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (39) Diversity is calculated similarly, with q defined as the centroid combiner of the M ensemble members in a given trial: diversity ≈ 1 D 1 M 1 N D � k=1 M � i=1 N � j=1 Bφ (q(xj, Dk), qi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Dk)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (40) 34 C Proofs and further explanations for Section 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 Bregman Ambiguity and Bregman Diversity The following sections will make use of Pfau (2013)’s bias-variance decomposition, which for referencing purposes we restate here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 15 (Bregman Bias-Variance Decomposition Pfau (2013)) Given a loss Bφ (Y, q(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)), the expectation of the risk with respect to D can be written, ED � EXY �� Bφ(Y, q(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)) �� = EX � EY|X � Bφ � Y, Y �� � �� � noise + Bφ � Y, q∗(X) � � �� � bias + ED [Bφ (q∗(X), q(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D))] � �� � variance � , where Y = EY|X[Y], the conditional mean of the vector Y, and q∗(x) def= arg min z ED � Bφ (z, q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)) � = [∇φ]−1 � ED [∇φ (q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D))] � (41) The centroid q∗ takes different forms dependent on the generator used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Examples are below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Loss Gradient η = ∇φ(q) Inverse Grad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q = [∇φ]−1 (η) Left Bregman Centroid q∗ = [∇φ]−1 � ED [∇φ (qD)] � Squared 2q 1 2η ED[qD] Itakura-Saito −1 q − 1 η 1 �� ED [1/qD] � Poisson loss ln q exp(η) exp �ED [ln qD] � KL-divergence ln q(c) 1−�k−1 c′=1 q(c′) exp(η(c)) 1+�k−1 c′=1 exp(η(c′)) 1 Z exp � ED [ln qD] � Table 5: Common losses (see Table 1) and their Bregman centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In the case of KL-divergence, Z is a normalizer to ensure a valid distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Whilst we assert that the ambiguity decomposition is a special case of this (which emerges trivially if we assume zero noise, and replace expectations by summations), it can be proven independently, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 3 (Bregman Ambiguity Decomposition) For a target label y ∈ S and a set of predictions q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , qM ∈ ri(S), Bφ (y, q) = 1 M M � i=1 Bφ (y, qi) − 1 M M � i=1 Bφ (q, qi) (16) where q = [∇φ]−1� 1 M � i ∇φ(qi) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 35 Proof 1 Take the average loss over the M models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' and subtract the loss of the ensemble: 1 M M � i=1 Bφ (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' qi) − Bφ (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q) = 1 M M � i=1 � φ(y) − φ(qi) − ⟨∇φ(qi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' y − qi⟩ � − � φ(y) − φ(q) − ⟨∇φ(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' y − q⟩ � = 1 M M � i=1 � φ(y) − φ(qi) − ⟨∇φ(qi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' y − qi⟩ − φ(y) + φ(q) + ⟨∇φ(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' y − q⟩ � = 1 M M � i=1 � φ(q) − φ(qi) − ⟨∇φ(qi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' y − qi⟩ + ⟨∇φ(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' y − q⟩ � Now expand the inner products and use the definition of q = [∇φ]−1� 1 M � i ∇φ(qi) �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' to note that ∇φ(q) = 1 M � i ∇φ(qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' = 1 M M � i=1 � φ(q) − φ(qi) − y · ∇φ(qi) + qi · ∇φ(qi) + y · 1 M M � i=1 ∇φ(qi) − q · 1 M M � i=1 ∇φ(qi) � = 1 M M � i=1 � φ(q) − φ(qi) + qi · ∇φ(qi) − q · 1 M M � i=1 ∇φ(qi) � = 1 M M � i=1 � φ(q) − φ(qi) − ∇φ(qi) · [q − qi] � = 1 M M � i=1 Bφ (q, qi) which after rearranging completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 4 (Bregman Bias-Variance-Diversity decomposition) For an ensemble q1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , qM(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D), let q∗ i be the left Bregman centroid of qi (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', q∗ i def= [∇φ]−1 (ED [∇φ(qi)])) and define q def= [∇φ]−1 � 1 M �M i=1 ∇φ (qi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Then we have the decomposition, ED [EXY [Bφ (Y, q)]] = EXY � Bφ � Y, Y � � �� � noise + 1 M M � i=1 Bφ � Y, q∗ i � � �� � average bias + 1 M M � i=1 ED [Bφ (q∗ i , qi)] � �� � average variance − ED � 1 M M � i=1 Bφ (q, qi) � � �� � diversity � , where Y = EY|X [Y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Proof 2 Starting with the ambiguity decomposition, we have ED [EXY [Bφ (Y, q)]] = ED � EXY � 1 M M � i=1 Bφ (Y, qi) �� − ED � EXY � 1 M M � i=1 Bφ (q, qi) �� (42) Applying Theorem 15 to the first term on the RHS, we have ED � EXY � 1 M M � i=1 Bφ(Y, qi) �� = EX � EY|X � Bφ � Y, Y �� + 1 M M � i=1 Bφ � Y, q∗ i � + 1 M M � i=1 ED [Bφ (q∗ i , qi)] � (43) Plugging Equation (43) into (42) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 36 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 The Importance of Parameter Encoding in the KL-divergence In order to apply our decomposition to the cross-entropy, we use the fact that when the target is one-hot encoded, the cross-entropy coincides with the KL-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' There are two ways in which we can express the KL-divergence as a Bregman divergence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' either using the full-length probability vectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' p ∈ Rk giving the generator φfull(p) = k � c=1 p(c) ln p(c) (44) or using the minimally parameterised vectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' �p ∈ Rk−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' where the last entry is omitted: φmin(�p) = k−1 � c=1 �p(c) ln �p(c) + (1 − k−1 � c′=1 �p(c′)) ln(1 − k−1 � c′=1 �p(c′)) (45) Given two probability vectors in the appropriate form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' either formulation gives a Bregman divergence is equivalent to the KL-divergence Nielsen & Nock (2009),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', Bφfull (p, q) = Bφmin (�p, �q) = K(p || q) However, we prefer the second as it exhibits desirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In particular, it is necessary to use the second to ensure that the Bregman centroid is always a valid distribution on the probability simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In this case, the centroid combiner is the normalised geometric mean, as we now demonstrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' To show this we consider M minimally parameterised vectors, qi ∈ Rk−1 (note that we have dropped tilde above q for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Our claim is that the centroid combiner is the normalised geometric mean q = [∇φ]−1 � 1 M �M i=1 ∇φ(qi) � , is of the form q(c) = [∇φmin]−1 � 1 M M � i=1 ∇φmin(qi) � = �M i=1 q(c) i 1 M �k c′=1 �M i=1 �q(c′) 1 M , (46) where �q denotes the extension of the k − 1 length vector into a full k length probability vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Plugging in the gradients from Table 5, we start with q(c) = exp � 1 M �M i=1 ln q(c) i 1−�k−1 c′=1 q(c′) i � 1 + �k−1 c′=1 exp � 1 M �M i=1 ln q(c′) i 1−�k−1 c′′=1 q(c′′) i �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Note that the numerator here can be rearranged: exp � 1 M M � i=1 ln q(c) i 1 − �k−1 c′=1 q(c′) i � = M � i=1 � 1 − k−1 � c′=1 q(c′) i �− 1 M M � i=1 � q(c) i � 1 M ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 37 and the denominator can be written 1 + k−1 � c′=1 exp � 1 M M � i=1 ln q(c′) m 1 − �k−1 c′′=1 q(c′′) m � = 1 + k−1 � c′=1 M � i=1 q(c′) i 1 M � 1 − k−1 � c′′=1 q(c′′) i �− 1 M = 1 + M � i′=1 � 1 − k−1 � c′′=1 q(c′′) i′ �− 1 M k−1 � c′=1 M � i=1 q(c′) i 1 M = M � i′=1 � 1 − k−1 � c′′=1 q(c′′) i′ �− 1 M � � � M � i=1 � 1 − k−1 � c′′=1 q(c′′) i � 1 M + k−1 � c′=1 M � i=1 q(c′) i 1 M � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Putting the numerator and denominator back into the second expression of Equation (46) and using the definition of �q, we find the first terms in both products cancel and we are left with the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Full length k Probability Vector If we do not use the minimally parameterized vectors, we would have the Bregman generator φ(p) = �k c=1 p(c) ln p(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This gives the geometric mean, rather than the normalised version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' To see this, we first note that (∇φ(p))(c) = 1 + ln p(c) = η(c) � [∇φ]−1 (η) �(c) = exp � η(c) − 1 � , and therefore the centroid combiner is � [∇φ]−1� 1 M M � m=1 ∇φ(qm) ��(c) = exp � 1 M M � i=1 1 + ln q(c) i − 1 � = exp � 1 M M � i=1 ln q(c) i � = M � i=1 q(c) i 1 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Note that this means that q is not necessarily a valid probability vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In fact, it is a valid probability vector only if q1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' = qM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='3 Decomposing the Cross-Entropy Theorem 5 Let y be a one-hot class vector of length k, and q ∈ Rk be a model’s prediction of the class distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Define a set of such models {qi}M i=1, and their combination q as their normalised geometric mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The following decomposition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' −ED [y · ln q] � �� � expected cross-entropy = − 1 M M � i=1 y · ln q∗ i � �� � average bias + 1 M M � i=1 ED [K(q∗ i || qi)] � �� � average variance − ED � 1 M M � i=1 K(q || qi) � � �� � diversity ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (18) Proof 3 From Theorem 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' using the generator φmin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' we have ED [Bφmin (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q)] = 1 M M � i=1 Bφmin (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q∗ i ) + 1 M M � i=1 ED [Bφmin (q∗ i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' qi)] + 1 M M � i=1 Bφmin (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' qi) 38 Using the equivalence with KL-divergences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' we have ED [K(y || q)] = 1 M M � i=1 K(y || q∗ i ) + 1 M M � i=1 ED [K(q∗ i || qi)] + 1 M M � i=1 K(q || qi) Since y is one-hot y · ln y = 0 and therefore for any vector q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' we can write K(y || q)) = −y · ln q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Applying this to the expected risk and average bias terms in the above gives ED [y · ln q] = − 1 M M � i=1 y · ln q∗ i + 1 M M � i=1 ED [K(q∗ i || qi)] − ED � 1 M M � i=1 K(q || qi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 Sources of Stochasticity in the Bias-Variance-Diversity Decomposition In this appendix, we clarify exactly what is meant when we write ED [·] when dealing ensembles and the bias-variance-diversity decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In particular, we show how D is constructed to account for the stochasticity in individual ensemble members and the interactions between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In the bias-variance decomposition of a single model, we consider q as dependent on a random variable D during its training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This might represent the random sample of training data, or initial weights in a neural net, or any other source of stochastic behavior during learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We can write the expected loss with respect to D, at a point as ED [L(y, q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' When considering an ensemble, the situation becomes more complicated, as there are multiple sources of stochasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For instance, if we use Bagging, we first consider the overall training set to be a sample of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' observations from P(X, Y ), then each ensemble member receives a random bootstrap from that same data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Hence, each ensemble member is influenced by a different— though not necessarily independent—random variable (determining the bootstrap sample they each receive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We write the random variable representing the training data of the ith ensemble member as Di, and therefore can write the ith mode output as q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Continuing the example, we can write the expected loss of this model as EDi[L(y, q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Di)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Furthermore, we can define D = (D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , DM), such that D is a random vector containing all M random variables of the individual ensemble members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Now, due to the law of total expectation we can write this using D instead of Di, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' ED[L(y, q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Di)] = EDi[L(y, q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Di)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' even when there dependencies between the individual Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We see therefore that the expectation over D reduces to Di for individual models, and the decomposition applies even with D being vector with dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 39 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5 Further Properties of the Bias-Variance-Diversity Decomposition This Appendix contains proofs for results in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Proposition 7 Assume a target probability vector, y ∈ Rk, and a set of models {qi}M i=1 combined by averaging, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' q† = 1 M �M i=1 qi, then the cross-entropy of q and y is −y · ln q† � �� � ensemble cross-entropy = − 1 M M � i=1 y · ln qi � �� � average cross-entropy − k � c=1 y(c) ln 1 M �M j=1 q(c) j ��M i=1 q(c) i � 1 M � �� � ambiguity (target-dependent) , (28) where the second term is non-negative, thus the ensemble loss is guaranteed less than or equal to the average individual loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Proof 4 Take the average cross-entropy, and subtract the ensemble cross entropy: − 1 M M � i=1 y · ln qi − � − y · ln q†� = k � c=1 y(c) ln q†(c) − 1 M M � i=1 k � c=1 y(c) ln q(c) i = k � c=1 y(c) ln q†(c) − k � c=1 y(c) ln � � i q(c) i �1/M = k � c=1 y(c) ln � � � � q†(c) �M i=1 � q(c) i � 1 M � � � � Using the definition of q† and rearranging completes the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' From the arithmetic-geometric mean inequality, q†(c) ≥ �M i=1 � q(c) i � 1 M , implying that the term inside the logarithm is greater or equal to 1, and the overall term is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Proposition 8 (Diversity for Averaged Probabilities is target-dependent) Let q† = 1 M �M i=1 qi, with qi ∈ [0, 1]k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The expected cross-entropy admits the decomposition: ED � −y · ln q†) � = − 1 M M � i=1 y · ln q∗ i � �� � average bias + 1 M M � i=1 ED [K(q∗ i || qi)] � �� � average variance − ED � ��� k � c=1 y(c) ln 1 M �M j=1 q(c) j ��M i=1 q(c) i � 1 M � ��� � �� � dependency .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Proof 5 Starting with Equation (28) and taking the expectation over D, we have −ED � y · ln q†� = ED � − 1 M M � i=1 y · ln qi � − ED � ��� k � c=1 y(c) ln 1 M �M j=1 q(c) j ��M i=1 q(c) i � 1 M � ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Now, applying the bias variance decomposition to each ensemble member we get the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 40 D Proofs and Additional Material for Section 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='1 Proofs for Section 6 In this section, we prove a theorem stated in Section 6 claiming that one cannot construct a bias- variance decomposition for the 0-1 loss, where the bias term is the 0-1 loss of some deterministic prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 9 (Non-existence of a Bias-Variance Decomposition for 0-1 loss) The 0-1 loss cannot be decomposed as a bias term plus a variance term, where the bias is the loss of some deterministic model q∗ derived from the distribution of q and the variance term is independent of the label y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', there is no V and q∗ such that in general ED � L0/1(y, q) � � �� � expected loss = L0/1(y, q∗) � �� � bias + V[q] � �� � variance .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' (30) Proof 6 Rearranging Equation (10), for a decomposition to hold, the 0-1 loss needs to satisfy V[q] = ED � L0/1(y, q) � − L0/1(y, q∗) (47) and therefore we require the right hand side to be independent of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We show that this cannot be the case by constructing an example such that there is no valid choice of q∗ that makes the right hand side independent of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The two class case: First, consider the case when there are two classes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', S = {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For a fixed x, let P(q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D) = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Since L0/1 : S × S → {0, 1} we necessarily have that q∗ ∈ S = {1, 2} in order for L0/1(y, q∗) to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We now show that for both possible q∗, the RHS above is dependent on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' When q∗ = 1, we have ED � L0/1(y, q) � − L0/1(y, q∗) = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 − 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 if y = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='6 − 1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 if y = 2 Alternatively, when q∗ = 2, ED � L0/1(y, q) � − L0/1(y, q∗) = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 − 1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='6 if y = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='6 − 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='6 if y = 2 For both possible values of q∗, the value of ED � L0/1(y, q) � − L0/1(y, q∗) is dependent on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The multiclass case: We set P(q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D) = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='6, and P(q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D) = 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4, and zero probability mass for all other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' From the two-class case we know that when q∗ ∈ {1, 2}, there is a dependency on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This persists for q∗ ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , k}, where we have ED � L0/1(y, q) � − L0/1(y, q∗) = � � � � � � � � � � � � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 − 1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='6 if y = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='6 − 1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='4 if y = 2 1 − 0 = 1 if y = q∗ 1 − 1 = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For all possible values of q∗, the expression ED � L0/1(y, q) � − L0/1(y, q∗) is dependent on the value of y, and therefore the bias-variance decomposition cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 41 Theorem 13 (Bias-Variance-Diversity effect decomposition) Given a loss function L : S × S → R+ and an ensemble of models q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , qM, where q is the majority vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' EY � ED � L0/1(Y, q) �� = EY � L0/1(Y, Y ∗) � � �� � noise + 1 M M � i=1 EY � L0/1(Y, q∗ i ) − L0/1(Y, Y ∗) � � �� � average bias-effect + 1 M M � i=1 ED � EY � L0/1(Y, qi) − L(Y, q∗ i ) �� � �� � average variance-effect − ED � EY � 1 M M � i=1 � L0/1(Y, qi) − L(Y, q) ��� � �� � diversity-effect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Proof 7 Note that several terms on the right cancel, reducing to the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='2 Diversity-Effect Decomposition for Weighted Ensembles In this section, we derive and justify the bias-variance-diversity-effect decomposition for weighted majority vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Given an ensemble of classification models q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , qM with f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , k}, and weights for those models α1(D), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , αM(D), we consider the ambiguity-effect decomposition for weighted plurality vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The weighted majority vote is q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D) = argmin c∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=',k} M � i=1 αi(D) �M j=1 αj(D)L0/1(c, qi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)), similarly, the central vote of an ensemble member is given by q∗ i (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D) = argmin c∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=',k} ED � αi(D) ED [αi(D)]L0/1(c, qi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)) � , with ties broken randomly (the tie break procedure can be thought of as part of the random variable D, since D implicitly contains all sources of stochasticity related to the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 16 (Ambiguity-Effect Decomposition for Weighted Majority Vote) With this, we can define a weighted effect decomposition as L0/1(y, q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)) = M � i=1 ai(D)L0/1(y, qi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D) � �� � weighted average loss − � ������ M � i=1 ai(D)L0/1(y, qi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)) − L0/1(y, q(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)) � �� � ambiguity-effect � ������ , where ai = αi �M j=1 αj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The validity of this theorem can be verified simply by cancelling terms on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' The theorem tells us that the loss of an ensemble can be decomposed into a positive term, giving the (weighted) average loss of the ensemble members, and an ambiguity-effect term, which quantifies how much better (or worse) the performance of the ensemble is over the average ensemble member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Using the same principle, we can also construct a bias-variance decomposition which takes into account a weighting α(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 42 Theorem 17 (Bias-variance-effect Decomposition for Weighted Majority Vote) ED � αL0/1(y, f) � = ED [α] L0/1(y, f∗) + � ED � αL0/1(y, f) � − ED [α] L0/1(y, f∗) � Again, the proof of the result is immediate from considering which terms on the right cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' However, it is worth considering how the decomposition works and what the terms mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Consider the decomposition when we replace α with a normalised version α ED[α], we get the following: ED � α ED [α]L(y, f) � = L0/1(y, f∗) + � ED � α ED [α]L0/1(y, f) � − L0/1(y, f∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' This is exactly bias-variance-effect decomposition that we have seen previously, but re-weighting the contributions of the different data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' In fact, the two are equivalent, with the weights defining a new probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Taking PD(D) as the probability density function over data sets, the decomposition above is exactly the bias-variance-effect decomposition with the new probability density function QD(D) = PD(D) α(D) ED[α(D)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We can also easily reintroduce label noise and expose a noise term, and turning the bias into a bias-effect: EY � ED � α ED [α]L(Y, f) �� = EY � L0/1(Y, Y ∗) � � �� � noise + EY � L0/1(Y, f∗) − L(Y, Y ∗) � � �� � weighted bias-effect � ED � EY � α ED [α]L0/1(Y, f) − L0/1(Y, f∗) ��� � �� � weighted variance-effect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We can now apply the double decomposition trick, getting the following bias-variance-diversity effect decomposition for weighted majority vote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 18 (Bias-Variance-Diversity-Effect for Weighted Voting) Given M classifiers q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , qM, where the ensemble is a weighted majority vote, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', q = argminz∈S �M i=1 αiL0/1(z, qi) for weights α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' , αM ∈ R+, the ensemble loss admits the following decomposition, where the normalised weight is ai = αi/�M j=1 αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' EY � ED � L0/1(Y, q) �� = EY � L0/1(Y, Y ∗) � � �� � noise + M � i=1 ED � ai EY � L0/1(Y, q∗ i ) − L0/1(Y, Y ∗) �� � �� � weighted average bias-effect + M � i=1 ED � ai EY � L0/1(Y, qi) − L0/1(Y, q∗ i ) �� � �� � weighted average variance-effect − ED � EY � M � i=1 ai L0/1(Y, qi) − L0/1(Y, q) �� � �� � diversity-effect � , where q∗ i = argminz∈S ED � αiL0/1(z, qi) � , noting that αi and qi are both dependent on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' 43 As before, the veracity of this decomposition can be seen by cancelling like terms on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' AdaBoost produces a set of binary classifiers hi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D) ∈ {−1, +1} and corresponding weights αi(D) ∈ R, so setting qi = hi allows immediate application of the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' LogitBoost does not produce classifier/weight pairs, but instead a set of regression models each gi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We can apply the decomposition by separating these into sign/magnitude components, giving a classification and weight: qi = sign(gi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)) and αi(D) = |gi(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' D)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Theorem 14 Assume an ensemble of models making independent errors on a k-class problem, with each model predicting the correct class with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' If p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='5, then the diversity-effect is guaranteed to be non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Proof 8 From the multi-class classification problem, we construct a binary classification problem and define the models qbin i such that qbin i = “y” when qi = y and qbin i = “not y” when qi ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' We define qbin as being the majority vote of qbin i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' “y” winning the majority vote is sufficient but not necessary for y to win the plurality vote in the multi-class setting, so we get P(q = y) ≥ P(qbin = “y”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' For the binary classification problem, we use Condorcet’s Jury Theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=', Berend & Sapir (2005)) to get that the probability of the ensemble being correct is greater than or equal to any given individual being correct and therefore P(q = y) ≥ P(qbin = “y”) ≥ P(qbin i = “y”) = P(qi = y), with the second inequality being strict when M ≥ 3 and P(qi = y) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Plugging this into the diversity-effect definition, for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' models we have DE = ED � 1 M M � i=1 L(y, qi) − L(y, q) � = 1 − P(qi = y) − 1 + P(q = y) = P(q = y) − P(qi = y) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' References K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19E2T4oBgHgl3EQfiwfE/content/2301.03962v1.pdf'} +page_content=' Audhkhasi, A.' 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b/1NAzT4oBgHgl3EQfRftD/content/tmp_files/2301.01216v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ba8017d8203b58fac6b3203a1b410bf00940724e --- /dev/null +++ b/1NAzT4oBgHgl3EQfRftD/content/tmp_files/2301.01216v1.pdf.txt @@ -0,0 +1,1690 @@ +1 +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY +An End-to-End Multi-Scale Network for Action +Prediction in Videos + +Xiaofa Liu, Jianqin Yin, Member, IEEE, Yuan Sun, Zhicheng Zhang, Jin Tang + + + Abstract—In this paper, we develop an efficient multi-scale +network to predict action classes in partial videos in an end-to- +end manner. Unlike most existing methods with offline feature +generation, our method directly takes frames as input and further +models motion evolution on two different temporal scales. +Therefore, we solve the complexity problems of the two stages of +modeling and the problem of insufficient temporal and spatial +information of a single scale. Our proposed End-to-End Multi- +Scale Network (E2EMSNet) is composed of two scales which are +named segment scale and observed global scale. The segment +scale leverages temporal difference over consecutive frames for +finer motion patterns by supplying 2D convolutions. For observed +global scale, a Long Short-Term Memory (LSTM) is incorporated +to capture motion features of observed frames. Our model +provides a simple and efficient modeling framework with a small +computational cost. Our E2EMSNet is evaluated on three +challenging datasets: BIT, HMDB51, and UCF101. The extensive +experiments demonstrate the effective-ness of our method for +action prediction in videos. + +Index terms: action prediction, multi-scale network, end-to- +end method. +I. +INTRODUCTION +HE goal of action prediction in videos is to predict the +class label of an ongoing action from an observed part +of it over temporal axis so far[1]. It is a subset of a +broader research domain on human activity analysis. Different +from conventional action recognition with fully executed +actions[2][3][4], it is more challenging to predict the action +label in ongoing actions due to the incompleteness of actions +and the continuous evolution of actions. It has attracted a lot of +research attention because of its wide application in some +scenarios with high real-time requirements, such as human- +machine interaction, security surveillance, etc. +Although the previous work has achieved promising results + +▪ This work was supported partly by the National Natural Science +Foundation of China (Grant No. 62173045, 61673192), partly by the +Fundamental Research Funds for the Central Universities (Grant No. 2020XD- +A04-3), and the Natural Science Foundation of Hainan Province (Grant No. +622RC675). (Corresponding author: Jianqin Yin). +▪ Xiaofa Liu is with the School of Modern Post, Beijing University of Posts +and +Telecom-munications, +Beijing +100876, +China +(e-mail: +liuxiaofamail@163.com ) +▪ Jianqin Yin, Zhicheng Zhang, and Jin Tang are with the school of Artificial +Intelligence, Beijing University of Posts and Telecommunications, Beijing +100876, +China +(e-mail: +jqyin@bupt.edu.cn, +zczhang@bupt.edu.cn, +tangjin@bupt.edu.cn ). +▪ Yuan Sun is with Electronic Engineering School, Beijing University of +Posts +and +Telecommunications, +Beijing +100876, +China +(e-mail: +sunyuan@bupt.edu.cn ). +by adopting a two-stage approach, there generally had +problems of complex modeling and feature redundancy. The +previous method separated feature extraction from predictive +modeling[5][6][7][8][9][10][11][12]. This separation operati- +on makes the spatio-temporal representation obtained may +deviate from the action prediction. Moreover, it complicates +the model design. Secondly, because the feature is generated +offline, the complete action must be divided into fixed +segments in advance, which not only results in the redundancy +of the feature in the time dimension, but also is not applicable +to the evolving action. +Therefore, in this paper, we propose an end-to-end method, +which effectively reduces the complexity of the model and +introduces more fine-grained spatio-temporal information. We +designed the end-to-end network from three aspects, sampling +method, local spatio-temporal information representation, and +long-term time sequence fusion. In order to adapt the end-to- +end structure to the evolving motion, we first changed the +preprocessing and feature generation method, which will be +described in Part 3. Second, to reduce computational +consumption to achieve end-to-end structure, we use 2D +convolution instead of two-stream networks or 3D +convolutions to extract local spatio-temporal features. Finally, +to enhance the temporal information of action evolution, we +present an observed global scale to fuse the historical evolution +information of actions. +Similar to the application of spatial multi-scale in image +field, multi-scale research in the temporal dimension is also +increasing in video analytics. Compared to images, the +variation of temporal scales in videos poses additional +challenges. How to effectively utilize the motion evolution +information at different time scales has gradually gained +attention in video motion analysis. Feichtenhofer[4] et al. +proposed SlowFast network for video recognition. Their +method utilizes two branches, a slow pathway with low frame +rate and a fast pathway with high frame rate, to capture spatial +semantics and motion at fine temporal resolution. Wang[13] et +al. proposed an efficient multi-scale model for action +recognition, which utilizes short-term and long-term temporal +difference modules to capture both short-term and long-term +motion information better. +Most of the existing action prediction methods are +insufficient to focus on multi-scale temporal, making them fail +to capture fine-grained temporal information. They use a fixed +frame rate to sample each partial video, and use a fixed +temporal scale for feature generation and modeling[1][5] +[6][7][8][9][11]. Although these methods simplify the +T + +2 +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY +processing of the input of feature generation and reduce the +computation to a certain extent, they ignore the evolution of +action. Too much fine-grained information will be lost, and the +spatio-temporal information in the video cannot be fully +utilized. +Our method takes both the local evolution information +between adjacent frames and the global evolution information +of the entire observed video sequence into account. Therefore, +we design two temporal scales to increase fine-grained timing +information. Firstly, the segment scale uses RGB frames with +temporal difference to capture temporal information in each +segment. Secondly, the observed global scale uses LSTM +module to fuse all the observed action evolution information. +Through modeling in short-term and long-term time scales, our +method can be mining more fine-grained temporal information +without increasing the computational load. +Our E2EMSNet provides a simple yet effective framework +for the problem of ongoing action prediction in videos. In +summary, our main contributions lie in the following three +aspects: +● We propose a simple end-to-end approach for action +prediction in videos. To the best of our knowledge, this is the +first work focusing on this problem. +● We investigate two scales in the temporal dimension to +model the evolution of actions, and propose a segment +summarization and propagation framework. The segment scale +is used to model the local evolution of the action, and the +observed global scale is used to model the global evolution of +the action. +● We achieve a trade-off of efficiency and effectiveness. +We achieve state-of-the-art performance on several datasets +while using only 2D convolutions framework and RGB format +of features. + +II. RELATED WORK +A. Action Recognition +Action recognition methods take fully observed videos as +input and output labels of human actions. Action recognition +has been extensively studied in past few years[2][3][4][13][14]. +These studies can be roughly divided into two categories. +Methods in the first category are two-stream CNNs, which was +first proposed in[15]. It used two inputs of RGB and optical +flow to model appearance and motion information separately +in videos with a late fusion. In addition, follow-up research has +adopted two RGB inputs sampled at different FPS or carefully +designed temporal modules for efficiency, including Non-local +Net[16], STM[17], SlowFast[4], and Correlation Net[18]. The +second method is to use 3D CNNs[19][20]. It proposed 3D +convolution and pooling to learn spatiotemporal features from +videos directly. Several variants adopted a 2D + 1D paradigm +to reduce the computation cost of 3D convolution, which +implement by decomposing 3D CNNs into a 2D convolution +and a 1D temporal convolution[21][22][23]. Several works +focused on designing more powerful and efficient temporal +modules, such as TSM[14], TAM[24], TEA[25], and TDN[13]. +More recent works tried clip-based architecture search for +video recognition, focusing on capturing appearance and +motion or context information in a more fine-grained and +efficient manner[13][26]. Although these methods mainly +learned features for the videos with full action executions, their +core ideas have certain reference significance for ongoing +action prediction in videos. + +B. Action Prediction +Action prediction methods were proposed to predict the +action given a partially observed video. [9] was the first work +along +these +lines, +they +formulated +the +problem +probabilistically and proposed a dynamic bag-of-words +approach, modeling how feature distributions of activities +change as observations increase. In the last decade, researchers +approach this task from various perspectives and can be +grouped into three major divisions[27]. The first method can +be formulated as one-shot mappings from partial observations +to groundtruth labels of full observations. The basic +assumption underlying these methods is that a partial +observation of an action video provides sufficient information +to define the appropriate overall action class regardless of the +unobserved part. Follow-up research work[28][29][6][30] +adopted more robust features, hierarchical extractions, and +learning-based classifiers to perform more fine-grained +analysis of an initial partial observation for better performance. +The second division is knowledge distillation-based methods. +These methods distill the information from the full +observations into partial observations[31][5][11][32]. These +methods attempted to lend power from unobserved data in +training to either enrich the feature representation of partial +data or encourage the classifiers to easily recognize partial data. +Another way to exploit future information is by propagating +the partial observation into the future in a temporal +extrapolation fashion[33][34] [12][35][36]. For example, [12] +learned to propagate frame-wise residuals in feature space to +complete partial observation. + + +Fig. 1. Relevant definitions in action prediction in videos: full video, partial video, segments, and observation ratio. + +Full video +X[1:T] +Segments +(K-10) +Partial video x[1:t] +k=2,observationratio:r=k/K +=2/10=0.23 +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY +C. Multiple temporal scales for action analysis in videos +Temporal sequence forecasting usually faces the following +situations for scenarios with insignificant periodic motion: +long-term forecasts need to consider trend information (long- +term dependencies), and short-term forecasts need to consider +fine-grained volatility (short-term dependencies). The current +difficulty is how to model long-term dynamic dependencies +and consider long-term and short-term dependencies. There +are two methods currently. The main existing method is +hierarchical modeling, which is achieved by establishing +hidden layers of different granularities[37][38][39][40][41] or +decomposing the original data to obtain data of different +granularities[42][43]. The second method is designing the gate +mechanism, which achieved by modifying the internal +structure of RNN[44]. We inherit this idea that both long-term +and short-term dependencies in video must be carefully +considered, and a trade-off approach is adopted. +III. OUR METHOD +In this section, we detail our approach to mining ongoing +action evolution information in videos using multiple scales in +an end-to-end fashion. Specifically, we first describe the +problem formulation. Then, we elaborate on our end-to-end +framework and method for multi-scale modeling of ongoing +action sequences. + +A. Problem formulation +Given a video containing human motion (the video may +contain arbitrary incomplete motion), the goal is to predict the +class label. We follow the problem formulation in the[31], +which has been widely adopted in subsequent work[5][7][11]. +As shown in Fig. 1, Given a full video +[1: ] +X +T with complete +action execution, 1 represents the first frame of the video, and +T represents the last frame. We use +[1, ], +[1, ] +x +t t +T + + to +simulate the action execution in video from 1 to t , defined as +partial video. In order to facilitate quantitative experiments, +we usually divide a full video into K segments, each +containing ( +/ +) +T +K frames. Assuming that the action is +executed to the +, +[1,2,..., +] +kth k +K += + segment, the observation +ratio is defined as +/ +r +k +K += +. As defined above, as shown in +Fig.1, the full video X , is divided into K segments. Among +them, the partial video marked with green has an observation +ratio +/ +2 /10 +0.2 +r +k +K += += += +, and it can be considered that its +action has been executed 20%. + +B. Data processing +We adopt a data processing method different from the +previous method. As shown in Fig. 2, the upper part is the data +processing method used in the previous method. They first +divided a complete video X into K segments, and combined +segments into partial videos to simulate action evolution. Then +the partial video is sampled to extract the spatio-temporal +representation. The problem caused by this is that each partial +video needs to be separately extracted for spatio-temporal +representation, which divides the continuous evolution of +action. The feature extraction of partial videos with higher +observation rates cannot use the previous partial videos with +lower observation rates. It will cause redundancy in the time +dimension. At the same time, with the increase in the +observation rate, the temporal information will become more +and more sparse. Compared with them, we directly extract the +local spatio-temporal representations of each segment. In this +way, the previous spatio-temporal information can be +continuously used with the evolution of actions. This makes +our model more robust to action duration, and more abundant +spatio-temporal information can be obtained. + + + +Fig. 2. Differences in data processing between our method and previous methods. The upper is the data processing method used +in the previous method, and the lower is the data processing strategy used in our method. + +LOTTE +Full +video X +Segments +artial video +Observation ratio=0.1 +Partial video I +Observation rafio=0.2 +Partial video k +Observation ratio=k/K +Partial video +Sampling and feature extraction +Feature of partial video +OTT +Full +video X +Segments +Sampling and feature extraction +Localfeature +Observed global feature +Obserred global feature 11 +Obserred global feature m +Feature of partial video +Observation ratio=0.1 +Obserration ratio=0.2 +Obserration ratio=0.34 +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY +C. Network architectures +In this subsection, we elaborate on our network structure. +Due to the data processing method mentioned in the previous +section and the design of network structure, we can model +action evolution in a finer-grained manner without increasing +the computational load. First, we introduce how to extract +short-term features for short time windows, which we call the +segment scale. Then, we introduce how to fuse the segment +scale to generate observed global features for the observed +local videos. +Segment scale. Compared with images, video is a dynamic +sequence of pictures arranged in time, so the temporal context +relationship of frames and the spatial relationship organization +of a single frame need to be considered simultaneously. For +extracting and fusion of two kinds of relations in local time +windows, directly stacking frames as input will bring a lot of +redundant information. This method is inefficient. Moreover, +it will introduce too much noise and reduce the robustness of +the model. If only a single image frame is used as input, the +dynamic information of the temporal window will be lost. +RGB temporal difference turned out to be an efficient +alternative modality to optical flow as motion representation +[45][13]. To extract the spatio-temporal features of each local +temporal window, we adopt the idea in[13] as a short-term +feature extraction module. Different from action recognition, +in the action prediction problem, we cannot get the spatio- +temporal information after the current frame, so we only keep +the short-term TDM (temporal difference module) in[13]. +Specifically, for each segment, we randomly sample 5 frames +2 +1 +1 +2 +[ +, +, +, +, +] +t +t +t +t +t +I +I +I +I I +I +− +− ++ ++ += +, then the RGB difference information of +these frames is down-sampled, and the 2D convolutions +network is used to obtain the depth feature +( ) +i +S I + , as +expressed in Equation (1). +( ) +( +( +( +( )))) +i +i +S I +Upsample CNN Downsample D I += + +(1) +At the same time, to preserve the original frame-level +representation as much as possible, we fuse the original +features +tI with +( ) +i +S I + after convolutions (in our actual +experiment, the original feature passes through a layer of 2D +CNN, as shown in Equation (2)). +( +) +( ) +( ) +i +t +S fuse +S I +CNN I += ++ + +(2) +The fused feature is fused again with the feature from RGB +difference (Equation (3)). Finally, the feature of each segment +is obtained, which is the representation of segment scale. +( +) +( ( +)) +( +( ( ))) +i +S out +CNN S fuse +CNN Downsample D I += ++ + +(3) +Observed global scale. In action prediction, the action +evolution of the human body is an ongoing sequence of +information, and we use the observation rate to simulate its +progress. Therefore, the segments are temporally sequential, +and the representative actions can only evolve from front to +back. In the previous section, we model the local spatio- +temporal action of each segment. More logically, as time +progresses, each segment’s local temporal window is added to +the historical sequence before it. Therefore, the crux of the +problem is how to effectively utilize all observed segments to +reconstruct the historical global evolution. + + +Fig. 3. Overview of End-to-End Multi-scale Network. Given a full video, split it into K segments. For each segment, a CNN-based +module extracts the local motion evolution to achieve more fine-grained modeling, which we call the segment scale. Then, temporal +modeling is performed on each segment in chronological order to model the observed global action evolution, which we call the +observed global scale. + +Full +video +ISegments +CNN-BasedArchitecture +Local +feature +X1 ++X2 +1x3 +Xi + X10 +RNN-Based +h1 +h2 +h3 +hi +Architecture +★Y1 +Y2 +y3 +vyi +VY10 +Global +feature +Action classification +Baslketball5 +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY +Moreover, in the actual scene, the evolution of the action +cannot know its end time and duration, which means that the +overall length of the history is uncertain. Therefore, it is natural +to use the variable-length input characteristics of LSTM to +model the global spatiotemporal characteristics of historical +observations, as shown in formula (4). +( ) +( ( +)) +Y i +L S out += + +(4) +As shown in Fig. 3, when the action evolves to the third +segment, the LSTM adds the short-term time window of the +third segment to the historical observation in the time +dimension. Implemented the observed global evolution to +model the first three segments progressively. In this way, the +spatiotemporal relationship in each segment can be modeled in +a more fine-grained manner, and the subsequent segments are +modeled in a progressive manner to model the historical global +history without additional computational consumption. +IV. EXPERIMENTS +In this section, we present the experiment results of our +framework. First, we describe the evaluation datasets and +implementation details. Then, we compare our E2EMSNet +with state-of-the-art methods. + +A. Datasets +We evaluate our method on three video datasets: BIT[46], +HMDB51[47] and UCF101[48]. BIT consists of 8 classes of +human interactions (bow, boxing, handshake, high-five, hug, +kick, pat, push), with 50 videos per class. Videos are captured +in realistic scenes with cluttered backgrounds, partially +occluded body parts, moving objects, and variations in subject +appearance, scale, illumination condition, and viewpoint. Even +though BIT has a limited number of classes and videos, it is a +complex dataset because of their backgrounds and the +similarity of the beginning and ending scenes. The ratio of +videos between training and testing is 17:8. HMDB51 is a +large-scale human action recognition dataset that comprises 51 +daily action categories. It contains some fine-grained human +facial motions, such as smiling, laughing, etc, in static +background windows, which are not seen in other comparable +datasets, and challenges the spatiotemporal modeling of +actions. There are 6766 video clips with at least 102 videos for +each class. There are three official data splits. UCF101 is a +dataset collected from Youtube and trimmed for action +recognition (each video contains exactly one action). It +includes 101 distinct action classes and 13320 overall video +clips with at least 100 videos for each category. All videos are +divided into 25 groups and updated with the setup of Three +Train/Test Splits. + +B. Implementation details +Thanks to our end-to-end network structure design, we can +easily generalize to various video datasets. In experiments, we +use ResNet50 with the short-term module in [13] to build +segment scale. On the three datasets, we simulated the action +evolution with the observation rate from 0.1 to 1, with a step +size of 0.1, to obtain ten segments, and use each segment as a +segment scale. Our network structure can use any length and +number of segments as the segment scale. For each segment, +we randomly sample 5 frames for computing RGB differential +information. We employ convolutional layers pre-trained on +kinetics400, and set dropout to reduce overfitting. We first +convert the video into video frames, and each video frame is +resized to have shorter side in [256, 320] and a crop of +224×224 is randomly cropped. We use two NVIDIA GeForce +RTX 3090s to train our model. On the BIT dataset, we follow +the official settings to divide the training set and test set. +Specifically, in each category, 34 videos are used as the +training set, and 16 videos are used as the test set. On the +HMDB51 dataset, we follow the standard evaluation protocol +using three training/testing splits, and report the average +accuracy over three splits. On the UCF101 dataset, we use the +first 15 groups of videos for model training, the following 3 +groups for model validation, and the remaining 7 groups for +testing. + +C. Comparison with the state of the art +In this subsection, we compare out E2EMSNet with those +state-of-the-art methods, including DBoW[9], MTSSVM[28], +MMAPM[31], Deep-SCN[5], AAPNet [49], RGN-KF[12], +RSPG + AS-GCN[8], AORAP[50], and AASE +JOLO- +GCN[51] on the BIT dataset, MTSSVM[28], Global-local[52], +AKT[7], STRR[30] on the HMDB51 dataset, MTSSVM[28], +DeepSCN[5], AAPNet[49], Teacher-Student[11], RGN-KF +[12], RSPG + AS-GCN[8], SPR-Net[53], JVS + JCC + +JFIP[32], STRR (ResNet18) [30], and Xinxiao Wu et al.[54] +on the UCF101 dataset. We reported the results of these +compared methods provided by authors. +TableⅠillustrates the accuracy of action prediction and +compares our method with several state-of-the-art methods on +the BIT dataset. As seen from the results, our method achieves +significant improvements in observation rates from 0.1 to 1. +This can be explained by the fact that our method can make +reliable predictions on actions as the actions evolve. + +TABLE I +THE ACCURACY (%) OF DIFFERENT ACTION PREDICTION METHODS ON BIT DATASET AT DIFFERENT +OBSERVATION RATIOS FROM 0.1 TO 1. NOTE THAT THE MISSING VALUE IS BECAUSE THE EXPERIMENTAL +RESULTS OF THE CORRESPONDING OBSERVATION RATE ARE NOT PROVIDED IN THE ORIGINAL PAPER. +Method +Input +Feature-dim +Observation Ratio +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Avg. +DBoW[9] + +Hand-crafted +22.66 +25.78 +40.63 +43.75 +46.88 +54.69 +55.47 +54.69 +55.47 +53.13 +45.31 + +6 +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY +MTSSVM[28] + +Hand-crafted +28.12 +32.81 +45.31 +55.45 +60.00 +61.72 +67.19 +70.31 +71.09 +76.56 +56.85 +MMAPM[31] + +Hand-crafted +32.81 +36.72 +53.90 +59.38 +67.97 +63.28 +68.75 +75.00 +75.78 +79.90 +61.32 +DeepSCN[5] +RGB +3D-CNN + +Hand-crafted +37.50 +44.53 +59.83 +71.88 +78.13 +85.16 +86.72 +87.50 +88.28 +90.63 +73.01 +AAPNet[49] +RGB +3D-CNN + +Hand-crafted +38.84 +45.31 +64.84 +73.40 +80.47 +88.28 +88.28 +89.06 +89.84 +91.40 +74.97 +RGN-KF[12] +RGB + Flow +2D-CNN +35.16 +46.09 +67.97 +75.78 +82.03 +88.28 +92.19 +92.28 +92.16 +92.16 +76.41 +RSPG+AS-GCN[8] +Skeleton +LSTM +55.70 + +77.30 + +91.00 + +93.00 + +93.00 +94.00 + +AORAP[50] +RGB + Flow +2D-CNN +40.16 + +71.48 + +92.89 + +96.8 + + +96.48 +79.56 +AASE + JOLO-GCN[51] +Skeleton +LSTM + + +80.20 + +92.40 + + + + + + +OCRL [6] +RGB +3D-CNN + + +65.6 + +84.4 + +90.6 + +89.1 + + +E2EMSNet (Ours) +RGB +2D-CNN + LSTM +82.81 +89.06 +96.88 +98.43 +98.43 +96.88 +100 +100 +100 +100 +96.25 +TableⅡshows the experimental results on the HMDB51 +dataset, and tableⅢshows the experimental results on the +UCF101 dataset. Thanks to the design of our segment scale, +action evolution can be modeled in a more fine-grained way. +As shown in the table, at 0.2 of observation rate, the accuracy +rate on HMDB51 dataset is increased by more than 10%, and +the accuracy rate on UCF101 in increased by more than 3% +except the results in[32]. This means that our method can better +predict its class in the early stages of the action. As the +observation rate increases, our method can achieve a more +competitive +performance, +although +the +performance +improvement is limited. +At the same time, we have to admit that on the HMDB51 +and UCF101datasets, although our method has achieved +relatively good performance when the observation rate is low, +as the action continues to evolve and the temporal scale +continues to grow, our model is limited in the later observation +ratios. We think that the modeling ability of observed global +scale for long time windows is insufficient. + +TABLE II +THE ACCURACY (%) OF DIFFERENT ACTION PREDICTION METHODS ON HMDB51 DATASET AT DIFFERENT +OBSERVATION RATIOS FROM 0.1 TO 1. NOTE THAT THE MISSING VALUE IS BECAUSE THE EXPERIMENTAL +RESULTS OF THE CORRESPONDING OBSERVATION RATE ARE NOT PROVIDED IN THE ORIGINAL PAPER. +Method +Input +Feature-dim +Observation Ratio +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Avg. +MTSSVM[28] + +Hand-crafted +13.60 + +26.70 + +33.80 + +37.80 + +38.80 + + +Global-local[52] + +Hand-crafted +38.80 +43.80 +49.10 +50.40 +52.60 +54.70 +56.30 +56.90 +57.30 +57.30 +51.72 +AKT[7] +RGB +3D-CNN +43.50 +48.40 +51.20 +54.20 +56.40 +58.40 +59.60 +60.20 +61.10 +61.80 +55.48 +STRR[30] +RGB +3D-CNN +45.10 + +52.35 + +56.73 + +5941 + +61.11 + + +E2EMSNet (Ours) +RGB +2D-CNN + LSTM +59.21 +60.52 +62.23 +64.47 +64.73 +64.86 +64.86 +65.26 +65.13 +65.39 +63.67 + +Table III +THE ACCURACY (%) OF DIFFERENT ACTION PREDICTION METHODS ON UCF101 DATASET AT DIFFERENT +OBSERVATION RATIOS FROM 0.1 TO 1. NOTE THAT THE MISSING VALUE IS BECAUSE THE EXPERIMENTAL +RESULTS OF THE CORRESPONDING OBSERVATION RATE ARE NOT PROVIDED IN THE ORIGINAL PAPER. +Method +Input +Feature-dim +Observation Ratio +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Avg. +MTSSVM[28] + +Hand-crafted +40.05 +72.83 +80.02 +82.18 +82.39 +83.12 +83.37 +83.51 +83.69 +82.82 +77.39 +DeepSCN[5] +RGB +3D-CNN + +Hand-crafted +45.02 +77.64 +82.95 +85.36 +85.75 +86.70 +87.10 +87.42 +87.50 +87.63 +81.30 + +7 +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY +AAPNet[49] +RGB +3D-CNN + +Hand-crafted +59.85 +80.85 +86.78 +86.47 +86.94 +88.34 +88.34 +89.85 +90.85 +91.99 +85.02 +Teacher-Student[11] +RGB +3D-CNN +83.32 +87.13 +88.92 +89.82 +90.85 +91.04 +91.28 +91.23 +91.31 +91.47 +89.63 +RGN-KF[12] +RGB + Flow +2D-CNN +83.12 +85.16 +88.44 +90.78 +91.42 +92.03 +92.00 +93.19 +93.13 +93.13 +90.24 +RSPG+AS-GCN[8] +Skeleton +LSTM + + +90.30 + +93.10 + + + + +94.70 + +SPR-Net[53] +RGB +3D-CNN +88.70 + + + +91.60 + + + + +91.40 + +JVS+JCC+JFIP[32] +RGB +(2D+1D)-CNN + +91.70 + + + + + + + + + +STRR (ResNet18)[30] +RGB +3D-CNN +80.86 + +88.61 + +89.31 + +90.31 + +89.82 + + +Xinxiao Wu et al.[54] +RGB + Flow +2D-CNN +82.36 +85.57 +88.97 + +91.32 + +92.41 + +93.02 + + +E2EMSNet (Ours) +RGB +2D-CNN + LSTM +88.77 +90.31 +90.94 +91.33 +91.96 +92.73 +93.11 +92.98 +92.98 +92.73 +91.78 + +D. Ablation study +Here, we provide more evaluation results on the UCF101 +dataset. +Influence of multi-scale architecture. TableⅣ. Illustrates +the results of the ablation study for different scale architecture. +First, we introduce the details of the ablation study. Then, we +analyze the effects of multi-scale architecture by comparing +the results with different settings. +TABLE IV +THE ACCURACY (%) AT DIFFERENT SCALE SETTINGS +ON THE UCF101 DATASET. +Observation +ratio +0.1 +0.3 +0.5 +0.9 +Avg. +The +segment +scale only +90.56 +91.58 +91.83 +91.45 +91.55 +The +segment +scale+observed +global scale +90.05 +90.82 +92.60 +92.47 +91.78 + +‘The segment scale only’ uses the CNN-based module for +action prediction. ‘The segment scale + observed global scale’ +uses the CNN-based and LSTM modules to learn different +scale information. In the first setting, for action clips with +different observation rates, we sample 5 frames and use the +segment scale only for prediction. In the second setting, we +adopt a complete structure with segment scale and observed +global scale. Even though the average accuracy difference is +insignificant, the multi-scale structure is essential for ongoing +action prediction. Results of ‘The segment scale only’ has little +discrimination under different observation rates, as shown in +Fig 4. This indicates that its feature representation and +discriminative degree for different observation rates are +insufficient. At the same time, due to the sparse sampling of +long-time scales, we believe this manner will perform worse +for complex actions and actions with long duration. Conversely, +adding observed global scale and changing the sampling +strategy will make the prediction process more cognitive (As +the observation rate increases, the confidence of the prediction +should be increasing.). Moreover, due to the more fine-grained +feature extraction for actions, it has better robustness to +complex and long-duration actions. + + +Fig. 4. Prediction accuracy (%) under two scale settings on +UCF101 dataset. +Influence of hyperparameters. Finally, we briefly +introduce the experimental results on UCF101 dataset under +different hyperparameter settings. To ensure a single variable, +we have conducted comparative experiments on the following +hyperparameters, and the results are shown in TableⅤ. + +E. Analysis of the performance of different actions +We follow the grouping of the UCF101 dataset and divide it +into five groups: Human-Object interaction, Body-Motion +only, Human-Human interaction, Playing musical instruments, +and Sports. We selected three action categories under each +group, for a total of fifteen action categories, to visually +analyze their classification results. We selected the following +action categories: Blowing Candles, Blow Dry Hair, Cutting In +Kitchen, Apply Eye Makeup, Baby Crawling, Pull Ups, +Haircut, Head Massage, Punch, Playing Guitar, Playing Piano, +Playing Violin, Basketball, Basketball Dunk, Biking. We keep +two modules, segment scale and observed global scale, and +only modify and retrain the last classification layer. The +confusion matrix of the results of 15 actions at progress level +of 20% is shown in Fig. 5. It can be seen intuitively from the +figure that our model still has stable prediction performance +for action prediction in different scenarios, even in the very +early stage of actions. Only a few actions (Haircut, Blow Dry +Hair, and Head Massage) with very similar external features +were mispredicted. As shown in Fig6, it is an appearance +comparison of Haircut, Blow Dry Hair, and Head Massage. It +can be seen that three actions are difficult to distinguish, +resulting in the problem of mispredicted. +89.5 +90 +90.5 +91 +91.5 +92 +92.5 +93 +0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 +Accuracy (%) +Observation ratio +Segment-scale +Two-scales + +8 +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY +TABLE V +THE ACCURACY (%) ON UCF101 DATASET UNDER SEVERAL HYPERPARAMETERS. (NOTE: LIMITED BY +RESOURCES AND TIME, OUR EXPERIMENTAL RESULTS DO NOT GUARANTEE THAT ALL HYPERPARAMETERS +HAVE BEEN ADJUSTED TO THE OPTIMUM.) + +Fig. 5. Confusion matrix of the result of 15 classes at +progress level of 20% on UCF101 dataset. + + + +Fig. 6. Appearance comparison of Haircut, Blow Dry Hair, +and Head Massage. +V. +CONCLUSION +In this paper, we have proposed a network model, +E2EMSNet, for action prediction in videos. We propose two +temporal scales, segment scale and observed global scale, to +model the evolution of actions, and fuse the two scales into an +end-to-end framework. A stack of 2D convolutional layers +with input of RGB difference is introduced to model the local +evolution of actions in a more fine-grained way. Next, the +LSTM layer fuses each segment scale in the temporal +dimension into an observed global scale to model the long- +term evolution of actions. After experimental validation and +analysis, our method possesses powerful local scale modeling +capability to model ongoing actions. However, due to the +growth of the time scale and the increasing noise, our observed +scale cannot achieve the global modeling ability we expected +for the evolving actions, which will also be the focus of our +future work. +Hyperparameter variables +Observation Ratios +0.1 +0.3 +0.5 +0.7 +0.9 +Avg. +Hidden size of LSTM +512 +90.05 +90.82 +92.60 +92.22 +92.48 +91.78 +1024 +88.77 +90.05 +90.82 +91.07 +91.20 +90.60 +2048 +88.23 +88.93 +89.95 +91.03 +91.73 +90.14 +Learning rate +0.0001 +82.14 +84.06 +85.97 +87.12 +88.01 +85.85 +0.0005 +90.05 +90.82 +92.60 +92.22 +92.48 +91.78 +0.001 +89.41 +90.31 +91.07 +90.82 +90.56 +90.57 +Decay step (decay +rate=0.1) +20, 80 +89.41 +90.05 +91.58 +91.84 +91.96 +91.09 +40, 100 +90.31 +90.18 +91.45 +92.09 +92.09 +91.28 +60, 100 +90.18 +91.07 +91.71 +92.35 +92.61 +91.78 + +Confusionmatrix +1.0 +Basketball +Haircut +CuttingInKitchen +Blow Dry Hair +0.8 +Pull Ups +True label +ApplyEyeMakeup +Playing Violin +0.6 +Punch +Biking +0.4 +BasketballDunk +BabyCrawling +HeadMassage +Playing Piano +0.2 +Blowing Candles +PlayingGuitar + Baby Crawling +0.0 +Basketball +Haircut +Kitchen +DryHair +Pull Ups +Makeup +Violin +Punch +Biking +Basketball Dunk +Head Massage + Playing Piano +Blowing Candles +Playing Guitar +M +Playing +Blow +Eye +Cutting +Apply +PredictedlabelHaircut +Blow Dry +Hair +Head +Massage10 +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY +References +[1] Liu J, Shahroudy A, Wang G, et al. 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Anticipating Future Relations via +Graph Growing for Action Prediction[C]// Proceedings of +the AAAI Conference on Artificial Intelligence. 2021, +35(4): 2952-2960. + + + +12 +IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY + + +Xiaofa Liu received the B.S. degree from +Hohai University, Nanjing, China, in 2017. +He is currently pursuing the M.S. degree +in mechanical engineering with the +School +of +Modern +Post, +Beijing +University +of +Posts +and +Telecom- +munications, Beijing, China. His research +interests include robotics, and computer +vision. + + +Jianqin Yin (Member, IEEE) received +the +Ph.D. +degree +from +Shandong +University, Jinan, China, in 2013. She +currently is a Professor with the School of +Artificial Intelligence, Beijing University +of Posts and Telecommunications, Beijing, +China. Her research interests include +service +robot, +pattern +recognition, +machine learning, and image processing. + + +Yuan Sun received the Ph.D. degree from +Beijing University of Aeronautics and +Astronautics, Beijing, China, in 2016. She +currently is an Assistant Professor with +Electronic Engineering School, Beijing +University +of +Posts +and +Telecommunications, Beijing, China. Her +research +interests +include +satellite +navigation +technology, +and +satellite +autonomous integrity. + + +Zhicheng Zhang received the Ph.D. +degree from Jilin University, Changchun, +China, in 2011. He currently is an +Associate Professor with the School of +Artificial Intelligence, Beijing University +of Posts and Telecommunications, Beijing, +China. His research interests include +Intelligent optimization and its application, +signal detection and estimation, machine learning. + + +Jin Tang received the Ph.D. degree from +Beijing Institute of Technology, Beijing, +China, in 2007. currently is an Assistant +Professor with Artificial Intelligence +School, Beijing University of Posts and +Telecommunications, Beijing, China. Her +research +interests +include +signal +processing, pattern recognition, and deep +learning. + diff --git a/1NAzT4oBgHgl3EQfRftD/content/tmp_files/load_file.txt b/1NAzT4oBgHgl3EQfRftD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2770985e888dca5dbac5bc19b73a98c7b44ce2da --- /dev/null +++ b/1NAzT4oBgHgl3EQfRftD/content/tmp_files/load_file.txt @@ -0,0 +1,880 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf,len=879 +page_content='1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY An End-to-End Multi-Scale Network for Action Prediction in Videos Xiaofa Liu, Jianqin Yin, Member, IEEE, Yuan Sun, Zhicheng Zhang, Jin Tang Abstract—In this paper, we develop an efficient multi-scale network to predict action classes in partial videos in an end-to- end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Unlike most existing methods with offline feature generation, our method directly takes frames as input and further models motion evolution on two different temporal scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Therefore, we solve the complexity problems of the two stages of modeling and the problem of insufficient temporal and spatial information of a single scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Our proposed End-to-End Multi- Scale Network (E2EMSNet) is composed of two scales which are named segment scale and observed global scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The segment scale leverages temporal difference over consecutive frames for finer motion patterns by supplying 2D convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' For observed global scale, a Long Short-Term Memory (LSTM) is incorporated to capture motion features of observed frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Our model provides a simple and efficient modeling framework with a small computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Our E2EMSNet is evaluated on three challenging datasets: BIT, HMDB51, and UCF101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The extensive experiments demonstrate the effective-ness of our method for action prediction in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Index terms: action prediction, multi-scale network, end-to- end method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' INTRODUCTION HE goal of action prediction in videos is to predict the class label of an ongoing action from an observed part of it over temporal axis so far[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' It is a subset of a broader research domain on human activity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Different from conventional action recognition with fully executed actions[2][3][4], it is more challenging to predict the action label in ongoing actions due to the incompleteness of actions and the continuous evolution of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' It has attracted a lot of research attention because of its wide application in some scenarios with high real-time requirements, such as human- machine interaction, security surveillance, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Although the previous work has achieved promising results ▪ This work was supported partly by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 62173045, 61673192), partly by the Fundamental Research Funds for the Central Universities (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 2020XD- A04-3), and the Natural Science Foundation of Hainan Province (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 622RC675).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' (Corresponding author: Jianqin Yin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' ▪ Xiaofa Liu is with the School of Modern Post, Beijing University of Posts and Telecom-munications, Beijing 100876, China (e-mail: liuxiaofamail@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='com ) ▪ Jianqin Yin, Zhicheng Zhang, and Jin Tang are with the school of Artificial Intelligence, Beijing University of Posts and Telecommunications, Beijing 100876, China (e-mail: jqyin@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='cn, zczhang@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='cn, tangjin@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='cn ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' ▪ Yuan Sun is with Electronic Engineering School, Beijing University of Posts and Telecommunications, Beijing 100876, China (e-mail: sunyuan@bupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='cn ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' by adopting a two-stage approach, there generally had problems of complex modeling and feature redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The previous method separated feature extraction from predictive modeling[5][6][7][8][9][10][11][12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' This separation operati- on makes the spatio-temporal representation obtained may deviate from the action prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Moreover, it complicates the model design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Secondly, because the feature is generated offline, the complete action must be divided into fixed segments in advance, which not only results in the redundancy of the feature in the time dimension, but also is not applicable to the evolving action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Therefore, in this paper, we propose an end-to-end method, which effectively reduces the complexity of the model and introduces more fine-grained spatio-temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We designed the end-to-end network from three aspects, sampling method, local spatio-temporal information representation, and long-term time sequence fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' In order to adapt the end-to- end structure to the evolving motion, we first changed the preprocessing and feature generation method, which will be described in Part 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Second, to reduce computational consumption to achieve end-to-end structure, we use 2D convolution instead of two-stream networks or 3D convolutions to extract local spatio-temporal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Finally, to enhance the temporal information of action evolution, we present an observed global scale to fuse the historical evolution information of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Similar to the application of spatial multi-scale in image field, multi-scale research in the temporal dimension is also increasing in video analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Compared to images, the variation of temporal scales in videos poses additional challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' How to effectively utilize the motion evolution information at different time scales has gradually gained attention in video motion analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Feichtenhofer[4] et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' proposed SlowFast network for video recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Their method utilizes two branches, a slow pathway with low frame rate and a fast pathway with high frame rate, to capture spatial semantics and motion at fine temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Wang[13] et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' proposed an efficient multi-scale model for action recognition, which utilizes short-term and long-term temporal difference modules to capture both short-term and long-term motion information better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Most of the existing action prediction methods are insufficient to focus on multi-scale temporal, making them fail to capture fine-grained temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' They use a fixed frame rate to sample each partial video, and use a fixed temporal scale for feature generation and modeling[1][5] [6][7][8][9][11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Although these methods simplify the T 2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY processing of the input of feature generation and reduce the computation to a certain extent, they ignore the evolution of action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Too much fine-grained information will be lost, and the spatio-temporal information in the video cannot be fully utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Our method takes both the local evolution information between adjacent frames and the global evolution information of the entire observed video sequence into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Therefore, we design two temporal scales to increase fine-grained timing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Firstly, the segment scale uses RGB frames with temporal difference to capture temporal information in each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Secondly, the observed global scale uses LSTM module to fuse all the observed action evolution information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Through modeling in short-term and long-term time scales, our method can be mining more fine-grained temporal information without increasing the computational load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Our E2EMSNet provides a simple yet effective framework for the problem of ongoing action prediction in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' In summary, our main contributions lie in the following three aspects: We propose a simple end-to-end approach for action prediction in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' To the best of our knowledge, this is the first work focusing on this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We investigate two scales in the temporal dimension to model the evolution of actions, and propose a segment summarization and propagation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The segment scale is used to model the local evolution of the action, and the observed global scale is used to model the global evolution of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We achieve a trade-off of efficiency and effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We achieve state-of-the-art performance on several datasets while using only 2D convolutions framework and RGB format of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Action Recognition Action recognition methods take fully observed videos as input and output labels of human actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Action recognition has been extensively studied in past few years[2][3][4][13][14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' These studies can be roughly divided into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Methods in the first category are two-stream CNNs, which was first proposed in[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' It used two inputs of RGB and optical flow to model appearance and motion information separately in videos with a late fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' In addition, follow-up research has adopted two RGB inputs sampled at different FPS or carefully designed temporal modules for efficiency, including Non-local Net[16], STM[17], SlowFast[4], and Correlation Net[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The second method is to use 3D CNNs[19][20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' It proposed 3D convolution and pooling to learn spatiotemporal features from videos directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Several variants adopted a 2D + 1D paradigm to reduce the computation cost of 3D convolution, which implement by decomposing 3D CNNs into a 2D convolution and a 1D temporal convolution[21][22][23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Several works focused on designing more powerful and efficient temporal modules, such as TSM[14], TAM[24], TEA[25], and TDN[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' More recent works tried clip-based architecture search for video recognition, focusing on capturing appearance and motion or context information in a more fine-grained and efficient manner[13][26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Although these methods mainly learned features for the videos with full action executions, their core ideas have certain reference significance for ongoing action prediction in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Action Prediction Action prediction methods were proposed to predict the action given a partially observed video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' [9] was the first work along these lines, they formulated the problem probabilistically and proposed a dynamic bag-of-words approach, modeling how feature distributions of activities change as observations increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' In the last decade, researchers approach this task from various perspectives and can be grouped into three major divisions[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The first method can be formulated as one-shot mappings from partial observations to groundtruth labels of full observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The basic assumption underlying these methods is that a partial observation of an action video provides sufficient information to define the appropriate overall action class regardless of the unobserved part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Follow-up research work[28][29][6][30] adopted more robust features, hierarchical extractions, and learning-based classifiers to perform more fine-grained analysis of an initial partial observation for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The second division is knowledge distillation-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' These methods distill the information from the full observations into partial observations[31][5][11][32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' These methods attempted to lend power from unobserved data in training to either enrich the feature representation of partial data or encourage the classifiers to easily recognize partial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Another way to exploit future information is by propagating the partial observation into the future in a temporal extrapolation fashion[33][34] [12][35][36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' For example, [12] learned to propagate frame-wise residuals in feature space to complete partial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Relevant definitions in action prediction in videos: full video, partial video, segments, and observation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Full video X[1:T] Segments (K-10) Partial video x[1:t] k=2,observationratio:r=k/K =2/10=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='23 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Multiple temporal scales for action analysis in videos Temporal sequence forecasting usually faces the following situations for scenarios with insignificant periodic motion: long-term forecasts need to consider trend information (long- term dependencies), and short-term forecasts need to consider fine-grained volatility (short-term dependencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The current difficulty is how to model long-term dynamic dependencies and consider long-term and short-term dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' There are two methods currently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The main existing method is hierarchical modeling, which is achieved by establishing hidden layers of different granularities[37][38][39][40][41] or decomposing the original data to obtain data of different granularities[42][43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The second method is designing the gate mechanism, which achieved by modifying the internal structure of RNN[44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We inherit this idea that both long-term and short-term dependencies in video must be carefully considered, and a trade-off approach is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' OUR METHOD In this section, we detail our approach to mining ongoing action evolution information in videos using multiple scales in an end-to-end fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Specifically, we first describe the problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Then, we elaborate on our end-to-end framework and method for multi-scale modeling of ongoing action sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Problem formulation Given a video containing human motion (the video may contain arbitrary incomplete motion), the goal is to predict the class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We follow the problem formulation in the[31], which has been widely adopted in subsequent work[5][7][11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 1, Given a full video [1: ] X T with complete action execution, 1 represents the first frame of the video, and T represents the last frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We use [1, ], [1, ] x t t T \uf0ce to simulate the action execution in video from 1 to t , defined as partial video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' In order to facilitate quantitative experiments, we usually divide a full video into K segments, each containing ( / ) T K frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Assuming that the action is executed to the , [1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=', ] kth k K = segment, the observation ratio is defined as / r k K = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' As defined above, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1, the full video X , is divided into K segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Among them, the partial video marked with green has an observation ratio / 2 /10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='2 r k K = = = , and it can be considered that its action has been executed 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Data processing We adopt a data processing method different from the previous method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 2, the upper part is the data processing method used in the previous method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' They first divided a complete video X into K segments, and combined segments into partial videos to simulate action evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Then the partial video is sampled to extract the spatio-temporal representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The problem caused by this is that each partial video needs to be separately extracted for spatio-temporal representation, which divides the continuous evolution of action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The feature extraction of partial videos with higher observation rates cannot use the previous partial videos with lower observation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' It will cause redundancy in the time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' At the same time, with the increase in the observation rate, the temporal information will become more and more sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Compared with them, we directly extract the local spatio-temporal representations of each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' In this way, the previous spatio-temporal information can be continuously used with the evolution of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' This makes our model more robust to action duration, and more abundant spatio-temporal information can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Differences in data processing between our method and previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The upper is the data processing method used in the previous method, and the lower is the data processing strategy used in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' LOTTE Full video X Segments artial video Observation ratio=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1 Partial video I Observation rafio=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='2 Partial video k Observation ratio=k/K Partial video Sampling and feature extraction Feature of partial video OTT Full video X Segments Sampling and feature extraction Localfeature Observed global feature Obserred global feature 11 Obserred global feature m Feature of partial video Observation ratio=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1 Obserration ratio=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='2 Obserration ratio=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='34 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Network architectures In this subsection, we elaborate on our network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Due to the data processing method mentioned in the previous section and the design of network structure, we can model action evolution in a finer-grained manner without increasing the computational load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' First, we introduce how to extract short-term features for short time windows, which we call the segment scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Then, we introduce how to fuse the segment scale to generate observed global features for the observed local videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Segment scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Compared with images, video is a dynamic sequence of pictures arranged in time, so the temporal context relationship of frames and the spatial relationship organization of a single frame need to be considered simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' For extracting and fusion of two kinds of relations in local time windows, directly stacking frames as input will bring a lot of redundant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' This method is inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Moreover, it will introduce too much noise and reduce the robustness of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' If only a single image frame is used as input, the dynamic information of the temporal window will be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' RGB temporal difference turned out to be an efficient alternative modality to optical flow as motion representation [45][13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' To extract the spatio-temporal features of each local temporal window, we adopt the idea in[13] as a short-term feature extraction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Different from action recognition, in the action prediction problem, we cannot get the spatio- temporal information after the current frame, so we only keep the short-term TDM (temporal difference module) in[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Specifically, for each segment, we randomly sample 5 frames 2 1 1 2 [ , , , , ] t t t t t I I I I I I − − + + = , then the RGB difference information of these frames is down-sampled, and the 2D convolutions network is used to obtain the depth feature ( ) i S I , as expressed in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' ( ) ( ( ( ( )))) i i S I Upsample CNN Downsample D I = (1) At the same time, to preserve the original frame-level representation as much as possible, we fuse the original features tI with ( ) i S I after convolutions (in our actual experiment, the original feature passes through a layer of 2D CNN, as shown in Equation (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' ( ) ( ) ( ) i t S fuse S I CNN I = + (2) The fused feature is fused again with the feature from RGB difference (Equation (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Finally, the feature of each segment is obtained, which is the representation of segment scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' ( ) ( ( )) ( ( ( ))) i S out CNN S fuse CNN Downsample D I = + (3) Observed global scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' In action prediction, the action evolution of the human body is an ongoing sequence of information, and we use the observation rate to simulate its progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Therefore, the segments are temporally sequential, and the representative actions can only evolve from front to back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' In the previous section, we model the local spatio- temporal action of each segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' More logically, as time progresses, each segment’s local temporal window is added to the historical sequence before it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Therefore, the crux of the problem is how to effectively utilize all observed segments to reconstruct the historical global evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Overview of End-to-End Multi-scale Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Given a full video, split it into K segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' For each segment, a CNN-based module extracts the local motion evolution to achieve more fine-grained modeling, which we call the segment scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Then, temporal modeling is performed on each segment in chronological order to model the observed global action evolution, which we call the observed global scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Full video ISegments CNN-BasedArchitecture Local feature X1 +X2 1x3 Xi X10 RNN-Based h1 h2 h3 hi Architecture ★Y1 Y2 y3 vyi VY10 Global feature Action classification Baslketball5 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY Moreover, in the actual scene, the evolution of the action cannot know its end time and duration, which means that the overall length of the history is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Therefore, it is natural to use the variable-length input characteristics of LSTM to model the global spatiotemporal characteristics of historical observations, as shown in formula (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' ( ) ( ( )) Y i L S out = (4) As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 3, when the action evolves to the third segment, the LSTM adds the short-term time window of the third segment to the historical observation in the time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Implemented the observed global evolution to model the first three segments progressively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' In this way, the spatiotemporal relationship in each segment can be modeled in a more fine-grained manner, and the subsequent segments are modeled in a progressive manner to model the historical global history without additional computational consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' EXPERIMENTS In this section, we present the experiment results of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' First, we describe the evaluation datasets and implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Then, we compare our E2EMSNet with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Datasets We evaluate our method on three video datasets: BIT[46], HMDB51[47] and UCF101[48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' BIT consists of 8 classes of human interactions (bow, boxing, handshake, high-five, hug, kick, pat, push), with 50 videos per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Videos are captured in realistic scenes with cluttered backgrounds, partially occluded body parts, moving objects, and variations in subject appearance, scale, illumination condition, and viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Even though BIT has a limited number of classes and videos, it is a complex dataset because of their backgrounds and the similarity of the beginning and ending scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The ratio of videos between training and testing is 17:8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' HMDB51 is a large-scale human action recognition dataset that comprises 51 daily action categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' It contains some fine-grained human facial motions, such as smiling, laughing, etc, in static background windows, which are not seen in other comparable datasets, and challenges the spatiotemporal modeling of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' There are 6766 video clips with at least 102 videos for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' There are three official data splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' UCF101 is a dataset collected from Youtube and trimmed for action recognition (each video contains exactly one action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' It includes 101 distinct action classes and 13320 overall video clips with at least 100 videos for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' All videos are divided into 25 groups and updated with the setup of Three Train/Test Splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Implementation details Thanks to our end-to-end network structure design, we can easily generalize to various video datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' In experiments, we use ResNet50 with the short-term module in [13] to build segment scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' On the three datasets, we simulated the action evolution with the observation rate from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1 to 1, with a step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1, to obtain ten segments, and use each segment as a segment scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Our network structure can use any length and number of segments as the segment scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' For each segment, we randomly sample 5 frames for computing RGB differential information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We employ convolutional layers pre-trained on kinetics400, and set dropout to reduce overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We first convert the video into video frames, and each video frame is resized to have shorter side in [256, 320] and a crop of 224×224 is randomly cropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We use two NVIDIA GeForce RTX 3090s to train our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' On the BIT dataset, we follow the official settings to divide the training set and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Specifically, in each category, 34 videos are used as the training set, and 16 videos are used as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' On the HMDB51 dataset, we follow the standard evaluation protocol using three training/testing splits, and report the average accuracy over three splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' On the UCF101 dataset, we use the first 15 groups of videos for model training, the following 3 groups for model validation, and the remaining 7 groups for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Comparison with the state of the art In this subsection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' we compare out E2EMSNet with those state-of-the-art methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' including DBoW[9],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' MTSSVM[28],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' MMAPM[31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Deep-SCN[5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' AAPNet [49],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' RGN-KF[12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' RSPG + AS-GCN[8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' AORAP[50],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' and AASE +JOLO- GCN[51] on the BIT dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' MTSSVM[28],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Global-local[52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' AKT[7],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' STRR[30] on the HMDB51 dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' MTSSVM[28],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' DeepSCN[5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' AAPNet[49],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Teacher-Student[11],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' RGN-KF [12],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' RSPG + AS-GCN[8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' SPR-Net[53],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' JVS + JCC + JFIP[32],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' STRR (ResNet18) [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' and Xinxiao Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' [54] on the UCF101 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We reported the results of these compared methods provided by authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' TableⅠillustrates the accuracy of action prediction and compares our method with several state-of-the-art methods on the BIT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' As seen from the results, our method achieves significant improvements in observation rates from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' This can be explained by the fact that our method can make reliable predictions on actions as the actions evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' TABLE I THE ACCURACY (%) OF DIFFERENT ACTION PREDICTION METHODS ON BIT DATASET AT DIFFERENT OBSERVATION RATIOS FROM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1 TO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' NOTE THAT THE MISSING VALUE IS BECAUSE THE EXPERIMENTAL RESULTS OF THE CORRESPONDING OBSERVATION RATE ARE NOT PROVIDED IN THE ORIGINAL PAPER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Method Input Feature-dim Observation Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='0 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' DBoW[9] Hand-crafted 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='66 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='78 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='63 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='75 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='88 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='69 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='47 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='69 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='47 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='13 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='31 6 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY MTSSVM[28] Hand-crafted 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='12 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='81 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='31 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='45 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='00 61.' metadata={'source': 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+page_content='81 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='06 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='88 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='43 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='43 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='88 100 100 100 100 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='25 TableⅡshows the experimental results on the HMDB51 dataset, and tableⅢshows the experimental results on the UCF101 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Thanks to the design of our segment scale, action evolution can be modeled in a more fine-grained way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' As shown in the table, at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='2 of observation rate, the accuracy rate on HMDB51 dataset is increased by more than 10%, and the accuracy rate on UCF101 in increased by more than 3% except the results in[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' This means that our method can better predict its class in the early stages of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' As the observation rate increases, our method can achieve a more competitive performance, although the performance improvement is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' At the same time, we have to admit that on the HMDB51 and UCF101datasets, although our method has achieved relatively good performance when the observation rate is low, as the action continues to evolve and the temporal scale continues to grow, our model is limited in the later observation ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We think that the modeling ability of observed global scale for long time windows is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' TABLE II THE ACCURACY (%) OF DIFFERENT ACTION PREDICTION METHODS ON HMDB51 DATASET AT DIFFERENT OBSERVATION RATIOS FROM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1 TO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' NOTE THAT THE MISSING VALUE IS BECAUSE THE EXPERIMENTAL RESULTS OF THE CORRESPONDING OBSERVATION RATE ARE NOT PROVIDED IN THE ORIGINAL PAPER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Method Input Feature-dim Observation Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1 0.' metadata={'source': 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65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='13 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='39 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='67 Table III THE ACCURACY (%) OF DIFFERENT ACTION PREDICTION METHODS ON UCF101 DATASET AT DIFFERENT OBSERVATION RATIOS FROM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1 TO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' NOTE THAT THE MISSING VALUE IS BECAUSE THE EXPERIMENTAL RESULTS OF THE CORRESPONDING OBSERVATION RATE ARE NOT PROVIDED IN THE ORIGINAL PAPER.' metadata={'source': 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+page_content='10 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='70 SPR-Net[53] RGB 3D-CNN 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='70 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='60 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='40 JVS+JCC+JFIP[32] RGB (2D+1D)-CNN 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='70 STRR (ResNet18)[30] RGB 3D-CNN 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='86 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='61 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='31 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='31 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='82 Xinxiao Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' [54] RGB + Flow 2D-CNN 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='36 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='57 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='97 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='32 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='41 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='02 E2EMSNet (Ours) RGB 2D-CNN + LSTM 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='77 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='31 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='94 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='33 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='96 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='73 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='11 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='98 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='98 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='73 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='78 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Ablation study Here, we provide more evaluation results on the UCF101 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Influence of multi-scale architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' TableⅣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Illustrates the results of the ablation study for different scale architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' First, we introduce the details of the ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Then, we analyze the effects of multi-scale architecture by comparing the results with different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' TABLE IV THE ACCURACY (%) AT DIFFERENT SCALE SETTINGS ON THE UCF101 DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Observation ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='9 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The segment scale only 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='56 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='58 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='83 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='45 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='55 The segment scale+observed global scale 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='05 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='82 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='60 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='47 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='78 ‘The segment scale only’ uses the CNN-based module for action prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' ‘The segment scale + observed global scale’ uses the CNN-based and LSTM modules to learn different scale information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' In the first setting, for action clips with different observation rates, we sample 5 frames and use the segment scale only for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' In the second setting, we adopt a complete structure with segment scale and observed global scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Even though the average accuracy difference is insignificant, the multi-scale structure is essential for ongoing action prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Results of ‘The segment scale only’ has little discrimination under different observation rates, as shown in Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' This indicates that its feature representation and discriminative degree for different observation rates are insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' At the same time, due to the sparse sampling of long-time scales, we believe this manner will perform worse for complex actions and actions with long duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Conversely, adding observed global scale and changing the sampling strategy will make the prediction process more cognitive (As the observation rate increases, the confidence of the prediction should be increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Moreover, due to the more fine-grained feature extraction for actions, it has better robustness to complex and long-duration actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Prediction accuracy (%) under two scale settings on UCF101 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Influence of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Finally, we briefly introduce the experimental results on UCF101 dataset under different hyperparameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' To ensure a single variable, we have conducted comparative experiments on the following hyperparameters, and the results are shown in TableⅤ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Analysis of the performance of different actions We follow the grouping of the UCF101 dataset and divide it into five groups: Human-Object interaction, Body-Motion only, Human-Human interaction, Playing musical instruments, and Sports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We selected three action categories under each group, for a total of fifteen action categories, to visually analyze their classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We selected the following action categories: Blowing Candles, Blow Dry Hair, Cutting In Kitchen, Apply Eye Makeup, Baby Crawling, Pull Ups, Haircut, Head Massage, Punch, Playing Guitar, Playing Piano, Playing Violin, Basketball, Basketball Dunk, Biking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We keep two modules, segment scale and observed global scale, and only modify and retrain the last classification layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' The confusion matrix of the results of 15 actions at progress level of 20% is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' It can be seen intuitively from the figure that our model still has stable prediction performance for action prediction in different scenarios, even in the very early stage of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Only a few actions (Haircut, Blow Dry Hair, and Head Massage) with very similar external features were mispredicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' As shown in Fig6, it is an appearance comparison of Haircut, Blow Dry Hair, and Head Massage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' It can be seen that three actions are difficult to distinguish, resulting in the problem of mispredicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='5 90 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='5 91 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='5 92 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='5 93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='0 Accuracy (%) Observation ratio Segment-scale Two-scales 8 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY TABLE V THE ACCURACY (%) ON UCF101 DATASET UNDER SEVERAL HYPERPARAMETERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' (NOTE: LIMITED BY RESOURCES AND TIME, OUR EXPERIMENTAL RESULTS DO NOT GUARANTEE THAT ALL HYPERPARAMETERS HAVE BEEN ADJUSTED TO THE OPTIMUM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=') Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Confusion matrix of the result of 15 classes at progress level of 20% on UCF101 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Appearance comparison of Haircut, Blow Dry Hair, and Head Massage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' CONCLUSION In this paper, we have proposed a network model, E2EMSNet, for action prediction in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' We propose two temporal scales, segment scale and observed global scale, to model the evolution of actions, and fuse the two scales into an end-to-end framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' A stack of 2D convolutional layers with input of RGB difference is introduced to model the local evolution of actions in a more fine-grained way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Next, the LSTM layer fuses each segment scale in the temporal dimension into an observed global scale to model the long- term evolution of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' After experimental validation and analysis, our method possesses powerful local scale modeling capability to model ongoing actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' However, due to the growth of the time scale and the increasing noise, our observed scale cannot achieve the global modeling ability we expected for the evolving actions, which will also be the focus of our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Hyperparameter variables Observation Ratios 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='9 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Hidden size of LSTM 512 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='05 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='82 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='60 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='22 92.' metadata={'source': 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29: 6017- 6031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' [54] Wu X, Zhao J, Wang R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Anticipating Future Relations via Graph Growing for Action Prediction[C]// Proceedings of the AAAI Conference on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 2021, 35(4): 2952-2960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' 12 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY Xiaofa Liu received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' degree from Hohai University, Nanjing, China, in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' He is currently pursuing the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' degree in mechanical engineering with the School of Modern Post, Beijing University of Posts and Telecom- munications, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' His research interests include robotics, and computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Jianqin Yin (Member, IEEE) received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' degree from Shandong University, Jinan, China, in 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' She currently is a Professor with the School of Artificial Intelligence, Beijing University of Posts and Telecommunications, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Her research interests include service robot, pattern recognition, machine learning, and image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Yuan Sun received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' degree from Beijing University of Aeronautics and Astronautics, Beijing, China, in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' She currently is an Assistant Professor with Electronic Engineering School, Beijing University of Posts and Telecommunications, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Her research interests include satellite navigation technology, and satellite autonomous integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Zhicheng Zhang received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' degree from Jilin University, Changchun, China, in 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' He currently is an Associate Professor with the School of Artificial Intelligence, Beijing University of Posts and Telecommunications, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' His research interests include Intelligent optimization and its application, signal detection and estimation, machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Jin Tang received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' degree from Beijing Institute of Technology, Beijing, China, in 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' currently is an Assistant Professor with Artificial Intelligence School, Beijing University of Posts and Telecommunications, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} +page_content=' Her research interests include signal processing, pattern recognition, and deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAzT4oBgHgl3EQfRftD/content/2301.01216v1.pdf'} diff --git a/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf b/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..58f6b297a2383e5b4ed6e9729af3e0ed7857f091 --- /dev/null +++ b/29AzT4oBgHgl3EQfuf0F/content/2301.01690v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5542bad206b71ce3a93e10f0f19d3aa166d1be2b0e784b2dc6e117e6228f5085 +size 315162 diff --git a/29AzT4oBgHgl3EQfuf0F/vector_store/index.faiss b/29AzT4oBgHgl3EQfuf0F/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..396126dea85a87d4ca98f40250c02074f2bbeaff --- /dev/null +++ b/29AzT4oBgHgl3EQfuf0F/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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a/3tAzT4oBgHgl3EQfD_po/content/tmp_files/2301.00985v1.pdf.txt b/3tAzT4oBgHgl3EQfD_po/content/tmp_files/2301.00985v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..85cbb98c89e1e2e1eb71f0f16b853af593debbba --- /dev/null +++ b/3tAzT4oBgHgl3EQfD_po/content/tmp_files/2301.00985v1.pdf.txt @@ -0,0 +1,2547 @@ +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +1 +More is Better: A Database for Spontaneous +Micro-Expression with High Frame Rates +Sirui Zhao, Huaying Tang, Xinglong Mao, Shifeng Liu, Hanqing Tao, Hao Wang, Tong Xu, Member, IEEE, +and Enhong Chen, Senior Member, IEEE, +Abstract—As one of the most important psychic stress reactions, micro-expressions (MEs), are spontaneous and transient facial +expressions that can reveal the genuine emotions of human beings. Thus, recognizing MEs (MER) automatically is becoming +increasingly crucial in the field of affective computing, and provides essential technical support in lie detection, psychological analysis +and other areas. However, the lack of abundant ME data seriously restricts the development of cutting-edge data-driven MER models. +Despite the recent efforts of several spontaneous ME datasets to alleviate this problem, it is still a tiny amount of work. To solve the +problem of ME data hunger, we construct a dynamic spontaneous ME dataset with the largest current ME data scale, called +DFME (Dynamic Facial Micro-expressions), which includes 7,526 well-labeled ME videos induced by 671 participants and annotated +by more than 20 annotators throughout three years. Afterwards, we adopt four classical spatiotemporal feature learning models on +DFME to perform MER experiments to objectively verify the validity of DFME dataset. In addition, we explore different solutions to the +class imbalance and key-frame sequence sampling problems in dynamic MER respectively on DFME, so as to provide a valuable +reference for future research. The comprehensive experimental results show that our DFME dataset can facilitate the research of +automatic MER, and provide a new benchmark for MER. DFME will be published via https://mea-lab-421.github.io. +Index Terms—Emotion recognition, facial micro-expression, micro-expression recognition, datasets +! +1 +INTRODUCTION +F +ACIAL expression is essential for humans to transmit +emotional information, accounting for 55% of our daily +communication [1]. As a particular facial expression, micro- +expression (ME) usually refers to the spontaneous and +subtle facial movements that appear instantaneously when +an individual tries to hide or suppress real emotions un- +der pressure. The concept of ME was first proposed in +1966 [2]. Subsequently, Ekman et al. [3] discovered a ME +case in a video of a psychiatrist and depressed patient +conversation in 1969. Concretely, throughout the pleasant +conversation, when the psychiatrist asked the patient about +her plans, a distressed expression quickly flashed across the +patient’s face, which was called ME by Ekman. As MEs +can effectively reveal the genuine emotions of individuals, +recognizing MEs can provide essential technical support in +• +Sirui Zhao is with the School of Computer Science and Technology, +University of Science and Technology of China, Hefei, Anhui 230027, +China, and also with the School of Computer Science and Technology, +Southwest University of Science and Technology, Mianyang 621010, +China. +E-mail: sirui@mail.ustc.edu.cn +• +Huaying Tang, Hanqing Tao are with the School of Computer Science and +Technology, University of Science and Technology of China, Hefei, Anhui +230027, China. +E-mail: {iamthy, hqtao}@mail.ustc.edu.cn +• +Xinglong Mao, Shifeng Liu, Hao Wang, Tong Xu and Enhong Chen are +with School of Data Science, University of Science and Technology of +China, Hefei, Anhui 230027, China. +E-mail: {maoxl, lsf0619}@mail.ustc.edu.cn, +{wanghao3, tongxu, cheneh}@ustc.edu.cn +This work has been submitted to the IEEE for possible publication. Copyright +may be transferred without notice, after which this version may no longer be +accessible. +Sirui Zhao, Huaying Tang, Xinglong Mao and Shifeng Liu contributed +equally. Corresponding authors: Enhong Chen and Tong Xu. +Manuscript received December xx, xx; revised xx xx, xx. +lie detection, psychological healing, and public safety [4], +[5], [6], [7]. +In essence, ME is a kind of psychic stress reaction. Com- +pared with ordinary facial expression (also called macro- +expression, MaE), ME has the characteristics of short dura- +tion (less than 0.5s), partial movement, and low movement +intensity, so it is challenging to recognize MEs accurately. +Figure 1 illustrates the comparison between a ME and a +MaE with the same emotion category. It shows vividly that +the MaE is obvious enough to be distinguished easily by +a single image, while the ME is subtle and can only be +observed through an image sequence. +The early research on ME recognition (MER) was mainly +based on manual analysis in the field of psychology. How- +ever, the manual analysis relies on expert experience, which +is time-consuming and labor-intensive, and has low recog- +nition accuracy. Therefore, it is urgent to use computers’ +powerful perception and computing power for automatic +MER. In recent years, lots of efforts in the fields of com- +puter vision and affective computing have been devoted +to automatic MER. For example, in order to extract the +spatial-temporal MEs, Pfister et al. [8] introduced a local +binary pattern from three orthogonal planes (LBP-TOP) [9] +for MER. Liu et al. [10] proposed Mian Directional Mean Op- +tical Flow (MDMO). Wang et al. [11] proposed Transferring +Long-term Convolutional Nerual Network (TLCNN). Zhao +et al. [12] proposed a novel two-stage learning (i.e., prior +learning and target learning) method based on a siamese 3D +convolutional neural network for MER. However, due to the +lack of support for a large number of well-labeled ME data, +the recognition accuracy and robustness of these methods +are challenging to meet the needs of actual scenarios. There- +fore, it is urgent to build a large-scale ME dataset. +arXiv:2301.00985v1 [cs.CV] 3 Jan 2023 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +2 +··· +··· +··· +··· +onset +0 +apex +1.25 +offset +2.08 +second +(a) An example of MaE with ”Happiness” emotion. +··· +··· +··· +··· +onset +0 +apex +0.19 +offset +0.36 +second +(b) An example of ME with “Happiness” emotion. +Fig. 1: Examples of MaE and ME from the same person with a timeline in seconds, both belong to the ”Happiness” emotion +category. Noteworthy, the onset frame and the offset frame denote the start and end time of an expression respectively, +and the apex frame represents the moment when an expression changes most dramatically. White arrows on the face of +the apex frame indicate the general directions of facial movements, and the longer and thicker the arrows, the greater the +intensity of facial movements. +Over +the +past +decade, +although +researchers +have +published +several +spontaneous +ME +datasets, +such +as +SMIC [13], CASME II [14], SAMM [15], MMEW [16] and +CAS(ME)3 [17], these datasets have a small sample size, +which still cannot completely meet the need of MER models +for large-scale ME samples. In fact, building a large-scale +spontaneous ME dataset is full of challenges, mainly from +three aspects: First, it is difficult to induce MEs because they +are facial movements that are disclosed after an individual +attempts to suppress them. Second, it is difficult to label and +distinguish ME fragments because the movement of ME is +weak and fast, which is hard for the naked eye to perceive. +Third, due to the short duration of MEs, high-speed cameras +are often needed to collect them. However, the data collected +by high-speed cameras are redundant, so labeling ME clips +is extremely time-consuming and labor-intensive. +In order to solve the challenge of ME data shortage, +this paper constructs the current largest ME dataset called +DFME (Dynamic Facial Micro-expressions) to advance the +development of MER. Specifically, our DFME includes 7,526 +well-labeled ME videos induced by 671 participants and +annotated by more than 20 annotators throughout three con- +secutive years. Subsequently, four popular spatiotemporal +video feature learning models were reproduced on DFME +to perform MER so as to objectively verify the availability +of the dataset and provide a benchmark for subsequent +research. In addition, aiming at the class imbalance and +key-frame sequence sampling problems existing in MER, +we explored different solutions to DFME. In general, the +contributions of this paper could be summarized as follows: +• +This paper focuses on solving the problem of lacking +abundant spontaneous ME data and builds a new +ME dataset called DFME containing 7,526 ME videos +across multiple high frame rates (i.e., 200fps, 300fps, +500fps). To the best of our knowledge, DFME has the +largest ME sample size at present. +• +We reproduced four spatiotemporal feature learning +models to carry out MER tasks in DFME, objectively +verifying the reliability of data quality, and providing +a benchmark for subsequent MER studies. +• +We explored and analyzed different solutions to the +class imbalance and key-frame sequence sampling +problems in dynamic MER respectively on DFME, +so as to provide a reference for future research. +The rest of this paper is organized as follows. First, we +summarize currently existing ME datasets and review re- +lated work on MER in the next section. In section 3, we elab- +orate on the building details and statistical properties of our +DFME dataset. Then the comprehensive dataset evaluation +is developed and discussed in Section 4. Finally, research +conclusions and future work are addressed in Section 5. +2 +RELATED WORK +In this section, we first review the existing public sponta- +neous ME datasets related to MER. Then, we summarize +some representative MER studies based on deep learning +technologies. +2.1 +Micro-expression Datasets +The premise of obtaining an automatic MER algorithm with +excellent performance is to hold a dataset with sufficient ME +samples whose labels are credible and whose visual features +are distinguishable. As an emerging field of affective com- +puting, the number of ME datasets is still relatively limited. + +香JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +3 +TABLE 1: Statistical Information of Current Spontaneous ME Datasets +ME Datasets +Participants +Samples of MEs +Annotation Labels +Number +Gender +(Male/Female) +Age +Number +Frame Rate +Resolution +Emotion +FACS AU +HS +16 +164 +100 +640×480 +Pos (51) Neg (70) Sur (43) +SMIC +VIS +8 +10/6 +Range: 22-34 +Mean=28.1 +71 +25 +640×480 +Pos (28) Neg (23) Sur (20) +No +NIR +8 +71 +25 +640×480 +Pos (28) Neg (23) Sur (20) +CASME +35 +22/13 +Mean=22.03 +195 +60 +640×480 +1280×720 +Amu (5) Dis (88) Fear (2) +Con (3) Sad (6) Tense (28) +Sur (20) Rep (40) +Yes +CASME II +35 +/ +Mean=22.03 +247 +200 +640×480 +Hap (33) Dis (60) Sur (25) +Rep (27) Oth (102) +Yes +CAS(ME)2 +22 +9/13 +Range: 19-26 +Mean=22.59 +57 +30 +640×480 +Pos (8) Neg (21) Sur (9) +Oth (19) +Yes +SAMM +32 +16/16 +Range: 19-57 +Mean=33.24 +159 +200 +2040×1088 +Hap (24) Dis (8) Fear (7) +Ang (20) Sur (13) Sad (3) +Oth (84) +Yes +MEVIEW +16 +/ +/ +29 +30 +1280×720 +Hap (5) Dis (1) Fear (3) +Ang (1) Sur (8) Con(4) +Unc (7) +Yes +MMEW +36 +/ +Mean=22.35 +300 +90 +1920×1080 +Hap (36) Dis (72) Fear (16) +Ang (8) Sur (89) Sad (13) +Oth (66) +Yes +CAS(ME)3 +PART A +100 +50/50 +/ +943 +30 +1280×720 +Hap (64) Dis (281) Fear (93) +Ang (70) Sur (201) Sad (64) +Oth (170) +Yes +PART C +31 +9/22 +Mean=23.5 +166 +30 +1280×720 +Pos (16) Neg(99) Sur (30) +Oth (20) +4DME +DI4D +65 +38/27 +Range: 22-57 +Mean=27.8 +267 +60 +1200×1600 +Pos (34) Neg (127) Sur (30) +Rep (6) PosSur (13) NegSur (8) +RepSur (3) PosRep(8) +NegRep(7) Oth(31) +Yes +Grayscale +267 +60 +640×480 +RGB +267 +30 +640×480 +Depth +267 +30 +640×480 +PART A +72 +31/41 +1118 +500 +1024×768 +Hap (111) Dis (321) Fear (143) +Ang (97) Con (77) Sur (187) +Sad (142) Oth (40) +DFME +PART B +92 +61/31 +Range: 17-40 +Mean=22.43 +969 +300 +1024×768 +Hap (78) Dis (406) Fear (115) +Ang (56) Con (45) Sur (143) +Sad (119) Oth (7) +Yes +PART C +492 +282/210 +5439 +200 +1024×768 +Hap (803) Dis (1801) Fear (634) +Ang (466) Con (279) Sur (878) +Sad (374) Oth (204) +1 Some datasets contain not only MEs but also MaEs, as well as long video clips for the detection task. But here we only show the information +about ME data. Note that all statistical data are from the corresponding original paper or downloaded datasets. +2 The number of participants was counted based on the data given in the corresponding original paper, but some participants were not +successfully induced to make MEs. +3 Pos: Positive; Neg: Negative; Sur: Surprise; Amu: Amusement; Hap: Happiness; Dis: Disgust; Rep: Repression; Ang: Anger; Sad: Sadness; +Con: Contempt; Unc: Unclear; Oth: Others; PosSur: Positively surprise; NegSur: Negatively surprise; RepSur: Repressively surprise; PosRep: +Positively repression; NegRep: Negatively repression. +Nevertheless, since more and more researchers have begun +to pay attention to ME analysis, some high-quality datasets +are gradually springing up. Table 1 clearly summarizes the +characteristics of these datasets. +As the two earliest proposed ME datasets, samples in +the USF-HD [18] and Polikovsky [19] datasets are all posed +MEs. The participants were first required to watch video +clips containing ME samples and then posed them by imi- +tation. However, naturally generated MEs strongly correlate +with emotions, while the posed ones are deliberately dis- +played and have nothing to do with the current emotional +state of the participants. Consequently, these two datasets +are rarely used by researchers for ME analysis. +The subsequent researchers proposed to induce spon- +taneous MEs with the neutralization paradigm. Under +this paradigm, several strong emotional stimuli were used +to elicit expressions, during which participants were in- +structed to keep a neutral face as much as possible, and +a certain degree of high-pressure mechanism was given +to them. Datasets adopting the neutralization paradigm +include SMIC [13], CASME [20], CASME II [14], CAS(ME)2 +[21], SAMM [15], MMEW [16], and 4DME [22], which will + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +4 +be introduced in turn below. +SMIC dataset [13] is the first published spontaneous ME +dataset, which consists of three parts: HS, VIS, and NIR. +The HS part includes 164 ME samples from 16 participants, +recorded by a high-speed camera with a frame rate of +100 frames per second (fps) and a resolution of 640×480. +Both the VIS and NIR parts contain 71 ME samples from +8 individuals, while the former part was recorded using a +standard visual camera and the latter using a near-infrared +camera. Two annotators classified each ME into three emo- +tion categories (positive, negative, and surprise) based on the +participants’ self-reports about the elicitation videos. Facial +action units (AUs) were not annotated in SMIC. +CASME series datasets are released by the Institute of +Psychology, Chinese Academy of Sciences. As the earliest +dataset in this series, CASME [20] contains a total of 195 +ME samples from 19 participants with a frame rate of +60fps. Two annotators labeled the facial AUs, together with +the corresponding onset, apex, and offset frames of each +ME sample frame by frame. According to the facial AUs, +participants’ self-reports, and the relevant video content, +MEs were divided into eight emotion categories: amuse- +ment, sadness, disgust, surprise, contempt, fear, repression, and +tense. CASME II [14] is an advanced version of CASME. +First, the number of ME samples in CASME II has been +expanded to 247 samples from 26 participants. Besides, +CASME II provides a higher frame rate of 200fps and facial +area resolution of 280×340 to capture more subtle changes +in expressions. Five emotion categories were labeled in +CASME II: happiness, disgust, surprise, repression, and others. +The CAS(ME)2 dataset [21] embodies two parts, both of +which were collected at 30fps and 640×480 pixels. Different +from all the other datasets above, there are 87 long video +clips containing both MaEs and MEs in the first part of +CAS(ME)2, which can be used to promote the research of +ME detection. The other part consists of 300 MaEs and 57 +MEs, which were labeled with four emotion tags, including +positive, negative, surprise, and others. +SAMM dataset [15] has the highest resolution of all +published spontaneous ME datasets, which includes 159 ME +samples generated by 32 participants, with a frame rate of +200fps and a resolution of 2040×1088. To achieve a better +elicitation effect, before the formal start of the collection, +participants were asked to fill in a scale, and then a series +of stimulus videos were customized for each participant +according to the scale. This is how SAMM differs from +other datasets. SAMM contains seven emotion categories: +happiness, disgust, surprise, fear, anger, sadness, and others. +Three coders annotated the AUs and key-frames in detail +for each ME sample. +MMEW dataset [16] consists of 300 ME and 900 MaE +samples from 36 participants, which were collected with 90 +fps and 1920×1080 resolution. Each expression sample is +marked with seven emotion labels (the same as SAMM), +AUs, and three key-frames. Compared with the previous +datasets, MMEW is more conducive to the models using +the MaE samples under the same parameter setting and +elicitated environment to assist in learning ME features. +To comprehensively capture the movement informa- +tion of ME in all directions as much as possible, 4DME +dataset [22] has made significant innovations in the record- +ing method. Each ME sample in this dataset has multi- +modality video data, including 4D facial data reconstructed +by 3D facial meshes sequences, traditional 2D frontal facial +grayscale, RGB and depth videos. 4DME contains 267 MEs +and 123 MaEs from 41 participants, thus 1,068 ME videos +of four forms and 492 MaE videos in total. In addition, +five emotion labels (positive, negative, surprise, repression, and +others) were annotated based on facial AUs only, noting that +each sample may have multiple emotion labels (up to two). +Unlike datasets with the neutralization paradigm, the +MEVIEW dataset [23] consists of video clips of two real +high-pressure scenes downloaded from the Internet. There +are 29 ME samples in total, with a frame rate of 30fps +and a resolution of 1280×720, divided into seven emotion +categories (the same as SAMM) with manual annotation. +Although these samples are from actual life scenarios and +have high ecological validity, there are many uncontrollable +factors, such as frequent camera shot switching, which re- +sults in fewer segments containing full human faces. +The CAS(ME)3 dataset [17] adopted the mock crime +paradigm to elicit MEs with high ecological validity. How- +ever, unlike MEVIEW, the collection was still controlled +in the laboratory environment, yielding 166 MEs and 347 +MaEs. CAS(ME)3 also contains two other parts: one consists +of 943 MEs and 3,143 MaEs collected using the neutraliza- +tion paradigm, respectively marked with AUs, key-frames, +and seven emotion labels (the same as SAMM) for each +sample; the other part contains 1,508 unlabeled long video +clips, which can be used for the self-supervised learning task +of ME detection and recognition. This dataset was collected +at a frame rate of 30fps with a resolution of 1280×720. +Despite more and more datasets striving to record the +movement characteristics of MEs more detailedly and com- +prehensively through various methods, these datasets are +still small-scale datasets. In automatic ME analysis, mod- +els based on deep learning have become mainstream by +practice. However, due to the insufficient sample size, the +complexity of the model can easily lead to overfitting in +the training process. Though we can alleviate this problem +by using data augmentation to increase the number of +samples, many uncontrollable noises might be introduced. +Some work has proposed using composite datasets to train +the model, but different datasets have different parameter +settings, and thus such a simple fusion is not reasonable. +In addition, due to the short duration and low intensity +of MEs, a higher frame rate may contribute to capturing +more details. Nevertheless, the highest frame rate of all +above datasets is only 200fps, and most are less than 100fps. +Therefore, it is necessary to establish a larger-scale ME +dataset with a higher frame rate. +2.2 +Micro-expression Recognition Approaches +In the past decade, MER has attracted more and more +attention from scholars in affective computing and com- +puter vision. The first attempt at automatic, spontaneous +MER dates back to 2011, Pfister et al. [8] utilized a local +binary pattern from three orthogonal planes (LBP-TOP) to +explore MER on the first spontaneous ME dataset SMIC. +Since then, more and more efforts have been devoted to +automatic MER. In general, the current MER methods can be + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +5 +roughly divided into hand-crafted feature based and deep +learning based methods. Typical hand-crafted ME features +include LBP-TOP [9], HOOF [24], 3DHOG [19], and their +variants [25], [26], [27]. However, the hand-crafted feature +based methods heavily rely on complex expert knowledge, +and the extracted ME features have limited discrimination. +Current MER methods mainly use deep neural networks for +high-level expression feature learning and emotion classifi- +cation, and focus on solving the challenges that ME is subtle +and ME data shortage for model training. Further, according +to whether the MER model considers the ME temporal +information or not, we divide the current deep learning +based MER methods into single frame based MER and video +sequence based MER. In the following subsections, we will +categorize and summarize these two types of MER methods. +2.2.1 +Single frame based MER methods. +The single frame based MER method usually only uses the +highest intensity frame, i.e., the apex frame with RGB or +optical-flow format in the ME video, as the input of neural +networks to learn the spatial ME features. After considering +the challenge of lacking sufficient ME samples, Peng et +al. [28] first selected ResNet-10 [29] pre-trained on a large- +scale image dataset as the backbone and then continued to +fine-tune the classification network on large MaE samples +for MER using apex frames. Encouragingly, the recognition +accuracy exceeds the hand-crafted methods based on LBP- +TOP, HOOF, and 3DHOG. Inspired by the success of capsule +models on image recognition, Quang et al. [30] proposed +a CapsuleNet for MER using only apex frames. Recently, +Wang et al. [31] proposed an expression-identity disentangle +network for MER by leveraging MaE databases as guidance. +Li et al. [32] first spotted the apex frame by estimating pixel- +level change rates in the frequency domain, then proposed a +joint feature learning architecture coupling local and global +information from the detected apex frames to recognize +MEs. At the same time, Liong et al. [33] explored the +effectiveness and superiority of using the optical flow of +the apex frame in ME video. Inspired by this work, Liu et +al. [34] first calculated the optical-flow image of the apex +frame to the onset frame in the ME clips and then used +the pre-trained ResNet-18 network to encode the optical- +flow image for MER. In particular, they introduced domain +adversarial training strategies to address the challenge of +lacking large-scale ME data for training and won first place +for MEGC2019. Furthermore, Zhou et al. [35] proposed +a novel Feature Refinement (FR) with expression-specific +feature learning and fusion for MER based on optical-flow +information of apex frames. Gong et al. [36] proposed a +meta-learning-based multi-model fusion network for MER. +Overall, the single frame based MER investigations are +conducted on apex frames of ME videos without temporal +information, which can reduce the complexity of the used +deep neural networks. In addition, the single frame based +MER method has the advantage of finding large-scale im- +ages for transfer learning to effectively solve the problem of +model overfitting with insufficient ME data. Nevertheless, +the single frame based MER discards the temporal informa- +tion in the ME video, which contains rich ME clues and is +an important feature that distinguishes MEs from MaEs. +2.2.2 +Video sequence based MER methods. +Unlike the single frame based MER, video sequence based +MER can learn spatiotemporal ME feature from the whole +ME video or its sub-sequence. Thus, the video sequence +based MER is preferred to the single frame based MER +for providing details. After fully considering the important +expression states in the ME video, Kim et al. [37] first +used CNN to encode the spatial feature of each expression +state (i.e., onset, onset to apex transition, apex, apex to +offset transition and offset), then adopted LSTM to learn the +temporal feature based on the encoded spatial ME feature. +Wang et al. [11] proposed Transferring Long-term Convo- +lutional Nerual Network (TLCNN) to solve the learning of +spatial-temporal ME feature under small sample ME data. +The TLCNN is also based on the CNN-LSTM structure and +transfers knowledge from large-scale expression data and +single frames of ME video clips. Khor et al. [38] proposed an +Enriched Long-term Recurrent Convolutional Network (EL- +RCN) that makes spatial and temporal enrichment by stack- +ing different input data and features. Unlike the CNN-LSTM +architecture, 3D convolution network (3DCNN) [39] can +simultaneously learn the spatial and temporal ME features. +Based on 3DCNN, Peng et al. [40] proposed a Dual Tempo- +ral Scale Convolutional Neural Network (DTSCNN), which +uses the optical-flow sequences of ME videos as model +input to obtain high-level ME features and can adapt to a +different frame rate of ME video clips. Wang et al. [41] pro- +posed a MER framework based on Eulerian motion based +3DCNN (EM-CED), which uses the pre-extracted Eulerian +motion feature maps as input and with a global attention +module to encode rich spatiotemporal information. Xia et +al. [42] proposed a deep recurrent convolutional networks +based MER approach, which modeled the spatiotemporal +ME deformations in views of facial appearance and geom- +etry separately. To solve the challenge of extracting high- +level ME features from the training model lacking sufficient +and class-balanced ME samples, Zhao et al. [12] extracted +the ME optical-flow sequence to express the original ME +video and proposed a novel two-stage learning (i.e., prior +learning and target learning) method based on a siamese +3D convolutional neural network for MER. Sun et al. [43] +proposed a knowledge transfer technique that distills and +transfers knowledge from action units for MER based on +crucial temporal sequences, where knowledge from a pre- +trained deep teacher neural network is distilled and trans- +ferred to a shallow student neural network. Zhao et al. [44] +proposed a deep prototypical learning framework on RGB +key-frame sequences, namely ME-PLAN, based on a 3D +residual prototypical network and a local-wise attention +module for MER. Recently, with the advancement of deep +learning technology, some excellent neural networks, such +as GCN [45] and transformers, have also been used for MER. +Although video sequence based MER makes full use +of spatial-temporal information of ME, the corresponding +model has higher structural complexity and faces seri- +ous over-fitting problems on the current small-scale ME +datasets. Therefore, building a large-scale ME dataset is still +the primary task of developing an automatic MER system, +which plays a pivotal role. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +6 + LED lights with +reflector umbrellas +Participant +Participant’s monitor +(playing elicitation +videos) +High-speed camera (1024×768, freely configurable frame rate ) +4T-sized high-speed +acquisition memory +Collector +Collector’s monitor +for recoding MEs +Collector’s monitor +for playing videos +10 Gigabit optical fiber +transmission line +Fig. 2: Experimental environment for eliciting MEs +3 +DFME +As the old saying goes, ”One can’t make bricks without +straw”. Similarly, it is difficult to design an automatic MER +model with high recognition rate and reliability without +sufficient training and testing samples of ME. However, due +to the short-duration, low-intensity, and local-movement +characteristics of ME, it is extremely challenging to construct +large-scale ME datasets. To solve the problem of ME data +hunger, we construct a dataset of spontaneous ME with the +largest sample size at present, called DFME. In the following +subsections, we will elaborate on the building details and +statistical properties of our DFME dataset. +3.1 +Participant and Equipment +In our DFME, 671 participants were recruited (381 males +and 290 females), mainly for college students and teaching +staff. Participants were age-distributed between 17 and 40 +years, with a mean age of 22.43 years (standard deviation = +2.54), and all from China. Before starting the formal exper- +iment, the participants were informed about the purpose, +experimental procedure, possible benefits and risks of our +research. On confirming their voluntary participation in the +experiment, participants would sign an informed consent +form and choose whether to allow their facial images and +videos used for the academic paper. +Considering the low intensity and short duration of MEs, +the recording process is easily disturbed by other factors, so +it is carried out in a well-controlled laboratory environment, +as shown in Fig. 2. In this environment, we set up three LED +lights with reflector umbrellas to ensure a bright and stable +light source on the participants’ faces during experiments. +In addition, we used a self-developed high-speed camera +(1024×768, freely configurable frame rates) to capture MEs, +and used a 10 Gigabit optical fiber transmission line to +connect the camera with a 4T-sized high-speed acquisition +memory to store the collected ME video clips in real-time. +3.2 +Elicitation Material and Procedure +At present, there are three generations of ME-eliciting +paradigms. Although the third generation has the highest +TABLE 2: Video clips for eliciting MEs +Video ID +During Time +Emotion Category +Mean Score(0-5) +02sa +3’44” +Sadness +4 +03sa +4’18” +Sadness +3.36 +06c +2’01” +Contempt +2.83 +07a +1’26” +Anger +3.49 +08su +1’26” +Surprise +2.16 +09f +2’22” +Fear +3.72 +10a +2’58” +Anger +4.33 +11d +1’24” +Disgust +3.95 +13f +2’14” +Fear +3.36 +14d +1’22” +Disgust +3.23 +17h +1’17” +Happiness +2.81 +18h +1’58” +Happiness +3.08 +20d +0’46” +Disgust +2.87 +21c +1’44” +Contempt +2.11 +23sa +1’44” +Sadness +3.25 +ecological validity, it is inevitable to interact and have +conversations with the participants when simulating the +natural scenes. These irrelevant body and mouth move- +ments caused by speaking are also a kind of noise for MEs. +Therefore, we still use the neutralization paradigm to elicit +MEs to avoid noise as much as possible and focus more +on the movement characteristics of MEs and facilitate the +operation, control, and implementation. The specific details +of the elicitation process will be introduced below. +The effectiveness of elicitation materials determines the +quantity and quality of MEs, so selecting the materials with +high emotional valence is very crucial [14]. The stimuli we +used were all video clips from the Internet, ranging in length +from 46 seconds to 258 seconds. In order to find more +effective stimulus materials, we recruited 50 volunteers to +evaluate 30 video clips collected previously. The evalua- +tion process was as follows: after watching each video, +volunteers were asked to choose only one emotion from +happiness, contempt, disgust, sadness, fear, surprise and +anger as the main emotion evoked by this video, and score +the stimulus level on a scale of 1 to 5, corresponding to +the intensity from weakest to strongest. Finally, we took the +emotion selected by more than half of the volunteers as the +emotional class of each video, and by ranking the average + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +7 +stimulus intensity values, we obtained the optimal 15 video +clips as elicitation materials adopted in our experiment. +Specific statistical details are shown in Table 2. +The collection took place in a configured laboratory +environment. Before start, each participant was taken to a +specific seat. By adjusting the height of the seat, the focal +length of the camera and the brightness of the LED lamps, +we ensured that the participant’s face appeared utterly, +clearly, and brightly in the centre of the screen. Then the +monitor in front of the participant would play ten randomly +selected elicitation videos covering all seven basic emotional +types that had been previously verified effective in turn. +While watching videos, participants were required to keep +a neutral face as far as possible and control the occurrence of +their facial expressions. If they failed and repeatedly showed +obvious expressions, they would have to complete an ex- +traordinarily long and boring questionnaire as punishment. +In addition, they were asked to keep their sitting posture +upright, without excessive head movements, and devote +their full attention to the video played. After watching each +video, participants would have a period of rest to ease their +emotions. During this procedure, they were also asked to +fill in an affective grade scale according to the emotional +experience generated just now, and form a self-report in- +cluding the timestamp where the expression occurred, emo- +tion category and intensity based on the video sequences +recorded by the high-speed camera, which would help the +subsequent annotators understand their MEs. Due to the +existence of cognitive differences, the emotional orientation +of the elicitation materials and the internal emotional expe- +rience of participants are sometimes not exactly consistent. +What’s more, external expressions of the same emotion are +also diverse on account of individual differences. Therefore, +it is worth noting that requiring participants to clarify their +true inner emotions when expressions appear in their self- +reports is necessary. +3.3 +ME Annotation +Building the DFME dataset required a two-stage annotation: +the sample selection stage as well as the coding and cate- +gories labeling stage. We clipped short fragments containing +valid expression samples from the collected long video +sequences in the first stage. The second stage included three +rounds of fine-grained annotation, through which we con- +firmed all MEs and labeled their key-frames, facial muscle +action units (AUs), and emotion categories. Furthermore, we +performed annotation agreement test to verify the reliability +of emotion labels. +3.3.1 +Sample Selection +In the sample selection stage, by taking a manual segmen- +tation roughly, the video sequences collected containing +participants’ facial information were segmented into sev- +eral shorter video fragments containing a single or more +MaEs or MEs. Using the self-developed video annotation +software, an experienced annotator checked through the +collected original video sequences frame by frame to locate +the fragments of facial muscle movements. With the guid- +ance of the self-reports from participants, the annotator was +able to effectively distinguish whether the facial movements +were expressions definitely related to emotion, or interfer- +ence data unrelated to emotion (such as violent blinking +caused by dry eyelids, habitual mouth opening, etc.), and +the former was retained while the latter was abandoned. +Besides, we also kept some fragments with blinking or eye +movements if they contained MaE or ME data. +3.3.2 +Coding and Categories Labeling +After the previous sample selection stage, three rounds of +fine-grained annotation were adopted successively in this +stage to determine the MEs together with their three key- +frames (i.e., onset frame, apex frame and offset frame), facial +muscle action unit (AU) labels and emotion category labels. +The apex frame is the frame corresponding to the mo- +ment when facial expression changes most dramatically. In +the first round of the fine-grained annotation, five annota- +tors independently marked out the onset, apex, and offset +frame of each expression clip, and the median value of their +annotation results was determined as the final result of the +three key-frames. Then we filtered the expressions whose +duration from onset to offset frame was less than 500ms or +from onset to apex frame was less than 250ms as the ME +samples, and those out of the time limit were considered as +the samples of MaEs. For instance, MEs collected at a frame +rate of 500fps should meet either foffset − fonset + 1 ≤ 250 +or fapex−fonset+1 ≤ 125, where fk represents the moment +index corresponding to the key-frame k. +In the second round of fine-grained annotation, we +mainly annotated the AUs that occurred in MEs using the +Facial Action Coding System (FACS) [46]. There may exist +one single AU (such as AU4) or a combination of more +different AUs (for example, AU6+AU12) in a ME. When +multiple categories of AUs appear, some obscure ones are +easily overlooked. To enhance the reliability and integrity of +the AU labels, two experienced annotators independently +labeled the AUs for all the MEs identified previously. Ac- +cording to the actual induction of the participants during +the experiments, and also referring to the AUs mainly in- +volved in the previously published ME datasets, we totally +included 24 different categories of AUs for annotation. Of +these AUs, six categories appear in the upper face, 13 in +the lower face, and the other five belong to miscellaneous +actions. Table 3 lists the specific AU numbers and their +corresponding face actions. Since the manually annotated +AU intensity is highly subjective, to avoid this shortcoming, +annotators merely indicated whether each AU appeared +during the annotation rather than defining the intensity of +its occurrence. +After labeling the AUs, the two annotators determined +the final AU label through crosscheck and discussion. The +reliability between the two annotators was 0.83, which was +calculated as +R = 2 × AU(A1) ∩ AU(A2) +AllAU +(1) +where AU(A1) ∩ AU(A2) means the number of AUs both +annotators agreed, and AllAU is the total number of AUs in +a ME labeled out by the two annotators. +In the third round of fine-grained labeling, we performed +the emotion labeling of MEs taking eight categories into + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +8 +TABLE 3: Key AUs Included in DFME +Upper Face Action Units +Lower Face Action Units +Miscellaneous Actions +AU1 +Inner Brow Raiser +AU9 +Nose Wrinkler +AU18 +Lip Pucker +AU31 +Jaw Clencher +AU2 +Outer Brow Raiser +AU10 +Upper Lip Raiser +AU20 +Lip Stretcher +AU38 +Nostril Dilator +AU4 +Brow Lowerer +AU12 +Lip Corner Puller +AU23 +Lip Tightener +AU39 +Nostril Compressor +AU5 +Upper Lid Raiser +AU14 +Dimpler +AU24 +Lip Presser +M57 +Head Forward +AU6 +Cheek Raiser +AU15 +Lip Corner Depressor +AU25 +Lips Part +M58 +Head Back +AU7 +Lid Tightener +AU16 +Lower Lip Depressor +AU28 +Lip Suck +AU17 +Chin Raiser +(a) Anger +(AU4+AU5) +(b) Contempt +(Left-AU6+ +Left-AU12) +(c) Disgust +(AU4+AU7+ +AU10) +(d) Fear +(AU1+AU4+ +AU7+AU20) +(e) Happiness +(AU6+AU12) +(f) Sadness +(AU17) +(g) Surprise +(AU1+AU2+ +AU5) +Fig. 3: Representative ME Samples of Seven Basic Emotion Categories in DFME +account: anger, contempt, disgust, fear, happiness, sadness, sur- +prise, and others. ’Others’ represents MEs that are difficult +to divide into the former seven prototypical emotion cat- +egories. Seven annotators independently gave the emotion +labels of all MEs, taking the emotion category that more +than half agreed with as the final label. +In previous spontaneous ME datasets, the reference basis +of emotion labeling was not precisely the same. In some +datasets, as represented by SMIC, emotion labels were de- +termined based on self-reports provided by participants. +Some other studies believed that seeing is believing, so their +annotation was based on the correspondence between AUs +and emotions. However, on the one hand, unlike MaEs, +only part of the AUs can appear simultaneously in MEs +due to their low intensity, and some AUs are shared by +different emotion categories, which may lead to category +confusion. On the other hand, we should not ignore the +differences in self-emotional cognition of different partici- +pants, which means that the self-reports given for the whole +piece of elicitation materials may be rough and inaccurate. +Therefore, in DFME, the emotion labels were determined +through a comprehensive analysis of facial AUs, self-reports +of participants, and elicitation material contents, which is +consistent with the method adopted by the CASME series. +It is worth mentioning that we obtained the participants’ +fine-grained self-reports in the data collection process, and +this is also the information that we recommend as a priority +for reference when determining emotion labels. We matched +the corresponding timestamps of MEs and elicitation ma- +terials through playback, enabling participants to report +their emotions for each time of successful ME induction, +which significantly improved the confidence of self-reports +in emotion labeling. Fig.3 shows some representative ME +samples of seven basic emotion categories in DFME. +3.3.3 +Annotation Agreement +Having reliable emotion categories of MEs is of vital sig- +nificance for a dataset. In this section, we utilized Fleiss’s +Kappa test [47] to evaluate the quality of our emotion +annotation encouraged by work [48]. Fleiss’s Kappa is a +measure of the agreement among three or more annotators, +testing the consistency of annotation results. Therefore, we +consider Fleiss’s Kappa as an excellent indicator to evaluate +the reliability of emotion annotation. +In DFME, seven annotators independently labeled each +ME sample based on facial AUs, an accurate self-report, and +the corresponding elicitation material content. The samples +were divided into eight emotion categories: {1: anger, 2: +contempt, 3: disgust, 4: fear, 5: happiness, 6: sadness, 7: +surprise, 8: others}. At this time, let n = 7 represent the +total number of annotation personnel, N indicate the total +number of ME video clips, K = 8 represent the number +of emotion categories. nij is the number of annotators who +assigned the i-th ME video clip to the j-th category, so we +can calculate pj, the proportion of all assignments which +were to the j-th emotion: +pj = +1 +N × n +N +� +i=1 +nij, +(2) +K +� +j=1 +pj = 1. +(3) +Then, the extent of agreement among the n annotators +for the i-th ME video clip indicated by Pi is calculated. In +other words, it can be indexed by the proportion of pairs +agreeing in their evaluation of the i-th ME out of all the +n(n − 1) possible pairs of agreement: +Pi = +1 +n × (n − 1)[( +K +� +j=1 +n2 +ij) − n], +(4) +The mean of Pi is therefore: +P = 1 +N +N +� +i=1 +Pi, +(5) + +二JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +9 +TABLE 4: AUs of High Occurrence in MEs of Seven Basic Emotion Categories +Anger +Contempt +Disgust +Fear +Happiness +Sadness +Surprise +AU +pct(%)1 +AU +pct(%) +AU +pct(%) +AU +pct(%) +AU +pct(%) +AU +pct(%) +AU +pct(%) +AU4 +72.5 +L/R-AU122 +78.7 +AU4 +73.6 +AU4 +54.1 +AU12 +79.8 +AU4 +42.2 +AU1 +65.6 +AU7 +29.1 +AU6 +19.2 +AU7 +40.4 +AU7 +35.3 +AU6 +61.6 +AU14 +26.1 +AU5 +60.2 +AU24 +16.3 +L/R-AU10 +10.6 +AU10 +11.8 +AU5 +16.2 +AU24 +12.1 +AU24 +19.2 +AU2 +60.0 +AU5 +7.6 +AU7 +7.8 +AU24 +8.4 +AU24 +14.5 +L/R-AU12 +10.1 +AU7 +16.5 +L/R-AU2 +25.6 +AU23 +5.6 +L/R-AU2 +5.7 +AU14 +6.7 +AU1 +11.1 +AU10 +6.2 +AU17 +10.8 +L/R-AU1 +17.8 +AU14 +5.6 +AU14 +5.7 +AU14 +8.8 +AU15 +6.9 +L/R-AU5 +10.7 +AU10 +5.2 +AU17 +6.0 +AU23 +5.1 +AU17 +4.8 +AU10 +4.8 +AU1 +4.8 +1 percentage(pct): the statistical range is all MEs from the first 300 participants. +2 L/R means the Left/Right half part of an AU. +And we also have Pe: +Pe = +K +� +j=1 +p2 +j, +(6) +Finally, we can calculate κ by: +κ = P − Pe +1 − Pe +. +(7) +Thus, we obtained κ = 0.72 through performing Fleiss’s +Kappa test in DFME. According to Table 5, we know that all +of our emotion annotators achieve substantial agreement, +meaning that our emotion labels are quite reliable. +TABLE 5: Interpretation of κ for Fleiss’Kappa Test +κ +Interpretation +≤ 0 +Poor agreement +0.01-0.20 +Slight agreement +0.21-0.40 +Fair agreement +0.41-0.60 +Moderate agreement +0.61-0.80 +Substantial agreement +0.81-1.00 +Almost perfect agreement +3.4 +Statistical Properties of DFME +The DFME dataset consists of three parts: PART A, PART +B, and PART C. The only difference between these three +parts is the frame rate setting of the high-speed camera in +the experiment. In PART A, all 1,118 ME samples from 72 +participants have a frame rate of 500fps. The frame rate of +PART B is 300fps with 969 ME samples from 92 participants. +PART C has the most data size with 5,439 ME samples +from 492 participants, whose frame rate is 200fps. Although +we recruited a total of 671 participants, 15 of them had +strong control over their facial expressions, from whom we +could not collect any ME sample. Therefore, the final DFME +dataset contains 7,526 ME samples from 656 participants, +and we gave each sample an emotion category label as well +as AU labels annotated according to FACS. Fig.4 describes +the distribution of ME samples detailedly. +Given that we have collected the fine-grained self- +reports and the AU labels with considerable reliability, this +is conducive to finding the emotion-AU correspondence +rule in MEs. Therefore, we counted the ratio of high- +occurrence AUs in each emotion (Table 4), which reflects the +existence preference of AU in MEs with different emotions, +not affected by the emotional category imbalance problem in +the dataset. We also matched the emotion and AU combina- +tions according to the statistical results, and the conclusions +are shown as Table 6. +TABLE 6: Matching Emotion and AU Combinations in MEs +Emotion Categories +AU Combinations +Anger +AU4+AU5, AU23 +Contempt +L/R-AU12, AU6+L/R-AU12 +Disgust +AU4+AU7+AU10, AU14 +Fear +AU14+AU24, AU1+AU4, AU4+AU5 +Happiness +AU6+AU12, AU12 +Sadness +AU14, AU17, AU15, AU14+AU24 +Surprise +AU1+AU2+AU5, AU1+AU2, AU5 +Shared1 +AU4, AU4+AU7, AU7, AU24 +1 Shared: the AU combinations commonly appearing in +Anger, Disgust, Fear and Sadness with high frequency. +Based on the statistical results presented in Table 4, we +have some findings to discuss: +• +In MaEs, AU9 (nose wrinkler) is highly associated +with disgust, and AU20 (lip stretcher) is related to +fear. These two AUs frequently appear in MaEs but +are not easily induced in MEs. We ought not to +conclude that these AUs’ association with their corre- +sponding emotions no longer exists in MEs. Instead, +when participants tried to restrain their emotions, +it was easier for them to control the movement of +certain facial muscles such as AU9 and AU20 rather +than others. +• +AU4 (brow lowerer), AU7 (lid tightener), and AU24 +(lip presser) simultaneously occur at high frequency +in different negative emotions (disgust, anger, fear, +sadness, etc.). Without the assistance of participants’ +fine-grained self-reports, it is definitely challenging +to distinguish MEs of negative emotions merely rely- +ing on these common AUs, which is also one of the +reasons why some models excessively confuse the +disgust MEs with those of other negative emotions in +the seven-classification automatic MER task. +• +In the positive emotion (i.e., happiness), some AUs +related to negative emotions can occur together with +AU6 or AU12, specifically, including AU10 (associ- +ated with disgust), AU24 (associated with negative +emotions), and Left/Right-AU12 (associated with +contempt). The appearance of these extra AUs is a + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +10 +Disgust +Surprise +Happiness +Fear +Sadness +Anger +Contempt +Others +PART A +321 +187 +111 +143 +142 +97 +77 +40 +PART B +406 +143 +78 +115 +119 +56 +45 +7 +PART C +1801 +878 +803 +634 +374 +466 +279 +204 +Combined +2528 +1208 +992 +892 +635 +619 +401 +251 +2528 +1208 +992 +892 +635 +619 +401 +251 +0 +500 +1000 +1500 +2000 +2500 +Positive +Surprise +Negative +Others +Fig. 4: Distribution of ME Samples in DFME. Each column represents the total sample number of an emotion category, and +the three pieces colored from light to deep show the proportion of samples in PART A, PART B, and PART C, respectively. +sign of participants trying to suppress their positive +feelings, hide their smiles and twist their expressions. +4 +DATASET EVALUATION +In this section, we conducted comprehensive experiments +to verify the effectiveness of our DFME dataset for auto- +matic MER task based on influential spatiotemporal feature +learning models. In addition, we specifically analyzed the +class imbalance problem in ME datasets, and explored two +kinds of strategies to solve the class imbalance problem +in our DFME. Furthermore, we explored the influence of +different sampling strategies of ME key-frame sequence on +MER. These experiments can provide reference for future +MER research using DFME dataset. +4.1 +Evaluation Dataset +The DFME dataset is described in detail in Section 3. For the +subsequent MER verification, we combined 7, 275 samples +with clear emotion labels in PART A, B and C of DFME +as our experimental dataset. The emotion labels include +disgust, surprise, happiness, fear, sadness, anger and contempt. +4.2 +Data Preprocessing +In facial expression recognition, many variables, such as +backgrounds, head poses and unequal video lengths, can +affect the final recognition results. Therefore, before formally +conducting automatic MER experiments, we need to prepro- +cess all ME videos in the following steps to minimize the +influence of irrelevant variables. +4.2.1 +Face Alignment +To eliminate the differences in pose and angle among all ME +samples, we need to perform face alignment. In this step, +we took the following operations for each ME sample. We +first selected a frontal face image as a reference and adopted +Style Aggregated Network (SAN) [49] to extract its facial +landmarks. Afterwards, we used Procrustes analysis [50] to +compute an affine transformation based on landmarks of +the onset frame and landmarks of the reference image. The +reason why we did not use landmarks of all frames in the +ME video is to avoid errors introduced by the calculation of +landmarks and transformations having a significant impact +on real MEs. Finally, the transformation was operated for +each frame to align the faces. Besides, some landmarks are +located in regions where MEs may appear, which may not +be stable enough for alignment. Thus, we excluded such +landmarks when performing the alignment. +4.2.2 +Face Cropping +Since the movement of MEs is mainly in the facial area, +face cropping is also a necessary step to eliminate the bias +caused by different backgrounds. After face alignment, we +chose RetinaFace [51] to crop the faces. For reasons similar to +face alignment, face cropping was based on the onset frame +instead of each frame of a sample. +4.2.3 +ME key-frame sequence sampling +Different ME videos have different lengths, while deep +learning models usually require a fixed input size, which +is shorter than ME sample lengths. Before inputting into +the model, we need to normalize the temporal length of all +ME videos. In general, video classification models usually +adopt the uniform sampling method to unify the video +length. However, this processing strategy is coarse-grained +for recognizing ME with local and subtle movements. Fol- +lowing previous studies [12], [44] and to be compatible with +popular video classification models, this work extracts 16 +key-frames from each ME video based on the annotated +three ME key-frames (i.e., onset frame, apex frame, and +offset frame) and temporal adaptive sampling strategy [44]. +4.3 +Evaluation Protocols and Metrics +Due to the small sample size of previous datasets such as +CASME II [14], SAMM [15], and SMIC [13], most MER stud- + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +11 +ies adopted the leave-one-subject-out strategy when evalu- +ating on them. Nevertheless, considering that the number of +ME clips in DFME is relatively large, this paper put to use a +simpler and more efficient 10-fold cross-validation strategy. +For each fold, 10% of the data were sampled as the test set, +and the remaining 90% as the training set. In addition, three +commonly used MEs classification indicators, namely Accu- +racy, Unweighted F1-Score and Unweighted Average Recall, +were used to evaluate the MER performance. Specifically, +before calculating them, we need to obtain the True Positive +(TPi), False Positive (FPi), and False Negative (FNi) for +each class i (K classes in total, and K = 7 in DFME). In the +end, we took the average results of ten experiments as the +final result. +4.3.1 +Accuracy (ACC) +Accuracy is one of the most common metrics, which can +evaluate the overall performance of the recognition method +on the dataset. It was calculated as follows: +ACC = +K +� +i=1 +TPi +Ni +, +(8) +where Ni is the number of samples of the i-th class. +4.3.2 +Unweighted F1-score (UF1) +Unweighted F1-score (UF1), also known as macro-averaged +F1-score, is defined as shown below: +UF1 = 1 +K +K +� +i=1 +UF1i, +(9) +where we have: +UF1i = +2 · TPi +2 · TPi + FPi + FNi +. +(10) +Class imbalance is an intractable problem in the MER task, +so introducing UF1 as an evaluation metric can better mea- +sure the method’s performance in all classes rather than in +some major classes. +4.3.3 +Unweighted Average Recall (UAR) +Unweighted Average Recall (UAR) is also a more reason- +able metric than accuracy in case of class imbalance. +UAR = 1 +K +K +� +i=1 +TPi +Ni +. +(11) +Both UF1 and UAR can effectively evaluate whether MER +methods give correct predictions in all classes. +4.4 +Evaluation Baseline Models +Although the spatiotemporal convolution models with +deeper layers and more parameters have achieved amazing +performance in the video classification tasks, due to the +scarcity of ME data, previous MER studies rarely use such +a model with a large number of parameters. In fact, both +time and space contain unique features of ME, and MER +should take into account both dimensions. To verify the +feasibility of applying large 3D models on our large-scale +dataset and to provide a reference for backbone selection of +MER methods based on extensive data, we have selected +the following standard backbone networks based on 3D +convolution architecture for validation experiments. +4.4.1 +3D-ResNet (R3D) +Hara et al. proposed 3D-ResNet (R3D) [52] for tasks such as +video classification and recognition. Since then, R3D is often +used as the backbone in approaches to video-related tasks. +The basic idea of this model is to replace the 2D convolu- +tional kernels with spatiotemporal 3D kernels according to +the 2D-ResNet [29] network structure. +4.4.2 +Pseudo-3D ResNet (P3D) +Pseudo-3D ResNet +(P3D) [53] is another 3D model back- +bone that has achieved good results in video tasks. It can be +considered as an improved version of R3D. The key point +of this model is the simulation of the 3×3×3 convolution +filter by using a 1×3×3 spatial domain convolution filter +and a 3×1×1 temporal domain convolution filter. Hence the +authors named it Pseudo-3D ResNet. This change controls +the model size and improves training efficiency and experi- +mental performance. +4.4.3 +3D-DenseNet (D3D) +DenseNet [54] has achieved excellent performance in image +tasks. It expanded the residual connection of ResNet. All +layers in DenseNet connect directly with each other. 3D- +DenseNet +(D3D) has also been widely used in the video +field. In the field of MER, Cai et al. [55] proposed a 3D- +DenseNet-based method. +4.4.4 +Inflated 3D ConvNet (I3D) +Inflated 3D ConvNet (I3D) [56] is based on 2D ConvNet in- +flation. The model size has increased significantly compared +to the 2D model. Therefore, the data requirements have +also increased significantly. For this reason, the authors also +published a large-scale video dataset Kinetics [56] simulta- +neously. The results on Kinetics demonstrate the excellent +performance of I3D when the amount of data is sufficient. +4.5 +Evaluation Implementation Settings +Our MER experiments were all conducted on 2 NVIDIA +GeForce RTX 3090 GPUs or a single NVIDIA A100-PCIE- +40GB GPU. Following the original settings, the length of +ME clips for all models was 16 frames, and for R3D, P3D, +D3D and I3D, the sizes of each input image were 224×224, +160×160, 224×224 and 224×224 respectively. +During training, cross-entropy loss and stochastic gradi- +ent descent (SGD) with a momentum of 0.9 were used to +optimize the model parameters, and the batch size was set +to 32 for all four models. For R3D, P3D, D3D, and I3D, the +initial learning rates were set to 0.005, 0.01, 0.05, and 0.005, +respectively, and learning rates were divided by 10 every 10 +epochs. +4.6 +Evaluation Baseline Results +To demonstrate the effectiveness of our DFME dataset for +automatic MER tasks, we conducted a comprehensive MER +experiment based on the above four baseline models. The +evaluation baseline results are shown in Table 7, and the +recognition confusion matrix of each baseline model is +shown in Figure 5. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +12 +anger +contempt +disgust +fear +happiness +sadness +surprise +Predicted label +anger +contempt +disgust +fear +happiness +sadness +surprise +True label +16.48% 3.39% 41.68% 15.99% 3.55% 12.60% 6.30% +7.48% 12.22% 25.44% 13.47% 18.45% 12.47% 10.47% +6.41% +3.44% 57.87% 14.72% 5.22% +7.16% +5.18% +5.61% +3.59% 34.30% 31.95% 3.92% +7.17% 13.45% +2.12% +8.67% 15.02% 11.59% 51.51% 6.05% +5.04% +8.66% +5.51% 24.57% 10.24% 3.46% 37.17% 10.39% +2.90% +3.89% +7.70% 15.07% 2.81% +6.37% 61.26% +R3D Model +10 +20 +30 +40 +50 +60 +(a) R3D +anger +contempt +disgust +fear +happiness +sadness +surprise +Predicted label +anger +contempt +disgust +fear +happiness +sadness +surprise +True label +23.75% 1.78% 42.97% 8.08% +4.20% 11.15% 8.08% +4.49% 12.22% 40.40% 9.23% 15.46% 5.99% 12.22% +6.25% +2.25% 61.59% 10.88% 5.34% +4.91% +8.78% +5.61% +2.69% 32.96% 32.74% 6.17% +4.04% 15.81% +1.31% +4.54% 21.57% 6.15% 54.33% 2.52% +9.58% +10.87% 2.83% 25.20% 8.03% +1.73% 39.53% 11.81% +3.64% +1.99% 25.08% 15.31% 7.04% +5.96% 40.98% +P3D model +10 +20 +30 +40 +50 +60 +(b) P3D +anger +contempt +disgust +fear +happiness +sadness +surprise +Predicted label +anger +contempt +disgust +fear +happiness +sadness +surprise +True label +16.48% 1.78% 49.60% 6.46% +6.14% 12.60% 6.95% +4.24% +6.23% 37.41% 6.48% 26.68% 10.72% 8.23% +4.47% +0.99% 68.99% 10.01% 5.54% +5.10% +4.91% +4.82% +1.35% 44.28% 23.99% 6.05% +5.72% 13.79% +1.31% +3.53% 16.43% 4.33% 65.32% 4.44% +4.64% +5.98% +3.31% 28.50% 8.98% +5.35% 38.90% 8.98% +1.57% +0.58% 12.67% 7.70% +4.80% +4.64% 68.05% +D3D Model +10 +20 +30 +40 +50 +60 +(c) D3D +anger +contempt +disgust +fear +happiness +sadness +surprise +Predicted label +anger +contempt +disgust +fear +happiness +sadness +surprise +True label +22.78% 1.62% 45.56% 12.76% 2.75% +8.72% +5.82% +5.99% 13.22% 32.67% 8.23% 20.45% 10.97% 8.48% +5.02% +2.37% 69.58% 10.76% 4.11% +4.31% +3.84% +4.60% +1.79% 38.79% 31.61% 4.82% +6.05% 12.33% +1.51% +5.04% 14.72% 5.75% 66.94% 2.62% +3.43% +6.30% +4.57% 25.67% 10.39% 2.52% 42.83% 7.72% +1.82% +2.15% 10.18% 9.93% +2.90% +2.81% 70.20% +I3D model +10 +20 +30 +40 +50 +60 +70 +(d) I3D +Fig. 5: Confusion matrices of R3D, P3D, D3D and I3D baseline models. +From Table 7, we can easily find that the I3D model +achieved the best performance among the four backbone +models with an average accuracy of 55.24%, an average UF1 +of 0.4576 and an average UAR of 0.4526, and the accuracy +is higher than the 47% achieved by naked eyes [57]. Besides, +the other three models were approximately as accurate as +the naked eye in DFME. The above experimental results +demonstrate the reliability of our DFME and provided a +reference for the selection of backbone models for future +works. Meanwhile, by observing the recognition confusion +matrices shown in Figure 5, we also find that all baseline +models present the same phenomenon, that is, these models +are more inclined to recognize the categories with more +samples. Obviously, this is mainly caused by the class im- +balance problem in DFME. Therefore, how to learn more +distinguishable spatiotemporal ME features from the ME +data with unbalanced classes is a vital exploration direction +of MER. Besides, confusion matrices shown in Figure 5 +illustrate that for all four backbone models, the disgust and +fear samples are the most difficult to distinguish. This result +is consistent with the statistics of the AU frequencies in Table +4. In both disgust and fear samples, the most frequent AUs +are AU4 and AU7, and AU10, AU14, and AU24 are also +found in both classes of samples. +TABLE 7: ME recognition performance of various baseline +models +Models +ACC +UF1 +UAR +R3D [52] +46.54% +0.3817 +0.3827 +P3D [53] +45.77% +0.3830 +0.3801 +D3D [55] +52.26% +0.4070 +0.4107 +I3D [56] +55.24% +0.4576 +0.4526 +4.7 +Evaluation Discussion +This section will focus on two key problems that are particu- +larly considered when using our DFME for MER, including +class imbalance problem and various key-frame sequence +sampling strategies. +4.7.1 +Class imbalance in DFME +Since the existence of individual differences of subjects and +the different inducing degrees of each category of ME, +the collected spontaneous ME dataset is hard to avoid the +problem of class imbalance. This is directly reflected in +the previous three datasets widely used in MER, including +SMIC, CASME II and SAMM, whose ratio of the most +category to the least category is 1.63, 3.52 and 6.13 [58], +respectively. Inevitably the class imbalance problem still +exists in our DFME dataset. +The statistic of emotion categories in DFME is shown +in Table 3, from which we can find that the number of +disgust samples is the largest among all emotion categories, +accounting for about 1/3 of the proportion, and the negative +samples (including disgust, fear, sadness, anger and contempt) +accounted for about 2/3 of the proportion. Moreover, the +confusion matrices in Figure 5 indicated the negative impact +of class imbalance on models. All four backbone models +tended to predict samples as disgust class more than others. +To solve the class imbalance problem, introducing a class +rebalancing strategy is an effective solution. In general, the +class rebalancing methods can be roughly divided into two +major categories: resampling and cost-sensitive reweighting. +TABLE 8: MER Performance with and without +Resampling. +Metrics +Resampling1 +ACC +UF1 +UAR +R3D +w/o +46.54% +0.3817 +0.3827 +w +47.05% +0.3823 +0.3659 +P3D +w/o +45.77% +0.3830 +0.3801 +w +42.02% +0.3949 +0.4078 +D3D +w/o +52.26% +0.4070 +0.4107 +w +48.37% +0.4489 +0.4656 +I3D +w/o +55.24% +0.4576 +0.4526 +w +53.91% +0.4902 +0.4924 +1 w/o: without resampling, w: with resampling +Resampling is one of the most widely used class rebal- +ancing methods. Moreover, uniform resampling is a fairly +common one of all resampling strategies, which is also used +in our experiments. Its main idea is to select each class of +samples with an equal probability when training models, +rather than sampling all samples uniformly. +Table 8 and Figure 6 show the comparison of the re- +sults with and without uniform resampling. The resampling +strategy improved UAR and UF1 on the three models except +for R3D, but the accuracy decreased. With the introduction +of the uniform resampling strategy, the model could better +learn the features of minor classes, but at the cost of weak- +ening the ability to predict major classes correctly. How to + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +13 +R3D +P3D +D3D +I3D +35.0 +37.5 +40.0 +42.5 +45.0 +47.5 +50.0 +52.5 +55.0 +ACC (%) +Accuracy (ACC) +without Resampling +with Resampling +(a) ACC +R3D +P3D +D3D +I3D +0.36 +0.38 +0.40 +0.42 +0.44 +0.46 +0.48 +0.50 +UF1 +Unweighted F1-Score (UF1) +without Resampling +with Resampling +(b) UF1 +R3D +P3D +D3D +I3D +0.36 +0.38 +0.40 +0.42 +0.44 +0.46 +0.48 +0.50 +UAR +Unweighted Average Recall (UAR) +without Resampling +with Resampling +(c) UAR +Fig. 6: Comparison of MER results with and without Resampling +reduce the information loss of the major classes in MER is a +problem that needs to be addressed in future works. +Reweighting approaches attempt to rebalance different +classes by reweighting their loss during training models. +Class-Balanced Loss +(CBLoss) [59] is a representative of +reweighting loss, which is simple and effective and, there- +fore, used extensively in different tasks. CBLoss proposed +the concept of effective number to estimate the actual impact +of samples of each class on the model. It can also be +combined with other losses, including Focal Loss [60], which +reweighted samples in different classes according to their +difficulty to be predicted. This feature further enhances the +adaptability of CBLoss to different domains. The losses we +calculated in our experiments are shown in Table 10. +The results of CBLoss are shown in Table 9. Similar to +uniform resampling, CBLoss also improved the UAR and +UF1 for all four models at the cost of ACC in our experi- +ments. This result demonstrates that CBLoss is compatible +with various models and suffers from similar problems as +resampling. Besides, CBLoss can be easily used for different +tasks with different models, but we should carefully fine- +tune it in various conditions to achieve better results. In +particular, the choice of β may need further study, which +controls the relationship between the effective number and +the actual number of samples. +TABLE 9: MER Performance with Different Losses +Metrics +Losses +ACC +UF1 +UAR +R3D +Cross Entropy Loss +46.54% +0.3817 +0.3827 +Class Balanced Loss +46.61% +0.3951 +0.3914 +P3D +Cross Entropy Loss +45.77% +0.3830 +0.3801 +Class Balanced Loss +43.23% +0.3921 +0.3955 +D3D +Cross Entropy Loss +52.26% +0.4070 +0.4107 +Class Balanced Loss +48.25% +0.4219 +0.4302 +I3D +Cross Entropy Loss +55.24% +0.4576 +0.4526 +Class Balanced Loss +54.56% +0.4789 +0.4777 +4.8 +ME key-frame sequence sampling Strategies +The key-frame sequence is a concise description of the +original video, which generally contains key information +about the content of the video. How to sample effective +ME key-frame sequence from the raw video is also an im- +portant factor for accurate recognition of ME. Video-related +TABLE 10: Cost-Sensitive Reweighting Losses. In this table, +py and ny are the softmax probability and the sample +number of the class y, and β is the hyperparameter in Class- +Balanced Loss (β = 0.999 in our experiments). +Loss +Equation +Cross Entropy Loss +Lce = −log(py) +Class-Balanced Loss [59] +Lcb = − +1−β +1−βny log(py) +recognition tasks usually adopt uniform sampling to obtain +a fixed-length key-frame sequence as model input, but the +instantaneously changing ME movements are often not +uniformly distributed in spatial-temporal space. Previous +studies [12], [44] have shown the superiority of key-frame +temporal adaptive sampling based on three key moments +of ME video, namely onset, apex and offset. Therefore, we +hereby compare and analyze the corresponding recognition +performance of these two sampling strategies (i.e., uniform +sampling and temporal adaptive sampling) in DFME using +baseline models. +TABLE 11: Comparison of MER Performace with +Different Key-Frame Sequence Sampling Strategies. +Metrics +Sampling Method1 +ACC +UF1 +UAR +R3D +adaptive +46.54% +0.3817 +0.3827 +uniform +46.49% +0.3710 +0.3715 +P3D +adaptive +45.77% +0.3830 +0.3801 +uniform +45.31% +0.3671 +0.3656 +D3D +adaptive +52.26% +0.4070 +0.4107 +uniform +52.62% +0.4124 +0.4203 +I3D +adaptive +55.24% +0.4576 +0.4526 +uniform +55.21% +0.4621 +0.4576 +1 adaptive: adaptive sampling in +[44], uniform: uniform +sampling +Table 11 and Fig 7 show the recognition performance of +uniform sampling and temporal adaptive sampling [44]. It is +clear that the temporal adaptive sampling strategy achieved +better results on R3D and P3D models while performing +worse on D3D. For I3D, the recognition performance of the +two sampling strategies is comparable. This result suggests +that different baseline models may require different sam- +pling approaches. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +14 +R3D +P3D +D3D +I3D +35.0 +37.5 +40.0 +42.5 +45.0 +47.5 +50.0 +52.5 +55.0 +ACC (%) +Accuracy (ACC) +Uniform Sampling +Adaptive Sampling +(a) ACC +R3D +P3D +D3D +I3D +0.36 +0.38 +0.40 +0.42 +0.44 +0.46 +UF1 +Unweighted F1-Score (UF1) +Uniform Sampling +Adaptive Sampling +(b) UF1 +R3D +P3D +D3D +I3D +0.36 +0.38 +0.40 +0.42 +0.44 +0.46 +UAR +Unweighted Average Recall (UAR) +Uniform Sampling +Adaptive Sampling +(c) UAR +Fig. 7: Comparison of MER results of Adaptive Key-frame Sampling and Uniform Key-frame Sampling. +5 +CONCLUSION AND FUTURE WORK +In this work, we focused on solving the problem of lacking +abundant spontaneous ME data for MER. To this end, we +built a new ME dataset called DFME containing 7,526 ME +videos across multiple frame rates. To the best of our knowl- +edge, DFME has the largest ME sample size at present. +Furthermore, to verify the feasibility and validity of DFME +dataset for MER task, we reproduced four spatiotemporal +visual feature learning models to carry out MER task in +DFME, objectively verifying the reliability of data quality, +and providing a benchmark for subsequent MER studies. +Particularly, we explored and analyzed two key problems +when using DFME for MER, including class imbalance and +key-frame sequence sampling, so as to provide directions +for future MER studies using DFME. +In the future, we will strive to expand the DFME dataset +to provide more abundant ME data for automatic ME +analysis research, including the collection of multimodal +ME data in multiple natural scenes. 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Belongie, “Class-balanced +loss based on effective number of samples,” 2019 IEEE/CVF Confer- +ence on Computer Vision and Pattern Recognition (CVPR), pp. 9260– +9269, 2019. +[60] T.-Y. Lin, P. Goyal, R. B. Girshick, K. He, and P. Doll´ar, “Focal loss +for dense object detection,” 2017 IEEE International Conference on +Computer Vision (ICCV), pp. 2999–3007, 2017. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2022 +16 +Sirui Zhao is currently working toward the +PhD degree with the Department of Com- +puter Science and Technology from University +of Science and Technology of China (USTC). +His research interests include automatic micro- +expressions analysis, human-computer interac- +tion (HCI) and affect computing. He has pub- +lished several papers in refereed conferences +and journals, including ACM Multimedia Confer- +ence, IEEE Transactions on Affective Comput- +ing, ACM TOMM, Neural Networks, etc. +Huaying Tang received the B.S. degree in the +School of Computer Science and Technology +from University of Science and Technology of +China (USTC), Hefei, China, in 2021. He is +currently pursuing the M.S. degree in computer +science and technology in USTC. His research +interests lie around automatic micro-expressions +analysis and affect computing. +Xinglong Mao received the B.S degree in +the School of Data Science from University +of Science and Technology of China (USTC), +Hefei, China. He is currently working toward +the M.S. degree from the School of Data Sci- +ence. His research interests include automatic +micro-expressions analysis and affect comput- +ing. He has published several conference papers +in ACM Multimedia Conference, etc. +Shifeng Liu received the B.S degree in the +School of Gifted Young from University of Sci- +ence and Technology of China (USTC), Hefei, +China. She is currently working toward the +M.S. degree from the School of Data Science. +Her research interests include automatic micro- +expressions analysis, human-computer interac- +tion (HCI) and affect computing. She has pub- +lished several papers in refereed conferences +and journals, including ACM Multimedia Confer- +ence, Neural Networks, etc. +Hanqing Tao is currently working toward the +Ph.D. degree in the Department of Computer +Science and Technology from University of Sci- +ence and Technology of China (USTC). His re- +search interests include data mining, deep learn- +ing, natural language processing and represen- +tation learning. He has published several papers +in referred journals and conference proceedings, +such as IEEE TKDE, IEEE TAC, AAAI, ICDM, +ICME etc. +Hao Wang received the PhD degree in computer +science from USTC. He is currently an associate +researcher with the School of Computer Science +and Technology, USTC. His main research inter- +ests include data mining, representation learn- +ing, network embedding and recommender sys- +tems. He has published several papers in re- +ferred conference proceedings, such as TKDE, +TOIS, NeuriPS, and AAAI.. +Tong Xu received the Ph.D. degree in University +of Science and Technology of China (USTC), +Hefei, China, in 2016. He is currently working +as an Associate Professor of the Anhui Province +Key Laboratory of Big Data Analysis and Ap- +plication, USTC. He has authored 50+ journal +and conference papers in the fields of social +network and social media analysis, including +IEEE TKDE, IEEE TMC, IEEE TMM, KDD, AAAI, +ICDM, etc. +Enhong Chen (Sensor Member, IEEE) received +the PhD degree from USTC. He is a professor +and vice dean with the School of Computer Sci- +ence, USTC. His general area of research in- +cludes data mining and machine learning, social +network analysis, and recommender systems. +He has published more than 100 papers in ref- +ereed conferences and journals, including IEEE +Transactions on Knowledge and Data Engineer- +ing, IEEE Transactions on Mobile Computing, +KDD, ICDM, NeurIPS, and CIKM. He was on +program committees of numerous conferences including KDD, ICDM, +and SDM. His research is supported by the National Science Foundation +for Distinguished Young Scholars of China. + diff --git a/3tAzT4oBgHgl3EQfD_po/content/tmp_files/load_file.txt b/3tAzT4oBgHgl3EQfD_po/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..55839194367b2113b7a6f7934c57bd5878b6de68 --- /dev/null +++ b/3tAzT4oBgHgl3EQfD_po/content/tmp_files/load_file.txt @@ -0,0 +1,1578 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf,len=1577 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 1 More is Better: A Database for Spontaneous Micro-Expression with High Frame Rates Sirui Zhao, Huaying Tang, Xinglong Mao, Shifeng Liu, Hanqing Tao, Hao Wang, Tong Xu, Member, IEEE, and Enhong Chen, Senior Member, IEEE, Abstract—As one of the most important psychic stress reactions, micro-expressions (MEs), are spontaneous and transient facial expressions that can reveal the genuine emotions of human beings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Thus, recognizing MEs (MER) automatically is becoming increasingly crucial in the field of affective computing, and provides essential technical support in lie detection, psychological analysis and other areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' However, the lack of abundant ME data seriously restricts the development of cutting-edge data-driven MER models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Despite the recent efforts of several spontaneous ME datasets to alleviate this problem, it is still a tiny amount of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' To solve the problem of ME data hunger, we construct a dynamic spontaneous ME dataset with the largest current ME data scale, called DFME (Dynamic Facial Micro-expressions), which includes 7,526 well-labeled ME videos induced by 671 participants and annotated by more than 20 annotators throughout three years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Afterwards, we adopt four classical spatiotemporal feature learning models on DFME to perform MER experiments to objectively verify the validity of DFME dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In addition, we explore different solutions to the class imbalance and key-frame sequence sampling problems in dynamic MER respectively on DFME, so as to provide a valuable reference for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The comprehensive experimental results show that our DFME dataset can facilitate the research of automatic MER, and provide a new benchmark for MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' DFME will be published via https://mea-lab-421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Index Terms—Emotion recognition, facial micro-expression, micro-expression recognition, datasets !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 1 INTRODUCTION F ACIAL expression is essential for humans to transmit emotional information, accounting for 55% of our daily communication [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' As a particular facial expression, micro- expression (ME) usually refers to the spontaneous and subtle facial movements that appear instantaneously when an individual tries to hide or suppress real emotions un- der pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The concept of ME was first proposed in 1966 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Subsequently, Ekman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [3] discovered a ME case in a video of a psychiatrist and depressed patient conversation in 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Concretely, throughout the pleasant conversation, when the psychiatrist asked the patient about her plans, a distressed expression quickly flashed across the patient’s face, which was called ME by Ekman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' As MEs can effectively reveal the genuine emotions of individuals, recognizing MEs can provide essential technical support in Sirui Zhao is with the School of Computer Science and Technology, University of Science and Technology of China, Hefei, Anhui 230027, China, and also with the School of Computer Science and Technology, Southwest University of Science and Technology, Mianyang 621010, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' E-mail: sirui@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='cn Huaying Tang, Hanqing Tao are with the School of Computer Science and Technology, University of Science and Technology of China, Hefei, Anhui 230027, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' E-mail: {iamthy, hqtao}@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='cn Xinglong Mao, Shifeng Liu, Hao Wang, Tong Xu and Enhong Chen are with School of Data Science, University of Science and Technology of China, Hefei, Anhui 230027, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' E-mail: {maoxl, lsf0619}@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='cn, {wanghao3, tongxu, cheneh}@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='cn This work has been submitted to the IEEE for possible publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Copyright may be transferred without notice, after which this version may no longer be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Sirui Zhao, Huaying Tang, Xinglong Mao and Shifeng Liu contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Corresponding authors: Enhong Chen and Tong Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Manuscript received December xx, xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' revised xx xx, xx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' lie detection, psychological healing, and public safety [4], [5], [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In essence, ME is a kind of psychic stress reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Com- pared with ordinary facial expression (also called macro- expression, MaE), ME has the characteristics of short dura- tion (less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5s), partial movement, and low movement intensity, so it is challenging to recognize MEs accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Figure 1 illustrates the comparison between a ME and a MaE with the same emotion category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' It shows vividly that the MaE is obvious enough to be distinguished easily by a single image, while the ME is subtle and can only be observed through an image sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The early research on ME recognition (MER) was mainly based on manual analysis in the field of psychology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' How- ever, the manual analysis relies on expert experience, which is time-consuming and labor-intensive, and has low recog- nition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, it is urgent to use computers’ powerful perception and computing power for automatic MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In recent years, lots of efforts in the fields of com- puter vision and affective computing have been devoted to automatic MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' For example, in order to extract the spatial-temporal MEs, Pfister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [8] introduced a local binary pattern from three orthogonal planes (LBP-TOP) [9] for MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [10] proposed Mian Directional Mean Op- tical Flow (MDMO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [11] proposed Transferring Long-term Convolutional Nerual Network (TLCNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [12] proposed a novel two-stage learning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=', prior learning and target learning) method based on a siamese 3D convolutional neural network for MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' However, due to the lack of support for a large number of well-labeled ME data, the recognition accuracy and robustness of these methods are challenging to meet the needs of actual scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' There- fore, it is urgent to build a large-scale ME dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='00985v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='CV] 3 Jan 2023 JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 2 ··· ··· ··· ··· onset 0 apex 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='25 offset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='08 second (a) An example of MaE with ”Happiness” emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' ··· ··· ··· ··· onset 0 apex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='19 offset 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='36 second (b) An example of ME with “Happiness” emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 1: Examples of MaE and ME from the same person with a timeline in seconds, both belong to the ”Happiness” emotion category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Noteworthy, the onset frame and the offset frame denote the start and end time of an expression respectively, and the apex frame represents the moment when an expression changes most dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' White arrows on the face of the apex frame indicate the general directions of facial movements, and the longer and thicker the arrows, the greater the intensity of facial movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Over the past decade, although researchers have published several spontaneous ME datasets, such as SMIC [13], CASME II [14], SAMM [15], MMEW [16] and CAS(ME)3 [17], these datasets have a small sample size, which still cannot completely meet the need of MER models for large-scale ME samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In fact, building a large-scale spontaneous ME dataset is full of challenges, mainly from three aspects: First, it is difficult to induce MEs because they are facial movements that are disclosed after an individual attempts to suppress them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Second, it is difficult to label and distinguish ME fragments because the movement of ME is weak and fast, which is hard for the naked eye to perceive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Third, due to the short duration of MEs, high-speed cameras are often needed to collect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' However, the data collected by high-speed cameras are redundant, so labeling ME clips is extremely time-consuming and labor-intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In order to solve the challenge of ME data shortage, this paper constructs the current largest ME dataset called DFME (Dynamic Facial Micro-expressions) to advance the development of MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Specifically, our DFME includes 7,526 well-labeled ME videos induced by 671 participants and annotated by more than 20 annotators throughout three con- secutive years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Subsequently, four popular spatiotemporal video feature learning models were reproduced on DFME to perform MER so as to objectively verify the availability of the dataset and provide a benchmark for subsequent research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In addition, aiming at the class imbalance and key-frame sequence sampling problems existing in MER, we explored different solutions to DFME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In general, the contributions of this paper could be summarized as follows: This paper focuses on solving the problem of lacking abundant spontaneous ME data and builds a new ME dataset called DFME containing 7,526 ME videos across multiple high frame rates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=', 200fps, 300fps, 500fps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' To the best of our knowledge, DFME has the largest ME sample size at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' We reproduced four spatiotemporal feature learning models to carry out MER tasks in DFME, objectively verifying the reliability of data quality, and providing a benchmark for subsequent MER studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' We explored and analyzed different solutions to the class imbalance and key-frame sequence sampling problems in dynamic MER respectively on DFME, so as to provide a reference for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' First, we summarize currently existing ME datasets and review re- lated work on MER in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In section 3, we elab- orate on the building details and statistical properties of our DFME dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Then the comprehensive dataset evaluation is developed and discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Finally, research conclusions and future work are addressed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 2 RELATED WORK In this section, we first review the existing public sponta- neous ME datasets related to MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Then, we summarize some representative MER studies based on deep learning technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 Micro-expression Datasets The premise of obtaining an automatic MER algorithm with excellent performance is to hold a dataset with sufficient ME samples whose labels are credible and whose visual features are distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' As an emerging field of affective com- puting, the number of ME datasets is still relatively limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 香JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 3 TABLE 1: Statistical Information of Current Spontaneous ME Datasets ME Datasets Participants Samples of MEs Annotation Labels Number Gender (Male/Female) Age Number Frame Rate Resolution Emotion FACS AU HS 16 164 100 640×480 Pos (51) Neg (70) Sur (43) SMIC VIS 8 10/6 Range: 22-34 Mean=28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 71 25 640×480 Pos (28) Neg (23) Sur (20) No NIR 8 71 25 640×480 Pos (28) Neg (23) Sur (20) CASME 35 22/13 Mean=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='03 195 60 640×480 1280×720 Amu (5) Dis (88) Fear (2) Con (3) Sad (6) Tense (28) Sur (20) Rep (40) Yes CASME II 35 / Mean=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='03 247 200 640×480 Hap (33) Dis (60) Sur (25) Rep (27) Oth (102) Yes CAS(ME)2 22 9/13 Range: 19-26 Mean=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='59 57 30 640×480 Pos (8) Neg (21) Sur (9) Oth (19) Yes SAMM 32 16/16 Range: 19-57 Mean=33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='24 159 200 2040×1088 Hap (24) Dis (8) Fear (7) Ang (20) Sur (13) Sad (3) Oth (84) Yes MEVIEW 16 / / 29 30 1280×720 Hap (5) Dis (1) Fear (3) Ang (1) Sur (8) Con(4) Unc (7) Yes MMEW 36 / Mean=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='35 300 90 1920×1080 Hap (36) Dis (72) Fear (16) Ang (8) Sur (89) Sad (13) Oth (66) Yes CAS(ME)3 PART A 100 50/50 / 943 30 1280×720 Hap (64) Dis (281) Fear (93) Ang (70) Sur (201) Sad (64) Oth (170) Yes PART C 31 9/22 Mean=23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 166 30 1280×720 Pos (16) Neg(99) Sur (30) Oth (20) 4DME DI4D 65 38/27 Range: 22-57 Mean=27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='8 267 60 1200×1600 Pos (34) Neg (127) Sur (30) Rep (6) PosSur (13) NegSur (8) RepSur (3) PosRep(8) NegRep(7) Oth(31) Yes Grayscale 267 60 640×480 RGB 267 30 640×480 Depth 267 30 640×480 PART A 72 31/41 1118 500 1024×768 Hap (111) Dis (321) Fear (143) Ang (97) Con (77) Sur (187) Sad (142) Oth (40) DFME PART B 92 61/31 Range: 17-40 Mean=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='43 969 300 1024×768 Hap (78) Dis (406) Fear (115) Ang (56) Con (45) Sur (143) Sad (119) Oth (7) Yes PART C 492 282/210 5439 200 1024×768 Hap (803) Dis (1801) Fear (634) Ang (466) Con (279) Sur (878) Sad (374) Oth (204) 1 Some datasets contain not only MEs but also MaEs, as well as long video clips for the detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' But here we only show the information about ME data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Note that all statistical data are from the corresponding original paper or downloaded datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 2 The number of participants was counted based on the data given in the corresponding original paper, but some participants were not successfully induced to make MEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 3 Pos: Positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Neg: Negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Sur: Surprise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Amu: Amusement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Hap: Happiness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Dis: Disgust;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Rep: Repression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Ang: Anger;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Sad: Sadness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Con: Contempt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Unc: Unclear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Oth: Others;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' PosSur: Positively surprise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' NegSur: Negatively surprise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' RepSur: Repressively surprise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' PosRep: Positively repression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' NegRep: Negatively repression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Nevertheless, since more and more researchers have begun to pay attention to ME analysis, some high-quality datasets are gradually springing up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Table 1 clearly summarizes the characteristics of these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' As the two earliest proposed ME datasets, samples in the USF-HD [18] and Polikovsky [19] datasets are all posed MEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The participants were first required to watch video clips containing ME samples and then posed them by imi- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' However, naturally generated MEs strongly correlate with emotions, while the posed ones are deliberately dis- played and have nothing to do with the current emotional state of the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Consequently, these two datasets are rarely used by researchers for ME analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The subsequent researchers proposed to induce spon- taneous MEs with the neutralization paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Under this paradigm, several strong emotional stimuli were used to elicit expressions, during which participants were in- structed to keep a neutral face as much as possible, and a certain degree of high-pressure mechanism was given to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Datasets adopting the neutralization paradigm include SMIC [13], CASME [20], CASME II [14], CAS(ME)2 [21], SAMM [15], MMEW [16], and 4DME [22], which will JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 4 be introduced in turn below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' SMIC dataset [13] is the first published spontaneous ME dataset, which consists of three parts: HS, VIS, and NIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The HS part includes 164 ME samples from 16 participants, recorded by a high-speed camera with a frame rate of 100 frames per second (fps) and a resolution of 640×480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Both the VIS and NIR parts contain 71 ME samples from 8 individuals, while the former part was recorded using a standard visual camera and the latter using a near-infrared camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Two annotators classified each ME into three emo- tion categories (positive, negative, and surprise) based on the participants’ self-reports about the elicitation videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Facial action units (AUs) were not annotated in SMIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' CASME series datasets are released by the Institute of Psychology, Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' As the earliest dataset in this series, CASME [20] contains a total of 195 ME samples from 19 participants with a frame rate of 60fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Two annotators labeled the facial AUs, together with the corresponding onset, apex, and offset frames of each ME sample frame by frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' According to the facial AUs, participants’ self-reports, and the relevant video content, MEs were divided into eight emotion categories: amuse- ment, sadness, disgust, surprise, contempt, fear, repression, and tense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' CASME II [14] is an advanced version of CASME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' First, the number of ME samples in CASME II has been expanded to 247 samples from 26 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Besides, CASME II provides a higher frame rate of 200fps and facial area resolution of 280×340 to capture more subtle changes in expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Five emotion categories were labeled in CASME II: happiness, disgust, surprise, repression, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The CAS(ME)2 dataset [21] embodies two parts, both of which were collected at 30fps and 640×480 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Different from all the other datasets above, there are 87 long video clips containing both MaEs and MEs in the first part of CAS(ME)2, which can be used to promote the research of ME detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The other part consists of 300 MaEs and 57 MEs, which were labeled with four emotion tags, including positive, negative, surprise, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' SAMM dataset [15] has the highest resolution of all published spontaneous ME datasets, which includes 159 ME samples generated by 32 participants, with a frame rate of 200fps and a resolution of 2040×1088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' To achieve a better elicitation effect, before the formal start of the collection, participants were asked to fill in a scale, and then a series of stimulus videos were customized for each participant according to the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' This is how SAMM differs from other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' SAMM contains seven emotion categories: happiness, disgust, surprise, fear, anger, sadness, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Three coders annotated the AUs and key-frames in detail for each ME sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' MMEW dataset [16] consists of 300 ME and 900 MaE samples from 36 participants, which were collected with 90 fps and 1920×1080 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Each expression sample is marked with seven emotion labels (the same as SAMM), AUs, and three key-frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Compared with the previous datasets, MMEW is more conducive to the models using the MaE samples under the same parameter setting and elicitated environment to assist in learning ME features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' To comprehensively capture the movement informa- tion of ME in all directions as much as possible, 4DME dataset [22] has made significant innovations in the record- ing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Each ME sample in this dataset has multi- modality video data, including 4D facial data reconstructed by 3D facial meshes sequences, traditional 2D frontal facial grayscale, RGB and depth videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4DME contains 267 MEs and 123 MaEs from 41 participants, thus 1,068 ME videos of four forms and 492 MaE videos in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In addition, five emotion labels (positive, negative, surprise, repression, and others) were annotated based on facial AUs only, noting that each sample may have multiple emotion labels (up to two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Unlike datasets with the neutralization paradigm, the MEVIEW dataset [23] consists of video clips of two real high-pressure scenes downloaded from the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' There are 29 ME samples in total, with a frame rate of 30fps and a resolution of 1280×720, divided into seven emotion categories (the same as SAMM) with manual annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Although these samples are from actual life scenarios and have high ecological validity, there are many uncontrollable factors, such as frequent camera shot switching, which re- sults in fewer segments containing full human faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The CAS(ME)3 dataset [17] adopted the mock crime paradigm to elicit MEs with high ecological validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' How- ever, unlike MEVIEW, the collection was still controlled in the laboratory environment, yielding 166 MEs and 347 MaEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' CAS(ME)3 also contains two other parts: one consists of 943 MEs and 3,143 MaEs collected using the neutraliza- tion paradigm, respectively marked with AUs, key-frames, and seven emotion labels (the same as SAMM) for each sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' the other part contains 1,508 unlabeled long video clips, which can be used for the self-supervised learning task of ME detection and recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' This dataset was collected at a frame rate of 30fps with a resolution of 1280×720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Despite more and more datasets striving to record the movement characteristics of MEs more detailedly and com- prehensively through various methods, these datasets are still small-scale datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In automatic ME analysis, mod- els based on deep learning have become mainstream by practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' However, due to the insufficient sample size, the complexity of the model can easily lead to overfitting in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Though we can alleviate this problem by using data augmentation to increase the number of samples, many uncontrollable noises might be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Some work has proposed using composite datasets to train the model, but different datasets have different parameter settings, and thus such a simple fusion is not reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In addition, due to the short duration and low intensity of MEs, a higher frame rate may contribute to capturing more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Nevertheless, the highest frame rate of all above datasets is only 200fps, and most are less than 100fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, it is necessary to establish a larger-scale ME dataset with a higher frame rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 Micro-expression Recognition Approaches In the past decade, MER has attracted more and more attention from scholars in affective computing and com- puter vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The first attempt at automatic, spontaneous MER dates back to 2011, Pfister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [8] utilized a local binary pattern from three orthogonal planes (LBP-TOP) to explore MER on the first spontaneous ME dataset SMIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Since then, more and more efforts have been devoted to automatic MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In general, the current MER methods can be JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 5 roughly divided into hand-crafted feature based and deep learning based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Typical hand-crafted ME features include LBP-TOP [9], HOOF [24], 3DHOG [19], and their variants [25], [26], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' However, the hand-crafted feature based methods heavily rely on complex expert knowledge, and the extracted ME features have limited discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Current MER methods mainly use deep neural networks for high-level expression feature learning and emotion classifi- cation, and focus on solving the challenges that ME is subtle and ME data shortage for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Further, according to whether the MER model considers the ME temporal information or not, we divide the current deep learning based MER methods into single frame based MER and video sequence based MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In the following subsections, we will categorize and summarize these two types of MER methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 Single frame based MER methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The single frame based MER method usually only uses the highest intensity frame, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=', the apex frame with RGB or optical-flow format in the ME video, as the input of neural networks to learn the spatial ME features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' After considering the challenge of lacking sufficient ME samples, Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [28] first selected ResNet-10 [29] pre-trained on a large- scale image dataset as the backbone and then continued to fine-tune the classification network on large MaE samples for MER using apex frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Encouragingly, the recognition accuracy exceeds the hand-crafted methods based on LBP- TOP, HOOF, and 3DHOG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Inspired by the success of capsule models on image recognition, Quang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [30] proposed a CapsuleNet for MER using only apex frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Recently, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [31] proposed an expression-identity disentangle network for MER by leveraging MaE databases as guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [32] first spotted the apex frame by estimating pixel- level change rates in the frequency domain, then proposed a joint feature learning architecture coupling local and global information from the detected apex frames to recognize MEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' At the same time, Liong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [33] explored the effectiveness and superiority of using the optical flow of the apex frame in ME video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Inspired by this work, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [34] first calculated the optical-flow image of the apex frame to the onset frame in the ME clips and then used the pre-trained ResNet-18 network to encode the optical- flow image for MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In particular, they introduced domain adversarial training strategies to address the challenge of lacking large-scale ME data for training and won first place for MEGC2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Furthermore, Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [35] proposed a novel Feature Refinement (FR) with expression-specific feature learning and fusion for MER based on optical-flow information of apex frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [36] proposed a meta-learning-based multi-model fusion network for MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Overall, the single frame based MER investigations are conducted on apex frames of ME videos without temporal information, which can reduce the complexity of the used deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In addition, the single frame based MER method has the advantage of finding large-scale im- ages for transfer learning to effectively solve the problem of model overfitting with insufficient ME data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Nevertheless, the single frame based MER discards the temporal informa- tion in the ME video, which contains rich ME clues and is an important feature that distinguishes MEs from MaEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 Video sequence based MER methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Unlike the single frame based MER, video sequence based MER can learn spatiotemporal ME feature from the whole ME video or its sub-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Thus, the video sequence based MER is preferred to the single frame based MER for providing details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' After fully considering the important expression states in the ME video, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [37] first used CNN to encode the spatial feature of each expression state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=', onset, onset to apex transition, apex, apex to offset transition and offset), then adopted LSTM to learn the temporal feature based on the encoded spatial ME feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [11] proposed Transferring Long-term Convo- lutional Nerual Network (TLCNN) to solve the learning of spatial-temporal ME feature under small sample ME data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The TLCNN is also based on the CNN-LSTM structure and transfers knowledge from large-scale expression data and single frames of ME video clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Khor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [38] proposed an Enriched Long-term Recurrent Convolutional Network (EL- RCN) that makes spatial and temporal enrichment by stack- ing different input data and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Unlike the CNN-LSTM architecture, 3D convolution network (3DCNN) [39] can simultaneously learn the spatial and temporal ME features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Based on 3DCNN, Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [40] proposed a Dual Tempo- ral Scale Convolutional Neural Network (DTSCNN), which uses the optical-flow sequences of ME videos as model input to obtain high-level ME features and can adapt to a different frame rate of ME video clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [41] pro- posed a MER framework based on Eulerian motion based 3DCNN (EM-CED), which uses the pre-extracted Eulerian motion feature maps as input and with a global attention module to encode rich spatiotemporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [42] proposed a deep recurrent convolutional networks based MER approach, which modeled the spatiotemporal ME deformations in views of facial appearance and geom- etry separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' To solve the challenge of extracting high- level ME features from the training model lacking sufficient and class-balanced ME samples, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [12] extracted the ME optical-flow sequence to express the original ME video and proposed a novel two-stage learning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=', prior learning and target learning) method based on a siamese 3D convolutional neural network for MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [43] proposed a knowledge transfer technique that distills and transfers knowledge from action units for MER based on crucial temporal sequences, where knowledge from a pre- trained deep teacher neural network is distilled and trans- ferred to a shallow student neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [44] proposed a deep prototypical learning framework on RGB key-frame sequences, namely ME-PLAN, based on a 3D residual prototypical network and a local-wise attention module for MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Recently, with the advancement of deep learning technology, some excellent neural networks, such as GCN [45] and transformers, have also been used for MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Although video sequence based MER makes full use of spatial-temporal information of ME, the corresponding model has higher structural complexity and faces seri- ous over-fitting problems on the current small-scale ME datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, building a large-scale ME dataset is still the primary task of developing an automatic MER system, which plays a pivotal role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 6 LED lights with reflector umbrellas Participant Participant’s monitor (playing elicitation videos) High-speed camera (1024×768, freely configurable frame rate ) 4T-sized high-speed acquisition memory Collector Collector’s monitor for recoding MEs Collector’s monitor for playing videos 10 Gigabit optical fiber transmission line Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 2: Experimental environment for eliciting MEs 3 DFME As the old saying goes, ”One can’t make bricks without straw”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Similarly, it is difficult to design an automatic MER model with high recognition rate and reliability without sufficient training and testing samples of ME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' However, due to the short-duration, low-intensity, and local-movement characteristics of ME, it is extremely challenging to construct large-scale ME datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' To solve the problem of ME data hunger, we construct a dataset of spontaneous ME with the largest sample size at present, called DFME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In the following subsections, we will elaborate on the building details and statistical properties of our DFME dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 Participant and Equipment In our DFME, 671 participants were recruited (381 males and 290 females), mainly for college students and teaching staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Participants were age-distributed between 17 and 40 years, with a mean age of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='43 years (standard deviation = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='54), and all from China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Before starting the formal exper- iment, the participants were informed about the purpose, experimental procedure, possible benefits and risks of our research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' On confirming their voluntary participation in the experiment, participants would sign an informed consent form and choose whether to allow their facial images and videos used for the academic paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Considering the low intensity and short duration of MEs, the recording process is easily disturbed by other factors, so it is carried out in a well-controlled laboratory environment, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In this environment, we set up three LED lights with reflector umbrellas to ensure a bright and stable light source on the participants’ faces during experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In addition, we used a self-developed high-speed camera (1024×768, freely configurable frame rates) to capture MEs, and used a 10 Gigabit optical fiber transmission line to connect the camera with a 4T-sized high-speed acquisition memory to store the collected ME video clips in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 Elicitation Material and Procedure At present, there are three generations of ME-eliciting paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Although the third generation has the highest TABLE 2: Video clips for eliciting MEs Video ID During Time Emotion Category Mean Score(0-5) 02sa 3’44” Sadness 4 03sa 4’18” Sadness 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='36 06c 2’01” Contempt 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='83 07a 1’26” Anger 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='49 08su 1’26” Surprise 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='16 09f 2’22” Fear 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='72 10a 2’58” Anger 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='33 11d 1’24” Disgust 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='95 13f 2’14” Fear 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='36 14d 1’22” Disgust 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='23 17h 1’17” Happiness 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='81 18h 1’58” Happiness 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='08 20d 0’46” Disgust 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='87 21c 1’44” Contempt 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='11 23sa 1’44” Sadness 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='25 ecological validity, it is inevitable to interact and have conversations with the participants when simulating the natural scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' These irrelevant body and mouth move- ments caused by speaking are also a kind of noise for MEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, we still use the neutralization paradigm to elicit MEs to avoid noise as much as possible and focus more on the movement characteristics of MEs and facilitate the operation, control, and implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The specific details of the elicitation process will be introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The effectiveness of elicitation materials determines the quantity and quality of MEs, so selecting the materials with high emotional valence is very crucial [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The stimuli we used were all video clips from the Internet, ranging in length from 46 seconds to 258 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In order to find more effective stimulus materials, we recruited 50 volunteers to evaluate 30 video clips collected previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The evalua- tion process was as follows: after watching each video, volunteers were asked to choose only one emotion from happiness, contempt, disgust, sadness, fear, surprise and anger as the main emotion evoked by this video, and score the stimulus level on a scale of 1 to 5, corresponding to the intensity from weakest to strongest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Finally, we took the emotion selected by more than half of the volunteers as the emotional class of each video, and by ranking the average JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 7 stimulus intensity values, we obtained the optimal 15 video clips as elicitation materials adopted in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Specific statistical details are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The collection took place in a configured laboratory environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Before start, each participant was taken to a specific seat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' By adjusting the height of the seat, the focal length of the camera and the brightness of the LED lamps, we ensured that the participant’s face appeared utterly, clearly, and brightly in the centre of the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Then the monitor in front of the participant would play ten randomly selected elicitation videos covering all seven basic emotional types that had been previously verified effective in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' While watching videos, participants were required to keep a neutral face as far as possible and control the occurrence of their facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' If they failed and repeatedly showed obvious expressions, they would have to complete an ex- traordinarily long and boring questionnaire as punishment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In addition, they were asked to keep their sitting posture upright, without excessive head movements, and devote their full attention to the video played.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' After watching each video, participants would have a period of rest to ease their emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' During this procedure, they were also asked to fill in an affective grade scale according to the emotional experience generated just now, and form a self-report in- cluding the timestamp where the expression occurred, emo- tion category and intensity based on the video sequences recorded by the high-speed camera, which would help the subsequent annotators understand their MEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Due to the existence of cognitive differences, the emotional orientation of the elicitation materials and the internal emotional expe- rience of participants are sometimes not exactly consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' What’s more, external expressions of the same emotion are also diverse on account of individual differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, it is worth noting that requiring participants to clarify their true inner emotions when expressions appear in their self- reports is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3 ME Annotation Building the DFME dataset required a two-stage annotation: the sample selection stage as well as the coding and cate- gories labeling stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' We clipped short fragments containing valid expression samples from the collected long video sequences in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The second stage included three rounds of fine-grained annotation, through which we con- firmed all MEs and labeled their key-frames, facial muscle action units (AUs), and emotion categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Furthermore, we performed annotation agreement test to verify the reliability of emotion labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 Sample Selection In the sample selection stage, by taking a manual segmen- tation roughly, the video sequences collected containing participants’ facial information were segmented into sev- eral shorter video fragments containing a single or more MaEs or MEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Using the self-developed video annotation software, an experienced annotator checked through the collected original video sequences frame by frame to locate the fragments of facial muscle movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' With the guid- ance of the self-reports from participants, the annotator was able to effectively distinguish whether the facial movements were expressions definitely related to emotion, or interfer- ence data unrelated to emotion (such as violent blinking caused by dry eyelids, habitual mouth opening, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' ), and the former was retained while the latter was abandoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Besides, we also kept some fragments with blinking or eye movements if they contained MaE or ME data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 Coding and Categories Labeling After the previous sample selection stage, three rounds of fine-grained annotation were adopted successively in this stage to determine the MEs together with their three key- frames (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=', onset frame, apex frame and offset frame), facial muscle action unit (AU) labels and emotion category labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The apex frame is the frame corresponding to the mo- ment when facial expression changes most dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In the first round of the fine-grained annotation, five annota- tors independently marked out the onset, apex, and offset frame of each expression clip, and the median value of their annotation results was determined as the final result of the three key-frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Then we filtered the expressions whose duration from onset to offset frame was less than 500ms or from onset to apex frame was less than 250ms as the ME samples, and those out of the time limit were considered as the samples of MaEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' For instance, MEs collected at a frame rate of 500fps should meet either foffset − fonset + 1 ≤ 250 or fapex−fonset+1 ≤ 125, where fk represents the moment index corresponding to the key-frame k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In the second round of fine-grained annotation, we mainly annotated the AUs that occurred in MEs using the Facial Action Coding System (FACS) [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' There may exist one single AU (such as AU4) or a combination of more different AUs (for example, AU6+AU12) in a ME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' When multiple categories of AUs appear, some obscure ones are easily overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' To enhance the reliability and integrity of the AU labels, two experienced annotators independently labeled the AUs for all the MEs identified previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Ac- cording to the actual induction of the participants during the experiments, and also referring to the AUs mainly in- volved in the previously published ME datasets, we totally included 24 different categories of AUs for annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Of these AUs, six categories appear in the upper face, 13 in the lower face, and the other five belong to miscellaneous actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Table 3 lists the specific AU numbers and their corresponding face actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Since the manually annotated AU intensity is highly subjective, to avoid this shortcoming, annotators merely indicated whether each AU appeared during the annotation rather than defining the intensity of its occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' After labeling the AUs, the two annotators determined the final AU label through crosscheck and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The reliability between the two annotators was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='83, which was calculated as R = 2 × AU(A1) ∩ AU(A2) AllAU (1) where AU(A1) ∩ AU(A2) means the number of AUs both annotators agreed, and AllAU is the total number of AUs in a ME labeled out by the two annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In the third round of fine-grained labeling, we performed the emotion labeling of MEs taking eight categories into JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' AUGUST 2022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='TABLE 3: Key AUs Included in DFME ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Upper Face Action Units ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Lower Face Action Units ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Miscellaneous Actions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Inner Brow Raiser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Nose Wrinkler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Lip Pucker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Jaw Clencher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Outer Brow Raiser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Upper Lip Raiser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Lip Stretcher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Nostril Dilator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Brow Lowerer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Lip Corner Puller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Lip Tightener ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Nostril Compressor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Upper Lid Raiser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Dimpler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Lip Presser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='M57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Head Forward ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Cheek Raiser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Lip Corner Depressor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Lips Part ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='M58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Head Back ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Lid Tightener ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Lower Lip Depressor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Lip Suck ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Chin Raiser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(a) Anger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(AU4+AU5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(b) Contempt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(Left-AU6+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Left-AU12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(c) Disgust ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(AU4+AU7+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(d) Fear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(AU1+AU4+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU7+AU20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(e) Happiness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(AU6+AU12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(f) Sadness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(AU17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(g) Surprise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='(AU1+AU2+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='AU5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 3: Representative ME Samples of Seven Basic Emotion Categories in DFME account: anger, contempt, disgust, fear, happiness, sadness, sur- prise, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' ’Others’ represents MEs that are difficult to divide into the former seven prototypical emotion cat- egories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Seven annotators independently gave the emotion labels of all MEs, taking the emotion category that more than half agreed with as the final label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In previous spontaneous ME datasets, the reference basis of emotion labeling was not precisely the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In some datasets, as represented by SMIC, emotion labels were de- termined based on self-reports provided by participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Some other studies believed that seeing is believing, so their annotation was based on the correspondence between AUs and emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' However, on the one hand, unlike MaEs, only part of the AUs can appear simultaneously in MEs due to their low intensity, and some AUs are shared by different emotion categories, which may lead to category confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' On the other hand, we should not ignore the differences in self-emotional cognition of different partici- pants, which means that the self-reports given for the whole piece of elicitation materials may be rough and inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, in DFME, the emotion labels were determined through a comprehensive analysis of facial AUs, self-reports of participants, and elicitation material contents, which is consistent with the method adopted by the CASME series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' It is worth mentioning that we obtained the participants’ fine-grained self-reports in the data collection process, and this is also the information that we recommend as a priority for reference when determining emotion labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' We matched the corresponding timestamps of MEs and elicitation ma- terials through playback, enabling participants to report their emotions for each time of successful ME induction, which significantly improved the confidence of self-reports in emotion labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3 shows some representative ME samples of seven basic emotion categories in DFME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3 Annotation Agreement Having reliable emotion categories of MEs is of vital sig- nificance for a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In this section, we utilized Fleiss’s Kappa test [47] to evaluate the quality of our emotion annotation encouraged by work [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Fleiss’s Kappa is a measure of the agreement among three or more annotators, testing the consistency of annotation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, we consider Fleiss’s Kappa as an excellent indicator to evaluate the reliability of emotion annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In DFME, seven annotators independently labeled each ME sample based on facial AUs, an accurate self-report, and the corresponding elicitation material content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The samples were divided into eight emotion categories: {1: anger, 2: contempt, 3: disgust, 4: fear, 5: happiness, 6: sadness, 7: surprise, 8: others}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' At this time, let n = 7 represent the total number of annotation personnel, N indicate the total number of ME video clips, K = 8 represent the number of emotion categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' nij is the number of annotators who assigned the i-th ME video clip to the j-th category, so we can calculate pj, the proportion of all assignments which were to the j-th emotion: pj = 1 N × n N � i=1 nij, (2) K � j=1 pj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' (3) Then, the extent of agreement among the n annotators for the i-th ME video clip indicated by Pi is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In other words, it can be indexed by the proportion of pairs agreeing in their evaluation of the i-th ME out of all the n(n − 1) possible pairs of agreement: Pi = 1 n × (n − 1)[( K � j=1 n2 ij) − n], (4) The mean of Pi is therefore: P = 1 N N � i=1 Pi, (5) 二JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 9 TABLE 4: AUs of High Occurrence in MEs of Seven Basic Emotion Categories Anger Contempt Disgust Fear Happiness Sadness Surprise AU pct(%)1 AU pct(%) AU pct(%) AU pct(%) AU pct(%) AU pct(%) AU pct(%) AU4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 L/R-AU122 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='7 AU4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='6 AU4 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 AU12 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='8 AU4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 AU1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='6 AU7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 AU6 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 AU7 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4 AU7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3 AU6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='6 AU14 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 AU5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 AU24 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3 L/R-AU10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='6 AU10 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='8 AU5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 AU24 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 AU24 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 AU2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='0 AU5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='6 AU7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='8 AU24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4 AU24 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 L/R-AU12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 AU7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 L/R-AU2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='6 AU23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='6 L/R-AU2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='7 AU14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='7 AU1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 AU10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 AU17 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='8 L/R-AU1 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='8 AU14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='6 AU14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='7 AU14 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='8 AU15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='9 L/R-AU5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='7 AU10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 AU17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='0 AU23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 AU17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='8 AU10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='8 AU1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='8 1 percentage(pct): the statistical range is all MEs from the first 300 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 2 L/R means the Left/Right half part of an AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' And we also have Pe: Pe = K � j=1 p2 j, (6) Finally, we can calculate κ by: κ = P − Pe 1 − Pe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' (7) Thus, we obtained κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='72 through performing Fleiss’s Kappa test in DFME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' According to Table 5, we know that all of our emotion annotators achieve substantial agreement, meaning that our emotion labels are quite reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' TABLE 5: Interpretation of κ for Fleiss’Kappa Test κ Interpretation ≤ 0 Poor agreement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='01-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='20 Slight agreement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='21-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='40 Fair agreement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='41-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='60 Moderate agreement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='61-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='80 Substantial agreement 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='81-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='00 Almost perfect agreement 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4 Statistical Properties of DFME The DFME dataset consists of three parts: PART A, PART B, and PART C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The only difference between these three parts is the frame rate setting of the high-speed camera in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In PART A, all 1,118 ME samples from 72 participants have a frame rate of 500fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The frame rate of PART B is 300fps with 969 ME samples from 92 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' PART C has the most data size with 5,439 ME samples from 492 participants, whose frame rate is 200fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Although we recruited a total of 671 participants, 15 of them had strong control over their facial expressions, from whom we could not collect any ME sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, the final DFME dataset contains 7,526 ME samples from 656 participants, and we gave each sample an emotion category label as well as AU labels annotated according to FACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4 describes the distribution of ME samples detailedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Given that we have collected the fine-grained self- reports and the AU labels with considerable reliability, this is conducive to finding the emotion-AU correspondence rule in MEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, we counted the ratio of high- occurrence AUs in each emotion (Table 4), which reflects the existence preference of AU in MEs with different emotions, not affected by the emotional category imbalance problem in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' We also matched the emotion and AU combina- tions according to the statistical results, and the conclusions are shown as Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' TABLE 6: Matching Emotion and AU Combinations in MEs Emotion Categories AU Combinations Anger AU4+AU5, AU23 Contempt L/R-AU12, AU6+L/R-AU12 Disgust AU4+AU7+AU10, AU14 Fear AU14+AU24, AU1+AU4, AU4+AU5 Happiness AU6+AU12, AU12 Sadness AU14, AU17, AU15, AU14+AU24 Surprise AU1+AU2+AU5, AU1+AU2, AU5 Shared1 AU4, AU4+AU7, AU7, AU24 1 Shared: the AU combinations commonly appearing in Anger, Disgust, Fear and Sadness with high frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Based on the statistical results presented in Table 4, we have some findings to discuss: In MaEs, AU9 (nose wrinkler) is highly associated with disgust, and AU20 (lip stretcher) is related to fear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' These two AUs frequently appear in MaEs but are not easily induced in MEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' We ought not to conclude that these AUs’ association with their corre- sponding emotions no longer exists in MEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Instead, when participants tried to restrain their emotions, it was easier for them to control the movement of certain facial muscles such as AU9 and AU20 rather than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' AU4 (brow lowerer), AU7 (lid tightener), and AU24 (lip presser) simultaneously occur at high frequency in different negative emotions (disgust, anger, fear, sadness, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Without the assistance of participants’ fine-grained self-reports, it is definitely challenging to distinguish MEs of negative emotions merely rely- ing on these common AUs, which is also one of the reasons why some models excessively confuse the disgust MEs with those of other negative emotions in the seven-classification automatic MER task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In the positive emotion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=', happiness), some AUs related to negative emotions can occur together with AU6 or AU12, specifically, including AU10 (associ- ated with disgust), AU24 (associated with negative emotions), and Left/Right-AU12 (associated with contempt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The appearance of these extra AUs is a JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 10 Disgust Surprise Happiness Fear Sadness Anger Contempt Others PART A 321 187 111 143 142 97 77 40 PART B 406 143 78 115 119 56 45 7 PART C 1801 878 803 634 374 466 279 204 Combined 2528 1208 992 892 635 619 401 251 2528 1208 992 892 635 619 401 251 0 500 1000 1500 2000 2500 Positive Surprise Negative Others Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4: Distribution of ME Samples in DFME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Each column represents the total sample number of an emotion category, and the three pieces colored from light to deep show the proportion of samples in PART A, PART B, and PART C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' sign of participants trying to suppress their positive feelings, hide their smiles and twist their expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4 DATASET EVALUATION In this section, we conducted comprehensive experiments to verify the effectiveness of our DFME dataset for auto- matic MER task based on influential spatiotemporal feature learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In addition, we specifically analyzed the class imbalance problem in ME datasets, and explored two kinds of strategies to solve the class imbalance problem in our DFME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Furthermore, we explored the influence of different sampling strategies of ME key-frame sequence on MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' These experiments can provide reference for future MER research using DFME dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 Evaluation Dataset The DFME dataset is described in detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' For the subsequent MER verification, we combined 7, 275 samples with clear emotion labels in PART A, B and C of DFME as our experimental dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The emotion labels include disgust, surprise, happiness, fear, sadness, anger and contempt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 Data Preprocessing In facial expression recognition, many variables, such as backgrounds, head poses and unequal video lengths, can affect the final recognition results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, before formally conducting automatic MER experiments, we need to prepro- cess all ME videos in the following steps to minimize the influence of irrelevant variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 Face Alignment To eliminate the differences in pose and angle among all ME samples, we need to perform face alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In this step, we took the following operations for each ME sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' We first selected a frontal face image as a reference and adopted Style Aggregated Network (SAN) [49] to extract its facial landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Afterwards, we used Procrustes analysis [50] to compute an affine transformation based on landmarks of the onset frame and landmarks of the reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The reason why we did not use landmarks of all frames in the ME video is to avoid errors introduced by the calculation of landmarks and transformations having a significant impact on real MEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Finally, the transformation was operated for each frame to align the faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Besides, some landmarks are located in regions where MEs may appear, which may not be stable enough for alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Thus, we excluded such landmarks when performing the alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 Face Cropping Since the movement of MEs is mainly in the facial area, face cropping is also a necessary step to eliminate the bias caused by different backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' After face alignment, we chose RetinaFace [51] to crop the faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' For reasons similar to face alignment, face cropping was based on the onset frame instead of each frame of a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3 ME key-frame sequence sampling Different ME videos have different lengths, while deep learning models usually require a fixed input size, which is shorter than ME sample lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Before inputting into the model, we need to normalize the temporal length of all ME videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In general, video classification models usually adopt the uniform sampling method to unify the video length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' However, this processing strategy is coarse-grained for recognizing ME with local and subtle movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Fol- lowing previous studies [12], [44] and to be compatible with popular video classification models, this work extracts 16 key-frames from each ME video based on the annotated three ME key-frames (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=', onset frame, apex frame, and offset frame) and temporal adaptive sampling strategy [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3 Evaluation Protocols and Metrics Due to the small sample size of previous datasets such as CASME II [14], SAMM [15], and SMIC [13], most MER stud- JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 11 ies adopted the leave-one-subject-out strategy when evalu- ating on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Nevertheless, considering that the number of ME clips in DFME is relatively large, this paper put to use a simpler and more efficient 10-fold cross-validation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' For each fold, 10% of the data were sampled as the test set, and the remaining 90% as the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In addition, three commonly used MEs classification indicators, namely Accu- racy, Unweighted F1-Score and Unweighted Average Recall, were used to evaluate the MER performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Specifically, before calculating them, we need to obtain the True Positive (TPi), False Positive (FPi), and False Negative (FNi) for each class i (K classes in total, and K = 7 in DFME).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In the end, we took the average results of ten experiments as the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 Accuracy (ACC) Accuracy is one of the most common metrics, which can evaluate the overall performance of the recognition method on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' It was calculated as follows: ACC = K � i=1 TPi Ni , (8) where Ni is the number of samples of the i-th class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 Unweighted F1-score (UF1) Unweighted F1-score (UF1), also known as macro-averaged F1-score, is defined as shown below: UF1 = 1 K K � i=1 UF1i, (9) where we have: UF1i = 2 · TPi 2 · TPi + FPi + FNi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' (10) Class imbalance is an intractable problem in the MER task, so introducing UF1 as an evaluation metric can better mea- sure the method’s performance in all classes rather than in some major classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3 Unweighted Average Recall (UAR) Unweighted Average Recall (UAR) is also a more reason- able metric than accuracy in case of class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' UAR = 1 K K � i=1 TPi Ni .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' (11) Both UF1 and UAR can effectively evaluate whether MER methods give correct predictions in all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4 Evaluation Baseline Models Although the spatiotemporal convolution models with deeper layers and more parameters have achieved amazing performance in the video classification tasks, due to the scarcity of ME data, previous MER studies rarely use such a model with a large number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In fact, both time and space contain unique features of ME, and MER should take into account both dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' To verify the feasibility of applying large 3D models on our large-scale dataset and to provide a reference for backbone selection of MER methods based on extensive data, we have selected the following standard backbone networks based on 3D convolution architecture for validation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 3D-ResNet (R3D) Hara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' proposed 3D-ResNet (R3D) [52] for tasks such as video classification and recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Since then, R3D is often used as the backbone in approaches to video-related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The basic idea of this model is to replace the 2D convolu- tional kernels with spatiotemporal 3D kernels according to the 2D-ResNet [29] network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='2 Pseudo-3D ResNet (P3D) Pseudo-3D ResNet (P3D) [53] is another 3D model back- bone that has achieved good results in video tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' It can be considered as an improved version of R3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The key point of this model is the simulation of the 3×3×3 convolution filter by using a 1×3×3 spatial domain convolution filter and a 3×1×1 temporal domain convolution filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Hence the authors named it Pseudo-3D ResNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' This change controls the model size and improves training efficiency and experi- mental performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3 3D-DenseNet (D3D) DenseNet [54] has achieved excellent performance in image tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' It expanded the residual connection of ResNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' All layers in DenseNet connect directly with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 3D- DenseNet (D3D) has also been widely used in the video field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In the field of MER, Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' [55] proposed a 3D- DenseNet-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4 Inflated 3D ConvNet (I3D) Inflated 3D ConvNet (I3D) [56] is based on 2D ConvNet in- flation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The model size has increased significantly compared to the 2D model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, the data requirements have also increased significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' For this reason, the authors also published a large-scale video dataset Kinetics [56] simulta- neously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The results on Kinetics demonstrate the excellent performance of I3D when the amount of data is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 Evaluation Implementation Settings Our MER experiments were all conducted on 2 NVIDIA GeForce RTX 3090 GPUs or a single NVIDIA A100-PCIE- 40GB GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Following the original settings, the length of ME clips for all models was 16 frames, and for R3D, P3D, D3D and I3D, the sizes of each input image were 224×224, 160×160, 224×224 and 224×224 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' During training, cross-entropy loss and stochastic gradi- ent descent (SGD) with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='9 were used to optimize the model parameters, and the batch size was set to 32 for all four models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' For R3D, P3D, D3D, and I3D, the initial learning rates were set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='05, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='005, respectively, and learning rates were divided by 10 every 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='6 Evaluation Baseline Results To demonstrate the effectiveness of our DFME dataset for automatic MER tasks, we conducted a comprehensive MER experiment based on the above four baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The evaluation baseline results are shown in Table 7, and the recognition confusion matrix of each baseline model is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 12 anger contempt disgust fear happiness sadness surprise Predicted label anger contempt disgust fear happiness sadness surprise True label 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='48% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='39% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='68% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='99% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='18% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='93% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='90% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='81% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='20% I3D model 10 20 30 40 50 60 70 (d) I3D Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 5: Confusion matrices of R3D, P3D, D3D and I3D baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' From Table 7, we can easily find that the I3D model achieved the best performance among the four backbone models with an average accuracy of 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='24%, an average UF1 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4576 and an average UAR of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4526, and the accuracy is higher than the 47% achieved by naked eyes [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Besides, the other three models were approximately as accurate as the naked eye in DFME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The above experimental results demonstrate the reliability of our DFME and provided a reference for the selection of backbone models for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Meanwhile, by observing the recognition confusion matrices shown in Figure 5, we also find that all baseline models present the same phenomenon, that is, these models are more inclined to recognize the categories with more samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Obviously, this is mainly caused by the class im- balance problem in DFME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, how to learn more distinguishable spatiotemporal ME features from the ME data with unbalanced classes is a vital exploration direction of MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Besides, confusion matrices shown in Figure 5 illustrate that for all four backbone models, the disgust and fear samples are the most difficult to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' This result is consistent with the statistics of the AU frequencies in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In both disgust and fear samples, the most frequent AUs are AU4 and AU7, and AU10, AU14, and AU24 are also found in both classes of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' TABLE 7: ME recognition performance of various baseline models Models ACC UF1 UAR R3D [52] 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='54% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3827 P3D [53] 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='77% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3830 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3801 D3D [55] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='26% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4107 I3D [56] 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='24% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4526 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='7 Evaluation Discussion This section will focus on two key problems that are particu- larly considered when using our DFME for MER, including class imbalance problem and various key-frame sequence sampling strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='1 Class imbalance in DFME Since the existence of individual differences of subjects and the different inducing degrees of each category of ME, the collected spontaneous ME dataset is hard to avoid the problem of class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' This is directly reflected in the previous three datasets widely used in MER, including SMIC, CASME II and SAMM, whose ratio of the most category to the least category is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='63, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='52 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='13 [58], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Inevitably the class imbalance problem still exists in our DFME dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The statistic of emotion categories in DFME is shown in Table 3, from which we can find that the number of disgust samples is the largest among all emotion categories, accounting for about 1/3 of the proportion, and the negative samples (including disgust, fear, sadness, anger and contempt) accounted for about 2/3 of the proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Moreover, the confusion matrices in Figure 5 indicated the negative impact of class imbalance on models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' All four backbone models tended to predict samples as disgust class more than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' To solve the class imbalance problem, introducing a class rebalancing strategy is an effective solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In general, the class rebalancing methods can be roughly divided into two major categories: resampling and cost-sensitive reweighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' TABLE 8: MER Performance with and without Resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Metrics Resampling1 ACC UF1 UAR R3D w/o 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='54% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3827 w 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='05% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3659 P3D w/o 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='77% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3830 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3801 w 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='02% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4078 D3D w/o 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='26% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4107 w 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='37% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4656 I3D w/o 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='24% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4526 w 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='91% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4924 1 w/o: without resampling, w: with resampling Resampling is one of the most widely used class rebal- ancing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Moreover, uniform resampling is a fairly common one of all resampling strategies, which is also used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Its main idea is to select each class of samples with an equal probability when training models, rather than sampling all samples uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Table 8 and Figure 6 show the comparison of the re- sults with and without uniform resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The resampling strategy improved UAR and UF1 on the three models except for R3D, but the accuracy decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' With the introduction of the uniform resampling strategy, the model could better learn the features of minor classes, but at the cost of weak- ening the ability to predict major classes correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' How to JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 13 R3D P3D D3D I3D 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='0 ACC (%) Accuracy (ACC) without Resampling with Resampling (a) ACC R3D P3D D3D I3D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='50 UF1 Unweighted F1-Score (UF1) without Resampling with Resampling (b) UF1 R3D P3D D3D I3D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='50 UAR Unweighted Average Recall (UAR) without Resampling with Resampling (c) UAR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 6: Comparison of MER results with and without Resampling reduce the information loss of the major classes in MER is a problem that needs to be addressed in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Reweighting approaches attempt to rebalance different classes by reweighting their loss during training models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Class-Balanced Loss (CBLoss) [59] is a representative of reweighting loss, which is simple and effective and, there- fore, used extensively in different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' CBLoss proposed the concept of effective number to estimate the actual impact of samples of each class on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' It can also be combined with other losses, including Focal Loss [60], which reweighted samples in different classes according to their difficulty to be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' This feature further enhances the adaptability of CBLoss to different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The losses we calculated in our experiments are shown in Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' The results of CBLoss are shown in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Similar to uniform resampling, CBLoss also improved the UAR and UF1 for all four models at the cost of ACC in our experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' This result demonstrates that CBLoss is compatible with various models and suffers from similar problems as resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Besides, CBLoss can be easily used for different tasks with different models, but we should carefully fine- tune it in various conditions to achieve better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In particular, the choice of β may need further study, which controls the relationship between the effective number and the actual number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' TABLE 9: MER Performance with Different Losses Metrics Losses ACC UF1 UAR R3D Cross Entropy Loss 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='54% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3827 Class Balanced Loss 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='61% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3914 P3D Cross Entropy Loss 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='77% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3830 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3801 Class Balanced Loss 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='23% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3955 D3D Cross Entropy Loss 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='26% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4107 Class Balanced Loss 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='25% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4302 I3D Cross Entropy Loss 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='24% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4526 Class Balanced Loss 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='56% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4789 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4777 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='8 ME key-frame sequence sampling Strategies The key-frame sequence is a concise description of the original video, which generally contains key information about the content of the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' How to sample effective ME key-frame sequence from the raw video is also an im- portant factor for accurate recognition of ME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Video-related TABLE 10: Cost-Sensitive Reweighting Losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In this table, py and ny are the softmax probability and the sample number of the class y, and β is the hyperparameter in Class- Balanced Loss (β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='999 in our experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Loss Equation Cross Entropy Loss Lce = −log(py) Class-Balanced Loss [59] Lcb = − 1−β 1−βny log(py) recognition tasks usually adopt uniform sampling to obtain a fixed-length key-frame sequence as model input, but the instantaneously changing ME movements are often not uniformly distributed in spatial-temporal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Previous studies [12], [44] have shown the superiority of key-frame temporal adaptive sampling based on three key moments of ME video, namely onset, apex and offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Therefore, we hereby compare and analyze the corresponding recognition performance of these two sampling strategies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=', uniform sampling and temporal adaptive sampling) in DFME using baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' TABLE 11: Comparison of MER Performace with Different Key-Frame Sequence Sampling Strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Metrics Sampling Method1 ACC UF1 UAR R3D adaptive 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='54% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3827 uniform 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='49% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3710 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3715 P3D adaptive 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='77% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3830 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3801 uniform 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='31% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3671 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='3656 D3D adaptive 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='26% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4107 uniform 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='62% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4203 I3D adaptive 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='24% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4526 uniform 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='21% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4621 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='4576 1 adaptive: adaptive sampling in [44], uniform: uniform sampling Table 11 and Fig 7 show the recognition performance of uniform sampling and temporal adaptive sampling [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' It is clear that the temporal adaptive sampling strategy achieved better results on R3D and P3D models while performing worse on D3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' For I3D, the recognition performance of the two sampling strategies is comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' This result suggests that different baseline models may require different sam- pling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 14 R3D P3D D3D I3D 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='0 ACC (%) Accuracy (ACC) Uniform Sampling Adaptive Sampling (a) ACC R3D P3D D3D I3D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='46 UF1 Unweighted F1-Score (UF1) Uniform Sampling Adaptive Sampling (b) UF1 R3D P3D D3D I3D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='46 UAR Unweighted Average Recall (UAR) Uniform Sampling Adaptive Sampling (c) UAR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 7: Comparison of MER results of Adaptive Key-frame Sampling and Uniform Key-frame Sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 5 CONCLUSION AND FUTURE WORK In this work, we focused on solving the problem of lacking abundant spontaneous ME data for MER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' To this end, we built a new ME dataset called DFME containing 7,526 ME videos across multiple frame rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' To the best of our knowl- edge, DFME has the largest ME sample size at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Furthermore, to verify the feasibility and validity of DFME dataset for MER task, we reproduced four spatiotemporal visual feature learning models to carry out MER task in DFME, objectively verifying the reliability of data quality, and providing a benchmark for subsequent MER studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Particularly, we explored and analyzed two key problems when using DFME for MER, including class imbalance and key-frame sequence sampling, so as to provide directions for future MER studies using DFME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' In the future, we will strive to expand the DFME dataset to provide more abundant ME data for automatic ME analysis research, including the collection of multimodal ME data in multiple natural scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Based on this, we will also study the high accuracy and robust MER models, such as self-supervised MER combined with more samples with uncertain labels, and apply them to actual scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 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+page_content=' Girshick, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Doll´ar, “Focal loss for dense object detection,” 2017 IEEE International Conference on Computer Vision (ICCV), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 2999–3007, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' 8, AUGUST 2022 16 Sirui Zhao is currently working toward the PhD degree with the Department of Com- puter Science and Technology from University of Science and Technology of China (USTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' His research interests include automatic micro- expressions analysis, human-computer interac- tion (HCI) and affect computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He has pub- lished several papers in refereed conferences and journals, including ACM Multimedia Confer- ence, IEEE Transactions on Affective Comput- ing, ACM TOMM, Neural Networks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Huaying Tang received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' degree in the School of Computer Science and Technology from University of Science and Technology of China (USTC), Hefei, China, in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He is currently pursuing the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' degree in computer science and technology in USTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' His research interests lie around automatic micro-expressions analysis and affect computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Xinglong Mao received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='S degree in the School of Data Science from University of Science and Technology of China (USTC), Hefei, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He is currently working toward the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' degree from the School of Data Sci- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' His research interests include automatic micro-expressions analysis and affect comput- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He has published several conference papers in ACM Multimedia Conference, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Shifeng Liu received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='S degree in the School of Gifted Young from University of Sci- ence and Technology of China (USTC), Hefei, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' She is currently working toward the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' degree from the School of Data Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Her research interests include automatic micro- expressions analysis, human-computer interac- tion (HCI) and affect computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' She has pub- lished several papers in refereed conferences and journals, including ACM Multimedia Confer- ence, Neural Networks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Hanqing Tao is currently working toward the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' degree in the Department of Computer Science and Technology from University of Sci- ence and Technology of China (USTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' His re- search interests include data mining, deep learn- ing, natural language processing and represen- tation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He has published several papers in referred journals and conference proceedings, such as IEEE TKDE, IEEE TAC, AAAI, ICDM, ICME etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Hao Wang received the PhD degree in computer science from USTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He is currently an associate researcher with the School of Computer Science and Technology, USTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' His main research inter- ests include data mining, representation learn- ing, network embedding and recommender sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He has published several papers in re- ferred conference proceedings, such as TKDE, TOIS, NeuriPS, and AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='. Tong Xu received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' degree in University of Science and Technology of China (USTC), Hefei, China, in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He is currently working as an Associate Professor of the Anhui Province Key Laboratory of Big Data Analysis and Ap- plication, USTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He has authored 50+ journal and conference papers in the fields of social network and social media analysis, including IEEE TKDE, IEEE TMC, IEEE TMM, KDD, AAAI, ICDM, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' Enhong Chen (Sensor Member, IEEE) received the PhD degree from USTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He is a professor and vice dean with the School of Computer Sci- ence, USTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' His general area of research in- cludes data mining and machine learning, social network analysis, and recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He has published more than 100 papers in ref- ereed conferences and journals, including IEEE Transactions on Knowledge and Data Engineer- ing, IEEE Transactions on Mobile Computing, KDD, ICDM, NeurIPS, and CIKM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' He was on program committees of numerous conferences including KDD, ICDM, and SDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} +page_content=' His research is supported by the National Science Foundation for Distinguished Young Scholars of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAzT4oBgHgl3EQfD_po/content/2301.00985v1.pdf'} diff --git a/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf b/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5ef0e0641acf8f2f205a01b799419103f02370ed --- /dev/null +++ b/49AyT4oBgHgl3EQfpPiQ/content/2301.00522v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ebd1f65e76542de74aea043e97551a159a006a3c62ebf1c04ee362c0f2a8a2e8 +size 231558 diff --git a/49AyT4oBgHgl3EQfpPiQ/vector_store/index.faiss b/49AyT4oBgHgl3EQfpPiQ/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ca13a30493bf538aec2f1f2112b4b9775d5032e2 --- /dev/null +++ b/49AyT4oBgHgl3EQfpPiQ/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b4c57b3825d10ba6dc6b867f4056323c9413b134ea31b032d5d942b848943616 +size 3473453 diff --git a/49AyT4oBgHgl3EQfpPiQ/vector_store/index.pkl b/49AyT4oBgHgl3EQfpPiQ/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..c16fb9511984636e44c7214e2933661036090f06 --- /dev/null +++ b/49AyT4oBgHgl3EQfpPiQ/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:84e4beedbd1afb1f3fc57beb238f4058b0e3a47317821c845670ff9a9e8c367a +size 132227 diff --git a/4dAyT4oBgHgl3EQf2Pkf/content/tmp_files/2301.00746v1.pdf.txt b/4dAyT4oBgHgl3EQf2Pkf/content/tmp_files/2301.00746v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..083e8ff0969b87becb58d4b0beed4956241135e5 --- /dev/null +++ b/4dAyT4oBgHgl3EQf2Pkf/content/tmp_files/2301.00746v1.pdf.txt @@ -0,0 +1,1615 @@ +NaQ: Leveraging Narrations as Queries to Supervise Episodic Memory +Santhosh Kumar Ramakrishnan1, Ziad Al-Halah1, Kristen Grauman1,2 +1UT Austin, 2Meta AI +Abstract +Searching long egocentric videos with natural language +queries (NLQ) has compelling applications in augmented +reality and robotics, where a fluid index into everything +that a person (agent) has seen before could augment human +memory and surface relevant information on demand. How- +ever, the structured nature of the learning problem (free- +form text query inputs, localized video temporal window +outputs) and its needle-in-a-haystack nature makes it both +technically challenging and expensive to supervise. We in- +troduce Narrations-as-Queries (NaQ), a data augmentation +strategy that transforms standard video-text narrations into +training data for a video query localization model. Vali- +dating our idea on the Ego4D benchmark, we find it has +tremendous impact in practice. NaQ improves multiple top +models by substantial margins (even doubling their accu- +racy), and yields the very best results to date on the Ego4D +NLQ challenge, soundly outperforming all challenge win- +ners in the CVPR and ECCV 2022 competitions and topping +the current public leaderboard. Beyond achieving the state- +of-the-art for NLQ, we also demonstrate unique properties +of our approach such as gains on long-tail object queries, +and the ability to perform zero-shot and few-shot NLQ. +1. Introduction +Human memory can fail us in day-to-day things in our +visual experience. We misplace objects in the house (where +is my passport?), we lose track of what tasks we have or +have not done (did I add the salt already?), we forget where +we did a given activity (where did I buy tickets last time?), +we do not notice the state of an object in our environment +(did I leave the garage door open?). First-person or “ego- +centric” perception on a wearable camera could reduce that +cognitive overload and provide us with a superhuman per- +sonal episodic memory—by seeing what we see, and index- +ing it in meaningful and easy-to-access ways. +This is the vision of the Natural Language Query (NLQ) +task in Ego4D’s Episodic Memory benchmark [12]. Given +a natural language question and a long egocentric video, the +NLQ task requires identifying the precise temporal window +. . . +. . . +Query: How many eggs did I break into the bowl? +Response +Figure 1. Episodic memory with natural language queries (NLQ) +aims to search long egocentric videos to identify the temporal +“response” window revealing the answer to a free-form question +about the camera wearer’s past visual experience. +in the camera wearer’s past video that reveals the answer. +See Figure 1. Such functionality could transform the every- +day experience of an augmented reality user with always- +on AR glasses. It could similarly play a role for a mobile +household robot, whom a user may wish to query about its +own visual history (have you seen my keys?). +The NLQ challenge has attracted substantial attention +in the research community over the last year [18, 19, 31] +as have related video-language efforts for question answer- +ing [23, 26–30]. +The technical challenges are striking. +Queries are free-form natural language, response windows +are tiny slivers (a few seconds or less) within a long stretch +of video, and wearable camera video is notoriously noisy +with its quick head motions and limited field of view. +Today’s most successful methods embrace the visual- +language aspect of the problem. In particular, inspired by +the growing success of visual-linguistic embeddings [17, +20,22,25,28], top competitors on NLQ perform large-scale +pretraining on ⟨video clip, text description⟩ pairs mined +from the Ego4D dataset’s provided narrations [18], which +are timestamped play-by-play descriptions of the camera- +wearer’s activity (see Figure 2). The result is a video back- +bone enhanced by the semantics of grounded language. +1 +arXiv:2301.00746v1 [cs.CV] 2 Jan 2023 + +C turns on the tap with her right hand +C opens a drawer +C cracks an egg into the bowl +C opens the third refrigerator door +Figure 2. Narration examples. “C” refers to the camera-wearer. +While it is important to have strong video and text repre- +sentations, the downstream query localization models that +search the video for a response are also crucial to NLQ, yet +relatively starved for data. This is a direct consequence of +the difficulty in annotating a query-response pair (which en- +tails posing a creative question and scrolling the long video +to mark the temporal response window) versus the relative +ease in narrating a video (which entails pausing the video +at regular intervals and writing down what happened). For +example, whereas Ego4D has 3,670 hours of data annotated +with narrations—more than 3.85M sentences in total—it of- +fers only 227 hours of NLQ query examples, for 19k total +text queries. Accordingly, existing methods risk failing to +learn the task-specific skills that are poorly represented in +training, such as responding to queries about objects in the +long-tail or performing complex reasoning for queries in- +volving interactions between multiple visual entities. +To address this issue, we introduce Narrations-as- +Queries (NaQ), a simple but exceptionally effective data +augmentation strategy for NLQ. NaQ is a novel strategy +that uses timestamped narrations to expand the supervision +available for training query-localization modules within an +episodic memory architecture. Our hypothesis is that nar- +rations provide descriptive information that is localizable in +long videos, and thus can benefit an episodic memory model +when used as training queries. +Specifically, we derive +⟨video, language query, temporal window response⟩ anno- +tations from timestamped narrations, and augment the con- +ventional query-response data with these pseudo-queries. +Importantly, this allows us to influence the localization +module—the workhorse responsible for finding a needle in +a haystack—with multimodal data, as opposed to just the +video and text encoders. +Empirically, our idea has tremendous impact. Demon- +strating NaQ on the Ego4D Episodic Memory benchmark, +we find our simple augmentation strategy successfully com- +plements multiple existing state-of-the-art episodic mem- +ory methods, achieving sizeable improvements (e.g., 32% +to 125% relative jumps in accuracy) across query types, +metrics, and methods. Notably, our gains hold even com- +pared to existing methods such as EgoVLP [18] that use the +same (or even more) narration annotations as our model, +meaning that NaQ’s success can be attributed to good mod- +eling, not more data. Moreover, to our knowledge, NaQ +yields the very best results to date on the NLQ chal- +lenge, strongly outperforming all the challenge winners +from Ego4D CVPR’22 and Ego4D ECCV’22 by a substan- +tial margin, and topping the current public leaderboard. Be- +yond achieving state-of-the-art results, we perform a thor- +ough analysis of the strengths and weaknesses of NaQ, and +demonstrate useful properties such as benefits on long-tail +object queries as well as zero-shot and few-shot NLQ. We +are the first to do so. +2. Related work +Egocentric video understanding. +Prior work has devel- +oped video datasets and methods for egocentric percep- +tion [4, 8, 10, 12, 14]. +Egocentric video offers a camera +wearer’s perspective of their activities over a long time +horizon and raises challenging research problems such as +human-object interactions [3, 5], activity recognition [14, +33], anticipation [1, 11], episodic memory [12], and video +summarization [6,16]. In this work, we tackle the episodic +memory task. We leverage the Ego4D dataset [12], which +consists of 3,670 hours of video of daily-life activity cap- +tured by 931 camera wearers around the world. +Vision-language +pretraining. +Vision-Language +Pre- +training (VLP) methods rely on large-scale video-text +datasets [2, 21] to learn transferable representations for +video-language tasks such as retrieval [7, 13], question- +answering [23,27] and video captioning [15,32]. VideoBert +learns joint video-text embeddings by discretizing video +frames +into +tokens +and +performing +BERT-like +pre- +training [25]. HERO improves over this with a hierarchical +encoding of multi-modal inputs to better capture long-term +structure [17]. MIL-NCE learns to match clips with tempo- +rally close captions to address video-text misalignment in +HowTo100M [20,21]. While these methods primarily focus +on third-person videos, EgoVLP [18] adapts the InfoNCE +objective to egocentric settings and uses video-narration +annotations from Ego4D [12] to learn video-text backbones +for egocentric video understanding tasks. +Just-Ask [28] +proposes a strategy to generate video question-answering +data consisting of (short clips, questions, text answers) +from narrated YouTube videos. +While we take inspiration from such methods, our idea is +very different. Unlike prior work that learns representations +or video-QA systems from short video clips and aligned +(possibly weak) text, we learn to temporally localize short +2 + +pJ(5)text queries in long untrimmed videos egocentric videos. +Whereas Just-Ask’s data generation procedure [28] outputs +questions with text responses for short video clips, ours out- +puts temporal windows in long videos. Rather than pretrain- +ing a video/text backbone [17,18,20,25], our model injects +multimodal supervision to train a query-localization mod- +ule. Overall, our idea is complementary to prior video-text +pretraining efforts, as we will demonstrate in the results. +Episodic memory. +The episodic memory benchmark’s +natural language queries (NLQ) task was first introduced +in the Ego4D dataset [12]. In NLQ, the goal is to tem- +porally localize the response to a natural language text +question. Existing video-language grounding methods like +2D-TAN [30] and VSLNet [29] have been adapted to per- +form this task. Our goal is to improve such methods via +large-scale data augmentation with narration-based queries. +More recently, ReLER [19] achieved the state-of-the-art for +NLQ by using a multi-scale and cross-model transformer +with video-level data augmentation and contrastive losses. +Our proposed strategy performs query-level augmentation +and is complementary to the video-level data augmentation +from [19]. As we will demonstrate in experiments, our ap- +proach stacks well when combined with prior NLQ meth- +ods [18,19,29]. +3. Approach +Our key insight is to leverage narrations as an additional +data source to improve a model’s ability to localize answers +in a long video when prompted with a natural language +query. To do this, we propose a strategy to convert narra- +tions and their timestamps into episodic memory queries. +Our strategy is automatic and simple which allows us to +scale the training data for episodic memory search by two +orders of magnitude. Furthermore, we generate the data in +a form that is compatible with the manually labeled NLQ +annotations, which allows an NLQ model to directly take +advantage of this additional data source and achieve signif- +icant improvements in performance without any modifica- +tions to the model itself. +Next, we define the episodic memory task (Sec. 3.1), +then describe our Narrations-as-Queries approach to con- +vert narrations into natural language queries (Sec. 3.2), and +finally describe our training strategy (Sec. 3.3). +3.1. Episodic memory with natural language query +The goal of episodic memory is to perform query-driven +reasoning about long-form egocentric videos. First intro- +duced in Ego4D [12], it is well-motivated by applications +discussed above, such as augmented reality assistants that +enable superhuman memory. The NLQ task has attracted +significant attention in the research community, with 10+ +teams from labs around the world competing on the bench- +mark over the last year [18, 19, 31], two organized chal- +lenges at CVPR’22 and ECCV’22, and an active public +leaderboard1. +More formally, given an egocentric video V capturing +a camera wearer’s past experiences and a natural language +query Q in the form of a question, the task requires tempo- +rally localizing where the answer can be seen in the video, +i.e., a response window R = [ts, te]. For example, the +query could be Q =“What vegetables did I put in the soup +the last time I made it?”, and the model needs to search a +given video V to identify the time window [ts, te] that con- +tains the answer, i.e., the type of vegetables in the soup. +A data sample for this task is of the form ⟨video, query, +response⟩. The video can be several minutes long, and the +response to the query can appear in a time window that is +shorter than a second, making this a very challenging task. +3.2. Narrations-as-Queries +Prior NLQ methods are limited in performance due to +the lack of large-scale NLQ annotations of the form ⟨video, +query, response⟩. We address this limitation by proposing +a method to automatically transform narrations associated +with egocentric videos to a compatible form for NLQ. Nar- +rations are free-form sentences describing the current ac- +tivity performed by the camera-wearer (see Fig. 2). They +are time-stamped and temporally dense (e.g., there are 13.2 +sentences per minute of video on average in Ego4D [12]). +These annotations are substantially cheaper to obtain +than NLQ annotations. For narrations, the annotators needs +to simply describe the activity that is seen in the video; +whereas for NLQ, first a meaningful, unambiguous ques- +tion needs to be formulated and then the annotator needs +to manually search the video back and forth to identify the +time window that shows the answer. Hence, narrations can +be annotated at a much larger scale compared to NLQ sam- +ples (e.g., Ego4D has 3.85M narrations compared to 19K +NLQ samples). +Our idea is to leverage this massive data source to aid the +learning in the NLQ task. We achieve this by first generat- +ing a temporal window associated with each narration that +approximately captures when the activity described by the +narration started and ended. Then, we use these samples +(narrations coupled with temporal windows) as additional +supervision to train an NLQ localization model to identify +where these narrations happen in the video (see Fig. 3). +Next, we formally describe our approach in detail. +1. Generating temporal windows for narrations. +Each +video narration consists of a textual sentence T , and a single +timestamp t marking the correspondence to the underlying +video (see Fig. 3, left). However, this is incompatible with +1NLQ challenge leaderboard: https://eval.ai/web/challenges/ +challenge-page/1629/leaderboard/3920 +3 + +Text +Encoder +Video +Encoder +queries +videos +Query Localization +Module +( Ƹ𝑡𝑠, Ƹ𝑡𝑒) +NLQ Model +. . . +. . . +NLQ Dataset +How many eggs did I break? +Narrations-as-Queries (NaQ ) +C takes the ingredients out of the shelf +𝑉𝑗 +𝑇𝑖 +𝑅𝑖 +𝑡𝑖 ++𝛽/2𝛼 +−𝛽/2𝛼 +Seed Temporal Window +Temporal Response Jittering +𝑡𝑠 +𝑡𝑒 +.. +.. +.. +responses +𝑉 +𝑄 +𝑅 +. . . +. . . +𝑡𝑖 +−𝑠Δ ++𝑠Δ +𝑡𝑐 −𝛿𝑡 +Δ +ത𝑅𝑖 +Figure 3. Narrations-as-Queries: We propose a simple-yet-effective data-augmentation strategy for natural language queries (NLQ). +The status-quo NLQ methods train in a supervised fashion on annotated (V: video, Q: query, R: response) tuples, where the response +is a (ts, te) temporal window (see right). This is severely limiting, since such task-specific data is expensive to obtain and is available +only on a small scale. We propose a narrations-as-queries pipeline to tackle this issue (see left). Our key idea is to leverage densely +annotated video narrations, where each narration Ti for video Vj is a textual description of the camera-wearer’s activity at time ti. We +propose “temporal response jittering”, a technique to convert timestamped narrations into natural language queries with temporal response +windows ⟨Vj, Ti, Ri⟩ and obtain the NaQ dataset, which contains 80× more samples when compared to the NLQ dataset. We then train +various NLQ models jointly on the NLQ and NaQ datasets to obtain significant gains across query types, architectures, and metrics. +NLQ task architectures which require queries and tempo- +ral response windows as supervision. To address this, we +propose temporal response jittering, a technique to convert +narration timestamps to temporal windows conditioned on +the video. +Temporal response jittering: Our goal is to convert a +narration timestamp ti from video Vj into a response win- +dow Ri = (ts, te). +First, we use “contextual variable- +length clip pairing strategy” introduced in EgoVLP [18] to +obtain a video-conditioned seed temporal window centered +around ti: +¯ +Ri = [ti − βi/2α, ti + βi/2α] +(1) +where βi captures the average temporal length between con- +secutive narrations in video Vj, and α is the average of all +βi across all videos (please see [18] for details). While this +offers a good starting point, it fails to address the inherent +noise in ¯ +Ri arising from the lack of explicit human annota- +tion. The responses generated are also typically short (less +than a second) and do not match the distribution over NLQ +response windows that are 10 seconds long on average. To +account for these factors, we transform ¯ +Ri = (¯ts, ¯te) fur- +ther using a randomized expansion and translation of the +response window: +Ri = [(¯tc − δt) − s∆, (¯tc − δt) + s∆], +(2) +where ∆ = (¯te − ¯ts)/2 is the half-width of ¯Ri, ¯tc = (¯ts + +¯te)/2 is the center of ¯Ri, s ∼ U[1, S] is an expansion factor, +and δt ∼ U[−T, T] is a translation factor. Intuitively, the +translation factor δt randomly shifts ¯R to model uncertainty +in its estimate, and the scaling factor s randomly expands ¯R +to match the distribution of NLQ response windows. S is a +hyperparameter selected through validation, and T is set as +(s − 1)∆ after sampling s to ensure that the seed temporal +window ¯ +Ri is contained within Ri. +Following this strategy, we can extract narrations and +their inferred temporal windows for all video clips with +available narrations (denoted by V) to obtain a dataset +D = +� +(N v +1 , · · · , N v +n) | ∀v ∈ V +� +, +(3) +where N v +i = +� +Ti, Ri +� +is the transformed sample that con- +sists of a narration and its corresponding response window. +We apply this method to the video clips from the train +split of the Ego4D Episodic Memory benchmark to create a +dataset D that contains 850k samples of transformed narra- +tions from 4,851 video clips. +2. Generating episodic memory queries. Given the pre- +vious dataset of narrations with associated temporal win- +dows D, we now convert these to a dataset of NLQ queries. +Specifically, given a video Vj, we sample a narration Ni +from Vj and obtain the task input X = (Vj, Ti), where +Ti is the narration text, and the label Y = Ri which rep- +resents the start and end times for a narration as defined +in Eq. (2). In other words, the narration Ti becomes the +query2 that effectively asks the model to locate in Vj where +2We found that simply using narration text as the query to work well. +4 + +the activity described by Ti can be found, i.e., the response +window (tstart +i +, tend +i +). This dataset of (X, Y ) pairs is our +Narrations-as-Queries (NaQ ) dataset. Next, we incorporate +this dataset into the NLQ training pipeline as a form of data +augmentation. +3.3. Narrations-as-Queries training for NLQ +Our NaQ is model-agnostic: it stands to benefit any NLQ +model out of the box without any model-specific modifica- +tions due to the direct compatibility of NaQ with the NLQ +data. We demonstrate the universal advantage of NaQ by +benchmarking several baselines with NaQ in experiments. +Specifically, for a given NLQ model M, we train it with +NaQ in two stages. Let us denote the NaQ dataset as DNaQ +and the NLQ train dataset as DNLQ. First, we jointly train +M with both DNaQ and DNLQ, effectively treating NaQ as a +query augmentation strategy. Since NaQ expands the train- +ing dataset significantly (by 2 orders of magnitude in size), +we rely on large batch training with 2048 batch size and an +appropriately large initial learning rate of 0.001 on 4-8 A40 +GPUs. We train in this large-batch setting for 200 epochs, +with early stopping when the validation performance satu- +rates. We then finetune the model on DNLQ with the default +small-batch training used for M, and perform a grid search +to determine the learning rate based on M performance on +the validation split. +4. Experiments +4.1. Experimental setup +We evaluate our approach on the NLQ task from the +episodic memory benchmark from Ego4D [12]. +This +benchmark has gained significant interest and has been the +subject of two Ego4D challenges held at CVPR 2022 and +ECCV 2022. +The NLQ task contains 11.3k/3.9k/4.0k +queries annotated over 136/45/46 hours of train / val / test +videos. Each video clip is 8.2 minutes on average, and the +ground-truth query response is 10.5 seconds on average in +the train dataset. That means the response window occupies +only 2% of the input video on average. +Evaluation metrics. We measure performance on NLQ us- +ing metrics from the video-language grounding literature +and adapted for NLQ in [12]. +We report the recall@k, +IoU=m metric, where k = {1, 5} and m = {0.3, 0.5}. This +measures the percentage of times where at least one of +the top-k predicted candidates have at least an intersection- +over-union (IoU) of m. +We expect this is due to the use of pretrained BERT query encoders in +NLQ models [18, 19, 29], which can effectively adapt to the difference +between using a “narrated text” vs. “natural language question” as the +query. However, it would be interesting to study techniques to transform +narrations to questions [28], which we reserve for future work. +Baselines. +We evaluate the impact of our NaQ data aug- +mentation strategy by combining it with 3 existing methods +in the literature. +(1) VSLNet treats natural-language grounding as a text- +based question answering problem [29]. It represents the +input video as a text passage and uses a span-based QA +framework [24] to localize responses to text queries. This +was adapted to perform the NLQ task in [12] by using Slow- +Fast features pretrained on Kinetics 400 [9]. +(2) EgoVLP proposes to pretrain video and text back- +bones on the EgoNCE pretraining task [18]. By leverag- +ing large-scale video + text narrations from Ego4D, they +successfully transfer features to a variety of tasks includ- +ing NLQ. It was the runner-up entry for the Ego4D NLQ +challenge at CVPR 2022. This method replaces the Slow- +Fast features from the VSLNet baseline with the EgoVLP +pretrained backbones. This baseline is complementary to +our own approach where we use narrations to augment the +localization training for NLQ task. +(3) ReLER adapts VSLNet to use a multi-scale cross- +modal transformer architecture [19]. +It also proposes to +augment the training data using video-level augmentation +strategies like randomly sampling a subset of the video to +try and mitigate overfitting. This was the winning entry of +the Ego4D NLQ challenge at CVPR 2022. We augment +this method with EgoVLP pretrained backbones to obtain +a stronger ‘ReLER∗’ baseline. Unlike this method, which +augments the data at the video level, we propose to augment +the data at the query level. We will demonstrate that NaQ is +complementary and boosts the performance of ReLER. +Note that both EgoVLP and ReLER∗ leverage the exact +same narration data as NaQ ; NaQ requires no greater super- +vision or data. +Implementation details. For each baseline, we adapt the +authors’ code bases to train with NaQ data augmentation. +For consistency, we report the results of each method as re- +produced using the provided code and instructions, in ad- +dition to reporting the official paper numbers. +We train +each method with NaQ augmentation for 200 epochs and +stop training early when the validation performance satu- +rates. We found that it was helpful to finetune for up to 30 +epochs on only the NLQ dataset. Please see Sec. S1 for +details. +4.2. Experimental results +We report results on the NLQ validation set in Tab. 1. +The poor performance of the VSLNet baseline on NLQ +highlights the difficulty of the task. It requires localizing re- +sponses typically shorter than 10 seconds in 8+ minute long +egocentric videos. The limited size of the training dataset +further exacerbates this problem, since there are only 11.3k +training queries. +However, when augmented with NaQ , +5 + +IoU=0.3 +IoU=0.5 +Method +Narrations +R@1 +R@5 +R@1 +R@5 +1. +VSLNet [29] + +5.45 +10.74 +3.12 +6.63 +2. +VSLNet† + +4.78 +10.14 +2.56 +6.12 +3. +VSLNet + NaQ + +10.14 +19.01 +5.78 +12.69 +absolute gain ++5.36 ++8.87 ++3.22 ++6.57 +4. +EgoVLP [18] + +10.84 +18.84 +6.81 +13.45 +5. +EgoVLP† + +10.43 +19.75 +6.55 +13.46 +6. +EgoVLP + NaQ + +15.90 +26.38 +9.46 +17.80 +absolute gain ++5.47 ++6.63 ++2.91 ++4.34 +7. +ReLER [19] + +10.79 +13.19 +6.74 +8.85 +8. +ReLER† + +10.25 +12.49 +6.27 +8.23 +9. +ReLER∗ + +14.48 +17.55 +8.52 +11.33 +10. +ReLER∗ + NaQ + +19.31 +23.59 +11.62 +15.51 +absolute gain ++4.83 ++6.04 ++3.10 ++4.18 +Table 1. Results on NLQ validation. +∗replace SlowFast with +EgoVLP features. †Results reproduced using authors’ code. +the performance across all metrics nearly doubles, indicat- +ing the effectiveness of NaQ in addressing these challenges. +This is a dramatic gain, though it comes at the cost of larger +narrations data that is not available to VSLNet. +When VSLNet is augmented with NaQ , it is already +competitive with EgoVLP, which pretrains video and text +backbones with Ego4D videos + narrations and uses the +same VSLNet query-localization architecture (rows 3 vs. +5). When NaQ is combined with EgoVLP, it further im- +proves the performance by 2.9 - 6.6 points across metrics +(row 5 vs. row 6). This confirms that NaQ augmentation +for query localization training complements the EgoVLP +pretraining of video-text backbones. Importantly, our gain +here comes at no additional cost in data or annotations. +ReLER [19] uses SlowFast + CLIP video features. For +a fair comparison, we replace the SlowFast features with +EgoVLP features to obtain ReLER∗. This improves by a +large margin as expected, and gives us a stronger baseline +to compare with (row 8 vs. row 9). Recall that ReLER∗ uses +video-level data augmentation using variable-length sliding +windows and video splicing [19]. When ReLER∗ is aug- +mented with NaQ , the performance increases by a signifi- +cant margin. This confirms the complementary nature of the +query-level augmentation we propose in NaQ with video- +level augmentation in ReLER. +Overall, we find that NaQ augmentation greatly improves +the performance of all methods across all metrics. The ab- +solute gains across metrics are remarkably consistent re- +gardless of the underlying method. When averaged across +the methods, NaQ improves the absolute recall@1 perfor- +mance by 5.22 at IoU=0.3 and 3.07 at IoU=0.5, and the ab- +solute recall@5 performance by 7.18 at IoU=0.3 and 5.03 +at IoU=0.5. This confirms the generality and effectiveness +of NaQ at expanding the limited NLQ annotations by boot- +strapping it with narrations, a relatively cheaper and more +abundant data source. More importantly, the insight in NaQ +Method +R@1 +IoU=0.3 +R@1 +IoU=0.5 +Mean +R@1† +R@5 +IoU=0.3 +R@5 +IoU=0.5 +NaQ (ours) +18.46 +10.74 +14.59 +21.50 +13.74 +Red Panda∗ +16.46 +10.06 +13.26 +22.95 +16.11 +Badgers@UW-Mad.∗ +15.71 +9.57 +12.64 +28.45 +18.03 +CONE∗ +15.26 +9.24 +12.25 +26.42 +16.51 +ReLER [19] +12.89 +8.14 +10.51 +15.41 +9.94 +EgoVLP [18] +10.46 +6.24 +8.35 +16.76 +11.29 +VSLNet [29] +5.42 +2.75 +4.08 +8.79 +5.07 +Table 2. Results on Ego4D NLQ challenge. †Primary metric for +the challenge. ∗Unpublished work. +is not simply that large-scale data benefits performance. +Rather, we emphasize how to use this data: we leverage nar- +rations as queries for query-localization network training. +This is evidenced by our experiments demonstrating major +gains on EgoVLP and ReLER∗, methods which also benefit +from large-scale pretraining on video-narrations data. +Ego4D NLQ challenge. We submitted our best perform- +ing method (ReLER∗ + NaQ ) to the Ego4D NLQ challenge +leaderboard, where the NLQ evaluation is performed on a +EvalAI server on a held-out set of test annotations [12]. +Note that while the videos are available to participants, the +annotations (including narrations) are not accessible. The +results are shown in Tab. 2. VSLNet is the baseline provided +by the organizers. ReLER and EgoVLP were the winning +and runner-up entries from the CVPR 2022 edition of the +challenge. Red Panda, Badgers@UW-Madison, and CONE +are the top three entries from the ECCV 2022 edition of the +challenge.3 As of the time of submission, NaQ is the lead- +ing entry among all methods on the leaderboard, including +those. Our approach has the best available results on this +challenge, by a healthy margin. +TRJ ablation. We study the impact of using temporal re- +sponse jittering (TRJ) (Sec. 3.2) in an ablation study. We +observe that using TRJ improves the performance by up to +0.7 points in recall @ 1 metrics and 1.7 in recall @ 5 met- +rics consistently across all methods. Please see Sec. S3 for +the complete results. +4.3. Performance analyses +In the previous section, we verified the effectiveness +of our approach through a careful comparison with recent +state-of-the-art methods. We now ascertain the strengths +and weaknesses of our approach through a series of quan- +titative studies and discuss qualitative results in Fig. 4. +For performing analysis-specific experiments, we adopt the +EgoVLP + NaQ method since it requires lower computa- +tional cost and time to train. +(1) How does performance scale with narrations? One +3The code+reports for these methods were unavailable at the time of +our experiments, so we could not compare with them outside the leader- +board. +6 + +Video +ReLER* +Ground truth +Ours +270 +276 +273 +272 +274 +276 +Query: How many funnels are on the shelf? +0 +9 +18 +Video +201 +207 +204 +202 +204 +207 +Query: Where was the brake pad before I took it? +104 +106 +108 +Video +180 +198 +189 +164 +166 +168 +Query: What color bottle is on the sink? +180 +190 +200 +𝑡 = 𝑇 +𝑡 = 0 +1 +𝑡 = 𝑇 +𝑡 = 0 +2 +3 +𝑡 = 𝑇 +𝑡 = 0 +Figure 4. Qualitative analysis. We show three examples of NLQ task predictions (one per column). In each column, the natural language +query is displayed at the top, the ground truth responses are in the central row, and the model predictions are on the first and last rows. The +temporal extents of the video and predicted time windows are shown right next to the images on each column. We compare ReLER∗ [19] +baseline (on the first row) against our NaQ method which augments the NLQ training for ReLER∗. Example 1: Our method successfully +identifies the response window showing how many funnels are on the shelf, while the baseline fails. The object ‘funnel’ is a low-shot +object with fewer than 10 training queries. This supports our experimental observation that NaQ has a strong advantage on low-shot objects +and counting-based queries (see Tabs. 3 and 4). Example 2: NaQ successfully recognizes the object ‘brake pad’ and is able to localize +where it was taken. ReLER* incorrectly identifies a spanner as the response. Example 3: This is a failure case for NaQ . While it correctly +identifies a sink, this particular sink does not contain the bottle and the model fails to respond. +Object / place queries +People queries +Method +Where is X +before/after +Y? +Where did +I put X? +Where +is X? +What did I +put in X? +How many +X’s? +In what +location did +I see X? +What X +did I Y? +What X +is Y? +State? +Who did I +interact with +during Y? +VSLNet +1.86 +0.96 +3.13 +2.94 +4.67 +2.39 +3.53 +1.96 +3.57 +2.94 ++NaQ +6.62 +3.58 +3.14 +5.76 +9.82 +2.60 +8.61 +5.86 +8.59 +6.52 +EgoVLP +5.26 +3.22 +3.62 +10.37 +14.39 +2.23 +9.27 +3.52 +8.59 +7.61 ++NaQ +10.70 +6.44 +4.83 +13.13 +15.79 +2.60 +11.59 +7.03 +12.88 +13.04 +ReLER* +9.78 +6.39 +5.82 +10.29 +14.33 +4.78 +11.54 +6.54 +10.12 +4.90 ++NaQ +13.98 +11.34 +6.26 +12.61 +20.67 +4.78 +15.38 +6.86 +14.29 +7.84 +Table 3. Performance over NLQ query types. We report recall@1 at IoU=0.5. We include query types with ≥ 100 val samples. We +highlight cases where NaQ improves recall by more than 0.5 points. +% of narrations as queries +Recall @ 1 +5 +10 +15 +20 +0 +25 +50 +75 +100 +IoU=0.3 +IoU=0.5 +% of narrations as queries +Recall @ 5 +5 +10 +15 +20 +25 +30 +0 +25 +50 +75 +100 +IoU=0.3 +IoU=0.5 +% of narrations as queries +Recall @ 1 +5 +10 +15 +20 +0 +25 +50 +75 +100 +IoU=0.3 +IoU=0.5 +% of narrations as queries +Recall @ 1 +5 +10 +15 +20 +0 +25 +50 +75 +100 +IoU=0.3 +IoU=0.5 +% of NaQ dataset +% of NaQ dataset +Figure 5. Data scaling analysis. We train EgoVLP + NaQ using +all NLQ and k% of NaQ dataset (k represented on the X-axis). +NLQ performance scales linearly with the size of the NaQ dataset. +of the key benefits of using narrations for pretraining is that +they are available on a large scale. We generated 850k nar- +rations as queries for the NLQ task, which is two orders +larger than the NLQ dataset containing 11.3k train queries. +We now study performance scaling as a function of the +amount of narrations used for training. For this, we addi- +tionally trained EgoVLP + NaQ with 10%, 25%, 50% of +the narrations. Fig. 5 shows the results on NLQ (val). The +0% performance represents EgoVLP and the 100% perfor- +mance represents the full EgoVLP + NaQ reported in Tab. 1. +When adding only 10% of our NaQ data, we already observe +good improvements on all metrics. The performance con- +tinues to linearly scale as we add more narrations for NaQ +augmentation, confirming the utility of our paradigm. +(2) What types of queries does NaQ benefit? +Next, we +break down the NLQ performance across query types, i.e., +the form of reasoning required by the query (e.g., where +did I put object X? who did I talk to while doing activity +Y?). The NLQ dataset was created by providing an ini- +tial set of 13 query templates [12]. For reliable evaluation, +we select 10 out of the 13 templates which contain 100 +or more samples in the validation split, and report results +7 + +High-shot +Mid-shot +Low-shot +Method +IoU=0.3 +IoU=0.5 +IoU=0.3 +IoU=0.5 +IoU=0.3 +IoU=0.5 +VSLNet +5.65 +2.82 +3.71 +2.48 +3.84 +2.30 ++NaQ +9.72 +5.53 +11.26 +7.00 +10.14 +5.57 +EgoVLP +11.32 +5.83 +10.96 +6.70 +9.63 +6.42 ++NaQ +16.59 +9.27 +16.13 +10.20 +16.05 +10.30 +ReLER∗ +17.07 +10.35 +17.74 +10.18 +13.21 +8.29 ++NaQ +21.37 +12.37 +21.87 +12.38 +17.20 +10.75 +Table 4. Performance breakdown across object types. For ob- +ject type queries, we categories objects into low-shot, mid-shot, +and high-shot objects based on their frequency of occurrence. We +report the recall@1 metric at IoU=0.3 and IoU=0.5. We highlight +cases where NaQ improves recall by over 0.5 points. +in Tab. 3. We observe that using NaQ leads to significant +improvements (marked in green) on 8/10 templates for at +least 2/3 methods. However, it only has a limited impact +for ‘Where is object X?’ and ‘In what location did I see X?’ +queries. These queries may require explicit spatial under- +standing to achieve better performance. Since all methods +perform poorly on those queries and do not benefit from +training on NaQ , it hints at the need to incorporate better +spatial understanding for video models. +(3) Does NaQ help respond about long-tail objects? The +NLQ dataset has a long-tail of objects that are the sub- +ject of queries due to the sparse nature of NLQ annota- +tions (1 query per 1.4 minutes of videos on average). How- +ever, since narrations are more densely annotated through- +out the video (20+ narrations per minute), they contain rich +information about objects that are rarely queried about. We +therefore study if pretraining NLQ localization models with +narrations can help respond to queries about long-tail ob- +jects. We divide objects from the NLQ train annotations +into 3 types (as shown in Fig. S1): 1. high-shot objects +which are queried more than 50 times (65 in total), 2. mid- +shot objects which are queried about 10 to 50 times (147 in +total), and 3. low-shot objects which are queried about be- +tween 2 to 10 times (967 in total). The results are in Tab. 4. +Overall, we observe that NaQ improves performance by a +large margin in most cases, and has the biggest gains on +mid-shot and low-shot objects. This indicates that using +narrations as queries helps mitigate some of the biases in +the NLQ data, and improves responses to queries about less- +frequently occurring objects. +(4) Does NaQ facilitate zero-shot / few-shot NLQ? Con- +sidering that NaQ enables better performance on long-tail +objects, we next study whether it can facilitate zero-shot or +few-shot learning for NLQ, i.e., given our large-scale NaQ +data and little to no NLQ task annotations, can we learn +good NLQ models? We are first to study this to the best of +our knowledge. We train EgoVLP + NaQ method with all of +% of narrations as queries +Recall @ 1 +0 +5 +10 +15 +0 +10 +20 +30 +IoU=0.3 +IoU=0.5 +% of narrations as queries +Recall @ 5 +10 +15 +20 +25 +0 +10 +20 +30 +IoU=0.3 +IoU=0.5 +% of narrations as queries +Recall @ 1 +0 +5 +10 +15 +0 +10 +20 +30 +IoU=0.3 +IoU=0.5 +% of narrations as queries +Recall @ 1 +0 +5 +10 +15 +0 +10 +20 +30 +IoU=0.3 +IoU=0.5 +% of NLQ dataset +% of NLQ dataset +Figure 6. Zero-shot and few-shot learning for NLQ. We train +EgoVLP + NaQ using all NaQ and k% of the NLQ train data (k +on the X-axis). The dotted horizontal lines represent the EgoVLP +performance with 100% NLQ and no NaQ augmentation. +NaQ and k% of NLQ train data, where k = {0, 10, 25, 35}. +k = 0 represents the zero-shot case, and the rest represent +few-shot learning. The results are in Fig. 6. The triangles +represent EgoVLP + NaQ with k% NLQ data, and the hor- +izontal line represents the EgoVLP baseline with no NaQ +data. It is interesting to observe that even with no NLQ +data, the model performs well using NaQ and matches the +EgoVLP performance on the R@5 metrics. When we inject +10% of the NLQ dataset, we get comparable or better per- +formances on 3/4 metrics. At 25% of NLQ data, it matches +or outperforms EgoVLP on all metrics. Finally, at 35%, +we comprehensively outperform EgoVLP. This study sug- +gests that we can leverage large-scale free-form narration +annotations using NaQ to compensate for the lack of NLQ +annotations. While these are not free to obtain, they are eas- +ier to annotate than NLQ and can also be used for various +purposes other than the NLQ task itself [12], meaning that +many research directions are likely to continue investing in +them. +5. Conclusions +In this work, we propose Narrations-as-Queries, a sim- +ple data augmentation technique that dramatically improves +state-of-the-art results on the Natural Language Queries +task in the Episodic Memory benchmark. Our key insight is +to convert timestamped narrations in egocentric videos into +natural language queries and use them as additional data +for training NLQ localization models. To convert times- +tamped narrations into a form compatible with NLQ, we +propose a temporal response jittering technique to convert a +single timestamp into temporal windows. We perform ex- +periments to demonstrate that our approach can be used as +a simple plug-in to existing methods, massively improves +multiple top methods for this task, and yields the very best +performance to-date on the Ego4D NLQ benchmark. We +hope that our approach serves as a useful tool for future +research on this problem. We will share code, data, and +models upon publication. +8 + +References +[1] Yazan Abu Farha, Alexander Richard, and Juergen Gall. +When will you do what?-anticipating temporal occurrences +of activities. +In Proceedings of the IEEE conference on +computer vision and pattern recognition, pages 5343–5352, +2018. 2 +[2] Max Bain, Arsha Nagrani, G¨ul Varol, and Andrew Zisser- +man. Frozen in time: A joint video and image encoder for +end-to-end retrieval. 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Long-tail of objects in NLQ. +Supplementary Materials +We now provide additional information about our exper- +imental settings, and qualitative and quantitative analyses to +support our experiments in the main paper. +S1. Implementation details +We perform joint NaQ + NLQ training with a large batch +sizes and high learning rates for accelerated convergence. +For VSLNet and EgoVLP methods, we use a batch size of +2048 and initial learning rate of 0.001 on 2 A40 GPUs with +a memory size of 46GB per GPU. For ReLER∗, we use a +batch size of 1536 and an initial learning rate of 0.001 on 8 +A40 GPUs since it has larger memory and compute require- +ments. We train each method for up to 200 epochs on NaQ ++ NLQ training data, and then finetune them for up to 30 +epochs on NLQ training data alone with a lower learning +rate. We found finetuning to be unnecessary for VSLNet. +For EgoVLP, we finetuned with the original hyperparame- +ter settings from [18] and a learning rate of 0.00001. For +ReLER∗, we finetuned with the original hyperparameter +setting from [19] and a learning rate of 0.0001. We per- +form early stopping in each case using the performance on +NLQ validation split. +For +temporal +random +jittering +(TRJ), +we +per- +formed a grid search with the expansion factor values +S={2.5, 5.0, 10.0, 20.0}. We found S=2.5 to work best for +EgoVLP and VSLNet, and S=5.0 to work best for ReLER∗ +based on their NLQ validation performance. +S2. Long-tail of objects in NLQ +Fig. S1 shows the long-tail of objects queried about in +NLQ, and the split of low-shot, mid-shot, and high-shot ob- +jects used in Sec. 4.3. Note that for a given point x on X- +axis, the Y-axis shows the number of objects that have x +queries in the NLQ train dataset. For example, there are +more than 1000 objects with only 1 training sample. +S3. Ablation study for Temporal Response Jit- +tering +We study the impact of using temporal response jittering +(TRJ) described in Eq. (2). In Tab. S1, we measure the per- +IoU=0.3 +IoU=0.5 +Method +TRJ +R@1 +R@5 +R@1 +R@5 +VSLNet + NaQ + +9.89 +18.02 +5.30 +10.99 +VSLNet + NaQ + +10.14 +19.01 +5.78 +12.69 +absolute gain ++0.25 ++0.99 ++0.48 ++1.70 +EgoVLP + NaQ + +15.27 +25.93 +9.07 +17.14 +EgoVLP + NaQ + +15.90 +26.38 +9.46 +17.80 +absolute gain ++0.63 ++0.45 ++0.39 ++0.66 +ReLER∗ + NaQ + +18.48 +23.26 +11.25 +15.44 +ReLER∗ + NaQ + +19.31 +23.59 +11.62 +15.51 +absolute gain ++0.83 ++0.33 ++0.37 ++0.07 +Table S1. Ablation study of temporal random jittering (TRJ). +formance of using NaQ with and without TRJ, where not us- +ing TRJ implies that the seed temporal window from Eq. (1) +is used. Overall, we observe a consistent improvement of up +to 0.83 in R@1 metrics and 1.70 in R@5 metrics. This in- +dicates that TRJ is able to address the limitations of the seed +temporal window. +S4. Few-shot analysis +We perform a more detailed analysis of the few-shot per- +formance discussed in Sec. 4.3 and Fig. 6. Specifically, we +analyze the zero-/few-shot performance across the various +query templates in Tab. S2. When tested zero-shot, NaQ +already competes with or outperforms the baseline on ob- +ject/place templates such as ‘where is X before/after Y?’, +‘where did I put X?’, ‘where is X?’, ‘In what location did +I see X?’, ‘what X is Y?’, and ‘object state’.4 As we in- +ject NLQ data into NaQ training, the performance improves +quickly on the remaining templates, and outperforms the +baseline on 8/10 templates. +S5. Qualitative examples +In supplementary.html shared here, we link to qual- +itative videos for the following: +• Comparing annotations for NLQ vs. Narrations +• NaQ benefits performance on most query templates +• NaQ benefits performance on queries about long-tail +objects +• NaQ facilitates zero-shot NLQ +4We provide video visualizations of the zero-shot performance on these +4 templates in supplementary.html. +11 + +Distribution over obiect freguencies +103 +Objects +101 +# +1 +2 +10 +50 +100 +1000 +# queries per objectObject / place queries +People queries +% NLQ +% NaQ +Where is X +before/after +Y? +Where did +I put X? +Where +is X? +What did I +put in X? +How many +X’s? +In what +location did +I see X? +What X +did I Y? +What X +is Y? +State? +Who did I +interact with +during Y? +100 +0 +5.26 +3.22 +3.62 +10.37 +14.39 +2.23 +9.27 +3.52 +8.59 +7.61 +0 +100 +4.41 +4.29 +2.90 +2.53 +5.26 +1.49 +4.30 +6.25 +7.36 +3.26 +10 +100 +8.15 +5.72 +2.66 +5.07 +5.96 +1.12 +3.64 +5.86 +6.13 +4.35 +25 +100 +10.70 +5.19 +3.38 +5.99 +8.07 +1.49 +5.30 +6.25 +6.13 +5.43 +35 +100 +9.51 +5.55 +3.86 +7.83 +14.04 +4.09 +7.62 +7.81 +7.98 +5.43 +100 +100 +10.70 +6.44 +4.83 +13.13 +15.79 +2.60 +11.59 +7.03 +12.88 +13.04 +Table S2. Few-shot analysis. We split the few-shot results from Fig. 6 in the main paper across the various query templates. We report +recall@1 at IoU=0.5. The first two columns show the percentage of the NLQ and NaQ data used for training. For example, the first row +with 100% NLQ and 0% NaQ is the baseline, the second row with 0% NLQ and 100% NaQ is our zero-shot setting, and so on. +12 + diff --git a/4dAyT4oBgHgl3EQf2Pkf/content/tmp_files/load_file.txt b/4dAyT4oBgHgl3EQf2Pkf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..18d18b44e97220835de9c6e8a2a6a432466847f7 --- /dev/null +++ b/4dAyT4oBgHgl3EQf2Pkf/content/tmp_files/load_file.txt @@ -0,0 +1,943 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf,len=942 +page_content='NaQ: Leveraging Narrations as Queries to Supervise Episodic Memory Santhosh Kumar Ramakrishnan1, Ziad Al-Halah1, Kristen Grauman1,2 1UT Austin, 2Meta AI Abstract Searching long egocentric videos with natural language queries (NLQ) has compelling applications in augmented reality and robotics, where a fluid index into everything that a person (agent) has seen before could augment human memory and surface relevant information on demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' How- ever, the structured nature of the learning problem (free- form text query inputs, localized video temporal window outputs) and its needle-in-a-haystack nature makes it both technically challenging and expensive to supervise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We in- troduce Narrations-as-Queries (NaQ), a data augmentation strategy that transforms standard video-text narrations into training data for a video query localization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Vali- dating our idea on the Ego4D benchmark, we find it has tremendous impact in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' NaQ improves multiple top models by substantial margins (even doubling their accu- racy), and yields the very best results to date on the Ego4D NLQ challenge, soundly outperforming all challenge win- ners in the CVPR and ECCV 2022 competitions and topping the current public leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Beyond achieving the state- of-the-art for NLQ, we also demonstrate unique properties of our approach such as gains on long-tail object queries, and the ability to perform zero-shot and few-shot NLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Introduction Human memory can fail us in day-to-day things in our visual experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We misplace objects in the house (where is my passport?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ), we lose track of what tasks we have or have not done (did I add the salt already?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ), we forget where we did a given activity (where did I buy tickets last time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ), we do not notice the state of an object in our environment (did I leave the garage door open?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' First-person or “ego- centric” perception on a wearable camera could reduce that cognitive overload and provide us with a superhuman per- sonal episodic memory—by seeing what we see, and index- ing it in meaningful and easy-to-access ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This is the vision of the Natural Language Query (NLQ) task in Ego4D’s Episodic Memory benchmark [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Given a natural language question and a long egocentric video, the NLQ task requires identifying the precise temporal window .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Query: How many eggs did I break into the bowl?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Response Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Episodic memory with natural language queries (NLQ) aims to search long egocentric videos to identify the temporal “response” window revealing the answer to a free-form question about the camera wearer’s past visual experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' in the camera wearer’s past video that reveals the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' See Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Such functionality could transform the every- day experience of an augmented reality user with always- on AR glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' It could similarly play a role for a mobile household robot, whom a user may wish to query about its own visual history (have you seen my keys?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The NLQ challenge has attracted substantial attention in the research community over the last year [18, 19, 31] as have related video-language efforts for question answer- ing [23, 26–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The technical challenges are striking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Queries are free-form natural language, response windows are tiny slivers (a few seconds or less) within a long stretch of video, and wearable camera video is notoriously noisy with its quick head motions and limited field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Today’s most successful methods embrace the visual- language aspect of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' In particular, inspired by the growing success of visual-linguistic embeddings [17, 20,22,25,28], top competitors on NLQ perform large-scale pretraining on ⟨video clip, text description⟩ pairs mined from the Ego4D dataset’s provided narrations [18], which are timestamped play-by-play descriptions of the camera- wearer’s activity (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The result is a video back- bone enhanced by the semantics of grounded language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='00746v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='CV] 2 Jan 2023 C turns on the tap with her right hand C opens a drawer C cracks an egg into the bowl C opens the third refrigerator door Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Narration examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' “C” refers to the camera-wearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' While it is important to have strong video and text repre- sentations, the downstream query localization models that search the video for a response are also crucial to NLQ, yet relatively starved for data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This is a direct consequence of the difficulty in annotating a query-response pair (which en- tails posing a creative question and scrolling the long video to mark the temporal response window) versus the relative ease in narrating a video (which entails pausing the video at regular intervals and writing down what happened).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For example, whereas Ego4D has 3,670 hours of data annotated with narrations—more than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='85M sentences in total—it of- fers only 227 hours of NLQ query examples, for 19k total text queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Accordingly, existing methods risk failing to learn the task-specific skills that are poorly represented in training, such as responding to queries about objects in the long-tail or performing complex reasoning for queries in- volving interactions between multiple visual entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' To address this issue, we introduce Narrations-as- Queries (NaQ), a simple but exceptionally effective data augmentation strategy for NLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' NaQ is a novel strategy that uses timestamped narrations to expand the supervision available for training query-localization modules within an episodic memory architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Our hypothesis is that nar- rations provide descriptive information that is localizable in long videos, and thus can benefit an episodic memory model when used as training queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Specifically, we derive ⟨video, language query, temporal window response⟩ anno- tations from timestamped narrations, and augment the con- ventional query-response data with these pseudo-queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Importantly, this allows us to influence the localization module—the workhorse responsible for finding a needle in a haystack—with multimodal data, as opposed to just the video and text encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Empirically, our idea has tremendous impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Demon- strating NaQ on the Ego4D Episodic Memory benchmark, we find our simple augmentation strategy successfully com- plements multiple existing state-of-the-art episodic mem- ory methods, achieving sizeable improvements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', 32% to 125% relative jumps in accuracy) across query types, metrics, and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Notably, our gains hold even com- pared to existing methods such as EgoVLP [18] that use the same (or even more) narration annotations as our model, meaning that NaQ’s success can be attributed to good mod- eling, not more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Moreover, to our knowledge, NaQ yields the very best results to date on the NLQ chal- lenge, strongly outperforming all the challenge winners from Ego4D CVPR’22 and Ego4D ECCV’22 by a substan- tial margin, and topping the current public leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Be- yond achieving state-of-the-art results, we perform a thor- ough analysis of the strengths and weaknesses of NaQ, and demonstrate useful properties such as benefits on long-tail object queries as well as zero-shot and few-shot NLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We are the first to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Related work Egocentric video understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Prior work has devel- oped video datasets and methods for egocentric percep- tion [4, 8, 10, 12, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Egocentric video offers a camera wearer’s perspective of their activities over a long time horizon and raises challenging research problems such as human-object interactions [3, 5], activity recognition [14, 33], anticipation [1, 11], episodic memory [12], and video summarization [6,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' In this work, we tackle the episodic memory task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We leverage the Ego4D dataset [12], which consists of 3,670 hours of video of daily-life activity cap- tured by 931 camera wearers around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Vision-language pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Vision-Language Pre- training (VLP) methods rely on large-scale video-text datasets [2, 21] to learn transferable representations for video-language tasks such as retrieval [7, 13], question- answering [23,27] and video captioning [15,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' VideoBert learns joint video-text embeddings by discretizing video frames into tokens and performing BERT-like pre- training [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' HERO improves over this with a hierarchical encoding of multi-modal inputs to better capture long-term structure [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' MIL-NCE learns to match clips with tempo- rally close captions to address video-text misalignment in HowTo100M [20,21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' While these methods primarily focus on third-person videos, EgoVLP [18] adapts the InfoNCE objective to egocentric settings and uses video-narration annotations from Ego4D [12] to learn video-text backbones for egocentric video understanding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Just-Ask [28] proposes a strategy to generate video question-answering data consisting of (short clips, questions, text answers) from narrated YouTube videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' While we take inspiration from such methods, our idea is very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Unlike prior work that learns representations or video-QA systems from short video clips and aligned (possibly weak) text, we learn to temporally localize short 2 pJ(5)text queries in long untrimmed videos egocentric videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Whereas Just-Ask’s data generation procedure [28] outputs questions with text responses for short video clips, ours out- puts temporal windows in long videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Rather than pretrain- ing a video/text backbone [17,18,20,25], our model injects multimodal supervision to train a query-localization mod- ule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Overall, our idea is complementary to prior video-text pretraining efforts, as we will demonstrate in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Episodic memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The episodic memory benchmark’s natural language queries (NLQ) task was first introduced in the Ego4D dataset [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' In NLQ, the goal is to tem- porally localize the response to a natural language text question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Existing video-language grounding methods like 2D-TAN [30] and VSLNet [29] have been adapted to per- form this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Our goal is to improve such methods via large-scale data augmentation with narration-based queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' More recently, ReLER [19] achieved the state-of-the-art for NLQ by using a multi-scale and cross-model transformer with video-level data augmentation and contrastive losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Our proposed strategy performs query-level augmentation and is complementary to the video-level data augmentation from [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' As we will demonstrate in experiments, our ap- proach stacks well when combined with prior NLQ meth- ods [18,19,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Approach Our key insight is to leverage narrations as an additional data source to improve a model’s ability to localize answers in a long video when prompted with a natural language query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' To do this, we propose a strategy to convert narra- tions and their timestamps into episodic memory queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Our strategy is automatic and simple which allows us to scale the training data for episodic memory search by two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Furthermore, we generate the data in a form that is compatible with the manually labeled NLQ annotations, which allows an NLQ model to directly take advantage of this additional data source and achieve signif- icant improvements in performance without any modifica- tions to the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Next, we define the episodic memory task (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='1), then describe our Narrations-as-Queries approach to con- vert narrations into natural language queries (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='2), and finally describe our training strategy (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Episodic memory with natural language query The goal of episodic memory is to perform query-driven reasoning about long-form egocentric videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' First intro- duced in Ego4D [12], it is well-motivated by applications discussed above, such as augmented reality assistants that enable superhuman memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The NLQ task has attracted significant attention in the research community, with 10+ teams from labs around the world competing on the bench- mark over the last year [18, 19, 31], two organized chal- lenges at CVPR’22 and ECCV’22, and an active public leaderboard1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' More formally, given an egocentric video V capturing a camera wearer’s past experiences and a natural language query Q in the form of a question, the task requires tempo- rally localizing where the answer can be seen in the video, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', a response window R = [ts, te].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For example, the query could be Q =“What vegetables did I put in the soup the last time I made it?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', and the model needs to search a given video V to identify the time window [ts, te] that con- tains the answer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', the type of vegetables in the soup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' A data sample for this task is of the form ⟨video, query, response⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The video can be several minutes long, and the response to the query can appear in a time window that is shorter than a second, making this a very challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Narrations-as-Queries Prior NLQ methods are limited in performance due to the lack of large-scale NLQ annotations of the form ⟨video, query, response⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We address this limitation by proposing a method to automatically transform narrations associated with egocentric videos to a compatible form for NLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Nar- rations are free-form sentences describing the current ac- tivity performed by the camera-wearer (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' They are time-stamped and temporally dense (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', there are 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='2 sentences per minute of video on average in Ego4D [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' These annotations are substantially cheaper to obtain than NLQ annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For narrations, the annotators needs to simply describe the activity that is seen in the video;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' whereas for NLQ, first a meaningful, unambiguous ques- tion needs to be formulated and then the annotator needs to manually search the video back and forth to identify the time window that shows the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Hence, narrations can be annotated at a much larger scale compared to NLQ sam- ples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', Ego4D has 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='85M narrations compared to 19K NLQ samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Our idea is to leverage this massive data source to aid the learning in the NLQ task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We achieve this by first generat- ing a temporal window associated with each narration that approximately captures when the activity described by the narration started and ended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Then, we use these samples (narrations coupled with temporal windows) as additional supervision to train an NLQ localization model to identify where these narrations happen in the video (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Next, we formally describe our approach in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Generating temporal windows for narrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Each video narration consists of a textual sentence T , and a single timestamp t marking the correspondence to the underlying video (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 3, left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' However, this is incompatible with 1NLQ challenge leaderboard: https://eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='ai/web/challenges/ challenge-page/1629/leaderboard/3920 3 Text Encoder Video Encoder queries videos Query Localization Module ( Ƹ𝑡𝑠, Ƹ𝑡𝑒) NLQ Model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' NLQ Dataset How many eggs did I break?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Narrations-as-Queries (NaQ ) C takes the ingredients out of the shelf 𝑉𝑗 𝑇𝑖 𝑅𝑖 𝑡𝑖 +𝛽/2𝛼 −𝛽/2𝛼 Seed Temporal Window Temporal Response Jittering 𝑡𝑠 𝑡𝑒 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='. responses 𝑉 𝑄 𝑅 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 𝑡𝑖 −𝑠Δ +𝑠Δ 𝑡𝑐 −𝛿𝑡 Δ ത𝑅𝑖 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Narrations-as-Queries: We propose a simple-yet-effective data-augmentation strategy for natural language queries (NLQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The status-quo NLQ methods train in a supervised fashion on annotated (V: video, Q: query, R: response) tuples, where the response is a (ts, te) temporal window (see right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This is severely limiting, since such task-specific data is expensive to obtain and is available only on a small scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We propose a narrations-as-queries pipeline to tackle this issue (see left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Our key idea is to leverage densely annotated video narrations, where each narration Ti for video Vj is a textual description of the camera-wearer’s activity at time ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We propose “temporal response jittering”, a technique to convert timestamped narrations into natural language queries with temporal response windows ⟨Vj, Ti, Ri⟩ and obtain the NaQ dataset, which contains 80× more samples when compared to the NLQ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We then train various NLQ models jointly on the NLQ and NaQ datasets to obtain significant gains across query types, architectures, and metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' NLQ task architectures which require queries and tempo- ral response windows as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' To address this, we propose temporal response jittering, a technique to convert narration timestamps to temporal windows conditioned on the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Temporal response jittering: Our goal is to convert a narration timestamp ti from video Vj into a response win- dow Ri = (ts, te).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' First, we use “contextual variable- length clip pairing strategy” introduced in EgoVLP [18] to obtain a video-conditioned seed temporal window centered around ti: ¯ Ri = [ti − βi/2α, ti + βi/2α] (1) where βi captures the average temporal length between con- secutive narrations in video Vj, and α is the average of all βi across all videos (please see [18] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' While this offers a good starting point, it fails to address the inherent noise in ¯ Ri arising from the lack of explicit human annota- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The responses generated are also typically short (less than a second) and do not match the distribution over NLQ response windows that are 10 seconds long on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' To account for these factors, we transform ¯ Ri = (¯ts, ¯te) fur- ther using a randomized expansion and translation of the response window: Ri = [(¯tc − δt) − s∆, (¯tc − δt) + s∆], (2) where ∆ = (¯te − ¯ts)/2 is the half-width of ¯Ri, ¯tc = (¯ts + ¯te)/2 is the center of ¯Ri, s ∼ U[1, S] is an expansion factor, and δt ∼ U[−T, T] is a translation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Intuitively, the translation factor δt randomly shifts ¯R to model uncertainty in its estimate, and the scaling factor s randomly expands ¯R to match the distribution of NLQ response windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' S is a hyperparameter selected through validation, and T is set as (s − 1)∆ after sampling s to ensure that the seed temporal window ¯ Ri is contained within Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Following this strategy, we can extract narrations and their inferred temporal windows for all video clips with available narrations (denoted by V) to obtain a dataset D = � (N v 1 , · · · , N v n) | ∀v ∈ V � , (3) where N v i = � Ti, Ri � is the transformed sample that con- sists of a narration and its corresponding response window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We apply this method to the video clips from the train split of the Ego4D Episodic Memory benchmark to create a dataset D that contains 850k samples of transformed narra- tions from 4,851 video clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Generating episodic memory queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Given the pre- vious dataset of narrations with associated temporal win- dows D, we now convert these to a dataset of NLQ queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Specifically, given a video Vj, we sample a narration Ni from Vj and obtain the task input X = (Vj, Ti), where Ti is the narration text, and the label Y = Ri which rep- resents the start and end times for a narration as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' In other words, the narration Ti becomes the query2 that effectively asks the model to locate in Vj where 2We found that simply using narration text as the query to work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 4 the activity described by Ti can be found, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', the response window (tstart i , tend i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This dataset of (X, Y ) pairs is our Narrations-as-Queries (NaQ ) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Next, we incorporate this dataset into the NLQ training pipeline as a form of data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Narrations-as-Queries training for NLQ Our NaQ is model-agnostic: it stands to benefit any NLQ model out of the box without any model-specific modifica- tions due to the direct compatibility of NaQ with the NLQ data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We demonstrate the universal advantage of NaQ by benchmarking several baselines with NaQ in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Specifically, for a given NLQ model M, we train it with NaQ in two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Let us denote the NaQ dataset as DNaQ and the NLQ train dataset as DNLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' First, we jointly train M with both DNaQ and DNLQ, effectively treating NaQ as a query augmentation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Since NaQ expands the train- ing dataset significantly (by 2 orders of magnitude in size), we rely on large batch training with 2048 batch size and an appropriately large initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='001 on 4-8 A40 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We train in this large-batch setting for 200 epochs, with early stopping when the validation performance satu- rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We then finetune the model on DNLQ with the default small-batch training used for M, and perform a grid search to determine the learning rate based on M performance on the validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Experimental setup We evaluate our approach on the NLQ task from the episodic memory benchmark from Ego4D [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This benchmark has gained significant interest and has been the subject of two Ego4D challenges held at CVPR 2022 and ECCV 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The NLQ task contains 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3k/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='9k/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='0k queries annotated over 136/45/46 hours of train / val / test videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Each video clip is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='2 minutes on average, and the ground-truth query response is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 seconds on average in the train dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' That means the response window occupies only 2% of the input video on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We measure performance on NLQ us- ing metrics from the video-language grounding literature and adapted for NLQ in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We report the recall@k, IoU=m metric, where k = {1, 5} and m = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This measures the percentage of times where at least one of the top-k predicted candidates have at least an intersection- over-union (IoU) of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We expect this is due to the use of pretrained BERT query encoders in NLQ models [18, 19, 29], which can effectively adapt to the difference between using a “narrated text” vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' “natural language question” as the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' However, it would be interesting to study techniques to transform narrations to questions [28], which we reserve for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We evaluate the impact of our NaQ data aug- mentation strategy by combining it with 3 existing methods in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' (1) VSLNet treats natural-language grounding as a text- based question answering problem [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' It represents the input video as a text passage and uses a span-based QA framework [24] to localize responses to text queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This was adapted to perform the NLQ task in [12] by using Slow- Fast features pretrained on Kinetics 400 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' (2) EgoVLP proposes to pretrain video and text back- bones on the EgoNCE pretraining task [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' By leverag- ing large-scale video + text narrations from Ego4D, they successfully transfer features to a variety of tasks includ- ing NLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' It was the runner-up entry for the Ego4D NLQ challenge at CVPR 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This method replaces the Slow- Fast features from the VSLNet baseline with the EgoVLP pretrained backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This baseline is complementary to our own approach where we use narrations to augment the localization training for NLQ task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' (3) ReLER adapts VSLNet to use a multi-scale cross- modal transformer architecture [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' It also proposes to augment the training data using video-level augmentation strategies like randomly sampling a subset of the video to try and mitigate overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This was the winning entry of the Ego4D NLQ challenge at CVPR 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We augment this method with EgoVLP pretrained backbones to obtain a stronger ‘ReLER∗’ baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Unlike this method, which augments the data at the video level, we propose to augment the data at the query level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We will demonstrate that NaQ is complementary and boosts the performance of ReLER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Note that both EgoVLP and ReLER∗ leverage the exact same narration data as NaQ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' NaQ requires no greater super- vision or data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For each baseline, we adapt the authors’ code bases to train with NaQ data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For consistency, we report the results of each method as re- produced using the provided code and instructions, in ad- dition to reporting the official paper numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We train each method with NaQ augmentation for 200 epochs and stop training early when the validation performance satu- rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We found that it was helpful to finetune for up to 30 epochs on only the NLQ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Please see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' S1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Experimental results We report results on the NLQ validation set in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The poor performance of the VSLNet baseline on NLQ highlights the difficulty of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' It requires localizing re- sponses typically shorter than 10 seconds in 8+ minute long egocentric videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The limited size of the training dataset further exacerbates this problem, since there are only 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3k training queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' However, when augmented with NaQ , 5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 Method Narrations R@1 R@5 R@1 R@5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' VSLNet [29] \x17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='45 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' VSLNet† \x17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='78 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='56 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' VSLNet + NaQ \x13 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='14 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='78 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='69 absolute gain +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='36 +8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='87 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='22 +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='57 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' EgoVLP [18] \x13 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='84 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='84 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='81 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='45 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' EgoVLP† \x13 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='43 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='55 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='46 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' EgoVLP + NaQ \x13 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='90 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='38 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='46 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='80 absolute gain +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='47 +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='63 +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='91 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='34 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ReLER [19] \x17 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='79 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='74 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='85 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ReLER† \x17 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='25 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='49 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='27 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='23 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ReLER∗ \x13 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='48 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='55 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='52 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='33 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ReLER∗ + NaQ \x13 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='31 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='59 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='62 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='51 absolute gain +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='83 +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='04 +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='10 +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='18 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Results on NLQ validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ∗replace SlowFast with EgoVLP features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' †Results reproduced using authors’ code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' the performance across all metrics nearly doubles, indicat- ing the effectiveness of NaQ in addressing these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This is a dramatic gain, though it comes at the cost of larger narrations data that is not available to VSLNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' When VSLNet is augmented with NaQ , it is already competitive with EgoVLP, which pretrains video and text backbones with Ego4D videos + narrations and uses the same VSLNet query-localization architecture (rows 3 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' When NaQ is combined with EgoVLP, it further im- proves the performance by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='9 - 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='6 points across metrics (row 5 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' row 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This confirms that NaQ augmentation for query localization training complements the EgoVLP pretraining of video-text backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Importantly, our gain here comes at no additional cost in data or annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ReLER [19] uses SlowFast + CLIP video features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For a fair comparison, we replace the SlowFast features with EgoVLP features to obtain ReLER∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This improves by a large margin as expected, and gives us a stronger baseline to compare with (row 8 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' row 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Recall that ReLER∗ uses video-level data augmentation using variable-length sliding windows and video splicing [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' When ReLER∗ is aug- mented with NaQ , the performance increases by a signifi- cant margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This confirms the complementary nature of the query-level augmentation we propose in NaQ with video- level augmentation in ReLER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Overall, we find that NaQ augmentation greatly improves the performance of all methods across all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The ab- solute gains across metrics are remarkably consistent re- gardless of the underlying method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' When averaged across the methods, NaQ improves the absolute recall@1 perfor- mance by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='22 at IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='07 at IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5, and the ab- solute recall@5 performance by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='18 at IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='03 at IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This confirms the generality and effectiveness of NaQ at expanding the limited NLQ annotations by boot- strapping it with narrations, a relatively cheaper and more abundant data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' More importantly, the insight in NaQ Method R@1 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 R@1 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 Mean R@1† R@5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 R@5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 NaQ (ours) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='46 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='74 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='59 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='50 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='74 Red Panda∗ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='46 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='06 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='26 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='95 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='11 Badgers@UW-Mad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='∗ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='71 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='57 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='64 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='45 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='03 CONE∗ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='26 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='24 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='25 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='42 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='51 ReLER [19] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='89 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='14 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='51 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='41 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='94 EgoVLP [18] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='46 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='35 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='76 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='29 VSLNet [29] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='08 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='79 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='07 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Results on Ego4D NLQ challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' †Primary metric for the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ∗Unpublished work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' is not simply that large-scale data benefits performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Rather, we emphasize how to use this data: we leverage nar- rations as queries for query-localization network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This is evidenced by our experiments demonstrating major gains on EgoVLP and ReLER∗, methods which also benefit from large-scale pretraining on video-narrations data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Ego4D NLQ challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We submitted our best perform- ing method (ReLER∗ + NaQ ) to the Ego4D NLQ challenge leaderboard, where the NLQ evaluation is performed on a EvalAI server on a held-out set of test annotations [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Note that while the videos are available to participants, the annotations (including narrations) are not accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The results are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' VSLNet is the baseline provided by the organizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ReLER and EgoVLP were the winning and runner-up entries from the CVPR 2022 edition of the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Red Panda, Badgers@UW-Madison, and CONE are the top three entries from the ECCV 2022 edition of the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 As of the time of submission, NaQ is the lead- ing entry among all methods on the leaderboard, including those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Our approach has the best available results on this challenge, by a healthy margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' TRJ ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We study the impact of using temporal re- sponse jittering (TRJ) (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='2) in an ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We observe that using TRJ improves the performance by up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='7 points in recall @ 1 metrics and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='7 in recall @ 5 met- rics consistently across all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Please see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' S3 for the complete results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Performance analyses In the previous section, we verified the effectiveness of our approach through a careful comparison with recent state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We now ascertain the strengths and weaknesses of our approach through a series of quan- titative studies and discuss qualitative results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For performing analysis-specific experiments, we adopt the EgoVLP + NaQ method since it requires lower computa- tional cost and time to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' (1) How does performance scale with narrations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' One 3The code+reports for these methods were unavailable at the time of our experiments, so we could not compare with them outside the leader- board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 6 Video ReLER* Ground truth Ours 270 276 273 272 274 276 Query: How many funnels are on the shelf?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 0 9 18 Video 201 207 204 202 204 207 Query: Where was the brake pad before I took it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 104 106 108 Video 180 198 189 164 166 168 Query: What color bottle is on the sink?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 180 190 200 𝑡 = 𝑇 𝑡 = 0 1 𝑡 = 𝑇 𝑡 = 0 2 3 𝑡 = 𝑇 𝑡 = 0 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Qualitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We show three examples of NLQ task predictions (one per column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' In each column, the natural language query is displayed at the top, the ground truth responses are in the central row, and the model predictions are on the first and last rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The temporal extents of the video and predicted time windows are shown right next to the images on each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We compare ReLER∗ [19] baseline (on the first row) against our NaQ method which augments the NLQ training for ReLER∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Example 1: Our method successfully identifies the response window showing how many funnels are on the shelf, while the baseline fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The object ‘funnel’ is a low-shot object with fewer than 10 training queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This supports our experimental observation that NaQ has a strong advantage on low-shot objects and counting-based queries (see Tabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Example 2: NaQ successfully recognizes the object ‘brake pad’ and is able to localize where it was taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' ReLER* incorrectly identifies a spanner as the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Example 3: This is a failure case for NaQ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' While it correctly identifies a sink, this particular sink does not contain the bottle and the model fails to respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Object / place queries People queries Method Where is X before/after Y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Where did I put X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Where is X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' What did I put in X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' How many X’s?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' In what location did I see X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' What X did I Y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' What X is Y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' State?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Who did I interact with during Y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' VSLNet 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='96 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='94 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='67 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='39 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='96 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='57 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='94 +NaQ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='76 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='60 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='61 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='86 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='59 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='52 EgoVLP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='62 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='37 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='23 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='52 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='59 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='61 +NaQ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='70 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='44 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='83 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='13 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='79 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='60 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='59 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='03 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='88 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='04 ReLER* 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='78 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='82 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='29 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='78 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='54 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='54 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='90 +NaQ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='98 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='34 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='26 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='61 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='67 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='78 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='86 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='29 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='84 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Performance over NLQ query types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We report recall@1 at IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We include query types with ≥ 100 val samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We highlight cases where NaQ improves recall by more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' % of narrations as queries Recall @ 1 5 10 15 20 0 25 50 75 100 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 % of narrations as queries Recall @ 5 5 10 15 20 25 30 0 25 50 75 100 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 % of narrations as queries Recall @ 1 5 10 15 20 0 25 50 75 100 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 % of narrations as queries Recall @ 1 5 10 15 20 0 25 50 75 100 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 % of NaQ dataset % of NaQ dataset Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Data scaling analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We train EgoVLP + NaQ using all NLQ and k% of NaQ dataset (k represented on the X-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' NLQ performance scales linearly with the size of the NaQ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' of the key benefits of using narrations for pretraining is that they are available on a large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We generated 850k nar- rations as queries for the NLQ task, which is two orders larger than the NLQ dataset containing 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3k train queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We now study performance scaling as a function of the amount of narrations used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For this, we addi- tionally trained EgoVLP + NaQ with 10%, 25%, 50% of the narrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 5 shows the results on NLQ (val).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The 0% performance represents EgoVLP and the 100% perfor- mance represents the full EgoVLP + NaQ reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' When adding only 10% of our NaQ data, we already observe good improvements on all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The performance con- tinues to linearly scale as we add more narrations for NaQ augmentation, confirming the utility of our paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' (2) What types of queries does NaQ benefit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Next, we break down the NLQ performance across query types, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', the form of reasoning required by the query (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', where did I put object X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' who did I talk to while doing activity Y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The NLQ dataset was created by providing an ini- tial set of 13 query templates [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For reliable evaluation, we select 10 out of the 13 templates which contain 100 or more samples in the validation split, and report results 7 High-shot Mid-shot Low-shot Method IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 VSLNet 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='82 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='71 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='30 +NaQ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='72 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='53 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='26 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='00 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='57 EgoVLP 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='32 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='83 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='96 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='70 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='63 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='42 +NaQ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='59 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='27 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='13 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='20 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='05 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='30 ReLER∗ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='07 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='35 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='74 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='18 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='21 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='29 +NaQ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='37 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='37 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='87 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='38 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='20 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='75 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Performance breakdown across object types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For ob- ject type queries, we categories objects into low-shot, mid-shot, and high-shot objects based on their frequency of occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We report the recall@1 metric at IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 and IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We highlight cases where NaQ improves recall by over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We observe that using NaQ leads to significant improvements (marked in green) on 8/10 templates for at least 2/3 methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' However, it only has a limited impact for ‘Where is object X?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' and ‘In what location did I see X?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' These queries may require explicit spatial under- standing to achieve better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Since all methods perform poorly on those queries and do not benefit from training on NaQ , it hints at the need to incorporate better spatial understanding for video models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' (3) Does NaQ help respond about long-tail objects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The NLQ dataset has a long-tail of objects that are the sub- ject of queries due to the sparse nature of NLQ annota- tions (1 query per 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='4 minutes of videos on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' How- ever, since narrations are more densely annotated through- out the video (20+ narrations per minute), they contain rich information about objects that are rarely queried about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We therefore study if pretraining NLQ localization models with narrations can help respond to queries about long-tail ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We divide objects from the NLQ train annotations into 3 types (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' S1): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' high-shot objects which are queried more than 50 times (65 in total), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' mid- shot objects which are queried about 10 to 50 times (147 in total), and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' low-shot objects which are queried about be- tween 2 to 10 times (967 in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The results are in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Overall, we observe that NaQ improves performance by a large margin in most cases, and has the biggest gains on mid-shot and low-shot objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This indicates that using narrations as queries helps mitigate some of the biases in the NLQ data, and improves responses to queries about less- frequently occurring objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' (4) Does NaQ facilitate zero-shot / few-shot NLQ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Con- sidering that NaQ enables better performance on long-tail objects, we next study whether it can facilitate zero-shot or few-shot learning for NLQ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', given our large-scale NaQ data and little to no NLQ task annotations, can we learn good NLQ models?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We are first to study this to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We train EgoVLP + NaQ method with all of % of narrations as queries Recall @ 1 0 5 10 15 0 10 20 30 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 % of narrations as queries Recall @ 5 10 15 20 25 0 10 20 30 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 % of narrations as queries Recall @ 1 0 5 10 15 0 10 20 30 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 % of narrations as queries Recall @ 1 0 5 10 15 0 10 20 30 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 % of NLQ dataset % of NLQ dataset Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Zero-shot and few-shot learning for NLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We train EgoVLP + NaQ using all NaQ and k% of the NLQ train data (k on the X-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The dotted horizontal lines represent the EgoVLP performance with 100% NLQ and no NaQ augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' NaQ and k% of NLQ train data, where k = {0, 10, 25, 35}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' k = 0 represents the zero-shot case, and the rest represent few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The results are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The triangles represent EgoVLP + NaQ with k% NLQ data, and the hor- izontal line represents the EgoVLP baseline with no NaQ data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' It is interesting to observe that even with no NLQ data, the model performs well using NaQ and matches the EgoVLP performance on the R@5 metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' When we inject 10% of the NLQ dataset, we get comparable or better per- formances on 3/4 metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' At 25% of NLQ data, it matches or outperforms EgoVLP on all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Finally, at 35%, we comprehensively outperform EgoVLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This study sug- gests that we can leverage large-scale free-form narration annotations using NaQ to compensate for the lack of NLQ annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' While these are not free to obtain, they are eas- ier to annotate than NLQ and can also be used for various purposes other than the NLQ task itself [12], meaning that many research directions are likely to continue investing in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Conclusions In this work, we propose Narrations-as-Queries, a sim- ple data augmentation technique that dramatically improves state-of-the-art results on the Natural Language Queries task in the Episodic Memory benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Our key insight is to convert timestamped narrations in egocentric videos into natural language queries and use them as additional data for training NLQ localization models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' To convert times- tamped narrations into a form compatible with NLQ, we propose a temporal response jittering technique to convert a single timestamp into temporal windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We perform ex- periments to demonstrate that our approach can be used as a simple plug-in to existing methods, massively improves multiple top methods for this task, and yields the very best performance to-date on the Ego4D NLQ benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We hope that our approach serves as a useful tool for future research on this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We will share code, data, and models upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 8 References [1] Yazan Abu Farha, Alexander Richard, and Juergen Gall.' metadata={'source': 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2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 2 10 Low-shot Mid-shot High-shot Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Long-tail of objects in NLQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Supplementary Materials We now provide additional information about our exper- imental settings, and qualitative and quantitative analyses to support our experiments in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Implementation details We perform joint NaQ + NLQ training with a large batch sizes and high learning rates for accelerated convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For VSLNet and EgoVLP methods, we use a batch size of 2048 and initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='001 on 2 A40 GPUs with a memory size of 46GB per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For ReLER∗, we use a batch size of 1536 and an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='001 on 8 A40 GPUs since it has larger memory and compute require- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We train each method for up to 200 epochs on NaQ + NLQ training data, and then finetune them for up to 30 epochs on NLQ training data alone with a lower learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We found finetuning to be unnecessary for VSLNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For EgoVLP, we finetuned with the original hyperparame- ter settings from [18] and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='00001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For ReLER∗, we finetuned with the original hyperparameter setting from [19] and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We per- form early stopping in each case using the performance on NLQ validation split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For temporal random jittering (TRJ), we per- formed a grid search with the expansion factor values S={2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='0, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='0, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We found S=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 to work best for EgoVLP and VSLNet, and S=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='0 to work best for ReLER∗ based on their NLQ validation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Long-tail of objects in NLQ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' S1 shows the long-tail of objects queried about in NLQ, and the split of low-shot, mid-shot, and high-shot ob- jects used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Note that for a given point x on X- axis, the Y-axis shows the number of objects that have x queries in the NLQ train dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For example, there are more than 1000 objects with only 1 training sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Ablation study for Temporal Response Jit- tering We study the impact of using temporal response jittering (TRJ) described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' S1, we measure the per- IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5 Method TRJ R@1 R@5 R@1 R@5 VSLNet + NaQ \x17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='89 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='30 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='99 VSLNet + NaQ \x13 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='14 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='78 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='69 absolute gain +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='25 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='99 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='48 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='70 EgoVLP + NaQ \x17 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='27 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='93 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='07 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='14 EgoVLP + NaQ \x13 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='90 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='38 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='46 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='80 absolute gain +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='63 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='45 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='39 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='66 ReLER∗ + NaQ \x17 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='48 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='26 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='25 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='44 ReLER∗ + NaQ \x13 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='31 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='59 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='62 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='51 absolute gain +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='83 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='33 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='37 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='07 Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Ablation study of temporal random jittering (TRJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' formance of using NaQ with and without TRJ, where not us- ing TRJ implies that the seed temporal window from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' (1) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Overall, we observe a consistent improvement of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='83 in R@1 metrics and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='70 in R@5 metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' This in- dicates that TRJ is able to address the limitations of the seed temporal window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Few-shot analysis We perform a more detailed analysis of the few-shot per- formance discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Specifically, we analyze the zero-/few-shot performance across the various query templates in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' When tested zero-shot, NaQ already competes with or outperforms the baseline on ob- ject/place templates such as ‘where is X before/after Y?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', ‘where did I put X?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', ‘where is X?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', ‘In what location did I see X?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', ‘what X is Y?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=', and ‘object state’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='4 As we in- ject NLQ data into NaQ training, the performance improves quickly on the remaining templates, and outperforms the baseline on 8/10 templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Qualitative examples In supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='html shared here, we link to qual- itative videos for the following: Comparing annotations for NLQ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Narrations NaQ benefits performance on most query templates NaQ benefits performance on queries about long-tail objects NaQ facilitates zero-shot NLQ 4We provide video visualizations of the zero-shot performance on these 4 templates in supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 11 Distribution over obiect freguencies 103 Objects 101 # 1 2 10 50 100 1000 # queries per objectObject / place queries People queries % NLQ % NaQ Where is X before/after Y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Where did I put X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Where is X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' What did I put in X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' How many X’s?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' In what location did I see X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' What X did I Y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' What X is Y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' State?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Who did I interact with during Y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 100 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='62 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='37 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='23 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='52 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='59 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='61 0 100 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='04 Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' Few-shot analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We split the few-shot results from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 6 in the main paper across the various query templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' We report recall@1 at IoU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' The first two columns show the percentage of the NLQ and NaQ data used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' For example, the first row with 100% NLQ and 0% NaQ is the baseline, the second row with 0% NLQ and 100% NaQ is our zero-shot setting, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} +page_content=' 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dAyT4oBgHgl3EQf2Pkf/content/2301.00746v1.pdf'} diff --git a/79AyT4oBgHgl3EQf2_lP/vector_store/index.faiss b/79AyT4oBgHgl3EQf2_lP/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b3236a0d05474f9b2f90aec8843e0444600a80d5 --- /dev/null +++ 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0000000000000000000000000000000000000000..0a289a71fd10495ef00920cebfe23b1676cf002b Binary files /dev/null and b/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf differ diff --git a/AdAzT4oBgHgl3EQfTPx3/content/tmp_files/2301.01246v1.pdf.txt b/AdAzT4oBgHgl3EQfTPx3/content/tmp_files/2301.01246v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a2cbf17fed1b8ce84b348f8432aece3f90ba4af --- /dev/null +++ b/AdAzT4oBgHgl3EQfTPx3/content/tmp_files/2301.01246v1.pdf.txt @@ -0,0 +1,694 @@ +arXiv:2301.01246v1 [cs.AI] 3 Jan 2023 +Efficient method for handling diverse agents in QDec-POMDPs +Nitsan Soffair +Ben Gurion University +soffair@post.bgu.ac.il +Abstract +The SOTA algorithms for addressing QDec- +POMDP issues, QDec-FP and QDec-FPS, are un- +able to effectively tackle problems that involve dif- +ferent types of sensing agents. We propose a new +algorithm that addresses this issue by requiring +agents to adopt the same plan if one agent is unable +to take a sensing action but the other can. Our algo- +rithm performs significantly better than both QDec- +FP and QDec-FPS in these types of situations. +1 +Introduction +Automated +planning +and +scheduling +[Wikipedia contributors, 2022a] +is +a +field +of +artificial +intelligence that deals with creating and implementing strate- +gies or action sequences for intelligent agents, autonomous +robots, and unmanned vehicles. +It involves finding and +optimizing solutions in complex multidimensional spaces +and is closely related to decision theory. +Planning can +be done offline in known environments, but in unknown +environments, the strategy may need to be revised online and +models and policies may need to be adapted. +2 +Background +2.1 +MDP +An +MDP +[Wikipedia contributors, 2022b] +is +a +4-tuple +(S, A, P, R) where S is the state space, A is the action space, +P is the probability that action a in state s will lead to the next +state, R is the immediate reward received after transforming +from a state to the next state. A policy function π is a map- +ping from state space to action space. +2.2 +POMDP +A POMDP [Wikipedia contributors, 2022c] is a 7-tuple +(S, A, T, R, Ω, O, γ) where S is the set of states, A is the +set of actions, T is a set of transition probabilities between +states, R is the reward function, Ω is a set of observations, O +is a set of observation probabilities, γ ∈ [0, 1] is the discount +factor. At each time period, the environment is in some state. +The agent takes an action a, which causes the environment to +transition to the next state with probability T (s|s′, a). At the +same time, the agent receives an observation o which depends +on the new state of the environment, and on the just taken ac- +tion a, with probability O(o|s′, a). Finally, the agent receives +a reward r equal to R(s′, a). +2.3 +Dec-POMDP +A Dec-POMDP [Wikipedia contributors, 2020] is a 7-tuple +(S, {Ai}, T, R, {Ωi}, O, γ) where S is the set of states, Ai is +the set of actions for agent i, {Ai} is the set of joint actions, +T is a set of transition probabilities between states, Ωi is a set +of observations for agent i, {Ωi} is the set of joint observa- +tions, O is a set of observation probabilities, γ ∈ [0, 1] is the +discount factor. At each time step, each agent takes an action +a, the state updates based on the transition function T , each +agent observes an observation based on the observation func- +tion O, and a reward is generated for the whole team based +on the reward function R. +2.4 +QDec-POMDP +A QDec-POMDP [Brafman et al., 2013] is a model for rep- +resenting the decision-making process of multiple agents in a +dynamic environment. It consists of a set of agents, states, ac- +tions, observations, and a goal. The QDec-POMDP uses pol- +icy trees to represent the local plans of each agent, with each +node labeled with an action and each branch labeled with an +observation. To execute the plan, the agent performs the ac- +tion at the root of the tree and then uses the subtree labeled +with the observation it obtains to guide future action selec- +tion. +2.5 +SDR +The SDR [Brafman and Shani, 2012] planner is a method for +planning under uncertainty in which a single state is chosen +from the current belief state and used to create a determin- +istic classical problem. The resulting plan is then executed +until a sensing action is performed, at which point the belief +state is updated and the process is repeated. This version of +SDR maintains and uses a complete, explicit description of +the belief state, though a modified version of the algorithm +uses sampling and lazy belief-state maintenance. +2.6 +CPOR +The CPOR [Maliah et al., 2014] algorithm repeatedly selects +and executes sensing actions in order to gather information +and achieve a goal. The planner uses a classical projection to + +plan for the preconditions of each observation action and then +executes the action. The selection of the next sensing action +is based on an estimation of the myopic value of information, +or the value that will be achieved from executing the action +without considering future observations. This value is calcu- +lated using the number of disjunctive action landmarks that +can be achieved following the sensing action. +2.7 +Factored planning +The algorithm [Shekhar et al., 2021a] first creates a single- +agent team problem by treating all actions and observations +as if they are performed by a single combined agent. This +results in a team solution tree, which is then projected to each +individual agent. Each agent then tries to generate a local +policy that includes the projected sub-tree as a solution. If all +agents are able to solve their local problems, the actions are +aligned and a solution is returned. If one of the agents cannot +solve their problem, a new team solution is generated and the +process is repeated. If no new team solution is possible, the +process fails. +2.8 +QDec-FP +QDec-FP [Shekhar et al., 2021b] is a three-stage process for +solving multi-agent problems. In the first stage, a team so- +lution is generated by treating all actions as if they were ex- +ecuted by a single meta-agent. In the second stage, the pro- +jection of the team solution is extended for each individual +agent. Finally, in the third stage, the single agent plan trees +are aligned. +2.9 +QDec-FPS +In QDec-FPS [Shekhar et al., 2021b] the SDR translation +maintains two propositions for each proposition, represent- +ing that the agent knowing that it is true or false. It also trans- +forms preconditions of actions into propositions that must be +known to be true in all possible worlds. In addition, QDec- +FPS allows for agents to communicate by signal to each other +by setting the value of a variable that can be sensed by other +agents, allowing them to reason about the value of a proposi- +tion they cannot sense. +3 +Algorithm +The algorithm consists of two steps. In the first step, we pre- +pare the environment by determining the sensory capabilities +of each agent. In the second step, we use QDec-FP to create a +team plan, ensuring that any actions that rely on observations +that an agent cannot make are eliminated. The subsequent +steps are identical to those in QDec-FP. +4 +Domains +4.1 +Box-pushing +There is a grid with boxes that need to be moved to different +locations outside of the column they are currently in. One +agent can push a light box, but two agents are required to +push a heavy box. The agents can vary in their abilities and +can be assigned to push different boxes. +4.2 +Table-mover +The system includes several tables and rooms that are con- +nected, and agents that can move between them. The exact +location of the tables is not known at the beginning, and the +agents must move them to their designated locations. The +agents can have different capabilities for sensing and manip- +ulating objects. All actions involving the manipulation of ta- +bles require the collaboration of at least two agents, including +moving, lifting, and dropping the tables. +5 +Results +The experiments were run on a computer with a 4-core pro- +cessor running at 2.40GHz. The domain of the experiment +could be either homogeneous or heterogeneous, represented +by HM and HT, respectively. The variables measured in the +experiments included the number of backtracks, time needed +for the planning process, maximum tree width, and maximum +tree height. The results are an average of 10 experiments. The +winner of the QDec versus the variant for each criterion is +noted in bold. If the solver was unable to solve the problem, +this is indicated by an asterisk. +5.1 +Box-pushing +Grid size 3 with 1 box is represented by B1(3). +QDec-FP +type +domain +#bts +time +width +height +HM +B1(3) +0 +4.3 +7.6 +20.1 +HM +B2(4) +0 +7.8 +15.6 +18.6 +HT +B4(3) +8.5 +14.1 +4 +8.4 +HT +B5(3) +11.1 +40.8 +7.4 +12.9 +HT +B6(3) +17.5 +77.6 +8 +14.4 +HT +B9(5) +36.5 +7.7M +27 +25 +HT +B10(5) +* +* +* +* +QDec-FP variant +type +domain +#bts +time +width +height +HM +B1(3) +0 +3.9 +7.2 +18.8 +HM +B2(4) +0 +8.1 +15.6 +20.4 +HT +B4(3) +7.7 +9.6 +4 +10.1 +HT +B5(3) +4.8 +14.4 +6 +11.8 +HT +B6(3) +18.2 +51.1 +8 +14.6 +HT +B9(5) +30.5 +1.5M +27.25 +26.75 +HT +B10(5) +19.5 +689K +27 +27.25 +The variant has no additional costs when there are no back- +tracks. However, when backtracking is necessary, the vari- +ant allows for faster planning and produces a higher quality +tree. This is because the variant focuses on the failing agent, +speeds up the backtracking process, ensures that branching +is equal among agents who cannot sense their surroundings, +and enables the creation of valid team plans through the use +of CPOR. + +QDec-FPS +type +domain +#bts +time +width +height +HM +B1(3) +0 +3.4 +6.1 +16.9 +HM +B2(4) +0 +7 +9 +17 +HT +B4(3) +0 +1.2 +4 +8.4 +HT +B5(3) +1.7 +10.6 +7.2 +13.3 +HT +B6(3) +0 +3.5 +7.6 +16 +HT +B7(4) +* +* +* +* +HT +B8(4) +* +* +* +* +HT +B9(4) +* +* +* +* +QDec-FPS variant +type +domain +#bts +time +width +height +HM +B1(3) +0 +4 +5.7 +18.2 +HM +B2(4) +0 +8.3 +11.7 +19.4 +HT +B4(3) +0.8 +2.2 +4 +8.8 +HT +B5(3) +1.3 +7 +6.4 +12.6 +HT +B6(3) +0 +4.2 +8 +16 +HT +B7(4) +0 +3.7 +6 +9.5 +HT +B8(4) +0.2 +5 +5.6 +9.5 +HT +B9(4) +0 +8.3 +13 +19 +In the case of no backtracks, the variant has a slower run- +ning time and lower quality trees. In the case of 1+ back- +tracks, the variant has a faster running time and higher quality +trees. This is because the variant has fewer agent constraints +and larger SDR problems, which makes the backtrack mech- +anism faster and allows for better team plans. +5.2 +Table-mover +T1(3) refers to a grid with a size of 3 and containing only 1 +table. +QDec-FP +type +domain +#bts +time +width +height +HM +T1(3) +0 +9.6 +8 +19.8 +HM +T3(4) +1.3 +37.4 +14.1 +33.6 +HT +T6(3) +8.7 +7.3 +2 +8 +HT +T9(3) +12.5 +66.1 +8 +21 +HT +T11(5) +39.5 +879K +13 +25 +QDec-FP variant +type +domain +#bts +time +width +height +HM +T1(3) +0 +4.4 +8 +20 +HM +T3(4) +1.1 +26.6K +14.2 +33.7 +HT +T6(3) +9 +11.1K +2 +7.6 +HT +T9(3) +10 +26K +8 +18.67 +HT +T11(5) +15 +176K +11.25 +24.25 +The QDec-FP variant is efficient in simple problems with +no added overhead. It also performs faster and more effi- +ciently in complex problems, using fewer backtracks and pro- +ducing smaller plan trees. +QDec-FPS +type +domain +#bts +time +width +height +HM +T1(3) +0 +4.5 +7.8 +18 +HM +T3(4) +0 +13.6 +13.8 +27.5 +HT +T6(3) +0 +0.7 +2 +6.4 +HT +T9(3) +0 +3.7 +8 +16 +HT +T11(5) +0 +15.5K +12 +33 +HT +T12(5) +* +* +* +* +HT +T13(5) +* +* +* +* +HT +T14(5) +* +* +* +* +QDec-FPS variant +type +domain +#bts +time +width +height +HM +T1(3) +0 +5.5 +7.9 +18 +HM +T3(4) +0 +13.7 +13.4 +17.4 +HT +T6(3) +0 +0.7 +2 +7.2 +HT +T9(3) +1 +7.6 +8 +25 +HT +T11(5) +3 +42K +14 +24.5 +HT +T12(5) +0 +32K +8 +20 +HT +T13(5) +5 +233K +16 +27 +HT +T14(5) +10 +140K +16 +39 +The QDec-FPS variant is able to handle more complex +problems, solve them faster, and has a small overhead when +dealing with simple problems. +6 +Conclusion +The QDec-FP variant is a planning algorithm that is efficient +in both simple and complex problems, producing high qual- +ity tree plans. In cases of backtracking, it speeds up the pro- +cess and creates better team plans. The QDec-FPS variant is +also able to handle complex problems efficiently, with a small +overhead in simple problems. +7 +Further work +The variant is not capable of addressing the need for complex +communication between agents in certain domains. +References +[Brafman and Shani, 2012] R. I. Brafman and G. Shani. Re- +planning in domains with partial information and sens- +ing actions. +Journal of Artificial Intelligence Research, +45:565–600, dec 2012. +[Brafman et al., 2013] Ronen I. Brafman, Guy Shani, and +Shlomo Zilberstein. Qualitative planning under partial ob- +servability in multi-agent domains. +Proceedings of the +AAAI Conference on Artificial Intelligence, 2013. +[Maliah et al., 2014] Shlomi Maliah, Ronen Brafman, Erez +Karpas, and Guy Shani. Partially observable online con- +tingent planning using landmark heuristics. +In Twenty- +Fourth International Conference on Automated Planning +and Scheduling, 2014. +[Shekhar et al., 2021a] Shashank Shekhar, Ronen I. Braf- +man, and Guy Shani. A factored approach to deterministic +contingent multi-agent planning. Proceedings of the Inter- +national Conference on Automated Planning and Schedul- +ing, 29(1):419–427, May 2021. + +[Shekhar et al., 2021b] Shashank Shekhar, Ronen I. Braf- +man, and Guy Shani. Improved knowledge modeling and +its use for signaling in multi-agent planning with partial +observability. Proceedings of the AAAI Conference on Ar- +tificial Intelligence, 35(13):11954–11961, May 2021. +[Wikipedia contributors, 2020] Wikipedia +contributors. +Decentralized +partially +observable +markov +deci- +sion process — Wikipedia, +the free encyclopedia. +https://en.wikipedia.org/w/index.php?title=Decentralized partially observable Markov decision process&oldid=992800884, +2020. [Online; accessed 1-December-2022]. +[Wikipedia contributors, 2022a] Wikipedia +contributors. +Automated planning and scheduling — Wikipedia, the +free encyclopedia, 2022. +[Online; accessed 2-January- +2023]. +[Wikipedia contributors, 2022b] Wikipedia +contributors. +Markov +decision +pro- +cess +— +Wikipedia, +the +free +encyclopedia. +https://en.wikipedia.org/w/index.php?title=Markov decision process&oldid=1124829194, +2022. [Online; accessed 1-December-2022]. +[Wikipedia contributors, 2022c] Wikipedia +contrib- +utors. +Partially +observable +markov +decision +process +— +Wikipedia, +the +free +encyclopedia. +https://en.wikipedia.org/w/index.php?title=Partially observable Markov decision process&oldid=1104376990, +2022. [Online; accessed 1-December-2022]. + diff --git a/AdAzT4oBgHgl3EQfTPx3/content/tmp_files/load_file.txt b/AdAzT4oBgHgl3EQfTPx3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..426ed18fb7b4f699f3857f68da35e370ffc98ede --- /dev/null +++ b/AdAzT4oBgHgl3EQfTPx3/content/tmp_files/load_file.txt @@ -0,0 +1,259 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf,len=258 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='01246v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='AI] 3 Jan 2023 Efficient method for handling diverse agents in QDec-POMDPs Nitsan Soffair Ben Gurion University soffair@post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='bgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='il Abstract The SOTA algorithms for addressing QDec- POMDP issues, QDec-FP and QDec-FPS, are un- able to effectively tackle problems that involve dif- ferent types of sensing agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' We propose a new algorithm that addresses this issue by requiring agents to adopt the same plan if one agent is unable to take a sensing action but the other can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' Our algo- rithm performs significantly better than both QDec- FP and QDec-FPS in these types of situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 1 Introduction Automated planning and scheduling [Wikipedia contributors, 2022a] is a field of artificial intelligence that deals with creating and implementing strate- gies or action sequences for intelligent agents, autonomous robots, and unmanned vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' It involves finding and optimizing solutions in complex multidimensional spaces and is closely related to decision theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' Planning can be done offline in known environments, but in unknown environments, the strategy may need to be revised online and models and policies may need to be adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 MDP An MDP [Wikipedia contributors, 2022b] is a 4-tuple (S, A, P, R) where S is the state space, A is the action space, P is the probability that action a in state s will lead to the next state, R is the immediate reward received after transforming from a state to the next state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' A policy function π is a map- ping from state space to action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='2 POMDP A POMDP [Wikipedia contributors, 2022c] is a 7-tuple (S, A, T, R, Ω, O, γ) where S is the set of states, A is the set of actions, T is a set of transition probabilities between states, R is the reward function, Ω is a set of observations, O is a set of observation probabilities, γ ∈ [0, 1] is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' At each time period, the environment is in some state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The agent takes an action a, which causes the environment to transition to the next state with probability T (s|s′, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' At the same time, the agent receives an observation o which depends on the new state of the environment, and on the just taken ac- tion a, with probability O(o|s′, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' Finally, the agent receives a reward r equal to R(s′, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='3 Dec-POMDP A Dec-POMDP [Wikipedia contributors, 2020] is a 7-tuple (S, {Ai}, T, R, {Ωi}, O, γ) where S is the set of states, Ai is the set of actions for agent i, {Ai} is the set of joint actions, T is a set of transition probabilities between states, Ωi is a set of observations for agent i, {Ωi} is the set of joint observa- tions, O is a set of observation probabilities, γ ∈ [0, 1] is the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' At each time step, each agent takes an action a, the state updates based on the transition function T , each agent observes an observation based on the observation func- tion O, and a reward is generated for the whole team based on the reward function R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 QDec-POMDP A QDec-POMDP [Brafman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=', 2013] is a model for rep- resenting the decision-making process of multiple agents in a dynamic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' It consists of a set of agents, states, ac- tions, observations, and a goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The QDec-POMDP uses pol- icy trees to represent the local plans of each agent, with each node labeled with an action and each branch labeled with an observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' To execute the plan, the agent performs the ac- tion at the root of the tree and then uses the subtree labeled with the observation it obtains to guide future action selec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 SDR The SDR [Brafman and Shani, 2012] planner is a method for planning under uncertainty in which a single state is chosen from the current belief state and used to create a determin- istic classical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The resulting plan is then executed until a sensing action is performed, at which point the belief state is updated and the process is repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' This version of SDR maintains and uses a complete, explicit description of the belief state, though a modified version of the algorithm uses sampling and lazy belief-state maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 CPOR The CPOR [Maliah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=', 2014] algorithm repeatedly selects and executes sensing actions in order to gather information and achieve a goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The planner uses a classical projection to plan for the preconditions of each observation action and then executes the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The selection of the next sensing action is based on an estimation of the myopic value of information, or the value that will be achieved from executing the action without considering future observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' This value is calcu- lated using the number of disjunctive action landmarks that can be achieved following the sensing action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='7 Factored planning The algorithm [Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=', 2021a] first creates a single- agent team problem by treating all actions and observations as if they are performed by a single combined agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' This results in a team solution tree, which is then projected to each individual agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' Each agent then tries to generate a local policy that includes the projected sub-tree as a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' If all agents are able to solve their local problems, the actions are aligned and a solution is returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' If one of the agents cannot solve their problem, a new team solution is generated and the process is repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' If no new team solution is possible, the process fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='8 QDec-FP QDec-FP [Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=', 2021b] is a three-stage process for solving multi-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' In the first stage, a team so- lution is generated by treating all actions as if they were ex- ecuted by a single meta-agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' In the second stage, the pro- jection of the team solution is extended for each individual agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' Finally, in the third stage, the single agent plan trees are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='9 QDec-FPS In QDec-FPS [Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=', 2021b] the SDR translation maintains two propositions for each proposition, represent- ing that the agent knowing that it is true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' It also trans- forms preconditions of actions into propositions that must be known to be true in all possible worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' In addition, QDec- FPS allows for agents to communicate by signal to each other by setting the value of a variable that can be sensed by other agents, allowing them to reason about the value of a proposi- tion they cannot sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 3 Algorithm The algorithm consists of two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' In the first step, we pre- pare the environment by determining the sensory capabilities of each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' In the second step, we use QDec-FP to create a team plan, ensuring that any actions that rely on observations that an agent cannot make are eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The subsequent steps are identical to those in QDec-FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 4 Domains 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 Box-pushing There is a grid with boxes that need to be moved to different locations outside of the column they are currently in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' One agent can push a light box, but two agents are required to push a heavy box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The agents can vary in their abilities and can be assigned to push different boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='2 Table-mover The system includes several tables and rooms that are con- nected, and agents that can move between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The exact location of the tables is not known at the beginning, and the agents must move them to their designated locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The agents can have different capabilities for sensing and manip- ulating objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' All actions involving the manipulation of ta- bles require the collaboration of at least two agents, including moving, lifting, and dropping the tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 5 Results The experiments were run on a computer with a 4-core pro- cessor running at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='40GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The domain of the experiment could be either homogeneous or heterogeneous, represented by HM and HT, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The variables measured in the experiments included the number of backtracks, time needed for the planning process, maximum tree width, and maximum tree height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The results are an average of 10 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The winner of the QDec versus the variant for each criterion is noted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' If the solver was unable to solve the problem, this is indicated by an asterisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 Box-pushing Grid size 3 with 1 box is represented by B1(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' QDec-FP type domain #bts time width height HM B1(3) 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 HM B2(4) 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 HT B4(3) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 HT B5(3) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='9 HT B6(3) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 HT B9(5) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='7M 27 25 HT B10(5) QDec-FP variant type domain #bts time width height HM B1(3) 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='8 HM B2(4) 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 HT B4(3) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 HT B5(3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='8 HT B6(3) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 HT B9(5) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5M 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='25 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='75 HT B10(5) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 689K 27 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='25 The variant has no additional costs when there are no back- tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' However, when backtracking is necessary, the vari- ant allows for faster planning and produces a higher quality tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' This is because the variant focuses on the failing agent, speeds up the backtracking process, ensures that branching is equal among agents who cannot sense their surroundings, and enables the creation of valid team plans through the use of CPOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' QDec-FPS type domain #bts time width height HM B1(3) 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='9 HM B2(4) 0 7 9 17 HT B4(3) 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='2 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 HT B5(3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='3 HT B6(3) 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 16 HT B7(4) HT B8(4) HT B9(4) QDec-FPS variant type domain #bts time width height HM B1(3) 0 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='2 HM B2(4) 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='7 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 HT B4(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='2 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='8 HT B5(3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='3 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 HT B6(3) 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='2 8 16 HT B7(4) 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='7 6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 HT B8(4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='2 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 HT B9(4) 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='3 13 19 In the case of no backtracks, the variant has a slower run- ning time and lower quality trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' In the case of 1+ back- tracks, the variant has a faster running time and higher quality trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' This is because the variant has fewer agent constraints and larger SDR problems, which makes the backtrack mech- anism faster and allows for better team plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='2 Table-mover T1(3) refers to a grid with a size of 3 and containing only 1 table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' QDec-FP type domain #bts time width height HM T1(3) 0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='8 HM T3(4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 HT T6(3) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='3 2 8 HT T9(3) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 8 21 HT T11(5) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 879K 13 25 QDec-FP variant type domain #bts time width height HM T1(3) 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 8 20 HM T3(4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='1 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6K 14.' metadata={'source': 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The QDec-FP variant is efficient in simple problems with no added overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' It also performs faster and more effi- ciently in complex problems, using fewer backtracks and pro- ducing smaller plan trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' QDec-FPS type domain #bts time width height HM T1(3) 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='8 18 HM T3(4) 0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='8 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 HT T6(3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='7 2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 HT T9(3) 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='7 8 16 HT T11(5) 0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5K 12 33 HT T12(5) HT T13(5) HT T14(5) QDec-FPS variant type domain #bts time width height HM T1(3) 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='9 18 HM T3(4) 0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='4 HT T6(3) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='7 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='2 HT T9(3) 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='6 8 25 HT T11(5) 3 42K 14 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='5 HT T12(5) 0 32K 8 20 HT T13(5) 5 233K 16 27 HT T14(5) 10 140K 16 39 The QDec-FPS variant is able to handle more complex problems, solve them faster, and has a small overhead when dealing with simple problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 6 Conclusion The QDec-FP variant is a planning algorithm that is efficient in both simple and complex problems, producing high qual- ity tree plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' In cases of backtracking, it speeds up the pro- cess and creates better team plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' The QDec-FPS variant is also able to handle complex problems efficiently, with a small overhead in simple problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' 7 Further work The variant is not capable of addressing the need for complex communication between agents in certain domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' References [Brafman and Shani, 2012] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' Brafman and G.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' In Twenty- Fourth International Conference on Automated Planning and Scheduling, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' [Shekhar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=', 2021a] Shashank Shekhar, Ronen I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' Braf- man, and Guy Shani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' A factored approach to deterministic contingent multi-agent planning.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='org/w/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='title=Decentralized partially observable Markov decision process&oldid=992800884, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' accessed 1-December-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' [Wikipedia contributors, 2022a] Wikipedia contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' Automated planning and scheduling — Wikipedia, the free encyclopedia, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' accessed 2-January- 2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' [Wikipedia contributors, 2022b] Wikipedia contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' Markov decision pro- cess — Wikipedia, the free encyclopedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' [Wikipedia contributors, 2022c] Wikipedia contrib- utors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' Partially observable markov decision process — Wikipedia, the free encyclopedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='org/w/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='php?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content='title=Partially observable Markov decision process&oldid=1104376990, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} +page_content=' accessed 1-December-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdAzT4oBgHgl3EQfTPx3/content/2301.01246v1.pdf'} diff --git a/C9AyT4oBgHgl3EQf4fqw/content/tmp_files/2301.00788v1.pdf.txt b/C9AyT4oBgHgl3EQf4fqw/content/tmp_files/2301.00788v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9a7977b9e791d5706cebbe6d40d506360116bb8 --- /dev/null +++ b/C9AyT4oBgHgl3EQf4fqw/content/tmp_files/2301.00788v1.pdf.txt @@ -0,0 +1,677 @@ +Graphical Abstract +Electrochemical Polishing of Chemical Vapor Deposited Niobium +Thin Films +Zeming Sun, Mingqi Ge, James T. Maniscalco, Victor Arrieta, Shawn R. +McNeal, Matthias U. Liepe +arXiv:2301.00788v1 [cond-mat.mtrl-sci] 2 Jan 2023 + +Chemical vapor deposition +Electrochemical polishing +10 um +Functional Nb surface for superconducting RFHighlights +Electrochemical Polishing of Chemical Vapor Deposited Niobium +Thin Films +Zeming Sun, Mingqi Ge, James T. Maniscalco, Victor Arrieta, Shawn R. +McNeal, Matthias U. Liepe +• Electrochemical polishing (EP) is demonstrated to effectively minimize +the surface roughness for chemical vapor deposited (CVD) niobium thin +films. +• CVD niobium films contain steps, kinks, and pyramidal features, re- +sulting in large surface roughness. EP polishing of these films involves +both macroscale and microscale smoothing. +• A probable dependence on crystal orientation during EP is observed, +indicating strong influences from locally enhanced current density and +thickness variations of oxide dielectrics. +• Obtaining the required surface conditions by a combined EP-CVD tech- +nology marks a feasible application of niobium thin films in supercon- +ducting RF. + +Electrochemical Polishing of Chemical Vapor Deposited +Niobium Thin Films +Zeming Suna,∗, Mingqi Gea,1, James T. Maniscalcoa,2, Victor Arrietab, +Shawn R. McNealb, Matthias U. Liepea,∗∗ +aCornell Laboratory for Accelerator-Based Sciences and +Education, Ithaca, 14853, NY, USA +bUltramet, Pacoima, 12173, CA, USA +Abstract +Combining chemical vapor deposition (CVD) with electrochemical polish +(EP) operations is a promising route to producing performance-capable su- +perconducting films for use in the fabrication of cost-effective components +for superconducting radiofrequency (SRF) particle accelerators and super- +conducting quantum computers. The post-deposition EP process enables a +critically necessary reduction in surface roughness of niobium thin films to +promote optimal superconducting surface conditions. In this work, surface +morphology, roughness, and crystal orientation of the CVD-grown and EP- +polished niobium films were investigated. The grain growth and polishing +mechanisms were analyzed. The CVD films were found to comprise steps, +kinks, and pyramidal features, resulting in undesirable large peak-to-valley +distances. The electrochemical polish was demonstrated to significantly di- +minish the height of pyramids and effectively minimize the overall surface +roughness. +In contrast to buffered chemical polishing (BCP), EP results +showed a probable dependence on crystal orientation, suggesting this process +was influenced by locally enhanced current density and thickness variations +of oxide dielectrics. These understandings identify the EP principles tied +to CVD-grown Nb films that allow further refinement of surface profiles for +film-based SRF applications. +∗zs253@cornell.edu +∗∗mul2@cornell.edu +1Now at Jefferson Lab +2Now at SLAC +Preprint submitted to Applied Surface Science +January 3, 2023 + +Keywords: +Electrochemical polishing, chemical vapor deposition, niobium, +thin film, surface roughness, crystal orientation +1. Introduction +Niobium (Nb) is an important superconducting material that finds use in +superconducting radio-frequency (SRF) cavities, the chamber containing the +electromagnetic field in modern particle accelerators [1], and in components +needed in the emerging technological field of quantum computers [2]. SRF +cavities are critical components in a wide range of applications, including +synchrotron and free-electron-laser light sources (e.g., Linac Coherent Light +Source (LCLS)) [3, 4], high energy physics such as in the search for dark mat- +ter [5], high-precision (< 5 nm) photolithography for semiconductor device +fabrication [6], and in biopharmaceutical and medical applications [7]. +Since the transition of accelerators from low-gradient normal-conducting +RF to high-gradient superconducting RF, bulk Nb remains as the dominant +cavity technology used to obtain high accelerating gradients. Bulk Nb cavi- +ties are comprised of high-purity Nb with a residual resistivity ratio (RRR) +exceeding 300 and require high-cost triple arc-melted RRR-500+ start ma- +terials for fabrication. One promising direction for realizing cost-effective +cavities for SRF applications is the use of thin-film Nb coatings applied to +low-cost, high-thermal-conducting copper (Cu) cavity substrates. The thin- +film technology is viable since the active region for an SRF cavity is dictated +by the field penetration depth, typically, tens to hundreds of nanometers at +the inner surface, e.g., ∼ 40 nm for Nb. Additionally, due to the improved +thermal conductance, the Nb-coated Cu cavity promises enhanced thermal +stability during operation. The structural Cu cavity wall enables the out- +ward diffusion and removal of waste heat, while the Nb film functions as the +critical component interacting with the RF field. Controlling cavity surface +roughness and mitigating surface defects are important for achieving high- +quality factors as localized heat generated by these features can result in the +cascading loss of the superconducting state on the cavity surface, an effect +known as “quench” [8]. +Chemical vapor deposition (CVD) of Nb films, in addition to sputter- +ing [9, 10, 11] and epitaxy [12], were studied on silicon-carbide and graphite +substrates using NbCl5 and NbBr5 precursors [13, 14, 15]. This vapor-based +technique is suitable for coating the inner surface of cavities with intricate +2 + +Figure 1: (a) Picture of a Cu SRF cavity coated with CVD Nb thin films at the inner +surface. (b) Cross-sectional EDS mapping of CVD Nb films on Cu. Samples were cut +from the cavity. Inserts show locations of Cu substrate and Nb films. +shapes. Ultramet developed advanced CVD processing to deposit high-RRR +(> 280) and used rapid CVD process capabilities to produce freestanding +testable bulk Nb 3.9 GHz cavities [17]. Ultramet, working with Cornell’s +SRF Group, adapted the advanced CVD process technology to vapor de- +posit thick-, and thin-film Nb on 5-inch diameter plates and then scaled the +process to form Nb films on the interior surface of 1.3 GHz elliptical Cu cav- +ities of the full-scale single-cell ILC design (Fig. 1a) [17, 16]. Thin-film CVD +Nb coatings produced by Ultramet in this work demonstrated a high-quality +factor above 1010 at 2 K and a low residual resistance of ∼ 5 nΩ [16]. Fig. 1b +shows the results of the elemental mapping via an energy-dispersive X-ray +spectroscope (EDS), over the cross-section of a sample cut from the Nb/Cu +cavity that had been electrochemically polished. The excellent Nb-Cu inter- +face in the image confirms the ∼ 400 µm Nb film is strongly bonded to the +Cu substrate, and no Cu inclusions are observed in the film. However, a large +thickness variation of ∼ 150 µm remains even after the electrochemical pol- +ishing operation. The surface roughness can locally enhance the magnetic +field and negatively impact the RF performance, due for example, to the +degradation of quality factors (Q0) at high accelerating gradients [18]. Also, +this type of field enhancement can cause a quench and limit the maximum +field capability due to the permanent loss of superconductivity. +As such, engineering a smooth RF surface is required. Previous investi- +gations on bulk Nb involved mechanical polish [19], the use of chemicals such +as buffered chemical polish (BCP) [20], and electrochemical polish (EP) [21]. +Among these methods, the EP process that employs 9-part concentrated +H2SO4 to 1-part 48% HF under a DC current is typically performed as a +critical surface finish yielding an encouraging result of 300 nm roughness on +3 + +Cu wall +Nb film +Cu +500 μm +500 μum +500 μm +(b) +Cu -NbO +Nb +(a)Figure 2: +(a,b) Mechanisms of electrochemical polishing on a niobium surface using +H2SO4/HF electrolytes: (a) macropolishing and (b) micropolishing. +(c) Schematic of +the electrochemical polishing system and (d) polishing current oscillation. +bulk Nb [22]. A review of the literature suggests that an investigation into +EP processing to condition Nb thin-film surfaces for SRF applications has +not yet been done. +Electrochemical polishing includes two categories regarding surface fea- +ture size, macropolishing and micropolishing. Landolt et al. [23, 24] and +Hryniewicz et al. [25] have reviewed the fundamental aspects of each. As +shown in Fig. 2a, the local current density is significantly enhanced at posi- +tions with a smaller radius of curvature as described via [26] +σ = +2ε∆V +R +exp( −2∆n +R ) − 1 ∆ n→0 +(1) +where σ is the surface charge density, R is the radius of curvature, ∆n is a +limited distance normal to the surface, ∆V is the potential difference between +two endpoints of the distance ∆n, and ε is electric permittivity. Thus, for a +surface with high roughness, the leveling of the peak and recessed regions via +macropolishing is primarily determined by their difference in their current +4 + +Normalized current density +(b) Micropolishing +(a)Macropolishing +Electrolyte +R2? +R1 +F- +F +个 +个 +HNbF6 +Viscous layer +Nb.O5 +Nb +Radius of curvature +(p) +(c) +Current density [A/cm?] +DC power supply +Current monitor +CVD Nb film on Mc +> substrate (anode) +--->Al cathode +9 HSO4/ 1 HF +0 +5 +10 +15 +20 +Time [s]density. In contrast, a submicrometer-roughness surface has large radius- +of-curvature features (closer to R0 in Fig. 2a), leading to a more uniform +electrical field between peak and recessed regions, and making the microp- +olishing dominant by way of controlling the mass transport of species such +as reactants (water, F−, SO2− +4 ) and products (HNbF6 and other complexes). +Numerous studies have been carried out to investigate the transport mech- +anism in play during polishing operations performed on bulk Nb surfaces +[21, 27, 28]. Tian et al. [21, 27] identified the limiting of the transport of +F- ions as one mechanism and validated the theoretical interface model, as +illustrated in Fig. 2b, showing a compact Nb2O5 film and an HNbF6 (and +other complexes) diffusion layer. A viscous layer and/or dielectric film is +formed between the bulk solid and liquid regions so that the reaction is facil- +itated at the peak region where random diffusion of species (F−) is feasible +as compared to the recessed region. +Limitations in applying EP to thin Nb films arise due to the distinctive +surface profile and structural properties induced by CVD, which are detailed +in this work. For example, a variety of feature sizes appear on the film surface +ranging from ∼ 100 µm, large pyramidal features to several nm-size kinks +and steps, and present the challenge of smoothing the surface at the limit +of allowed polish thickness. Moreover, crystal defects such as dislocations, +impurities, and vacancies together with intrinsic stress in the film are more +common than bulk Nb. Owing to the defective sites, there is concern over +the formation of compact dielectric films as well as a desirable distribution +of electric fields. Cu EP studies have reported failure of dielectric formation +on a film sample and hence, a negative polish result, as compared to a bulk +sample [29]. These challenges motivate us to investigate EP on Nb thin films. +Here we analyze new phenomena tied to the EP treatment of CVD-grown +Nb films and to further advance the EP-CVD combined technology, paving +the way for film-based Nb RF cavities and other superconducting applica- +tions. We focus on comparing the characteristics between as-deposited and +electrochemically polished films. +Specifically, we investigate surface mor- +phology, roughness, and grain orientation. Also, we discuss the CVD growth +mode since these unique surface features observed are critical for determin- +ing the mechanism of a subsequent EP process. Moreover, the EP results to +date indicate a probable dependence on crystal orientation; and analysis is +provided in comparison with the chemically-controlled BCP treatment. +5 + +Figure 3: Comparison of surface SEM images for CVD Nb films on the Mo substrate (a,c) +before and (b,d) after EP under different fields of width: (a,b) 100 µm, (c,d) 500 µm. +2. Experimental section +Thin films (> 100 µm) of Nb on the molybdenum (Mo) substrates were +prepared by a low-temperature CVD process. The CVD Nb thin films were +provided by Ultramet and the recipes are not disclosed. The as-deposited +films were electrochemically polished by nominally 10 µm in thickness using +a 2-electrode system (Fig. 2c) consisting of the CVD Nb/Mo as an anode, +Al as a cathode, and the electrolyte of 98% H2SO4 and 48% HF at a 9:1 +volume ratio. The 2-electrode system is commonly used in the cavity polish +at Cornell, FNAL, KEK, and other accelerator laboratories [16, 22, 30]. The +current oscillation regime (Fig. 2d) was monitored to facilitate the genera- +tion and subsequent removal of compact Nb2O5 dielectrics. For reference to +EP, samples were polished in a standard BCP (buffered chemical polishing) +solution with 48% hydrofluoric, 70% nitric, and 85% phosphoric acids at a +volume ratio of 1:1:1. +To evaluate the surface morphology change, surface and cross-sectional +imaging were performed using a Zeiss Gemini scanning electron microscope +(SEM) equipped with an in-lens detector under low voltage regimes (1 – 5 +6 + +(a) +(b) +10 μm +no +(c) +(d)Figure 4: Comparison of cross-sectional SEM images for the largest pyramidal features +observed (a) before and (b) after EP. Inserts show closer inspections of (a) the CVD +pyramid and (b) the relatively smooth regions after EP. +kV). Electron dispersive x-ray spectroscopy (EDS) was used to determine +the chemical information. The surface roughness of films was measured via +an atomic force microscope (AFM, Asylum MFP-3D) but the high (> 100 +µm) pyramids affected the measurement, so the AFM results only compared +the relatively smooth regions. +To obtain effective comparison, films were +vertically placed under the SEM, and the cross-sections of the highest pyra- +mids were imaged and compared. Moreover, high-resolution X-ray diffraction +(XRD, Rigaku SmartLab) patterns were collected for analyzing grain orien- +tations. A Cu Kα radiation with a wavelength of 0.154 nm was used. +3. Results and discussion +3.1. Surface morphology +Fig. 3 shows the surface morphology of as-deposited and EP’ed films. As- +deposited films (Fig. 3a), although uniformly covering the substrate surface, +exhibit features of facets and steps. Also notably, pyramid-like structures +are widely observed on the surface as inspected under large fields of width +(Fig. 3c). +The cross-section of the largest pyramid observed is presented +in Fig. 4a. To summarize, there are two sources of surface roughness: (1) +pyramids as high as 100 µm; (2) step-kink structures appearing both in +the relatively flat regions and on the pyramids. Note that small but sharp +features, e.g., steps, would negatively affect the RF performance due to strong +local field enhancement. Hence, polishing the film surface is necessary to +improve the surface condition. +7 + +(a) +10 μm +(b) +20 μm +Nb film +0 μm +Nb pyramid +Nb pyramidFigure 5: Atomic models showing the terrace-step-kink formation on the Nb (110) plane. +Blue, red, and green atoms indicate the 1st, 2nd, and 3rd atomic layers, respectively. +Regarding the step-kink and pyramid formation, we analyze the film +growth mechanism. +Based on a typical terrace-step-kink model [31], the +nucleation events occur on multiple sites and a subsequent island growth +mode forms the pyramid structure. As shown in Fig. 5, the Nb atoms, as +a result of the chemical reactions of precursors, are adsorbed on a terrace +(the flat surface) and then diffuse to a kink site (the site at the terrace edge) +where the surface energy is typically low. If the lateral diffusion of adatoms +(adsorbed atoms) on the terrace is not sufficient, these adatoms build up to +pyramid islands together with the appearance of steps. Such effects are fur- +ther enhanced once islands are largely formed since adatoms cannot diffuse +to and join existing islands. Consequently, the terrace-step-kink and pyramid +structures predominate on the CVD Nb surface. +After CVD, EP polishing was conducted to alter the surface morphology +regarding two aspects, i.e., removing or smoothing large pyramid structures, +and eliminating surface steps and kinks. As demonstrated in Fig. 3b and 3d, +the edges and sharp features are greatly rounded after EP. Closer inspection +of the cross-sections (Fig. 4b) shows the regions that were relatively flat upon +deposition are further smoothed; small islands are completely dissolved, while +some large islands as high as 50 µm exist but their surfaces are also smoothed. +This infers that kink and step sites, regardless of their locations, favor the +onset of polishing, leading to a smooth and less-edged surface. +Due to the ex situ challenge, we compare the height of the highest pyra- +mids observed before and after EP. For example, the pyramid height prior to +polishing is as high as ∼ 100 µm, whereas the highest observed after polishing +is ∼ 50 µm. This empirical comparison suggests the pyramids are polished by +more than half in height, owing to intense macropolishing at these pyramids +8 + +Normal stack +Terrace-step-kink formationFigure 6: Representative AFM images taken on the relatively flat regions (a) before and +(b) after EP. +with a small radius of curvature (closer to R2 in Fig. 2a). +High-magnification images taken on the CVD pyramid (insert Fig. 4a) +show the pyramid consists of small nuclei (5 – 10 µm) and exhibits a similar +morphology of steps and kinks as other relatively flat regions. +After EP +(Fig. 4b), these features disappear resulting in a smooth pyramid surface. +This observation indicates micropolishing is also involved through leveling +the height difference at steps and kinks and dissolving the small nuclei. Note +that our primary motivation is to diminish the sharp features; while the +existence of tall pyramids is not ideal, the smoothed pyramids would less +severely impact the field enhancement. +3.2. Surface roughness +The quantification of surface roughness using AFM on a >10 µm uneven +surface is challenging owing to the instrumental capability of the depth of +field. The cross-sectional SEM images in Fig. 4 provide an empirical compar- +ison of height change for pyramid structures before and after EP. Here, the +AFM images were taken, as indications of roughness change, on the relatively +flat regions. +As shown in Fig. 6, the smooth areas (denoted in red) are prominently +enlarged after EP in the representative 202 µm2 areas. Taking account of +some inescapable small islands, the as-deposited samples have a large peak- +to-valley distance of 4.2 µm. In contrast, the EP’ed samples exhibit a reduced +9 + +(a)20 +(b) 20 +um +1.5 +15. +15 +1.0 +0.5 +10 +10 +0 +-0.5 +5 +-1.0 +5 +-1.5 +μm 0 +μmo +0 +5 +10 +15 +20 +0 +5 +10 +15 +20 +μm +μmFigure 7: XRD patterns of (a) as-deposited, (b) EP’ed, and (c) BCP’ed CVD Nb films. +Intensities are normalized to their highest diffraction limit as referenced to as-deposited +films. +value of 2.6 µm. Other surface parameters again indicate ∼ 50% reduction +of surface roughness, e.g., mean deviation (Ra) from 590 nm to 270 nm, and +root mean square (Rq) from 740 nm to 390 nm. Ra values from EP-smoothed +regions on the film are close to the typical value (∼ 300 nm) from an EP’ed +bulk surface, which indicates the effectiveness of EP polishing when applied +to thin films. Future work should focus on the removal of the remaining +pyramid features. +3.3. Crystal orientation +The X-ray diffraction characteristics of electrochemically (EP) and chem- +ically (BCP) polished CVD Nb films were compared (Fig. 7). +The as- +deposited films exhibit a predominant (110) peak, epitaxy from the cubic +Mo substrate, along with (100) and (211) diffractions. Fig. 8 illustrates the +formation mechanisms of (100) and (211) planes in addition to the (110) +epitaxy. In a body-centered cubic (bcc) structure, the [111] direction is the +closest packed, and (110) planes could easily slip along this direction yield- +ing (100) planes (Fig. 8a). The Burgers vector of dislocations in between +(100) and (110) planes is a/2 [111]. Additionally, rotating around the [111] +axis by 70.5 degrees, the (211) and (110) planes can form the twin structure +(Fig. 8b). These twin structures are extensively observed under SEM which +are marked by dashed lines in Fig. 3a. +Moreover, we observed an orientation dependence during EP. For exam- +ple, as shown in Fig. 7, the highest diffraction peak changed to (100) planes +from the initial highest (110) planes. Intensities were then normalized to that +10 + +(a) As-deposited +Intensity [arb. unit] +(b) After EP +c) After BCP +(200) +(211) +(110) +35 +45 +55 +65 +75 +20 [degrees]Figure 8: Atomic models showing the formation mechanisms of (a) (100) and (b) (211) +planes in addition to (110) planes. The lattice constant is denoted as “a”, and the Burgers +vector is denoted as “b”. +of (100) planes. Indeed, the (110) intensity reduced by half, and the (211) +intensity likewise dropped exceeding half. (The shifting to smaller diffraction +angles after EP indicates the compressive stress in the film is relieved.) +The orientation-dependence behaviors, however, do pose some subtle +questions for the conventional interpretation; the suppression of influences +from crystal orientation is expected in micropolishing. In general, electropol- +ishing is controlled by electrical, reaction, and diffusion processes. In mi- +cropolishing, the limiting factor nevertheless is the mass transport instead of +charge transfer [23]. The diffusion of species is a random motion and hence is +believed to be orientation-independent, whereas the reaction-controlled pol- +ishing is typically orientation-dependent since the planer density that char- +acterizes the average atoms in certain planes differs as summarized in Table +1. +Table 1: Planer density and plane spacing of (110), (100), and (211) planes in Nb. The +lattice constant (a) is 330 pm. +Plane orientation +(110) +(100) +(211) +Planer density +√ +2 +a2 +1 +a2 +√ +6 +3a2 +Plane spacing +√ +2a +2 +a +2 +√ +6a +6 +To test whether the orientation dependence during EP arises from a +reaction-controlled process, we carried out BCP polishing that underwent +similar chemical reactions as EP [31]. From XRD (Fig. 7), the (100) and +(211) planes that have small planer densities show a pronounced reduction +in intensity after BCP as compared to the (110) planes. This BCP behavior +significantly differs from the EP results; it supports the theory that EP is +less reaction-controlled. +We further analyze the possible mechanisms that induce an orientation +11 + +(a) +(b) +(121) +b =/ a[1-11] +[1-11] +(110) +(110) +(100)dependence. Our results have suggested that both macropolishing and mi- +cropolishing are involved in the EP process. Local electrical fields depending +on geometry factors play a major role at the pyramids where local polishing- +current densities are intensified resulting in large polishing rates. Upon as- +suming the statistical distribution of pyramids is uniform, the dominant pop- +ulation of (110)-structured pyramids are indicated by their highest intensity +in as-deposited films (Fig. 7a), and thus the global reduction of pyramids +would exhibit a preference in the (110) plane. For example, comparing the +pyramid cross-sections in Fig. 4, the FWHM (full width at half maximum) +remains the same value of 80 µm after EP, while the height reduces from 100 +µm to 50 µm, suggesting the polishing substantially occurs in the perpendic- +ular direction, say [110] orientation. +Another possible mechanism is based on the conventional theory (i.e., +mass transport controls EP); although the diffusion of species is orientation- +independent, the oxide growth during EP (Fig. 2b) varies in orientation. The +large local polishing current produces thicker oxide layers and hence larger +polishing rates – this scenario would produce a similar outcome discussed +above. Regardless of influences from the local polishing current, the oxide +growth rate on the (110) plane is found to be higher than other planes [33, 34]. +A thicker oxide layer on the (110) plane would induce a larger amount of re- +moval on this plane during EP. Overall, preferential polishing is critical since +it might provide selective polishing capabilities, and further investigations +are necessary to confirm the mechanisms indicated by this work. +4. Conclusions +In summary, electrochemical polishing (EP) was successfully performed +on the chemical vapor deposited (CVD) Nb films to reduce the surface rough- +ness, and compared with buffered chemical polishing (BCP). The character- +istics of surface morphology, roughness, and crystal orientation have been +analyzed to reveal the CVD growth and EP polishing mechanisms. +As-deposited films consist of relatively flat and pyramid-structured re- +gions, which cause a large peak-to-valley distance of > 100 µm. The obser- +vation of steps and kinks suggests that a terrace-step-kink model is respon- +sible for the generation of pyramids. Also, the CVD crystals exhibit a large +amount of (110) planes and some slip-induced (100) planes as well as the +(211) twinning planes. +12 + +EP is demonstrated to effectively minimize the mean surface roughness +on the relatively flat regions and significantly reduce the height of pyramids, +i.e., by more than half. These smoothening behaviors are critical to enhanc- +ing the RF performance of CVD Nb-based cavities. Besides the reduction +of pyramid height, the steps and kinks are found to disappear on the pyra- +mids, indicating the involvement of both macroscale and microscale smooth- +ing during the EP polish. The reaction-controlled mechanism is negligible in +EP as suggested by a comparison with chemical polishing (BCP). The local +enhanced current density and thickness variation of oxide dielectrics might +be the controlling factors in the CVD-film polishing, leading to the crystal +orientation dependence observed in this work. Overall, EP proceeds with +more complex scenarios for CVD Nb films which contain the removal of both +beyond and below-micrometer-scale sharp features. +Our demonstration of the EP-CVD technology represents a viable appli- +cation of Nb thin films for emerging superconducting applications. +Data availability statement +The data that support the findings of this study are available upon rea- +sonable request from the authors. +Conflicts of interest +V.A. and S.R.M. work at Ultramet. +Z.S., M.G., J.T.M., and M.U.L. +declare no competing financial interests. +Acknowledgments +This work is funded by the U.S. Department of Energy SBIR phase-II +award DE- SC0015727 and also supported by the National Science Founda- +tion under Grant No. PHY-1549132, the Center for Bright Beams. +References +[1] H. Padamsee, 50 years of success for srf accelerators – a review, Super- +cond. Sci. Technol. 30 (2017) 053003. doi:10.1088/1361-6668/aa6376. +[2] A. Blais, A. L. Grimsmo, S. M. Girvin, A. 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Sci. 80 (2016) +1. doi:10.1016/j.pmatsci.2016.02.001. +17 + diff --git a/C9AyT4oBgHgl3EQf4fqw/content/tmp_files/load_file.txt b/C9AyT4oBgHgl3EQf4fqw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbeb2bdc3ee9c8b8c318c3c5236838c3518ece42 --- /dev/null +++ b/C9AyT4oBgHgl3EQf4fqw/content/tmp_files/load_file.txt @@ -0,0 +1,651 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf,len=650 +page_content='Graphical Abstract Electrochemical Polishing of Chemical Vapor Deposited Niobium Thin Films Zeming Sun, Mingqi Ge, James T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Maniscalco, Victor Arrieta, Shawn R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' McNeal, Matthias U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Liepe arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='00788v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='mtrl-sci] 2 Jan 2023 Chemical vapor deposition Electrochemical polishing 10 um Functional Nb surface for superconducting RFHighlights Electrochemical Polishing of Chemical Vapor Deposited Niobium Thin Films Zeming Sun, Mingqi Ge, James T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Maniscalco, Victor Arrieta, Shawn R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' McNeal, Matthias U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Liepe Electrochemical polishing (EP) is demonstrated to effectively minimize the surface roughness for chemical vapor deposited (CVD) niobium thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' CVD niobium films contain steps, kinks, and pyramidal features, re- sulting in large surface roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' EP polishing of these films involves both macroscale and microscale smoothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' A probable dependence on crystal orientation during EP is observed, indicating strong influences from locally enhanced current density and thickness variations of oxide dielectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Obtaining the required surface conditions by a combined EP-CVD tech- nology marks a feasible application of niobium thin films in supercon- ducting RF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Electrochemical Polishing of Chemical Vapor Deposited Niobium Thin Films Zeming Suna,∗, Mingqi Gea,1, James T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Maniscalcoa,2, Victor Arrietab, Shawn R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' McNealb, Matthias U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Liepea,∗∗ aCornell Laboratory for Accelerator-Based Sciences and Education, Ithaca, 14853, NY, USA bUltramet, Pacoima, 12173, CA, USA Abstract Combining chemical vapor deposition (CVD) with electrochemical polish (EP) operations is a promising route to producing performance-capable su- perconducting films for use in the fabrication of cost-effective components for superconducting radiofrequency (SRF) particle accelerators and super- conducting quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The post-deposition EP process enables a critically necessary reduction in surface roughness of niobium thin films to promote optimal superconducting surface conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' In this work, surface morphology, roughness, and crystal orientation of the CVD-grown and EP- polished niobium films were investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The grain growth and polishing mechanisms were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The CVD films were found to comprise steps, kinks, and pyramidal features, resulting in undesirable large peak-to-valley distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The electrochemical polish was demonstrated to significantly di- minish the height of pyramids and effectively minimize the overall surface roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' In contrast to buffered chemical polishing (BCP), EP results showed a probable dependence on crystal orientation, suggesting this process was influenced by locally enhanced current density and thickness variations of oxide dielectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' These understandings identify the EP principles tied to CVD-grown Nb films that allow further refinement of surface profiles for film-based SRF applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' ∗zs253@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='edu ∗∗mul2@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='edu 1Now at Jefferson Lab 2Now at SLAC Preprint submitted to Applied Surface Science January 3, 2023 Keywords: Electrochemical polishing, chemical vapor deposition, niobium, thin film, surface roughness, crystal orientation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Introduction Niobium (Nb) is an important superconducting material that finds use in superconducting radio-frequency (SRF) cavities, the chamber containing the electromagnetic field in modern particle accelerators [1], and in components needed in the emerging technological field of quantum computers [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' SRF cavities are critical components in a wide range of applications, including synchrotron and free-electron-laser light sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=', Linac Coherent Light Source (LCLS)) [3, 4], high energy physics such as in the search for dark mat- ter [5], high-precision (< 5 nm) photolithography for semiconductor device fabrication [6], and in biopharmaceutical and medical applications [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Since the transition of accelerators from low-gradient normal-conducting RF to high-gradient superconducting RF, bulk Nb remains as the dominant cavity technology used to obtain high accelerating gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Bulk Nb cavi- ties are comprised of high-purity Nb with a residual resistivity ratio (RRR) exceeding 300 and require high-cost triple arc-melted RRR-500+ start ma- terials for fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' One promising direction for realizing cost-effective cavities for SRF applications is the use of thin-film Nb coatings applied to low-cost, high-thermal-conducting copper (Cu) cavity substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The thin- film technology is viable since the active region for an SRF cavity is dictated by the field penetration depth, typically, tens to hundreds of nanometers at the inner surface, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=', ∼ 40 nm for Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Additionally, due to the improved thermal conductance, the Nb-coated Cu cavity promises enhanced thermal stability during operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The structural Cu cavity wall enables the out- ward diffusion and removal of waste heat, while the Nb film functions as the critical component interacting with the RF field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Controlling cavity surface roughness and mitigating surface defects are important for achieving high- quality factors as localized heat generated by these features can result in the cascading loss of the superconducting state on the cavity surface, an effect known as “quench” [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Chemical vapor deposition (CVD) of Nb films, in addition to sputter- ing [9, 10, 11] and epitaxy [12], were studied on silicon-carbide and graphite substrates using NbCl5 and NbBr5 precursors [13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' This vapor-based technique is suitable for coating the inner surface of cavities with intricate 2 Figure 1: (a) Picture of a Cu SRF cavity coated with CVD Nb thin films at the inner surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' (b) Cross-sectional EDS mapping of CVD Nb films on Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Samples were cut from the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Inserts show locations of Cu substrate and Nb films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Ultramet developed advanced CVD processing to deposit high-RRR (> 280) and used rapid CVD process capabilities to produce freestanding testable bulk Nb 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='9 GHz cavities [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Ultramet, working with Cornell’s SRF Group, adapted the advanced CVD process technology to vapor de- posit thick-, and thin-film Nb on 5-inch diameter plates and then scaled the process to form Nb films on the interior surface of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='3 GHz elliptical Cu cav- ities of the full-scale single-cell ILC design (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 1a) [17, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Thin-film CVD Nb coatings produced by Ultramet in this work demonstrated a high-quality factor above 1010 at 2 K and a low residual resistance of ∼ 5 nΩ [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 1b shows the results of the elemental mapping via an energy-dispersive X-ray spectroscope (EDS), over the cross-section of a sample cut from the Nb/Cu cavity that had been electrochemically polished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The excellent Nb-Cu inter- face in the image confirms the ∼ 400 µm Nb film is strongly bonded to the Cu substrate, and no Cu inclusions are observed in the film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' However, a large thickness variation of ∼ 150 µm remains even after the electrochemical pol- ishing operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The surface roughness can locally enhance the magnetic field and negatively impact the RF performance, due for example, to the degradation of quality factors (Q0) at high accelerating gradients [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Also, this type of field enhancement can cause a quench and limit the maximum field capability due to the permanent loss of superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' As such, engineering a smooth RF surface is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Previous investi- gations on bulk Nb involved mechanical polish [19], the use of chemicals such as buffered chemical polish (BCP) [20], and electrochemical polish (EP) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Among these methods, the EP process that employs 9-part concentrated H2SO4 to 1-part 48% HF under a DC current is typically performed as a critical surface finish yielding an encouraging result of 300 nm roughness on 3 Cu wall Nb film Cu 500 μm 500 μum 500 μm (b) Cu -NbO Nb (a)Figure 2: (a,b) Mechanisms of electrochemical polishing on a niobium surface using H2SO4/HF electrolytes: (a) macropolishing and (b) micropolishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' (c) Schematic of the electrochemical polishing system and (d) polishing current oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' bulk Nb [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' A review of the literature suggests that an investigation into EP processing to condition Nb thin-film surfaces for SRF applications has not yet been done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Electrochemical polishing includes two categories regarding surface fea- ture size, macropolishing and micropolishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Landolt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' [23, 24] and Hryniewicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' [25] have reviewed the fundamental aspects of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 2a, the local current density is significantly enhanced at posi- tions with a smaller radius of curvature as described via [26] σ = 2ε∆V R exp( −2∆n R ) − 1 ∆ n→0 (1) where σ is the surface charge density, R is the radius of curvature, ∆n is a limited distance normal to the surface, ∆V is the potential difference between two endpoints of the distance ∆n, and ε is electric permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Thus, for a surface with high roughness, the leveling of the peak and recessed regions via macropolishing is primarily determined by their difference in their current 4 Normalized current density (b) Micropolishing (a)Macropolishing Electrolyte R2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' R1 F- F 个 个 HNbF6 Viscous layer Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='O5 Nb Radius of curvature (p) (c) Current density [A/cm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='] DC power supply Current monitor CVD Nb film on Mc > substrate (anode) --->Al cathode 9 HSO4/ 1 HF 0 5 10 15 20 Time [s]density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' In contrast, a submicrometer-roughness surface has large radius- of-curvature features (closer to R0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 2a), leading to a more uniform electrical field between peak and recessed regions, and making the microp- olishing dominant by way of controlling the mass transport of species such as reactants (water, F−, SO2− 4 ) and products (HNbF6 and other complexes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Numerous studies have been carried out to investigate the transport mech- anism in play during polishing operations performed on bulk Nb surfaces [21, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' [21, 27] identified the limiting of the transport of F- ions as one mechanism and validated the theoretical interface model, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 2b, showing a compact Nb2O5 film and an HNbF6 (and other complexes) diffusion layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' A viscous layer and/or dielectric film is formed between the bulk solid and liquid regions so that the reaction is facil- itated at the peak region where random diffusion of species (F−) is feasible as compared to the recessed region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Limitations in applying EP to thin Nb films arise due to the distinctive surface profile and structural properties induced by CVD, which are detailed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' For example, a variety of feature sizes appear on the film surface ranging from ∼ 100 µm, large pyramidal features to several nm-size kinks and steps, and present the challenge of smoothing the surface at the limit of allowed polish thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Moreover, crystal defects such as dislocations, impurities, and vacancies together with intrinsic stress in the film are more common than bulk Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Owing to the defective sites, there is concern over the formation of compact dielectric films as well as a desirable distribution of electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Cu EP studies have reported failure of dielectric formation on a film sample and hence, a negative polish result, as compared to a bulk sample [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' These challenges motivate us to investigate EP on Nb thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Here we analyze new phenomena tied to the EP treatment of CVD-grown Nb films and to further advance the EP-CVD combined technology, paving the way for film-based Nb RF cavities and other superconducting applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' We focus on comparing the characteristics between as-deposited and electrochemically polished films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Specifically, we investigate surface mor- phology, roughness, and grain orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Also, we discuss the CVD growth mode since these unique surface features observed are critical for determin- ing the mechanism of a subsequent EP process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Moreover, the EP results to date indicate a probable dependence on crystal orientation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' and analysis is provided in comparison with the chemically-controlled BCP treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 5 Figure 3: Comparison of surface SEM images for CVD Nb films on the Mo substrate (a,c) before and (b,d) after EP under different fields of width: (a,b) 100 µm, (c,d) 500 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Experimental section Thin films (> 100 µm) of Nb on the molybdenum (Mo) substrates were prepared by a low-temperature CVD process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The CVD Nb thin films were provided by Ultramet and the recipes are not disclosed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The as-deposited films were electrochemically polished by nominally 10 µm in thickness using a 2-electrode system (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 2c) consisting of the CVD Nb/Mo as an anode, Al as a cathode, and the electrolyte of 98% H2SO4 and 48% HF at a 9:1 volume ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The 2-electrode system is commonly used in the cavity polish at Cornell, FNAL, KEK, and other accelerator laboratories [16, 22, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The current oscillation regime (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 2d) was monitored to facilitate the genera- tion and subsequent removal of compact Nb2O5 dielectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' For reference to EP, samples were polished in a standard BCP (buffered chemical polishing) solution with 48% hydrofluoric, 70% nitric, and 85% phosphoric acids at a volume ratio of 1:1:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' To evaluate the surface morphology change, surface and cross-sectional imaging were performed using a Zeiss Gemini scanning electron microscope (SEM) equipped with an in-lens detector under low voltage regimes (1 – 5 6 (a) (b) 10 μm no (c) (d)Figure 4: Comparison of cross-sectional SEM images for the largest pyramidal features observed (a) before and (b) after EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Inserts show closer inspections of (a) the CVD pyramid and (b) the relatively smooth regions after EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' kV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Electron dispersive x-ray spectroscopy (EDS) was used to determine the chemical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The surface roughness of films was measured via an atomic force microscope (AFM, Asylum MFP-3D) but the high (> 100 µm) pyramids affected the measurement, so the AFM results only compared the relatively smooth regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' To obtain effective comparison, films were vertically placed under the SEM, and the cross-sections of the highest pyra- mids were imaged and compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Moreover, high-resolution X-ray diffraction (XRD, Rigaku SmartLab) patterns were collected for analyzing grain orien- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' A Cu Kα radiation with a wavelength of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='154 nm was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Results and discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Surface morphology Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 3 shows the surface morphology of as-deposited and EP’ed films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' As- deposited films (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 3a), although uniformly covering the substrate surface, exhibit features of facets and steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Also notably, pyramid-like structures are widely observed on the surface as inspected under large fields of width (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The cross-section of the largest pyramid observed is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' To summarize, there are two sources of surface roughness: (1) pyramids as high as 100 µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' (2) step-kink structures appearing both in the relatively flat regions and on the pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Note that small but sharp features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=', steps, would negatively affect the RF performance due to strong local field enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Hence, polishing the film surface is necessary to improve the surface condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 7 (a) 10 μm (b) 20 μm Nb film 0 μm Nb pyramid Nb pyramidFigure 5: Atomic models showing the terrace-step-kink formation on the Nb (110) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Blue, red, and green atoms indicate the 1st, 2nd, and 3rd atomic layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Regarding the step-kink and pyramid formation, we analyze the film growth mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Based on a typical terrace-step-kink model [31], the nucleation events occur on multiple sites and a subsequent island growth mode forms the pyramid structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 5, the Nb atoms, as a result of the chemical reactions of precursors, are adsorbed on a terrace (the flat surface) and then diffuse to a kink site (the site at the terrace edge) where the surface energy is typically low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' If the lateral diffusion of adatoms (adsorbed atoms) on the terrace is not sufficient, these adatoms build up to pyramid islands together with the appearance of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Such effects are fur- ther enhanced once islands are largely formed since adatoms cannot diffuse to and join existing islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Consequently, the terrace-step-kink and pyramid structures predominate on the CVD Nb surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' After CVD, EP polishing was conducted to alter the surface morphology regarding two aspects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=', removing or smoothing large pyramid structures, and eliminating surface steps and kinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' As demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 3b and 3d, the edges and sharp features are greatly rounded after EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Closer inspection of the cross-sections (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 4b) shows the regions that were relatively flat upon deposition are further smoothed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' small islands are completely dissolved, while some large islands as high as 50 µm exist but their surfaces are also smoothed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' This infers that kink and step sites, regardless of their locations, favor the onset of polishing, leading to a smooth and less-edged surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Due to the ex situ challenge, we compare the height of the highest pyra- mids observed before and after EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' For example, the pyramid height prior to polishing is as high as ∼ 100 µm, whereas the highest observed after polishing is ∼ 50 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' This empirical comparison suggests the pyramids are polished by more than half in height, owing to intense macropolishing at these pyramids 8 Normal stack Terrace-step-kink formationFigure 6: Representative AFM images taken on the relatively flat regions (a) before and (b) after EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' with a small radius of curvature (closer to R2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' High-magnification images taken on the CVD pyramid (insert Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 4a) show the pyramid consists of small nuclei (5 – 10 µm) and exhibits a similar morphology of steps and kinks as other relatively flat regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' After EP (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 4b), these features disappear resulting in a smooth pyramid surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' This observation indicates micropolishing is also involved through leveling the height difference at steps and kinks and dissolving the small nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Note that our primary motivation is to diminish the sharp features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' while the existence of tall pyramids is not ideal, the smoothed pyramids would less severely impact the field enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Surface roughness The quantification of surface roughness using AFM on a >10 µm uneven surface is challenging owing to the instrumental capability of the depth of field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The cross-sectional SEM images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 4 provide an empirical compar- ison of height change for pyramid structures before and after EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Here, the AFM images were taken, as indications of roughness change, on the relatively flat regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 6, the smooth areas (denoted in red) are prominently enlarged after EP in the representative 202 µm2 areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Taking account of some inescapable small islands, the as-deposited samples have a large peak- to-valley distance of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' In contrast, the EP’ed samples exhibit a reduced 9 (a)20 (b) 20 um 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='5 10 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='5 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='0 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='5 μm 0 μmo 0 5 10 15 20 0 5 10 15 20 μm μmFigure 7: XRD patterns of (a) as-deposited, (b) EP’ed, and (c) BCP’ed CVD Nb films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Intensities are normalized to their highest diffraction limit as referenced to as-deposited films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Other surface parameters again indicate ∼ 50% reduction of surface roughness, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=', mean deviation (Ra) from 590 nm to 270 nm, and root mean square (Rq) from 740 nm to 390 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Ra values from EP-smoothed regions on the film are close to the typical value (∼ 300 nm) from an EP’ed bulk surface, which indicates the effectiveness of EP polishing when applied to thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Future work should focus on the removal of the remaining pyramid features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Crystal orientation The X-ray diffraction characteristics of electrochemically (EP) and chem- ically (BCP) polished CVD Nb films were compared (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The as- deposited films exhibit a predominant (110) peak, epitaxy from the cubic Mo substrate, along with (100) and (211) diffractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 8 illustrates the formation mechanisms of (100) and (211) planes in addition to the (110) epitaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' In a body-centered cubic (bcc) structure, the [111] direction is the closest packed, and (110) planes could easily slip along this direction yield- ing (100) planes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The Burgers vector of dislocations in between (100) and (110) planes is a/2 [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Additionally, rotating around the [111] axis by 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='5 degrees, the (211) and (110) planes can form the twin structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' These twin structures are extensively observed under SEM which are marked by dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Moreover, we observed an orientation dependence during EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' For exam- ple, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 7, the highest diffraction peak changed to (100) planes from the initial highest (110) planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Intensities were then normalized to that 10 (a) As-deposited Intensity [arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' unit] (b) After EP c) After BCP (200) (211) (110) 35 45 55 65 75 20 [degrees]Figure 8: Atomic models showing the formation mechanisms of (a) (100) and (b) (211) planes in addition to (110) planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The lattice constant is denoted as “a”, and the Burgers vector is denoted as “b”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' of (100) planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Indeed, the (110) intensity reduced by half, and the (211) intensity likewise dropped exceeding half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' (The shifting to smaller diffraction angles after EP indicates the compressive stress in the film is relieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=') The orientation-dependence behaviors, however, do pose some subtle questions for the conventional interpretation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' the suppression of influences from crystal orientation is expected in micropolishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' In general, electropol- ishing is controlled by electrical, reaction, and diffusion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' In mi- cropolishing, the limiting factor nevertheless is the mass transport instead of charge transfer [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The diffusion of species is a random motion and hence is believed to be orientation-independent, whereas the reaction-controlled pol- ishing is typically orientation-dependent since the planer density that char- acterizes the average atoms in certain planes differs as summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Table 1: Planer density and plane spacing of (110), (100), and (211) planes in Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The lattice constant (a) is 330 pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Plane orientation (110) (100) (211) Planer density √ 2 a2 1 a2 √ 6 3a2 Plane spacing √ 2a 2 a 2 √ 6a 6 To test whether the orientation dependence during EP arises from a reaction-controlled process, we carried out BCP polishing that underwent similar chemical reactions as EP [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' From XRD (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 7), the (100) and (211) planes that have small planer densities show a pronounced reduction in intensity after BCP as compared to the (110) planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' This BCP behavior significantly differs from the EP results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' it supports the theory that EP is less reaction-controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' We further analyze the possible mechanisms that induce an orientation 11 (a) (b) (121) b =/ a[1-11] [1-11] (110) (110) (100)dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Our results have suggested that both macropolishing and mi- cropolishing are involved in the EP process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Local electrical fields depending on geometry factors play a major role at the pyramids where local polishing- current densities are intensified resulting in large polishing rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Upon as- suming the statistical distribution of pyramids is uniform, the dominant pop- ulation of (110)-structured pyramids are indicated by their highest intensity in as-deposited films (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 7a), and thus the global reduction of pyramids would exhibit a preference in the (110) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' For example, comparing the pyramid cross-sections in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 4, the FWHM (full width at half maximum) remains the same value of 80 µm after EP, while the height reduces from 100 µm to 50 µm, suggesting the polishing substantially occurs in the perpendic- ular direction, say [110] orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Another possible mechanism is based on the conventional theory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=', mass transport controls EP);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' although the diffusion of species is orientation- independent, the oxide growth during EP (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 2b) varies in orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The large local polishing current produces thicker oxide layers and hence larger polishing rates – this scenario would produce a similar outcome discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Regardless of influences from the local polishing current, the oxide growth rate on the (110) plane is found to be higher than other planes [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' A thicker oxide layer on the (110) plane would induce a larger amount of re- moval on this plane during EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Overall, preferential polishing is critical since it might provide selective polishing capabilities, and further investigations are necessary to confirm the mechanisms indicated by this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Conclusions In summary, electrochemical polishing (EP) was successfully performed on the chemical vapor deposited (CVD) Nb films to reduce the surface rough- ness, and compared with buffered chemical polishing (BCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The character- istics of surface morphology, roughness, and crystal orientation have been analyzed to reveal the CVD growth and EP polishing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' As-deposited films consist of relatively flat and pyramid-structured re- gions, which cause a large peak-to-valley distance of > 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The obser- vation of steps and kinks suggests that a terrace-step-kink model is respon- sible for the generation of pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Also, the CVD crystals exhibit a large amount of (110) planes and some slip-induced (100) planes as well as the (211) twinning planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' 12 EP is demonstrated to effectively minimize the mean surface roughness on the relatively flat regions and significantly reduce the height of pyramids, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=', by more than half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' These smoothening behaviors are critical to enhanc- ing the RF performance of CVD Nb-based cavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Besides the reduction of pyramid height, the steps and kinks are found to disappear on the pyra- mids, indicating the involvement of both macroscale and microscale smooth- ing during the EP polish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The reaction-controlled mechanism is negligible in EP as suggested by a comparison with chemical polishing (BCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' The local enhanced current density and thickness variation of oxide dielectrics might be the controlling factors in the CVD-film polishing, leading to the crystal orientation dependence observed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Overall, EP proceeds with more complex scenarios for CVD Nb films which contain the removal of both beyond and below-micrometer-scale sharp features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Our demonstration of the EP-CVD technology represents a viable appli- cation of Nb thin films for emerging superconducting applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Data availability statement The data that support the findings of this study are available upon rea- sonable request from the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Conflicts of interest V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' work at Ultramet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' declare no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content=' Acknowledgments This work is funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AyT4oBgHgl3EQf4fqw/content/2301.00788v1.pdf'} +page_content='S.' metadata={'source': 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b/D9E4T4oBgHgl3EQfGQye/content/tmp_files/2301.04893v1.pdf.txt @@ -0,0 +1,2527 @@ +Stripe Helical Magnetism and Two Regimes of Anomalous Hall Effect in NdAlGe +Hung-Yu Yang,1, ∗ Jonathan Gaudet,2, 3 Rahul Verma,4 Santu Baidya,5, 6 Faranak Bahrami,1 +Xiaohan Yao,1 Cheng-Yi Huang,7 Lisa DeBeer-Schmitt,8 Adam A. Aczel,8 +Guangyong Xu,2 Hsin Lin,9 Arun Bansil,7 Bahadur Singh,4 and Fazel Tafti1 +1Department of Physics, Boston College, Chestnut Hill, MA 02467, USA +2NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA +3Department of Materials Science and Eng., University of Maryland, College Park, MD 20742-2115 +4Department of Condensed Matter Physics and Materials Science, +Tata Institute of Fundamental Research, Colaba, Mumbai 400005, India +5Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-8019, USA +6Department of Physics and Materials Science, Jaypee University of Information Technology, +Waknaghat, Solan, Himachal Pradesh 173234, India +7Department of Physics, Northeastern University, Boston, MA 02115, USA +8Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA +9Institute of Physics, Academia Sinica, Taipei 11529, Taiwan +(Dated: January 13, 2023) +We report the magnetic and electronic transport properties of the inversion and time-reversal +symmetry breaking Weyl semimetal NdAlGe. This material is analogous to NdAlSi, whose helical +magnetism presents a rare example of a Weyl-mediated collective phenomenon, but with a larger +spin-orbit coupling. Our neutron diffraction experiments revealed that NdAlGe, similar to NdAlSi, +supports an incommensurate Ising spin density wave (Tinc = 6.8 K) with a small helical spin canting of +3° and a long-wavelength of ∼ 35 nm, which transitions to a commensurate ferrimagnetic state below +Tcom = 5.1 K. Using small-angle neutron scattering, we showed that the zero-field cooled ferrimagnetic +domains form stripes in real space with characteristic length scales of 18 nm and 72 nm parallel +and perpendicular to the [110] direction, respectively. Interestingly, for the transport properties, +NdAlSi does not exhibit an anomalous Hall effect (AHE) that is commonly observed in magnetic +Weyl semimetals. In contrast to NdAlSi, we identify two different AHE regimes in NdAlGe that are +respectively governed by intrinsic Berry curvature and extrinsic disorders/spin fluctuations. Our +study suggests that Weyl-mediated magnetism prevails in this group of noncentrosymmetric magnetic +Weyl semimetals NdAlX, but transport properties including AHE are affected by material-specific +extrinsic effects such as disorders, despite the presence of prominent Berry curvature. +I. +INTRODUCTION +To establish a Weyl semimetal phase, one needs to +break either inversion or time-reversal symmetry to split +Weyl nodes of opposite chirality, which may then lead to +interesting topological properties [1, 2]. Both routes have +been explored through candidate materials that break +either the inversion symmetry, such as the noncentrosym- +metric TaAs [3–7], or time-reversal symmetry, such as the +ferromagnetic (FM) Co3Sn2S2 [8, 9]. Weyl semimetals +that break both inversion and time-reversal symmetries +remain largely unexplored, despite theoretical predictions +of Weyl-mediated interactions with rich phase diagrams +and topological magnetic textures [10–14]. For instance, +in the RAlX (R = rare-earths, X = Ge/Si) material +family of double-symmetry-breaking Weyl semimetals [15– +17], a variety of rich magnetic orders have been found. +These include collinear FM order [17–20] and noncollinear +FM order [21–23], both of which are relatively common. +More unusual spin structures were also observed such as +∗ Present address: Department of Electrical and Computer Engi- +neering, University of California, Los Angeles, California 90095, +USA; hungyuyang@g.ucla.edu +a topological multi-⃗k structure in CeAlGe [24], a spiral +order in SmAlSi [25], and a helical incommensurate spin +density wave (SDW) in NdAlSi [26]. In particular, for +NdAlSi, its helical magnetism was shown to be stabilized +by bond-oriented Dzyaloshinskii-Moriya (DM) interaction +predicted to arise from Weyl-mediated Ruderman-Kittel- +Kasuya-Yosida (RKKY) coupling, owing to the presence +of itinerant Weyl electrons, local magnetic moments, and +broken inversion symmetry [14, 26]. +Despite the comprehensive characterization of Weyl- +mediated magnetism in NdAlSi [26] and some studies on +NdAlGe [27], their transport properties such as anoma- +lous Hall effect (AHE) remain unexplored. AHE has been +extensively investigated in FM Weyl semimetals where the +intrinsic Berry curvature may contribute to pronounced +AHE [8, 9, 28], but recently it has been shown that Berry +curvature is not always the dominant source of AHE in +FM Weyl semimetals, and extrinsic disorders can also play +a major role [20]. With a chiral magnetism established +in NdAlSi, NdAlX provides a unique system to study +AHE in helimagnetic Weyl semimetals. In this work, we +aim to first establish the magnetic structure of NdAlGe, +which is not obvious (not necessarily the same as NdAlSi) +considering the behavior in other materials in RAlX fam- +ily [20, 21, 24, 29], and study the AHE of NdAlX with +a focus on NdAlGe to understand the interplay among +arXiv:2301.04893v1 [cond-mat.mtrl-sci] 12 Jan 2023 + +2 +topology, magnetism, and electrical transport. +We investigate the magnetism and electrical transport +of NdAlGe with SQUID magnetometry, heat capacity +measurements, neutron scattering experiments, resistivity +measurements, and DFT calculations. Similar to NdAlSi, +we found a high temperature (5.1 K < T < 6.8 K) heli- +cal incommensurate SDW in NdAlGe characterized by a +multi-k structure with ordering vectors kAFM1 = (2/3 + +δ(T), 2/3 + δ(T), 0), kAFM2 = (1/3 − δ(T), 1/3 − δ(T), 0), +and kFM = (3δ(T), 3δ(T), 0), which evolves to a commen- +surate (δ = 0) helical ferrimagnetic state at low temper- +atures (T < 5.1 K). In this state, the small-angle neutron +scattering (SANS) can be modeled by an anisotropic +Lorentzian-squared function, which signifies stripes of fer- +rimagnetic domains with real space characteristic length +scales of 18(5) nm and 72(8) nm parallel and perpendic- +ular to the [110] ordering vector direction, respectively. +Surprisingly, we found AHE responses as Hall resistivity +plateaus under a magnetic field in NdAlGe but not in +NdAlSi, despite that both materials show clear plateaus in +their magnetization. Furthermore, as the field increases, +the AHE in NdAlGe shows a transition from intrinsic +AHE in the ferrimagnetic phase to an extrinsic AHE in +the polarized FM phase. Finally, we calculate the elec- +tronic band structure, Weyl-nodes, and Fermi surface of +NdAlGe. We also calculated the anomalous Hall conduc- +tivity of NdAlGe, which shows a reasonable agreement +with the observed intrinsic AHE. Our findings of helical +magnetism and two regimes of AHE in NdAlGe suggest +that Weyl-mediated magnetism is robust in these materi- +als provided that the itinerant Weyl electrons and nesting +Fermi pockets are intact, while AHE can be largely modi- +fied by extrinsic disorders and spin fluctuations despite +significant intrinsic Berry curvature. +II. +METHODS +Single crystals of NdAlGe and NdAlSi were grown by +a self-flux method. The starting materials are elemen- +tal chunks of Nd, Al, and Ge, weighed in a ratio of +Nd:Al:Ge=1:10:1 or 1:15:1, and mixed in an alumina cru- +cible. Single crystals made with a 1:10:1 recipe were used +in this study unless specified. NdAlSi single crystals were +grown with the 1:10:1 ratio. The crucibles were sealed +in an evacuated quartz tube, heated up to 1050◦C at +3◦C/min, dwelt for 12 hours, cooled down to 700◦C at +0.1◦C/min, and dwelt for another 12 hours. After the +heating sequence, the tube was centrifuged to remove +the excess Al flux. Plate-like single crystals were found +isolated and attached to the bottom of crucibles. X-ray +diffraction (PXRD) measurement was performed with a +Bruker D8 ECO instrument with a copper x-ray source +(Cu Kα) and a one-dimensional LINXEYE-XE detector +at room temperature. Rietveld refinement on the PXRD +patterns was performed using the FullProf suite [30]. The +elemental analysis of NdAlSi/Ge was determined by En- +ergy Dispersive X-ray Spectroscopy (EDX), carried out +with FEI Scios, operated at an acceleration voltage of 20 +kV and a current of 0.4 nA. +Electrical resistivity and heat capacity were measured +in a Quantum Design Physical Property Measurement +System (PPMS) Dynacool with the standard four-probe +technique and relaxation time method, respectively. DC +magnetization experiments were performed on the vibrat- +ing sample magnetometer in a Quantum Design MPMS3. +We performed neutron diffraction using both the HB- +1A thermal triple-axis spectrometer at ORNL and the +cold triple-axis spectrometer SPINS at NIST. The (HHL) +scattering plane of NdAlGe was probed for both experi- +ments at a base temperature of ∼1.5 K. 14.5 meV incident +neutrons filtered with pyrolytic graphite (PG) were used +on HB-1A. For SPINS, we used 3.7 meV incident neutrons +with cooled Be filters employed both before and after the +sample. +The HFIR GP-SANS instrument was utilized to probe +the SANS of NdAlGe with a base temperature of ∼2 K. +Two SANS configurations were used; 1) Uncollimated 12 ˚A +incident neutrons were used with the scattered neutrons +detected at a distance of 19 m away from the sample, 2) +4.75 ˚A neutrons collimated by 3 guides were incident to +the sample and detected at a distance of 8 m from it. A +circular aperture with a diameter of 8 mm was placed at +the sample position. A flat-plate sample was aligned such +that the [001] axis is parallel to the neutron beam. SANS +patterns were collected using an 11 T horizontal magnet +applied both parallel (H ∥ [001]) and perpendicular (H ∥ +[110]) to the incident neutron beam. Error bars associated +with all neutron diffraction intensities reported in this +work correspond to 1 standard deviation. +We performed electronic structure calculations within +the framework of density functional theory (DFT) based +on the projected augmented wave (PAW) method as imple- +mented in the Vienna ab-initio simulation package (VASP) +[31–33]. Generalized gradient approximation (GGA) [34] +was used to include exchange-correlation effects and spin- +orbit coupling (SOC) was added self-consistently. We +added an on-site Coulomb interaction with Ueff = 8 eV +for the Nd f electrons within the GGA+U scheme [35, 36] +to include strong electron-correlation effects. We consid- +ered the kinetic energy cut-off of 450 eV for the plane-wave +basis set and used Γ-centered 11 × 11 × 11 k-mesh [37] +for bulk Brillouin zone sampling. The tolerance of the +electronic energy minimization was set to 10−6 eV. We +generated material-specific tight-binding Hamiltonian us- +ing the VASP2WANNIER90 interface [38]. We included +Nd d, f, Al s, p and Ge s, p orbitals in construction of +the Wannier functions. The topological properties were +calculated using the WannierTools package [39]. +III. +RESULTS +Table I summarizes the main properties studied in this +paper, including helical magnetism and anomalous Hall +effect. When comparing NdAlGe and NdAlSi together, + +3 +TABLE I. Summary of magnetic and transport properties +of NdAlSi and NdAlGe, including the onset of incommen- +surate order (Tinc) and commensurate order (Tcom), satu- +rated moment Msat, residual resistivity ratio (RRR, defined as +ρ(300K)/ρ(2K), single-band carrier concentration (nh where +holes are the dominant electric carriers in both materials) and +mobility (µh), and anomalous Hall effect (AHE). +NdAlGe +NdAlSi [26] +Tinc +6.8(2) K +7.2 K +Tcom +5.1(1) K +3.3 K +Msat at 2 K, 6 T +2.8(1) µB +2.9 µB +RRR +2.0(1) +6.0 +nh at 2 K +1.06 × 1021 cm−3 +6.66 × 1019 cm−3a +µh at 2 K +134 cm2V−1s−1 +11008 cm2V−1s−1a +AHE +duu state: intrinsic +No clear Hall +FM state: extrinsic +resistivity plateaub +a See Appendix B. +b See Appendix D. +we find that their magnetic properties are similar to each +other, but their transport properties are quite different. +Incommensurate and commensurate magnetic orders are +found in both materials at close temperatures, and their +refined spin structures are also similar to each other [26]. +However, when it comes down to the transport properties, +the residual resistivity ratio (RRR) of NdAlGe is 3 times +smaller than that of NdAlSi, which suggests a higher +disorder level in NdAlGe. The lower RRR in NdAlGe is +also manifested in the single-band analysis, which shows a +higher hole concentration but a much lower hole mobility +compared to NdAlSi. As we will see, in spite of similar +Fermi surfaces, quantum oscillations are not observed +in NdAlGe, but are pronounced in NdAlSi [26] at the +same temperatures and magnetic fields; this distinction +also suggests a shorter mean free path and lower mobility +in NdAlGe. More interestingly, both intrinsic and ex- +trinsic AHE were observed in NdAlGe as magnetic field +increases, but no clear sign of AHE was found in NdAlSi. +In the following sections, we will present and interpret the +structural, magnetic, and electrical transport properties +of NdAlGe in details. +A. +Crystal structure and disorder in NdAlGe +The inset of Fig. 1 shows the crystal structure of +NdAlGe, which belongs to the same I41md space group +as the archetypal Weyl semimetal TaAs [3–6]. The com- +bination of the noncentrosymmetric crystal structure and +the collective magnetism hosted by Nd3+ f-orbitals at low +temperatures makes NdAlGe a double-symmetry-breaking +Weyl semimetal [15]. Based on previous second harmonic +generation experiments across different RAlX compounds, +it is now clear that the RAlX material family resides in the +noncentrosymmetric space group I41md [20, 21, 25, 26]. +We will then use the I41md space group as our starting +point for the nuclear structure refinement of NdAlGe. +Nd +Al +Ge +FIG. 1. Powder x-ray diffraction pattern and Rietveld refine- +ment of NdAlGe. The occupancy of each atom was refined. +The crystal structure of NdAlGe is presented on the top right +inset. +Another important point is the stoichiometry of RAlX +single crystals that may vary depending on the growth +methods. In terms of the growth methods, it has been +shown that single crystals grown by a floating-zone furnace +typically has a stoichiometric ratio much closer to 1:1:1 +compared to those grown by flux methods, which tend +to be Al-rich and have vacancies on the Ge/Si sites [24]. +Furthermore, current literature seems to suggest the Ge +variant is more prone to off-stoichiometry as compared +to their Si analogue. For example, CeAlSi crystals grown +by the flux method show little deviation from a 1:1:1 +stoichiometric ratio [21]; however, for CeAlGe, the crystals +grown by the flux method are predominantly Al-rich +and have significant vacancies on the Ge sites, while +those synthesized with a floating-zone growth show a +stoichiometric ratio close to 1:1:1 [24]. +Considering RAlGe grown by flux methods are prone +to be Al-rich, we have performed EDX measurements to +confirm the stoichiometry in our NdAlX crystals grown +by flux method (Table II). Our measurements show that +NdAlSi crystals are close to a 1:1:1 stoichiometry, while +NdAlGe crystals are predominantly Al-rich and show +larger variations in the stoichiometry. We also refined +the atomic occupancy for both materials by Rietveld +refinement, and the results show a similar trend (see +Appendix A). Our characterizations are consistent and +suggest that, compared to NdAlSi, the flux-grown NdAlGe +crystals show variation in their atomic compositions and +have a higher level of disorder. +B. +Resistivity, magnetic susceptibility, and heat +capacity +Figure 2(a,b) shows a typical resistivity (ρxx) curve +obtained for our NdAlGe crystals at T = 2 − 300 K and +T = 2 − 10 K, respectively. From the magnitude of ρxx +at T = 300 K and T = 2 K, we calculate the residual +resistivity ratio RRR = ρxx(300 K) +ρxx(2 K) +to be 1.9, much lower + +Occupancy +NdAIGe +Al +Ge +Nd +8 +Counts) +T= 295K +NdAIGe +Cu Kα +I41md +obs +Intensity (2 × 10° +obs +20 +40 +60 +80 +100 +20 (degrees)NdAlGe +1 41 mdobs. +calc. +obs. +calcc +b +a4 +than the RRR of NdAlSi, which is 6.0 (see Appendix B). A +lower RRR usually suggests the disorder level is higher so +that the resistivity is anchored at a higher value near zero +temperature. We attribute the lower RRR for NdAlGe to +its off-stoichiometry characterized in the previous section. +We measured the magnetic susceptibility of NdAlGe +(χ) with a magnetic field applied along the c-axis (χc) +and the a-axis (χa). The ratio χc/χa is plotted as a solid +line in Fig. 2(c) (left y-axis), while 1/χc is plotted as +squares in the same plot (right y-axis). We also plotted +the temperature dependence of χc in Fig. 2(d). At high +temperatures, the magnetic susceptibility shows isotropic +paramagnetic spins with a Curie-Weiss temperature of +18.0(7) K and an effective magnetic moment of 3.5(2) µB. +Upon cooling, similar to NdAlSi [26], an out-of-plane Ising +anisotropy gradually develops such that the ratio χc/χa +reaches as high as 80 at T = 2 K. The Ising anisotropy is +also visible in the low-temperature in-field magnetization +of NdAlGe (inset of Fig. 2(c)) where the magnetization +along the c-axis reaches saturation near 3 T at a value of +2.8(1)µB/Nd, while the magnetization along the a-axis is +still unsaturated and weak at 6 T. +The heat capacity (Cp) of NdAlGe is plotted in +Fig. 2(e,f). The main panel of Fig. 2(e) shows the mag- +netic contribution of the heat capacity Cmag +p +of NdAlGe, +which was obtained by subtracting the heat capacity of the +non-magnetic analogue compound LaAlGe. As expected, +the Cp attains the Dulong-Petit limit (Cp = 3R × Nions +where R is the gas constant and Nions is the number of +ions in the material and equals to 3 in NdAlGe) near the +room temperature (inset of Fig. 2(e)). Upon cooling, a +Schottky-like anomaly centered at 18(1) K is visible in +both Cp and Cmag +p +. This high-temperature anomaly can +be reproduced with an Nd3+ single-ion energy scheme +comprised of a doublet ground state separated by 3 ex- +cited doublets between 3 to 9 meV. At lower temperatures, +Cmag +p +shows two additional anomalies at Tcom = 5.1(1) K +and Tinc = 6.8(2) K, which originate from the collective +magnetism of the Nd3+ moments (Fig. 2(f)). The pres- +ence of two anomalies is also present in NdAlSi where the +phase transition at Tinc signifies the onset of an incom- +mensurate modulated spin density wave that transitions +into a commensurate ferrimagnetic state below Tcom. +The incommensurate and commensurate magnetic +phase transitions in NdAlSi were both observed to impact +its electric transport and bulk thermodynamic properties. +More specifically, the onset of the commensurate order +TABLE II. EDX measurements of NdAlSi and NdAlGe. The +occupancy was normalized by that of Al. The uncertainty was +defined by the standard deviation of all measurements, which +were taken from 2-3 different regions of several crystals for +each material. +NdAlSi +Occupancy +NdAlGe +Occupancy +Nd +0.99 ± 0.01 +Nd +0.94 ± 0.04 +Al +1 ± 0.01 +Al +1 ± 0.07 +Si +0.99 ± 0.01 +Ge +0.93 ± 0.04 +NdAlGe +Tinc +Tcom +Tinc +Tcom +Tinc +Tcom +RRR = 1.90 +(a) +(b) +(c) +(e) +(f) +(d) +FC +ZFC +Sample T7 +0 +2 +4 +6 +H (T) +0 +1 +2 +3 +M (𝜇B/Nd) +Mc +Ma +0 100 200 300 +T (K) +0 +25 +50 +75 +Cp +FIG. 2. (a,b) Resistivity as a function of temperature, plotted +from 0 to 300 K and 0 to 10 K, respectively. Tcom = 5.1 K and +Tinc = 6.8 K respectively indicates the transition temperature +of the commensurate and incommensurate orders. (c) The +ratio of magnetic susceptibility, which was measured with field +H = 100 Oe along the c-axis (χc), to that measured with +the same field along the a-axis (χa). Both χc and χa are +measured after cooling down to 2 K in zero magnetic field +(ZFC). 1/χc is also plotted in the same panel, and the black +line shows the result of a Curie-Weiss fit to the data above 150 +K. The inset shows the magnetization of NdAlGe measured at +T = 2 K with the magnetic field applied along the c-axis (Mc) +and a-axis (Ma). (d) χc measured while the sample is cooled +under field H = 100 Oe (FC) and χc measured under ZFC. (e) +Temperature dependence of the magnetic heat capacity Cmag +p +of NdAlGe; the inset shows the total heat capacity Cp. Cmag +p +was obtained by subtracting the heat capacity of LaAlGe from +the heat capacity of NdAlGe. The dashed black line in the +main panel is the predicted ”Schottky-like” Cmag +p +anomaly +calculated assuming that the (2J +1) spin-orbit levels of Nd3+ +(J += +7/2) are split by crystal electric field (CEF) effects +into a doublet ground state, 2 excited doublets at 4 meV, and +another excited doublet at 9 meV. (f) The total heat capacity +below T = 10 K. +in NdAlSi can be seen as discontinuity occurring at Tcom +in ρxx(T), χ(T), and Cp(T). For the incommensurate +order, however, it is less obvious, but NdAlSi shows a +drop in ρxx(T), an arguable change of slope in χ(T), and +a clear peak in Cp(T) at Tinc [26]. For comparison, we +also looked for similar effects in NdAlGe. In Fig. 2(b,d,f), +we again see clear features at Tcom as a drop in ρxx, a split +of FC and ZFC data in χc, and a peak in Cp. Anomalies +associated with the incommensurate order of NdAlGe are + +5 +still subtle in ρxx(T) and χ(T), where a mild upturn and +a mild change of slope are observed at Tinc, respectively. +In the heat capacity data, however, there is a peak that +starts at Tinc and one can argue the presence of two transi- +tions in Cp(T). The low temperature Cp peaks of NdAlGe +are broader than the ones observed for NdAlSi, which +have sharp discontinuities occurring exactly at Tcom and +Tinc. This observation suggests a similar magnetic phase +diagram for both NdAlSi and NdAlGe, but with more +disorder in NdAlGe. +C. +Neutron diffraction +To gain insights into the collective magnetism of +NdAlGe, we have performed single-crystal neutron diffrac- +tion to determine its temperature-dependent spin struc- +ture. Below Tinc, we found incommensurate magnetism in +NdAlGe that is characterized by strong magnetic Bragg +peaks (Qmag) indexed with an ordering vector kAFM1 = +(2/3 + δ(T), 2/3 + δ(T), 0), as well as weaker Bragg peaks +indexed with kAFM2 = (1/3 − δ(T), 1/3 − δ(T), 0). +As shown in Fig. 3(a), we determined the incommen- +surability δ(T) by tracking the temperature dependence +of the Qmag = (2/3 + δ(T), 2/3 + δ(T), 0) peak center +observed in an (HH0) scan. The temperature depen- +dence of δ(T) is plotted in Fig. 3(b) where a mild change +of δ(T) between Tcom < T < Tinc is observed, but a +transition to commensurate magnetism (δ = 0) arises +for T < Tcom. For comparison, we note that Fig. 3(b) +also includes data from our SANS analysis, which will be +presented in the next section. The order parameter of +the Qmag = (2/3 + δ(T), 2/3 + δ(T), 0) peak (Fig. 3(c)) +correlates with Tinc. +In addition to the antiferromagnetic kAFM1 and kAFM2, +we also observed ferromagnetism in NdAlGe. To prove +this, we acquired an order parameter at Q = (200) +(Fig. 3(c)), which shows it onsets slightly above Tcom. +In the next section, we will see that ferromagnetism actu- +ally onsets exactly at Tcom. The fact that we see magnetic +intensity at Q = (200) above Tcom is from the onset of +an incommensurate k = (δFM(T), δFM(T), 0) wave that +is a precursor to the δFM(T) = 0 ferromagnetism compo- +nent. Our neutron diffraction experiment simply could +not resolve this incommensurability, but we could do so +using small-angle neutron scattering presented in the next +section. +The commensurate magnetic phase of NdAlGe is thus +described by a multi-k spin structure including two differ- +ent antiferromagnetic components (kAFM1 and kAFM2), +as well as a ferromagnetic component (k = 0). +The +diffraction pattern of NdAlGe is practically identical to +the one observed in NdAlSi [26]. The possible magnetic +basis vectors describing this spin structure were obtained +by symmetry analysis and consist of the xyz components +of a SDW propagating along the [110] or [1¯10] directions. +The SDW can either have parallel or anti-parallel spins +sitting on the primitive Nd3+ sites at r1 += +(0,0,0) +(a) +a +b +c +NdAlGe +(d) +(b) +(c) +(e) +(f) +b +AB8XicbVDLSsNAFL2pr1pfVZduBovgqiQi6LoxmUF+8A2lMl0g6dTMLMjVBC/8KNC0Xc+jfu/BsnbRbaemDgcM69zLknSKQw6 +LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wST29zvPHFtRKwecJpwP6IjJULBKFrpsR9RHAdhFswG1Zpbd+cgq8QrSA0KNAfVr/4wZmnEFTJjel5boJ+RjUKJvms0k8NTyib0BHvWapoxI2fzRPyJlVhiSMtX0KyVz9vZHRyJhpFNjJPKFZ9nLxP6+XYnjtZ0IlKXLFh+FqSQYk/x8MhSaM5RTSyjTwmYlbEw1ZWhLqtgSvOWTV0n7ou65de/+sta4Keowmcwjl4cAUNu +IMmtICBgmd4hTfHOC/Ou/OxGC05xc4x/IHz+QPby5EG +AB8XicbVDLSsNAFL2pr1pfVZduBovgqiQi6LoxmUF+8A2lMl0g6dTMLMjVBC/8KNC0Xc+jfu/BsnbRbaemDgcM69zLknSKQw6 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+gcM69zLknSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlslrHuBtRwKRvoUDJu4nmNAok7wST29zvPHFtRKwecJpwP6IjJULBKFrpsR9RHAdhRmeDas2tu3OQVeIVpAYFmoPqV38YszTiCpmkxvQ8N0E/oxoFk3xW6aeGJ5RN6Ij3LFU04sbP5oln5MwqQxLG2j6FZK7+3shoZMw0CuxkntAse7n4n9dLMbz2M6GSFLli4/CVBKMSX4+GQrNGcqpJZRpYbMSNqaMrQl +VWwJ3vLJq6R9Ufcund/WvcFHWU4QRO4Rw8uIG3ETWsBAwTO8wptjnBfn3flYjJacYucY/sD5/AHaRpEF +FIG. 3. (a) Neutron diffraction scans collected along the recip- +rocal (HH0) space direction at various temperatures between +1.5 K to 8 K centered around Q = (2/3, 2/3, 0). (b) The tem- +perature dependence of the (δ(T), δ(T), 0) incommensurability +of the kAFM1 = (2/3, 2/3, 0) SDW is plotted in red. The green +curve shows 1/3 of the (δFM(T), δFM(T), 0) incommensurabil- +ity of the ferromagnetic component whereas the blue curve is +the total SANS collected using the 12 ˚A data. (c) The order +parameter of Q = (200) and Q = (2/3 + δ(T), 2/3 + δ(T), 0) +Bragg peaks. (d) The main panel shows the rocking scans +at Q = (2/3, 2/3, 0) and Q = (1/3, 1/3, 0) for T = 1.5 K. +We note that the source of the elevated background for the +Q = (2/3, 2/3, 0) rocking scan comes from proximity to an +Al Bragg peak. The inset of panel (d) shows the Q = (0, 0, 4) +rocking scans collected for both T = 10 K and T = 1.5 K. +(f) and (f) show the refined spin structure for the magnetic +commensurate phase of NdAlGe. Red (blue) arrows are used +to represent the up (down) spins. The tilting of the spins +within the ab plane was amplified by a factor of 4 to allow for +better visualization. +and r2 += +(1/2,0,1/4). +Anti-parallel (parallel) spin +components produce scattering at Bragg peaks indexed +by the magnetic ordering vector kAFM1 = (2/3, 2/3, 0) +(kAFM2 = (1/3, 1/3, 0)). As seen in the main panel of +Fig. 3(d), the Bragg peaks with kAFM1 = (2/3, 2/3, 0) +have almost two orders of magnitude greater intensi- +ties than the kAFM2 = (1/3, 1/3, 0) ones. This indicates +a dominant anti-parallel spin component for the SDW, +which is augmented by a weak parallel one. The anti- + +T +T +Inc : +u. +0.8 +0.6 +Int +0.4 +Mag. +Q=(200) +0.2 +Q=(2/3+8,2/3+8,0): +0 +- +-0.2 +- +2 +3 +4 +5 +6 +7 +T(K)1.8K +3.4 K +Intensity (cts/s +40 +4.6 K +5.0 K +5.4 K +5.8 K +20 +6.2 K +7.0K +0.65 +0.66 +0.67 +0.68 +0.69 +(HHO)0.02 +(a.u. +0 +(8,8,0) (r.1.u) +AEM +M,0)/3 +d +FM +FM? +O SANS Int. +0.01 +Inc.( +二 +6 +8 +T(K)c +ac +a1.5 K +10K +Q = (2/3,2/3,0) +10 +(004) +Q= (1/3,1/3,0) +(cts/s +5 +104 +0 +Intensity +102 +100 +-0.5 +0 +0.5 +Φ(°)6 +parallel spin component was refined to an Ising one, while +the parallel component to a weak in-plane spin canting +that is transverse to the propagation of the SDW. The +spin structure refinement of NdAlGe is shown in Appendix +E. +The k = 0 ferromagnetic part of the spin structure +was refined to a c-axis magnetized state. This is due +to the fact that we did not observe magnetic Bragg in- +tensity at nuclear-allowed Q = (0, 0, L) Bragg positions +(see top right inset of Fig. 3(d)), while we observed mag- +netic scattering at nuclear-allowed Bragg positions such as +Q = (2, 0, 0) (Fig. 3(c)). The magnetization was refined +to 0.9(1)µB/Nd, which is consistent with the value of the +low-field magnetization plateau reported in the inset of +Fig. 2(c). +Finally, adding all spin components together, the spin +structure of NdAlGe is an Ising down-up-up (duu) fer- +rimagnetic SDW propagating along the [110] or [1¯10] +direction that is augmented by a weak in-plane chiral +component lying transverse to its propagation (see sketch +of the spin structures in Fig. 3(e,f)). Mostly pointing +along the c-axis, the moment on each Nd sites was re- +fined to 3.0(2)µb with an in-plane tilting angle of 3(1)°. +The high temperature incommensurate spin structure is +similar to the commensurate one, but convoluted with an +amplitude-modulated wave that has a spatial wavelength +of ∼ 35 nm. The resulting spin structure of NdAlGe is +practically identical to the NdAlSi one, but we note that +the incommensurate to commensurate phase transition in +NdAlGe is much broader in temperatures than the one in +NdAlSi [26]. For example, the 5 K (HH0) scan presented +in Fig. 3(a) shows the presence of both a commensurate +and an incommensurate peak, which signifies inhomogene- +ity within the NdAlGe crystal. This is consistent with a +range of different critical temperatures Tcom coexisting +within the same crystal of NdAlGe, which likely arises +from a variation of the stoichiometry across the whole +sample. +Such a conclusion corroborates the fact that +NdAlGe has more disorder than NdAlSi. +D. +Small-Angle neutron scattering (SANS) +So far, we have shown that the details of the magnetism +of NdAlGe is impacted by the disorder. +In order to +characterize this further, we have performed a SANS +experiment to probe its magnetized inhomogeneities on a +spatial length scale of ∼ 1 to 500 nm. +We first collected field and temperature-dependent +SANS data with the c-axis parallel to the incident neu- +tron beam so we could probe the in-plane scattering +vectors. Representative data acquired with 4.75 ˚A in- +cident neutrons within the paramagnetic, incommensu- +rate, and commensurate phase of NdAlGe are respec- +tively shown in Fig. 4(a,b,c). +As expected, no coher- +ent magnetic scattering is detected in the paramagnetic +state. +In the incommensurate phase, Bragg peaks at +symmetry-related Q = (δFM(T), δFM(T), 0) positions are +observed corresponding to an SDW with a spatial modu- +lation of 116(7) ˚A at 5.4 K. This incommensurate SDW +could not be resolved in our neutron diffraction exper- +iment and is a precursor to the commensurate ferro- +magnetic spin component of NdAlGe occurring below +Tcom. The temperature dependence of the incommen- +surability δFM(T) is reported in Fig. 3(b). δFM(T) fol- +lows a δ(T) = δFM(T)/3 relationship indicating that the +kFM = (δFM(T), δFM(T), 0) SDW is the third harmonic of +the main kAFM1 = (2/3+δ(T), 2/3+δ(T), 0) wave. This is +typical of incommensurate magnetism in rare-earth metal- +lic systems where odd harmonics emerge from ”squaring- +up” of the main wave, which is expected upon cooling as +the magnetization becomes more constant through the +lattice [40, 41]. +Within the commensurate phase, the incommensurate +Bragg peaks disappear such that the SANS of NdAlGe +is now centered at |Q| = 0 and is shaped like a cross +extending along the <110> directions (Fig. 4(c)). This +is different from the isotropic in-plane SANS pattern ob- +served in the isostructural Weyl semimetal PrAlGe [19], +and we will argue that the in-plane anisotropic cross pat- +tern of NdAlGe arises from the finite size of the magnetic +domains forming the multi-domains state. Using a field +of 2 T, Fig. 4(d) shows that field-cooling (FC) NdAlGe +within its commensurate magnetic state significantly de- +pletes the cross pattern. +To probe the temperature dependence of the cross pat- +tern, we collected SANS data with 12 ˚A incident neutrons, +which exclude the magnetic incommensurate Bragg peaks +such that the SANS scattering from the cross pattern +can be isolated (see appendix F). The temperature de- +pendence of the cross pattern extracted this way shows +that it onsets below Tcom (blue circles in Fig. 3(b)) and +is indeed a feature of the commensurate order. +We then studied the field evolution of the cross pattern +by acquiring 4.75 ˚A SANS data for various in-plane and +out-of-plane fields. As reported in Fig. 4(e), an out-of- +plane field of only 0.30(5) T is enough to completely +suppress the SANS intensity associated with the cross +pattern, whereas a similar in-plane field strength does not +significantly affect the scattering. This is consistent with +the Ising anisotropy of the bulk magnetization (inset of +Fig. 2(c)) and shows that the cross is only observed when +the time-reversal symmetric domains coexist (i.e. only +when both up-down-down (udd) and duu domains are +present). +We found that the SANS Q = 0 cross pattern could be +modeled using an anisotropic Lorentzian-squared function +of the form S(Q) = +A +((ϵ∥Q∥)2+(ϵ⊥Q⊥)2+1)2 , which is often +used to phenomenologically describe the SANS of inhomo- +geneous magnetized systems such as spin glasses [42–45]. +In this expression, A is a scale factor, while ϵ∥ and ϵ⊥ +are the spatial correlation lengths of the ferrimagnetic +domains parallel and perpendicular to the magnetic order- +ing vector direction [110], respectively. The momentum +transfer Q is also expressed into a component that is +either parallel (Q∥) or perpendicular (Q⊥) to the order- + +7 +Para. +T= 8 K +ZFC +Q(HH0) +Q(H-H0) +0 +0 +0.1 +-0.1 +0.1 +-0.1 +0 +2 +SANS Intensity (arb. units) +(a) +4.75 Å +Inc. +T= 5.4 K +ZFC +Q(HH0) +0 +0.1 +-0.1 +0 +2 +SANS Intensity (arb. units) +(b) +Q(H-H0) +0 +0.1 +-0.1 +4.75 Å +Com. +T= 2 K +ZFC +Q(HH0) +0 +0.1 +-0.1 +0 +2 +SANS Intensity (arb. units) +(c) +Q(H-H0) +0 +0.1 +-0.1 +4.75 Å +Q[H,H,0] +FC +Com. +T= 2 K +Q(HH0) +Q(H-H0) +0 +0 +0.1 +-0.1 +-0.1 +0 +2 +SANS Intensity (arb. units) +0.1 +H||c +(d) +4.75 Å +H(T) +0 +1 +(e) +Q(HH0) +T= 2 K +(f) +Theory +Q(HH0) +0 +-0.1 +0 +2 +SANS Intensity (arb. units) +(g) +0 +0.1 +-0.1 +Q(H-H0) +Single Domain (FC) +Stripe Domains (ZFC) +(h) +ε[110] = 18(5) nm +0.1 +T= 2 K +2 +0 +1 +FIG. 4. Panels (a),(b) and (c) correspond to the zero-field cooled (ZFC) 4.75 ˚A SANS data respectively collected within the +paramagnetic state (8 K), the incommensurate phase (5.4 K), and the commensurate phase (2 K). Panel (d) is the 4.75 ˚A SANS +data collected within the commensurate phase (2 K) using a field-cooled (FC) protocol (2 T). Panel (e) shows the total SANS +intensity observed as a function of both an in-plane (blue) and out-of-plane (green) magnetic field using 4.75 ˚A neutrons at +T = 2 K. Panel (f) shows the T = 2 K total SANS scattering intensity as a function of the momentum transfer Q measured +along the [H,H,0] direction for both ZFC and FC. This plot includes the SANS data collected within both the 4.75 ˚A and +12 ˚A configurations as well as their appropriate fit to a Lorentzian-squared function. Panel (g) is the calculated SANS pattern +assuming an anisotropic Lorentzian-squared function with ϵ∥ = 18 nm and ϵ⊥ = 72 nm. Panel (h) shows a sketch of the 1D +representation of the commensurate spin structure of NdAlGe (single domain). This 1D representation is also used to represent +the stripe ferrimagnetic domains observed via SANS in a ZFC process. +ing vector. We note that there’s also the presence of +magnetic domains with ordering vector propagation along +the [1¯10] direction so we included a Lorentzian-squared +function where ϵ∥ and ϵ⊥ are swapped. The combination +of two Lorentzian-squared functions that represent the +SDW along [110] and [1¯10] accounts for the two branches +of the cross pattern. The scattering function S(Q) was +then convoluted to the 2D resolution function ellipsoid +of our SANS instrument and fitted to the 2D zero field +SANS data of NdAlGe acquired at T = 2 K (Fig. 4(c)). +Following this procedure, we obtained ϵ∥ = 18(5) nm +and ϵ⊥ = 72(8) nm. A quantitative comparison between +the fit and the data is shown in Fig. 4(f) for the momentum +transfer along the [H,H,0] direction, while the resulting +2D fit is presented in Fig. 4(g) and can be compared to +Fig. 4(c). Our result indicates the SANS observed within +the commensurate phase of NdAlGe originates from fer- +rimagnetic stripes domains that have a shorter spatial +length scale parallel to the SDW and a longer one perpen- +dicular to the SDW. The ferrimagnetic stripes domains of +NdAlGe are sketched in Fig. 4(h). The anisotropic shape +of the magnetic domains in NdAlGe may be a consequence +of anisotropic exchange interactions, dipolar interactions, +or Dzyaloshinskii-Moriya interactions. +We associate the origin of the magnetic stripes in +NdAlGe to the finite sizes of its bulk domains. +This +scenario is consistent with the observed field dependence +of the Q = 0 cross pattern. Indeed, contrary to an in- +plane field, a field parallel to the c-axis promotes one +time-reversal domain over the other. In this case, a multi- +domain sample is not preferred and the spatial dimensions +of the energetically favoured domain then diverge. The +same phenomenology explains the longer length scale ob- +served in the low-temperature FC SANS data (Fig. 4(d)), +which is not expected if the cross pattern originates purely +from domain wall scattering. +A fit to the FC SANS +data against the Lorentzian-squared scattering function +(Fig. 4(f)) shows that ϵ∥ = ϵ⊥ = 270(30) nm. +E. +Anomalous Hall effect +We now turn to the anomalous Hall effect (AHE) of +NdAlGe, and show that its duu and FM states host dif- +ferent types of AHE. Fig. 5(a) shows the field dependence +of the electrical resistivity ρxx(H) of NdAlGe measured +below the transition temperature Tinc. ρxx(H) curves +taken at opposite field-sweeping directions show a mild + +SANS Intensity (arb. units) +T=2K +H 「110l(In-plane) +H l[0011 (Out-of-plane) +0 +0.5 +1.5 +2 +H(T)[110] k +b +u-d-d +va +Cd-u-u +u-d-d +1.1..1.1. +d-u-u +: 11.o-1-1(arb. units +SANS Intensity +0.01 +ZFC += 18(5) nm +[110] +FC +10- += 270(30) nm +[110] +10 +0.01 +0.1 +IQI +[H.H,0十SANS Intensity (arb. units +0.01 +ZFC +8[110] += 18(5) nm +FC +10- +{110j = 270(30) nm +10-3 +0.01 +0.1 +IQI +[H,H,0]0.1 +0.08 +0.06 +0.04 +0.02 +0 +-0.02 +-0.04 +-0.06 +-0.08 +0.1 +0.1 +-0.08 +-0.06 +-0.04 +-0.02 +0 +0.02 +0.04 +0.06 +0.08 +0.18 +(a) +(b) +(c) +(d) +𝜌!" +#,%&& 𝜌!" +#,'( +𝜎"! +#,%&& +𝜎"! +#,'( +2 K +3 K +4 K +5 K +6 K +7 K +H ∥ c +H ∥ c +Sample T7 +2 K +7 K +FIG. 5. (a) Resistivity ρxx of the sample T7 as a function of +external magnetic field H below Tinc. The current is applied +along the a-axis (x) and the field is along the c-axis (z). For +each temperature, the data represented by a solid line are +measured while the field is swept from H = 6 T to H = −6 +T, while the dashed line is recorded in the opposite field- +sweeping direction. The same convention applies to panel (b). +(b) Hall resistivity ρyx(H) of the sample T7 collected at the +same temperatures as in panel (a). The data taken at each +temperature were antisymmetrized and shifted by 0.5 µΩ cm +from each other for visibility. The anomalous part of the Hall +resistivity ρA,duu +yx +and ρA,FM +yx +are extracted from the y-intercept +of a linear line fitted to the plateaus of duu (0.2 T < H < 1 +T) and FM (H > 4 T) states, respectively. (c) Anomalous +Hall conductivity (AHC) as a function of temperature in the +duu (σA,duu +xy +, solid lines) and FM state (σA,FM +xy +, dashed lines) +of seven samples (shown as different symbols and colors). +The data of sample T7 are plotted with black circles. (d) +Normalized AHC plotted as a function of temperature. The +light gray and dark gray stripes correspond to the re-scaled +magnetization in the duu state (Mduu) and FM state (MFM), +which are extracted from the y-intercept of a linear line fitted +to the plateaus of duu (0.2 T < H < 1 T) and FM (H > 4 T) +states, respectively (see Appendix C). +hysteresis below H∗ ≃ 3 T, which is the transition field +from the duu to the FM state (see Mc(H) data in top in- +set of Fig. 2(c)). The hysteresis starts from T = Tcom and +persists as the temperature decreases. Another feature +related to the transition field H∗ is the local maximum +of ρxx(H). At T = 2 K, ρxx(H) first increases with the +field in the duu state, peaks at H∗, and then starts to de- +crease as the system is going through a smooth transition +from the duu to the FM state. Then, ρxx(H) reaches a +minimum at the end of the smooth transition, and finally +starts to increase again when it is in the FM state. Such +a non-monotonic behavior of ρxx(H) can also be seen +in some of the half-Heusler compounds such as DyPtBi, +which shows multiple field-induced phase transitions and +has relatively low mobility (< 1000 cm2V−1s−1) [46]. We +note that the local maximum at H = H∗ persists above +Tcom and Tinc where the transition between duu and FM +states no longer exists; such non-monotonic magnetore- +sistance above TC has also been reported in DyPtBi and +other half-Heusler compounds [47–49]. We may quali- +tatively understand the behavior of ρxx(H) in terms of +the two-current model [50, 51], which suggests that the +resistivity in the duu state (ρduu) is larger than within +the FM state (ρFM). Assuming that both an up spin +(ρ↑) and a down spin (ρ↓) contribute to the current in +parallel, and also that ρ↑ ≫ ρ↓, we may then express +ρFM = (1/ρ↑ + 1/ρ↓)−1 ∼ ρ↓, which is a relatively low +value. In the duu state, since the up and down spins ad- +mix as the spin wave propagates, both ρ↑ and ρ↓ approach +an averaged value of the two. As a result, the resistivity +in the duu state ρduu has a significant contribution from +ρ↑ and is thus larger than ρFM. +Figure. 5(b) shows the Hall resistivity ρyx(H) measured +at T < Tinc. At T = 2 K, there are two plateaus in +ρyx(H), both of which correlate with the magnetization +plateaus observed in the duu and FM states. By fitting +each plateau to a linear line, we extracted the anomalous +part of ρyx(H) in the duu state (ρA,duu +yx +) and FM state +(ρA,FM +yx +) using the y-intercept of their respective fitting line. +ρA,duu +yx +was extracted only for T ≤ Tcom since beyond that +temperature there is no duu state, but the spins are still +polarized at high fields and high temperatures so ρA,FM +yx +was calculated up to T ∼ Tinc. From the information +extracted from Figs. 5(a,b), we calculated the anomalous +Hall conductivity (AHC) in the duu state (σA,duu +xy +) and +the FM state (σA,FM +xy +) at each temperature using ρ0 ≡ +ρxx(H = 0) and ρA +yx as σA +xy = +ρA +yx +(ρA +yx)2+ρ2 +0 . The results are +plotted in Fig. 5(c) for seven different samples. Each +sample is uniquely represented by a specific color and +symbol; for example, the data of the sample T7 is plotted +with black circles. The solid line is used for σA,duu +xy +and +the dashed line is for σA,FM +xy +. At first glance, the data are +all over the place and it seems difficult to draw a clear +conclusion. +However, since the resistivity is calculated from the +resistance that depends on the geometric factors of each +sample, the uncertainty in these sample-dependent factors +may have contributed to the “randomness” of the data +in Fig. 5(c). To eliminate the trivial effect of geometric +factors and extract the intrinsic properties of NdAlGe, +we divided both σA,duu +xy +and σA,FM +xy +of each sample by +its own σA,FM +xy +measured at 2 K (σA,FM +xy +(2K)). Assum- +ing ρxx ≫ ρyx [52], we show that the normalized AHC +σA +xy(T)/σA,FM +xy +(2K) is free of geometric factors (the super- +scripts are omitted below for simplicity): +σxy(T)/σxy(2K) = +ρyx(T ) +ρ2yx(T )+ρ2xx(T ) +ρyx(2K) +ρ2 +yx(2K)+ρ2 +xx(2K) +≃ +ρyx(T ) +ρ2xx(T ) +ρyx(2K) +ρ2 +xx(2K) += ρyx(T) +ρyx(2K) +ρ2 +xx(2K) +ρ2xx(T) . +Since geometric factors do not depend on T, the geomet- + +9 +ric factor of ρyx (involving sample thickness) and ρxx +(sample length, width, and thickness) are all eliminated in +normalized AHC. We plotted the results in Fig. 5(d) and +found interesting characteristics of σA,duu +xy +and σA,FM +xy +. +As shown in Fig. 5(d), for σA,duu +xy +, the normalized AHC +of different samples all collapsed onto a single curve. In +addition, as T decreases, the normalized σA,duu +xy +scales +with the magnetization of the duu state (Mduu, light +gray stripe) such that it saturates at low temperatures. +The convergence of the data from different samples, the +scaling between σA,duu +xy +and Mduu, and the saturation of +AHC at low T are strong evidence for an intrinsic AHE +[8, 20, 53–56]. +However, in sharp contrast to σA,duu +xy +, the normalized +σA,FM +xy +of different samples in Fig. 5(d) do not collapse +but diverge into several curves. Indeed, as T decreases, +the normalized AHC does not follow the magnetization +of the FM state (MFM, dark gray stripe). +Since the +NdAlGe crystals grown by flux method are prone to Ge +vacancies, we expect that extrinsic disorders vary in each +crystal and govern the variance in σA,FM +xy +. +When the +different disorder levels among all samples are taken into +account, the convergence of σA,duu +xy +among them becomes +quite nontrivial, and strongly suggests a robust intrinsic +contribution to the AHE due to Berry curvature. The +clear distinction between σA,duu +xy +and σA,FM +xy +marks two +regimes of AHE in NdAlGe: an intrinsic AHE in the duu +state and an extrinsic AHE in the FM state. +IV. +DISCUSSION +To better understand and interpret the magnetism and +AHE in NdAlGe, we calculate band structure, Fermi +surface, Weyl nodes, and AHC due to Berry curvature. +Fig. 6(a) shows the Brillouin zone and high-symmetry k- +points; a k-path along these high-symmetry k-points was +selected to plot the band structure of NdAlGe in Fig. 6(b). +At the first glance, the band structure of NdAlGe does +not look much different from that of NdAlSi [26], but +the similarities and differences are more visible when we +look at the Fermi surface. From Fig. 6(c), we can see the +butterfly-shaped hole pockets along the Γ − X k-path, +similar to the ones in NdAlSi near Q = (± 1 +3, ± 1 +3, l) [26]. +These pockets fulfill the nesting condition for the incom- +mensurate magnetic order to appear (see kAFM1 vector in +Fig. 6(c)). Besides, when looking at both Fig. 6(c) and +(e) together, we find that the nesting butterfly-shaped +hole pockets are also Weyl-like and Weyl nodes near dif- +ferent pockets are separated by the nesting wave vector. +The inter-node scatterings between these Weyl nodes can +provide the Weyl-mediated RKKY interactions and ac- +count for the chiral component in the duu ferrimagnetic +order (helical magnetism) [14, 26]. The similarities be- +tween NdAlSi and NdAlGe in the nesting Fermi pockets +and the distribution of Weyl nodes provide a reasonable +explanation for their similar magnetic orders. +On the other hand, there are differences in the Fermi +5 +5.5 +6 +6.5 +7 +Energy(eV) +-400 -200 +0 +200 400 +G S N S Z G X +(a) +(b) +(c) +(d) +(e) +(f) +𝜎! (102 Ω-1cm-1) +Energy (eV) +0 +2 +4 +-4 +-2 +6 +5.5 +5 +6.5 +7 +kAFM1 +kAFM1 +h+ +e- +FIG. 6. (a) Brillouin zone of NdAlGe and high-symmetry +k-points. (b) Band structure of NdAlGe. The dashed line +marks the Fermi level calculated by DFT. (c) Fermi surfaces +of NdAlGe. The blue pockets are electron pockets, while the +red ones represent hole pockets. (d) Side view of the Fermi +surfaces of NdAlGe. (e) The distribution of 56 Weyl nodes +in the Brillouin zone found for the FM state of NdAlGe. (f) +Anomalous Hall conductivity (AHC), calculated for different +energy relative to the Fermi level (marked by the dashed line). +surfaces of NdAlSi and NdAlGe that distinguish their +transport properties. The most pronounced difference lies +in the diminished electron pockets in NdAlGe. In NdAlSi, +in addition to the elongated electron pockets along the +Z − Σ1 k-path at high kz, an octagon-like network of +electron pockets that extend to lower kz and connect +the elongated pockets is also present [26]. However, in +NdAlGe, this network of electron pockets is diminished +and only the elongated electron pockets at high kz remain +(Fig. 6(c,d)). Without this network, not only the number +of electron carriers are reduced, but also the momentum +dispersion of electrons is limited to a narrower range; both +factors may explain the dominant role of hole carriers in +NdAlGe. +In Fig. 6(f), we report the AHC contributed by intrinsic +Berry curvature at different energies [57]. At the Fermi +level determined by our DFT calculations (indicated by +the dashed line), σA +xy ≃ 200 Ω−1cm−1 agrees with both +the sign and the order of magnitude of σA,duu +xy +, which is +≃ 400 Ω−1cm−1 for sample T7. Although the calculation +in Fig. 6(f) is done in the FM state, we expect it to re- +flect the AHC in the duu state because of the similarity + +1.5 +1.0 +0.5 +ev +0.0 +E-Ef +-0.5 +-1.0 +-1.5 +N> +Z +XZ +N +X +y +T +30 +xy +yz +0 +Zx10 +in the net moment along z in both states. For a more +quantitative comparison, additional scaling analysis is +required to determine the intrinsic AHC in σA,duu +xy +. We +tried to perform the scaling analysis proposed by Tian +et al. [58]; although the data points do follow the scaling +(σA,duu +xy +∝ σ2 +xx), the extracted intrinsic AHC seems unrea- +sonably large, likely due to the narrow temperature range +of the fitting, which is limited by Tcom [59]. However, +we argue that σA,duu +xy +should be dominated by intrinsic +Berry curvature because 1) the collapse of data taken +from samples of different disorders [20, 58], 2) the linear +dependence of σA,duu +xy +on Mduu [60], 3) the saturation of +AHC at low temperatures [54, 56], 4) the conductivity +σxx falls in the regime where intrinsic AHE usually domi- +nates [53, 54], and 5) the reasonable agreement between +σA,duu +xy +and σxy calculated by DFT. +Previously, the transition from intrinsic to extrinsic +AHE in RAlX family was only observed among materi- +als of different chemical compositions, and it was mainly +driven by enhanced disorders [20]. Here, we argue that +spin fluctuations may play a key role in such a transi- +tion in NdAlGe. In ferromagnets, it has been proposed +that carriers scattering in a fluctuating spin background +may lead to a chirality-driven AHE [60, 61], and a devi- +ation of AHC from its scaling with M was shown to be +a manifestation of such behavior in experiments [56]. In +NdAlGe, σA,FM +xy +also deviates from MFM as T increases +(see Fig. 5(d)). Besides, the magnetic fluctuations in the +FM state seem to be strong, as suggested by the slow +and smooth transition from the duu to the FM state, +instead of the sharp and steep one in NdAlSi [26]. As a +result, we infer that the enhanced spin fluctuations as the +magnetic field increases, which possibly intensified with +disorders, may be the key factor driving the transition +from intrinsic to extrinsic AHE in NdAlGe. A complete +description of σA,FM +xy +, however, could be quite complicated +and would require a combination of inherent Berry cur- +vature from band structure, chirality-driven AHE due to +spin fluctuations, and extrinsic disorders through skew +scatterings. +V. +CONCLUSION +In conclusion, we report incommensurate magnetism in +NdAlGe that onsets at Tinc = 6.8(2) K and consists of an +Ising modulated SDW with a small helical chiral spin cant- +ing of 3(1)°. The spin system transitions into a commen- +surate ferrimagnetic state below Tcom = 5.1(1) K where +the spins form a duu Ising spin structure while keeping +the helical spin canting. Similar to NdAlSi, we found that +the periodicity of the incommensurate SDW of NdAlGe +matches the nesting wave vector between the two of its +topologically non-trivial Fermi pockets, which confirms +the possibility that Weyl-mediated RKKY interactions +could also drive the collective magnetism of NdAlGe. In +contrast to NdAlSi, however, NdAlGe has a higher level +of disorders, which has a minor effect on its magnetic +properties but greatly modifies the transport ones. Ef- +fects of disorders in NdAlGe are manifested through the +anisotropic ferrimagnetic domains of finite size as well +as broad features in the temperature dependence of its +magnetic heat capacity, magnetic order parameters, and +electrical resistance. In terms of transport, the carrier +concentration, mobility, and AHE of NdAlGe are all dras- +tically different from those of NdAlSi. In particular, we +characterized an intrinsic as well as an extrinsic AHE +regime in NdAlGe that are both absent in the Si analogue. +In NdAlGe, we argued that the intrinsic AHE results +mainly from its intrinsic Berry curvature, while the ex- +trinsic AHE is tied to disorders and spin fluctuations. The +lack of AHE in NdAlSi may be due to differences in the +strength of spin-orbit coupling between Ge 4p and Si 3p +electrons and an interplay between different mechanisms. +Our work thus suggests that Weyl-mediated magnetism is +a robust feature of non-centrosymmetric Weyl semimetals +NdAlX, while the transport properties including AHE in +Weyl semimetals can be strongly impacted by extrinsic +effects despite the presence of prominent Berry curvature. +ACKNOWLEDGMENTS +H.-Y.Y. thanks Chunli Huang, +Hiroaki Ishizuka, +Christopher Eckberg, Allan MacDonald, Inti Sodemann, +Yaroslav Tserkovnyak, and Collin Broholm for fruitful +discussions. This material is based upon work supported +by the Air Force Office of Scientific Research under Award +No. FA2386-21-1-4059. The work at TIFR Mumbai was +supported by the Department of Atomic Energy of the +government of India under Project No. 12- R&D-TFR- +5.10-0100. We acknowledge the support of the National In- +stitute of Standards and Technology, U.S. Department of +Commerce. The work at Northeastern University was sup- +ported by the US Department of Energy (DOE), Office of +Science, Basic Energy Sciences Grant No. DE-SC0022216 +and benefited from Northeastern University’s Advanced +Scientific Computation Center and the Discovery Cluster +and the National Energy Research Scientific Computing +Center through DOE Grant No. DE-AC02-05CH11231. +The identification of any commercial product or trade +name does not imply endorsement or recommendation by +the National Institute of Standards and Technology. A +portion of this research used resources at the High Flux +Isotope Reactor, a DOE Office of Science User Facility +operated by the Oak Ridge National Laboratory. +Appendix A: Crystal structure refinement of +NdAlGe +In addition to the powder XRD refinement shown in +the main text Fig. 1 (NdAlGe made by 10 Al recipe with +a refined ratio 1.01:1:0.97), Fig. 7 shows the refinement of +NdAlSi made by 10 Al recipe and NdAlGe made by a 15 +Al recipe. The refined ratio for NdAlSi is essentially 1:1:1, + +11 +�� +�� +�� +�� +��� +�� +� +� +� +Intensity ��� +� ������� +1G��6� +I 41 m d +� � ���. +�� .� +����� +������ +����� � ������ +�� +�� +�� +�� +��� +2Θ �G������� +�� +� +� +� +Intensity ��� +� ������� +1G���� ��� ��� +I 41 m d +����� +������ +����� � ������ +(a) +(b) +FIG. 7. +Powder XRD refinement of (a) NdAlSi, and (b) +NdAlGe samples made with additional Al flux. +and for NdAlGe (15 Al) is Nd : Al : Ge = 0.90 : 1 : 0.82. +We note that it is known from neutron scattering that +the stoichiometry of NdAlSi single crystals is not exactly +1:1:1 [26], and we interpret our powder XRD refinement +results shown in Fig. 1 and Fig. 7 as follows: relatively +speaking, NdAlGe is more non-stoichiometric compared +to NdAlSi, and using more Al flux to grow NdAlGe single +crystals may result in a higher deficiency in the Nd and +Ge sites. +(a) +(b) +T = 2 K +FIG. 8. (a) ρxx(T), and (b) ρyx(H) of NdAlGe (blue dashed +lines, left y-axis) and NdAlSi (red solid line, right y-axis). The +high-field part of the data of both materials is fitted to a linear +expression (black line in panel (b)) to extract single-band +carrier concentrations. +(a) +(b) +2 K +3 K +4 K +5 K +6 K +7 K +2 K +7 K +NdAlGe +NdAlGe +FIG. 9. +(a) M(H) data of NdAlGe recorded at different +temperatures. The data at each temperature are vertically +shifted away from each other by 2.2 µB/Nd for visibility. (b) +The same data as in panel (a), but not shifted. +Appendix B: Comparison between the transport +properties of NdAlSi and NdAlGe +Figure 8 compares ρxx(T) and ρyx(H) of NdAlGe and +NdAlSi. From Fig. 8(a), it can be seen that at base +temperature T = 2 K, ρxx of NdAlGe is much closer to +its room-temperature value compared to that of NdAlSi, +hence the higher RRR reported in Table I. Figure 8(b) +shows typical ρyx(H) curves for both materials. +For +NdAlGe, two plateaus corresponding to the duu and +FM magnetic phases can be seen. For NdAlSi, however, +although there are also two steps in its magnetization just +like NdAlGe [26], its ρyx(H) curve is smooth for the first +transition and only shows a small discontinuity at the +transition field to the FM state. The ρyx(H) curve is also +mildly nonlinear overall such that it is difficult to argue an +AHE in NdAlSi (see also Appendix D). We obtained the +single-band carrier concentrations by fitting a linear line +to the high-field part of ρyx(H) curves for both materials, +and used them to calculate their respective single-band +mobility from ρxx at 2 K (see Table I). +Appendix C: M(H) of NdAlGe at different +temperatures +Figure 9 shows the details of M(H) data of NdAlGe at +different temperatures below Tcom and Tinc. The plateaus +in the duu and FM states are evident and correspond +to the plateaus in ρyx(H). At high magnetic field, the +magnetization converges to the saturated value. +The +decent saturation of magnetization even at T = 7 K +allows us to extract ρA, FM +yx +from the high-field plateaus +in ρyx(H) data. We note that the transition field from +duu to FM state is not always the same and varies among +samples, while the plateaus are persistent before and after +the transition. + +12 +(a) +(b) +1.8 K +3 K +4 K +5 K +6 K +7 K +NdAlSi +NdAlSi +8 K +7 K +8 K +1.8 K +3 K4 K +5 K +6 K +𝜎! (Ω-1cm-1) +𝐸 − 𝐸" (eV) +Γ +Σ N Σ ! +Z +Σ +X +(c) +FIG. 10. (a) ρyx(H) data of NdAlSi recorded at different +temperatures. (b) M(H) data of NdAlSi. (c) Left panel: Band +structure of NdAlSi in FM state. Right panel: Anomalous +Hall conductivity of NdAlSi, calculated by DFT considering +intrinsic Berry curvature. +Appendix D: ρyx(H) and M(H) of NdAlSi at different +temperatures +Figure 10(a) and (b) compare ρyx(H) and M(H) data +of NdAlSi side by side to reveal the absence of Hall resis- +tivity plateaus despite clear plateaus in M(H). Similar +to NdAlGe, M(H) of NdAlSi has a low-field transition to +duu state and a high-field transition to FM state, each +of which results in a plateau in M(H). In the Hall data, +however, near H = 0 T, ρyx(H) is smooth and featureless; +no feature can be associated with the transition to duu +state in ρyx(H). At the transition to FM state, there is +a concomitant jump in ρyx(H) as shown in the inset of +Fig. 10(a). It is tempting to subtract a smooth back- +ground from ρyx(H) and interpret such a jump as AHE, +but this is not feasible here since the ρyx data in the +FM state (H > 7 T) below Tcom actually lie on top of +the ρyx data above Tcom in the same field range. We +note that M significantly drops (more than 30%) as T +changes from 1.8 K to 8 K, so if there is a finite AHE, the +Hall data recorded at these two temperatures should be +quite different from each other due to the proportionality +between M and ρA +yx [60]. As a result, we conclude that +no Hall plateaus were observed in NdAlSi. The absence +of AHE in NdAlSi is intresting because it is in conflict +with the AHC calculated by DFT. We performed AHC +calculations for NdAlSi in Fig. 10(c) by considering the +intrinsic Berry curvature, similar to the one shown in +Fig. 6(f) for NdAlGe. Near EF , there is a persistent +AHC (σxy ∼ 500 Ω−1cm−1), in contrast to the absence +of Hall plateaus in Fig. 10(a). It would require further +theoretical investigation to understand this discrepancy. +Appendix E: Details on the spin structure +refinement of NdAlGe +We did magnetic refinement of our neutron diffraction +data to determine the antiferromagnetic (AFM) spin struc- +ture component of the commensurate phase of NdAlGe. +To do so, rocking scans at 32 symmetrically nonequivalent +Bragg positions were collected at T = 1.5 K within the +manifold of Bragg peaks Q+ = G ± ( 1 +3 + δ, 1 +3 + δ, 0) and +Q− = G ± ( 2 +3 + δ, 2 +3 + δ, 0). Here G refers to all nuclear +allowed Bragg peaks. +We performed representational +analysis of the NdAlGe commensurate magnetism using +SARAh refine [62] and found six possible basis vectors +divided into two different irreps (Γ1 and Γ2) [63], whose +real parts are shown in Fig. 11(a). The two primitive Nd +ions located at r1=(0,0,0) and r2=(1/2,0,1/4) within the +unit cell have spins anti-parallel to each other for spin +structures described by ⃗ψ1+ ⃗ψ2, ⃗ψ4- ⃗ψ5, and ⃗ψ3. These +anti-parallel spin structures lead to strong Q− peaks and +no intensity at Q+ peaks. On the other hand, spin struc- +tures described by ⃗ψ1- ⃗ψ2, ⃗ψ4+ ⃗ψ5, or ⃗ψ6 have parallel Nd +spins at r1 and r2. This situation leads to strong Q+ +peaks and no intensity at Q− peaks. As seen in Fig. 3(e) +of the main text, we observed intensities at Q− positions +that are two order of magnitude greater than at Q+ so +the spin structure is dominantly anti-parallel and was re- +fined to an Ising anisotropy ( ⃗ψ3). We also detected weak +intensity at Q− positions so there’s also a weak parallel +spin component, which originates from an in-plane spin +component µxy. We did a magnetic refinement against +both the Γ1 and Γ2 manifold and found our refined χ2 is +∼100 times smaller for the Γ1 manifold so we concluded +that Γ1 is the appropriate irrep for NdAlGe. The final +refinement is shown in Fig. 11(c) where the best solution +was obtained with ⃗ψ1=- ⃗ψ2 = 0.14(2)µB and ⃗ψ3=3.8(4)µB. +Furthermore, Fig. 11(d) shows the χ2 dependence on both +the angle direction of the spin canting within the ab plane +(θxy), and its magnitude (µxy). In this plot, θxy = 0 is +defined to be along the ordering vector direction [1,1,0]. +From Fig. 11(d), we deduced an in-plane spin canting of +0.14(2)µB/Nd oriented 90(20)° away from the ordering +vector direction, which produces an helical spin canting. +We determined the spin polarization of the ferromag- +netic (FM) k = (0, 0, 0) magnetic structure of NdAlSi +by collecting rocking scans at 24 symmetrically non- +equivalent k = (0, 0, 0) Bragg positions covering the +(H, H, L) plane. The nuclear and magnetic contributions +to the Bragg diffraction were distinguished by collecting +rocking scans within both the paramagnetic phase at 10 K +and in the commensurate phase at 1.5 K. Symmetry anal- +ysis reveals three possible irreducible representations (ir- +reps) to describe the k = (0, 0, 0) magnetic structure [63]: +Γ1 and Γ3 that respectively correspond to ferromagnetic +and antiferromagnetic structures where the spins are ori- +ented along the c axis, and Γ5 that describes structures +where the spins lie in the ab plane. The Ising ferromag- +netic Γ1 is the only irrep that matches the symmetry + +0.5 +xy +yz +0 +Zx +500 +0 +500 +1000.1500 +200013 +�1( 1) +AB+XicbVBNS8NAEJ34We +tX1KOXxSLUS8mKoMeiBz1WsB/QhLDZbtulu0nY3RK6D/x4kE +Rr/4Tb/4bt20O2vpg4PHeDPzolRwbTzv21lb39jc2i7tlHf3 +9g8O3aPjlk4yRVmTJiJRnYhoJnjMmoYbwTqpYkRGgrWj0d3M +b4+Z0jyJn8wkZYEkg5j3OSXGSqHr+vdEShLiqp9qHuKL0K14N +W8OtEpwQSpQoBG6X34voZlksaGCaN3FXmqCnCjDqWDTsp9plh +I6IgPWtTQmkukgn18+RedW6aF+omzFBs3V3xM5kVpPZGQ7JTF +DvezNxP+8bmb6N0HO4zQzLKaLRf1MIJOgWQyoxWjRkwsIVRx +eyuiQ6INTasg0BL7+8SlqXNezV8ONVpX5bxFGCUziDKmC4h +jo8QAOaQGEMz/AKb07uvDjvzseidc0pZk7gD5zPHxZtkqM= +AB+XicbVBNS8NAEJ34We +tX1KOXxSLUS8mKoMeiBz1WsB/QhLDZbtulu0nY3RK6D/x4kE +Rr/4Tb/4bt20O2vpg4PHeDPzolRwbTzv21lb39jc2i7tlHf3 +9g8O3aPjlk4yRVmTJiJRnYhoJnjMmoYbwTqpYkRGgrWj0d3M +b4+Z0jyJn8wkZYEkg5j3OSXGSqHr+vdEShLiqp9qHuKL0K14N +W8OtEpwQSpQoBG6X34voZlksaGCaN3FXmqCnCjDqWDTsp9plh +I6IgPWtTQmkukgn18+RedW6aF+omzFBs3V3xM5kVpPZGQ7JTF +DvezNxP+8bmb6N0HO4zQzLKaLRf1MIJOgWQyoxWjRkwsIVRx 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+XbgaLUDclEVGXpW5cVrAXaEOYTCft0JkzEWMoU/ixoUibn0Ud76NkzYLbf1h4OM/53DO/EHCqFSO82V1tY3NrfK25Wd3b39qn1w2JWxFph0cMxi0Q+QJIxGpKOoYqSfCIJ4wEgvmN7k9d4DEZLG0b1KE+Jx +NI5oSDFSxvLt6pBrP3tMZ/UcWme+XMazlxwFdwCaqBQ27e/hqMYa04ihRmScuA6ifIyJBTFjMwqQy1JgvAUjcnAYIQ4kV42P3wGT40zgmEszIsUnLu/JzLEpUx5YDo5UhO5XMvN/2oDrcJrL6NRohWJ8GJRqB +lUMcxTgCMqCFYsNYCwoOZWiCdIKxMVhUTgrv85VXonjfcy4Z7d1Frto4yuAYnIA6cMEVaIJb0AYdgIEGz+AVvFlP1ov1bn0sWktWMXME/sj6/AGC25L9µxy(µB) +0 +40 +80 +120 160 +AB/HicbVBNS8NAEN3Ur1q/oj16CRahXkoioh6LXjxWsB/QxLDZbtqlm +03YnYgh1L/ixYMiXv0h3vw3btsctPXBwO9GWbmBQlnCmz72yitrK6tb5Q3K1vbO7t75v5BR +8WpJLRNYh7LXoAV5UzQNjDgtJdIiqOA024wvp763QcqFYvFHWQJ9SI8FCxkBIOWfLPqwogC9 +vPHbFK/dwmT5MQ3a3bDnsFaJk5BaqhAyze/3EFM0ogKIBwr1XfsBLwcS2CE0nFTRVNMBnjIe +1rKnBElZfPjp9Yx1oZWGEsdQmwZurviRxHSmVRoDsjDCO16E3F/7x+CuGlzORpEAFmS8KU2 +5BbE2TsAZMUgI80wQTyfStFhlhiQnovCo6BGfx5WXSOW045w3n9qzWvCriKNDdITqyEXqI +luUAu1EUEZekav6M14Ml6Md+Nj3loyipkq+gPj8wfAvpTV✓xy(�) +FIG. 11. (a) shows the real part of the magnetic basis vectors +for the AFM spin component of NdAlGe obtained from symme- +try analysis. (b) shows rocking scans collected at Q = (1,1,0), +which is a higher harmonic originating from the addition of +the magnetic Q=(2/3,2/3,0) and Q=(1/3,1/3,0) Bragg peaks. +The scattered intensity is 100 to 1000 weaker than the mag- +netic Q=(2/3,2/3,0) and Q=(1/3,1/3,0) Bragg peaks. (c) is +the final refinement of the spin structure of NdAlGe against +both the ferromagnetic (FM) k=(0,0,0) spin component and +the antiferromagnetic (AFM) spin component described by +kAFM1 and kAFM2. |F|2 is the neutron structure factor. (d) is +the χ2 value of the neutron diffraction refinement against an +in-plane moment µxy and its direction within the ab plane θxy. +θ=0 lies along the magnetic ordering vector [1,1,0] direction. +of the antiferromagnetic spin component. Furthermore, +Γ3 and Γ5 respectively produce magnetic Bragg reflec- +tions at Q = G ± (1, 1, 0) and Q = (0, 0, L) positions. +As seen in Fig. 3(e), we did not observe any scattering +at (0, 0, L) Bragg positions. Also, we only observed ex- +tremely weak intensity on the Q = G ± (1, 1, 0) peaks +(such as Q = (1, 1, 0) in Fig. 11(b)), but these can be +understood as arising from the higher harmonics of the +AFM order or from the presence of small Nd vacancies. +Our final refinement, which is shown in Fig. 11(c), thus +leads to the Γ1 structure with µFM = 0.9(1)µB. +Related to the phase factor of the spin structure at +which neutron diffraction is insensitive, we note that for +the Ising component, the spatial variation of the Nd +moments is expressed as: +µAFM1(r) = 1.9(2)[exp (i[(2 +3 +2 +30) · r + iθ]) + c.c.]. +(E1) +where c.c. stands for complexe conjugate. This expression +includes both the k = ( 2 +3, 2 +3, 0), and the k = (¯2 +3, ¯2 +3, 0) com- +ponents as required for the magnetic moment to be real +for all r. While the diffraction pattern is independent of θ, +the real space spin structure does depend on θ. For θ = π, +the spin structure can be described as (0-up-down) where +0 means there is no net magnetization on this site, whereas +a θ += 0 phase shift leads to an (up-down-down) spin +structure. Within the commensurate phase, once the FM +component of the structure is added (µFM = 0.9(1)µB), +θ = 0(6)° is the only phase that allows for all the Nd +moments to not exceed the 2.80(5)µB saturated moment +determined by the out-of-plane magnetization data. +Appendix F: SANS data with 12 ˚A neutrons +As discussed in the main text, we collected SANS pat- +tern with 12 ˚A incident neutrons for various temperatures +ranging from 10 to 2 K. Representative SANS pattern col- +lected within both the magnetic incommensurate and the +magnetic commensurate phase of NdAlGe are respectively +shown in Fig. 12(a) and (b). As seen from the absence of +SANS scattering within the incommensurate phase, the +12 ˚A data set was used to isolate the SANS scattering +from the Q = 0 cross pattern by summing over all the +scattering detected in such data set. The temperature +dependence of the cross pattern extracted this way is +presented in Fig. 3(b) of the main text (blue dots). + +8 +Q= (110) +Intensity (cts/s +10 K +1.5 K +4 +2 +0 +0.5 +10.5 +(°)AFM-I +0.6 +FM- +0.4 +2 +obs! +F +x2 = 5.7 +0.2 +0 +0 +0.2 +0.4 +0.6 +2 +F +calc0.9 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +20 +40 +60 +80 +100 +120 +140 +160 +18014 +Inc. +T= 5.8 K +ZFC +[H,H,0] +0 +0.01 +-0.01 +0 +SANS Intensity (arb. Units) +(a) +[H,-H,0] +Com. +T= 2 K +ZFC +[H,H,0] +0 +0.01 +-0.01 +0 +SANS Intensity (arb. Units) +(b) +[H,-H,0] +0 +0.01 +-0.01 +0 +0.01 +-0.01 +FIG. 12. 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Rodr´ıguez-Carvajal, CEA/Saclay, France (2001). + diff --git a/D9E4T4oBgHgl3EQfGQye/content/tmp_files/load_file.txt b/D9E4T4oBgHgl3EQfGQye/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3b8e6e51a6d3c5e97ab6a20ae5275c80bfc1e79b --- /dev/null +++ b/D9E4T4oBgHgl3EQfGQye/content/tmp_files/load_file.txt @@ -0,0 +1,1902 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf,len=1901 +page_content='Stripe Helical Magnetism and Two Regimes of Anomalous Hall Effect in NdAlGe Hung-Yu Yang,1, ∗ Jonathan Gaudet,2, 3 Rahul Verma,4 Santu Baidya,5, 6 Faranak Bahrami,1 Xiaohan Yao,1 Cheng-Yi Huang,7 Lisa DeBeer-Schmitt,8 Adam A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Aczel,8 Guangyong Xu,2 Hsin Lin,9 Arun Bansil,7 Bahadur Singh,4 and Fazel Tafti1 1Department of Physics, Boston College, Chestnut Hill, MA 02467, USA 2NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA 3Department of Materials Science and Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' University of Maryland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' College Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' MD 20742-2115 4Department of Condensed Matter Physics and Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Tata Institute of Fundamental Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Colaba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Mumbai 400005,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' India 5Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Rutgers University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Piscataway,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' New Jersey 08854-8019,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' USA 6Department of Physics and Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Jaypee University of Information Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Waknaghat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Solan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Himachal Pradesh 173234,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' India 7Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Northeastern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' MA 02115,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' USA 8Neutron Scattering Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Oak Ridge National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Oak Ridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Tennessee 37831,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' USA 9Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Academia Sinica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Taipei 11529,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Taiwan (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2023) We report the magnetic and electronic transport properties of the inversion and time-reversal symmetry breaking Weyl semimetal NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This material is analogous to NdAlSi, whose helical magnetism presents a rare example of a Weyl-mediated collective phenomenon, but with a larger spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Our neutron diffraction experiments revealed that NdAlGe, similar to NdAlSi, supports an incommensurate Ising spin density wave (Tinc = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8 K) with a small helical spin canting of 3° and a long-wavelength of ∼ 35 nm, which transitions to a commensurate ferrimagnetic state below Tcom = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Using small-angle neutron scattering, we showed that the zero-field cooled ferrimagnetic domains form stripes in real space with characteristic length scales of 18 nm and 72 nm parallel and perpendicular to the [110] direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Interestingly, for the transport properties, NdAlSi does not exhibit an anomalous Hall effect (AHE) that is commonly observed in magnetic Weyl semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In contrast to NdAlSi, we identify two different AHE regimes in NdAlGe that are respectively governed by intrinsic Berry curvature and extrinsic disorders/spin fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Our study suggests that Weyl-mediated magnetism prevails in this group of noncentrosymmetric magnetic Weyl semimetals NdAlX, but transport properties including AHE are affected by material-specific extrinsic effects such as disorders, despite the presence of prominent Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' INTRODUCTION To establish a Weyl semimetal phase, one needs to break either inversion or time-reversal symmetry to split Weyl nodes of opposite chirality, which may then lead to interesting topological properties [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Both routes have been explored through candidate materials that break either the inversion symmetry, such as the noncentrosym- metric TaAs [3–7], or time-reversal symmetry, such as the ferromagnetic (FM) Co3Sn2S2 [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Weyl semimetals that break both inversion and time-reversal symmetries remain largely unexplored, despite theoretical predictions of Weyl-mediated interactions with rich phase diagrams and topological magnetic textures [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' For instance, in the RAlX (R = rare-earths, X = Ge/Si) material family of double-symmetry-breaking Weyl semimetals [15– 17], a variety of rich magnetic orders have been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' These include collinear FM order [17–20] and noncollinear FM order [21–23], both of which are relatively common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' More unusual spin structures were also observed such as ∗ Present address: Department of Electrical and Computer Engi- neering, University of California, Los Angeles, California 90095, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' hungyuyang@g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='edu a topological multi-⃗k structure in CeAlGe [24], a spiral order in SmAlSi [25], and a helical incommensurate spin density wave (SDW) in NdAlSi [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In particular, for NdAlSi, its helical magnetism was shown to be stabilized by bond-oriented Dzyaloshinskii-Moriya (DM) interaction predicted to arise from Weyl-mediated Ruderman-Kittel- Kasuya-Yosida (RKKY) coupling, owing to the presence of itinerant Weyl electrons, local magnetic moments, and broken inversion symmetry [14, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Despite the comprehensive characterization of Weyl- mediated magnetism in NdAlSi [26] and some studies on NdAlGe [27], their transport properties such as anoma- lous Hall effect (AHE) remain unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' AHE has been extensively investigated in FM Weyl semimetals where the intrinsic Berry curvature may contribute to pronounced AHE [8, 9, 28], but recently it has been shown that Berry curvature is not always the dominant source of AHE in FM Weyl semimetals, and extrinsic disorders can also play a major role [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' With a chiral magnetism established in NdAlSi, NdAlX provides a unique system to study AHE in helimagnetic Weyl semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In this work, we aim to first establish the magnetic structure of NdAlGe, which is not obvious (not necessarily the same as NdAlSi) considering the behavior in other materials in RAlX fam- ily [20, 21, 24, 29], and study the AHE of NdAlX with a focus on NdAlGe to understand the interplay among arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='04893v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='mtrl-sci] 12 Jan 2023 2 topology, magnetism, and electrical transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We investigate the magnetism and electrical transport of NdAlGe with SQUID magnetometry, heat capacity measurements, neutron scattering experiments, resistivity measurements, and DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Similar to NdAlSi, we found a high temperature (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 K < T < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8 K) heli- cal incommensurate SDW in NdAlGe characterized by a multi-k structure with ordering vectors kAFM1 = (2/3 + δ(T), 2/3 + δ(T), 0), kAFM2 = (1/3 − δ(T), 1/3 − δ(T), 0), and kFM = (3δ(T), 3δ(T), 0), which evolves to a commen- surate (δ = 0) helical ferrimagnetic state at low temper- atures (T < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In this state, the small-angle neutron scattering (SANS) can be modeled by an anisotropic Lorentzian-squared function, which signifies stripes of fer- rimagnetic domains with real space characteristic length scales of 18(5) nm and 72(8) nm parallel and perpendic- ular to the [110] ordering vector direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Surprisingly, we found AHE responses as Hall resistivity plateaus under a magnetic field in NdAlGe but not in NdAlSi, despite that both materials show clear plateaus in their magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Furthermore, as the field increases, the AHE in NdAlGe shows a transition from intrinsic AHE in the ferrimagnetic phase to an extrinsic AHE in the polarized FM phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Finally, we calculate the elec- tronic band structure, Weyl-nodes, and Fermi surface of NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We also calculated the anomalous Hall conduc- tivity of NdAlGe, which shows a reasonable agreement with the observed intrinsic AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Our findings of helical magnetism and two regimes of AHE in NdAlGe suggest that Weyl-mediated magnetism is robust in these materi- als provided that the itinerant Weyl electrons and nesting Fermi pockets are intact, while AHE can be largely modi- fied by extrinsic disorders and spin fluctuations despite significant intrinsic Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' METHODS Single crystals of NdAlGe and NdAlSi were grown by a self-flux method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The starting materials are elemen- tal chunks of Nd, Al, and Ge, weighed in a ratio of Nd:Al:Ge=1:10:1 or 1:15:1, and mixed in an alumina cru- cible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Single crystals made with a 1:10:1 recipe were used in this study unless specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' NdAlSi single crystals were grown with the 1:10:1 ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The crucibles were sealed in an evacuated quartz tube, heated up to 1050◦C at 3◦C/min, dwelt for 12 hours, cooled down to 700◦C at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1◦C/min, and dwelt for another 12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' After the heating sequence, the tube was centrifuged to remove the excess Al flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Plate-like single crystals were found isolated and attached to the bottom of crucibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' X-ray diffraction (PXRD) measurement was performed with a Bruker D8 ECO instrument with a copper x-ray source (Cu Kα) and a one-dimensional LINXEYE-XE detector at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Rietveld refinement on the PXRD patterns was performed using the FullProf suite [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The elemental analysis of NdAlSi/Ge was determined by En- ergy Dispersive X-ray Spectroscopy (EDX), carried out with FEI Scios, operated at an acceleration voltage of 20 kV and a current of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='4 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Electrical resistivity and heat capacity were measured in a Quantum Design Physical Property Measurement System (PPMS) Dynacool with the standard four-probe technique and relaxation time method, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' DC magnetization experiments were performed on the vibrat- ing sample magnetometer in a Quantum Design MPMS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We performed neutron diffraction using both the HB- 1A thermal triple-axis spectrometer at ORNL and the cold triple-axis spectrometer SPINS at NIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The (HHL) scattering plane of NdAlGe was probed for both experi- ments at a base temperature of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 meV incident neutrons filtered with pyrolytic graphite (PG) were used on HB-1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' For SPINS, we used 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='7 meV incident neutrons with cooled Be filters employed both before and after the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The HFIR GP-SANS instrument was utilized to probe the SANS of NdAlGe with a base temperature of ∼2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Two SANS configurations were used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 1) Uncollimated 12 ˚A incident neutrons were used with the scattered neutrons detected at a distance of 19 m away from the sample, 2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='75 ˚A neutrons collimated by 3 guides were incident to the sample and detected at a distance of 8 m from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' A circular aperture with a diameter of 8 mm was placed at the sample position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' A flat-plate sample was aligned such that the [001] axis is parallel to the neutron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' SANS patterns were collected using an 11 T horizontal magnet applied both parallel (H ∥ [001]) and perpendicular (H ∥ [110]) to the incident neutron beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Error bars associated with all neutron diffraction intensities reported in this work correspond to 1 standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We performed electronic structure calculations within the framework of density functional theory (DFT) based on the projected augmented wave (PAW) method as imple- mented in the Vienna ab-initio simulation package (VASP) [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Generalized gradient approximation (GGA) [34] was used to include exchange-correlation effects and spin- orbit coupling (SOC) was added self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We added an on-site Coulomb interaction with Ueff = 8 eV for the Nd f electrons within the GGA+U scheme [35, 36] to include strong electron-correlation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We consid- ered the kinetic energy cut-off of 450 eV for the plane-wave basis set and used Γ-centered 11 × 11 × 11 k-mesh [37] for bulk Brillouin zone sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The tolerance of the electronic energy minimization was set to 10−6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We generated material-specific tight-binding Hamiltonian us- ing the VASP2WANNIER90 interface [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We included Nd d, f, Al s, p and Ge s, p orbitals in construction of the Wannier functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The topological properties were calculated using the WannierTools package [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' RESULTS Table I summarizes the main properties studied in this paper, including helical magnetism and anomalous Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' When comparing NdAlGe and NdAlSi together, 3 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Summary of magnetic and transport properties of NdAlSi and NdAlGe, including the onset of incommen- surate order (Tinc) and commensurate order (Tcom), satu- rated moment Msat, residual resistivity ratio (RRR, defined as ρ(300K)/ρ(2K), single-band carrier concentration (nh where holes are the dominant electric carriers in both materials) and mobility (µh), and anomalous Hall effect (AHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' NdAlGe NdAlSi [26] Tinc 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8(2) K 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='2 K Tcom 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1(1) K 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='3 K Msat at 2 K, 6 T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8(1) µB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='9 µB RRR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='0(1) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='0 nh at 2 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='06 × 1021 cm−3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='66 × 1019 cm−3a µh at 2 K 134 cm2V−1s−1 11008 cm2V−1s−1a AHE duu state: intrinsic No clear Hall FM state: extrinsic resistivity plateaub a See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' b See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' we find that their magnetic properties are similar to each other, but their transport properties are quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Incommensurate and commensurate magnetic orders are found in both materials at close temperatures, and their refined spin structures are also similar to each other [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' However, when it comes down to the transport properties, the residual resistivity ratio (RRR) of NdAlGe is 3 times smaller than that of NdAlSi, which suggests a higher disorder level in NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The lower RRR in NdAlGe is also manifested in the single-band analysis, which shows a higher hole concentration but a much lower hole mobility compared to NdAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' As we will see, in spite of similar Fermi surfaces, quantum oscillations are not observed in NdAlGe, but are pronounced in NdAlSi [26] at the same temperatures and magnetic fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' this distinction also suggests a shorter mean free path and lower mobility in NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' More interestingly, both intrinsic and ex- trinsic AHE were observed in NdAlGe as magnetic field increases, but no clear sign of AHE was found in NdAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In the following sections, we will present and interpret the structural, magnetic, and electrical transport properties of NdAlGe in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Crystal structure and disorder in NdAlGe The inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 1 shows the crystal structure of NdAlGe, which belongs to the same I41md space group as the archetypal Weyl semimetal TaAs [3–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The com- bination of the noncentrosymmetric crystal structure and the collective magnetism hosted by Nd3+ f-orbitals at low temperatures makes NdAlGe a double-symmetry-breaking Weyl semimetal [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Based on previous second harmonic generation experiments across different RAlX compounds, it is now clear that the RAlX material family resides in the noncentrosymmetric space group I41md [20, 21, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We will then use the I41md space group as our starting point for the nuclear structure refinement of NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Nd Al Ge FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Powder x-ray diffraction pattern and Rietveld refine- ment of NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The occupancy of each atom was refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The crystal structure of NdAlGe is presented on the top right inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Another important point is the stoichiometry of RAlX single crystals that may vary depending on the growth methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In terms of the growth methods, it has been shown that single crystals grown by a floating-zone furnace typically has a stoichiometric ratio much closer to 1:1:1 compared to those grown by flux methods, which tend to be Al-rich and have vacancies on the Ge/Si sites [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Furthermore, current literature seems to suggest the Ge variant is more prone to off-stoichiometry as compared to their Si analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' For example, CeAlSi crystals grown by the flux method show little deviation from a 1:1:1 stoichiometric ratio [21];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' however, for CeAlGe, the crystals grown by the flux method are predominantly Al-rich and have significant vacancies on the Ge sites, while those synthesized with a floating-zone growth show a stoichiometric ratio close to 1:1:1 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Considering RAlGe grown by flux methods are prone to be Al-rich, we have performed EDX measurements to confirm the stoichiometry in our NdAlX crystals grown by flux method (Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Our measurements show that NdAlSi crystals are close to a 1:1:1 stoichiometry, while NdAlGe crystals are predominantly Al-rich and show larger variations in the stoichiometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We also refined the atomic occupancy for both materials by Rietveld refinement, and the results show a similar trend (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Our characterizations are consistent and suggest that, compared to NdAlSi, the flux-grown NdAlGe crystals show variation in their atomic compositions and have a higher level of disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Resistivity, magnetic susceptibility, and heat capacity Figure 2(a,b) shows a typical resistivity (ρxx) curve obtained for our NdAlGe crystals at T = 2 − 300 K and T = 2 − 10 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' From the magnitude of ρxx at T = 300 K and T = 2 K, we calculate the residual resistivity ratio RRR = ρxx(300 K) ρxx(2 K) to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='9, much lower Occupancy NdAIGe Al Ge Nd 8 Counts) T= 295K NdAIGe Cu Kα I41md obs Intensity (2 × 10° obs 20 40 60 80 100 20 (degrees)NdAlGe 1 41 mdobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' calc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' calcc b a4 than the RRR of NdAlSi, which is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='0 (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' A lower RRR usually suggests the disorder level is higher so that the resistivity is anchored at a higher value near zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We attribute the lower RRR for NdAlGe to its off-stoichiometry characterized in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We measured the magnetic susceptibility of NdAlGe (χ) with a magnetic field applied along the c-axis (χc) and the a-axis (χa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The ratio χc/χa is plotted as a solid line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2(c) (left y-axis), while 1/χc is plotted as squares in the same plot (right y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We also plotted the temperature dependence of χc in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' At high temperatures, the magnetic susceptibility shows isotropic paramagnetic spins with a Curie-Weiss temperature of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='0(7) K and an effective magnetic moment of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5(2) µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Upon cooling, similar to NdAlSi [26], an out-of-plane Ising anisotropy gradually develops such that the ratio χc/χa reaches as high as 80 at T = 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The Ising anisotropy is also visible in the low-temperature in-field magnetization of NdAlGe (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2(c)) where the magnetization along the c-axis reaches saturation near 3 T at a value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8(1)µB/Nd, while the magnetization along the a-axis is still unsaturated and weak at 6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The heat capacity (Cp) of NdAlGe is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2(e,f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The main panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2(e) shows the mag- netic contribution of the heat capacity Cmag p of NdAlGe, which was obtained by subtracting the heat capacity of the non-magnetic analogue compound LaAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' As expected, the Cp attains the Dulong-Petit limit (Cp = 3R × Nions where R is the gas constant and Nions is the number of ions in the material and equals to 3 in NdAlGe) near the room temperature (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Upon cooling, a Schottky-like anomaly centered at 18(1) K is visible in both Cp and Cmag p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This high-temperature anomaly can be reproduced with an Nd3+ single-ion energy scheme comprised of a doublet ground state separated by 3 ex- cited doublets between 3 to 9 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' At lower temperatures, Cmag p shows two additional anomalies at Tcom = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1(1) K and Tinc = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8(2) K, which originate from the collective magnetism of the Nd3+ moments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The pres- ence of two anomalies is also present in NdAlSi where the phase transition at Tinc signifies the onset of an incom- mensurate modulated spin density wave that transitions into a commensurate ferrimagnetic state below Tcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The incommensurate and commensurate magnetic phase transitions in NdAlSi were both observed to impact its electric transport and bulk thermodynamic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' More specifically, the onset of the commensurate order TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' EDX measurements of NdAlSi and NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The occupancy was normalized by that of Al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The uncertainty was defined by the standard deviation of all measurements, which were taken from 2-3 different regions of several crystals for each material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' NdAlSi Occupancy NdAlGe Occupancy Nd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='01 Nd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='04 Al 1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='01 Al 1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='07 Si 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='01 Ge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='04 NdAlGe Tinc Tcom Tinc Tcom Tinc Tcom RRR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='90 (a) (b) (c) (e) (f) (d) FC ZFC Sample T7 0 2 4 6 H (T) 0 1 2 3 M (𝜇B/Nd) Mc Ma 0 100 200 300 T (K) 0 25 50 75 Cp FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (a,b) Resistivity as a function of temperature, plotted from 0 to 300 K and 0 to 10 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Tcom = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 K and Tinc = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8 K respectively indicates the transition temperature of the commensurate and incommensurate orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (c) The ratio of magnetic susceptibility, which was measured with field H = 100 Oe along the c-axis (χc), to that measured with the same field along the a-axis (χa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Both χc and χa are measured after cooling down to 2 K in zero magnetic field (ZFC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 1/χc is also plotted in the same panel, and the black line shows the result of a Curie-Weiss fit to the data above 150 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The inset shows the magnetization of NdAlGe measured at T = 2 K with the magnetic field applied along the c-axis (Mc) and a-axis (Ma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (d) χc measured while the sample is cooled under field H = 100 Oe (FC) and χc measured under ZFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (e) Temperature dependence of the magnetic heat capacity Cmag p of NdAlGe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' the inset shows the total heat capacity Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Cmag p was obtained by subtracting the heat capacity of LaAlGe from the heat capacity of NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The dashed black line in the main panel is the predicted ”Schottky-like” Cmag p anomaly calculated assuming that the (2J +1) spin-orbit levels of Nd3+ (J = 7/2) are split by crystal electric field (CEF) effects into a doublet ground state, 2 excited doublets at 4 meV, and another excited doublet at 9 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (f) The total heat capacity below T = 10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' in NdAlSi can be seen as discontinuity occurring at Tcom in ρxx(T), χ(T), and Cp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' For the incommensurate order, however, it is less obvious, but NdAlSi shows a drop in ρxx(T), an arguable change of slope in χ(T), and a clear peak in Cp(T) at Tinc [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' For comparison, we also looked for similar effects in NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2(b,d,f), we again see clear features at Tcom as a drop in ρxx, a split of FC and ZFC data in χc, and a peak in Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Anomalies associated with the incommensurate order of NdAlGe are 5 still subtle in ρxx(T) and χ(T), where a mild upturn and a mild change of slope are observed at Tinc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In the heat capacity data, however, there is a peak that starts at Tinc and one can argue the presence of two transi- tions in Cp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The low temperature Cp peaks of NdAlGe are broader than the ones observed for NdAlSi, which have sharp discontinuities occurring exactly at Tcom and Tinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This observation suggests a similar magnetic phase diagram for both NdAlSi and NdAlGe, but with more disorder in NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Neutron diffraction To gain insights into the collective magnetism of NdAlGe, we have performed single-crystal neutron diffrac- tion to determine its temperature-dependent spin struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Below Tinc, we found incommensurate magnetism in NdAlGe that is characterized by strong magnetic Bragg peaks (Qmag) indexed with an ordering vector kAFM1 = (2/3 + δ(T), 2/3 + δ(T), 0), as well as weaker Bragg peaks indexed with kAFM2 = (1/3 − δ(T), 1/3 − δ(T), 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(a), we determined the incommen- surability δ(T) by tracking the temperature dependence of the Qmag = (2/3 + δ(T), 2/3 + δ(T), 0) peak center observed in an (HH0) scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The temperature depen- dence of δ(T) is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(b) where a mild change of δ(T) between Tcom < T < Tinc is observed, but a transition to commensurate magnetism (δ = 0) arises for T < Tcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' For comparison, we note that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(b) also includes data from our SANS analysis, which will be presented in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The order parameter of the Qmag = (2/3 + δ(T), 2/3 + δ(T), 0) peak (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(c)) correlates with Tinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In addition to the antiferromagnetic kAFM1 and kAFM2, we also observed ferromagnetism in NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' To prove this, we acquired an order parameter at Q = (200) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(c)), which shows it onsets slightly above Tcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In the next section, we will see that ferromagnetism actu- ally onsets exactly at Tcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The fact that we see magnetic intensity at Q = (200) above Tcom is from the onset of an incommensurate k = (δFM(T), δFM(T), 0) wave that is a precursor to the δFM(T) = 0 ferromagnetism compo- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Our neutron diffraction experiment simply could not resolve this incommensurability, but we could do so using small-angle neutron scattering presented in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The commensurate magnetic phase of NdAlGe is thus described by a multi-k spin structure including two differ- ent antiferromagnetic components (kAFM1 and kAFM2), as well as a ferromagnetic component (k = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The diffraction pattern of NdAlGe is practically identical to the one observed in NdAlSi [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The possible magnetic basis vectors describing this spin structure were obtained by symmetry analysis and consist of the xyz components of a SDW propagating along the [110] or [1¯10] directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The SDW can either have parallel or anti-parallel spins sitting on the primitive Nd3+ sites at r1 = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='NdAlGe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='(d) ' metadata={'source': 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at various temperatures between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 K to 8 K centered around Q = (2/3, 2/3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (b) The tem- perature dependence of the (δ(T), δ(T), 0) incommensurability of the kAFM1 = (2/3, 2/3, 0) SDW is plotted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The green curve shows 1/3 of the (δFM(T), δFM(T), 0) incommensurabil- ity of the ferromagnetic component whereas the blue curve is the total SANS collected using the 12 ˚A data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (c) The order parameter of Q = (200) and Q = (2/3 + δ(T), 2/3 + δ(T), 0) Bragg peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (d) The main panel shows the rocking scans at Q = (2/3, 2/3, 0) and Q = (1/3, 1/3, 0) for T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We note that the source of the elevated background for the Q = (2/3, 2/3, 0) rocking scan comes from proximity to an Al Bragg peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The inset of panel (d) shows the Q = (0, 0, 4) rocking scans collected for both T = 10 K and T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (f) and (f) show the refined spin structure for the magnetic commensurate phase of NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Red (blue) arrows are used to represent the up (down) spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The tilting of the spins within the ab plane was amplified by a factor of 4 to allow for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' and r2 = (1/2,0,1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Anti-parallel (parallel) spin components produce scattering at Bragg peaks indexed by the magnetic ordering vector kAFM1 = (2/3, 2/3, 0) (kAFM2 = (1/3, 1/3, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' As seen in the main panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(d), the Bragg peaks with kAFM1 = (2/3, 2/3, 0) have almost two orders of magnitude greater intensi- ties than the kAFM2 = (1/3, 1/3, 0) ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This indicates a dominant anti-parallel spin component for the SDW, which is augmented by a weak parallel one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The anti- T T Inc : u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='6 Int 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='4 Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Q=(200) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='2 Q=(2/3+8,2/3+8,0): 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='2 2 3 4 5 6 7 T(K)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8K 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='4 K Intensity (cts/s 40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='6 K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='0 K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='4 K 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8 K 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='2 K 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='0K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='69 (HHO)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='02 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 0 (8,8,0) (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='u) AEM M,0)/3 d FM FM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' O SANS Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='01 Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='( 二 6 8 T(K)c ac a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 K 10K Q = (2/3,2/3,0) 10 (004) Q= (1/3,1/3,0) (cts/s 5 104 0 Intensity 102 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 Φ(°)6 parallel spin component was refined to an Ising one, while the parallel component to a weak in-plane spin canting that is transverse to the propagation of the SDW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The spin structure refinement of NdAlGe is shown in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The k = 0 ferromagnetic part of the spin structure was refined to a c-axis magnetized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This is due to the fact that we did not observe magnetic Bragg in- tensity at nuclear-allowed Q = (0, 0, L) Bragg positions (see top right inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(d)), while we observed mag- netic scattering at nuclear-allowed Bragg positions such as Q = (2, 0, 0) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The magnetization was refined to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='9(1)µB/Nd, which is consistent with the value of the low-field magnetization plateau reported in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Finally, adding all spin components together, the spin structure of NdAlGe is an Ising down-up-up (duu) fer- rimagnetic SDW propagating along the [110] or [1¯10] direction that is augmented by a weak in-plane chiral component lying transverse to its propagation (see sketch of the spin structures in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(e,f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Mostly pointing along the c-axis, the moment on each Nd sites was re- fined to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='0(2)µb with an in-plane tilting angle of 3(1)°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The high temperature incommensurate spin structure is similar to the commensurate one, but convoluted with an amplitude-modulated wave that has a spatial wavelength of ∼ 35 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The resulting spin structure of NdAlGe is practically identical to the NdAlSi one, but we note that the incommensurate to commensurate phase transition in NdAlGe is much broader in temperatures than the one in NdAlSi [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' For example, the 5 K (HH0) scan presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(a) shows the presence of both a commensurate and an incommensurate peak, which signifies inhomogene- ity within the NdAlGe crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This is consistent with a range of different critical temperatures Tcom coexisting within the same crystal of NdAlGe, which likely arises from a variation of the stoichiometry across the whole sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Such a conclusion corroborates the fact that NdAlGe has more disorder than NdAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Small-Angle neutron scattering (SANS) So far, we have shown that the details of the magnetism of NdAlGe is impacted by the disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In order to characterize this further, we have performed a SANS experiment to probe its magnetized inhomogeneities on a spatial length scale of ∼ 1 to 500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We first collected field and temperature-dependent SANS data with the c-axis parallel to the incident neu- tron beam so we could probe the in-plane scattering vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Representative data acquired with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='75 ˚A in- cident neutrons within the paramagnetic, incommensu- rate, and commensurate phase of NdAlGe are respec- tively shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 4(a,b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' As expected, no coher- ent magnetic scattering is detected in the paramagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In the incommensurate phase, Bragg peaks at symmetry-related Q = (δFM(T), δFM(T), 0) positions are observed corresponding to an SDW with a spatial modu- lation of 116(7) ˚A at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This incommensurate SDW could not be resolved in our neutron diffraction exper- iment and is a precursor to the commensurate ferro- magnetic spin component of NdAlGe occurring below Tcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The temperature dependence of the incommen- surability δFM(T) is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' δFM(T) fol- lows a δ(T) = δFM(T)/3 relationship indicating that the kFM = (δFM(T), δFM(T), 0) SDW is the third harmonic of the main kAFM1 = (2/3+δ(T), 2/3+δ(T), 0) wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This is typical of incommensurate magnetism in rare-earth metal- lic systems where odd harmonics emerge from ”squaring- up” of the main wave, which is expected upon cooling as the magnetization becomes more constant through the lattice [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Within the commensurate phase, the incommensurate Bragg peaks disappear such that the SANS of NdAlGe is now centered at |Q| = 0 and is shaped like a cross extending along the <110> directions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This is different from the isotropic in-plane SANS pattern ob- served in the isostructural Weyl semimetal PrAlGe [19], and we will argue that the in-plane anisotropic cross pat- tern of NdAlGe arises from the finite size of the magnetic domains forming the multi-domains state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Using a field of 2 T, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 4(d) shows that field-cooling (FC) NdAlGe within its commensurate magnetic state significantly de- pletes the cross pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' To probe the temperature dependence of the cross pat- tern, we collected SANS data with 12 ˚A incident neutrons, which exclude the magnetic incommensurate Bragg peaks such that the SANS scattering from the cross pattern can be isolated (see appendix F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The temperature de- pendence of the cross pattern extracted this way shows that it onsets below Tcom (blue circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(b)) and is indeed a feature of the commensurate order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We then studied the field evolution of the cross pattern by acquiring 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='75 ˚A SANS data for various in-plane and out-of-plane fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' As reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 4(e), an out-of- plane field of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='30(5) T is enough to completely suppress the SANS intensity associated with the cross pattern, whereas a similar in-plane field strength does not significantly affect the scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This is consistent with the Ising anisotropy of the bulk magnetization (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2(c)) and shows that the cross is only observed when the time-reversal symmetric domains coexist (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' only when both up-down-down (udd) and duu domains are present).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We found that the SANS Q = 0 cross pattern could be modeled using an anisotropic Lorentzian-squared function of the form S(Q) = A ((ϵ∥Q∥)2+(ϵ⊥Q⊥)2+1)2 , which is often used to phenomenologically describe the SANS of inhomo- geneous magnetized systems such as spin glasses [42–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In this expression, A is a scale factor, while ϵ∥ and ϵ⊥ are the spatial correlation lengths of the ferrimagnetic domains parallel and perpendicular to the magnetic order- ing vector direction [110], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The momentum transfer Q is also expressed into a component that is either parallel (Q∥) or perpendicular (Q⊥) to the order- 7 Para.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' T= 8 K ZFC Q(HH0) Q(H-H0) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0 2 SANS Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' units) (a) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='75 Å Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' T= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='4 K ZFC Q(HH0) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0 2 SANS Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' units) (b) Q(H-H0) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='75 Å Com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' T= 2 K ZFC Q(HH0) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0 2 SANS Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' units) (c) Q(H-H0) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='75 Å Q[H,H,0] FC Com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' T= 2 K Q(HH0) Q(H-H0) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0 2 SANS Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 H||c (d) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='75 Å H(T) 0 1 (e) Q(HH0) T= 2 K (f) Theory Q(HH0) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0 2 SANS Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' units) (g) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 Q(H-H0) Single Domain (FC) Stripe Domains (ZFC) (h) ε[110] = 18(5) nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 T= 2 K 2 0 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Panels (a),(b) and (c) correspond to the zero-field cooled (ZFC) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='75 ˚A SANS data respectively collected within the paramagnetic state (8 K), the incommensurate phase (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='4 K), and the commensurate phase (2 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Panel (d) is the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='75 ˚A SANS data collected within the commensurate phase (2 K) using a field-cooled (FC) protocol (2 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Panel (e) shows the total SANS intensity observed as a function of both an in-plane (blue) and out-of-plane (green) magnetic field using 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='75 ˚A neutrons at T = 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Panel (f) shows the T = 2 K total SANS scattering intensity as a function of the momentum transfer Q measured along the [H,H,0] direction for both ZFC and FC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This plot includes the SANS data collected within both the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='75 ˚A and 12 ˚A configurations as well as their appropriate fit to a Lorentzian-squared function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Panel (g) is the calculated SANS pattern assuming an anisotropic Lorentzian-squared function with ϵ∥ = 18 nm and ϵ⊥ = 72 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Panel (h) shows a sketch of the 1D representation of the commensurate spin structure of NdAlGe (single domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This 1D representation is also used to represent the stripe ferrimagnetic domains observed via SANS in a ZFC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' ing vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We note that there’s also the presence of magnetic domains with ordering vector propagation along the [1¯10] direction so we included a Lorentzian-squared function where ϵ∥ and ϵ⊥ are swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The combination of two Lorentzian-squared functions that represent the SDW along [110] and [1¯10] accounts for the two branches of the cross pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The scattering function S(Q) was then convoluted to the 2D resolution function ellipsoid of our SANS instrument and fitted to the 2D zero field SANS data of NdAlGe acquired at T = 2 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Following this procedure, we obtained ϵ∥ = 18(5) nm and ϵ⊥ = 72(8) nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' A quantitative comparison between the fit and the data is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 4(f) for the momentum transfer along the [H,H,0] direction, while the resulting 2D fit is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 4(g) and can be compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Our result indicates the SANS observed within the commensurate phase of NdAlGe originates from fer- rimagnetic stripes domains that have a shorter spatial length scale parallel to the SDW and a longer one perpen- dicular to the SDW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The ferrimagnetic stripes domains of NdAlGe are sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 4(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The anisotropic shape of the magnetic domains in NdAlGe may be a consequence of anisotropic exchange interactions, dipolar interactions, or Dzyaloshinskii-Moriya interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We associate the origin of the magnetic stripes in NdAlGe to the finite sizes of its bulk domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This scenario is consistent with the observed field dependence of the Q = 0 cross pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Indeed, contrary to an in- plane field, a field parallel to the c-axis promotes one time-reversal domain over the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In this case, a multi- domain sample is not preferred and the spatial dimensions of the energetically favoured domain then diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The same phenomenology explains the longer length scale ob- served in the low-temperature FC SANS data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 4(d)), which is not expected if the cross pattern originates purely from domain wall scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' A fit to the FC SANS data against the Lorentzian-squared scattering function (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 4(f)) shows that ϵ∥ = ϵ⊥ = 270(30) nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Anomalous Hall effect We now turn to the anomalous Hall effect (AHE) of NdAlGe, and show that its duu and FM states host dif- ferent types of AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 5(a) shows the field dependence of the electrical resistivity ρxx(H) of NdAlGe measured below the transition temperature Tinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' ρxx(H) curves taken at opposite field-sweeping directions show a mild SANS Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' units) T=2K H 「110l(In-plane) H l[0011 (Out-of-plane) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 2 H(T)[110] k b u-d-d va Cd-u-u u-d-d 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='.1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' d-u-u : 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='o-1-1(arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' units SANS Intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='01 ZFC = 18(5) nm [110] FC 10- = 270(30) nm [110] 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 IQI [H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='H,0十SANS Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' units 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='01 ZFC 8[110] = 18(5) nm FC 10- {110j = 270(30) nm 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 IQI [H,H,0]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='18 (a) (b) (c) (d) 𝜌!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='" #,%&& 𝜌!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='" #,\'( 𝜎"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' #,%&& 𝜎"!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=" #,'( 2 K 3 K 4 K 5 K 6 K 7 K H ∥ c H ∥ c Sample T7 2 K 7 K FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (a) Resistivity ρxx of the sample T7 as a function of external magnetic field H below Tinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The current is applied along the a-axis (x) and the field is along the c-axis (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' For each temperature, the data represented by a solid line are measured while the field is swept from H = 6 T to H = −6 T, while the dashed line is recorded in the opposite field- sweeping direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The same convention applies to panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (b) Hall resistivity ρyx(H) of the sample T7 collected at the same temperatures as in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The data taken at each temperature were antisymmetrized and shifted by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 µΩ cm from each other for visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The anomalous part of the Hall resistivity ρA,duu yx and ρA,FM yx are extracted from the y-intercept of a linear line fitted to the plateaus of duu (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='2 T < H < 1 T) and FM (H > 4 T) states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (c) Anomalous Hall conductivity (AHC) as a function of temperature in the duu (σA,duu xy , solid lines) and FM state (σA,FM xy , dashed lines) of seven samples (shown as different symbols and colors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The data of sample T7 are plotted with black circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (d) Normalized AHC plotted as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The light gray and dark gray stripes correspond to the re-scaled magnetization in the duu state (Mduu) and FM state (MFM), which are extracted from the y-intercept of a linear line fitted to the plateaus of duu (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='2 T < H < 1 T) and FM (H > 4 T) states, respectively (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' hysteresis below H∗ ≃ 3 T, which is the transition field from the duu to the FM state (see Mc(H) data in top in- set of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The hysteresis starts from T = Tcom and persists as the temperature decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Another feature related to the transition field H∗ is the local maximum of ρxx(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' At T = 2 K, ρxx(H) first increases with the field in the duu state, peaks at H∗, and then starts to de- crease as the system is going through a smooth transition from the duu to the FM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Then, ρxx(H) reaches a minimum at the end of the smooth transition, and finally starts to increase again when it is in the FM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Such a non-monotonic behavior of ρxx(H) can also be seen in some of the half-Heusler compounds such as DyPtBi, which shows multiple field-induced phase transitions and has relatively low mobility (< 1000 cm2V−1s−1) [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We note that the local maximum at H = H∗ persists above Tcom and Tinc where the transition between duu and FM states no longer exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' such non-monotonic magnetore- sistance above TC has also been reported in DyPtBi and other half-Heusler compounds [47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We may quali- tatively understand the behavior of ρxx(H) in terms of the two-current model [50, 51], which suggests that the resistivity in the duu state (ρduu) is larger than within the FM state (ρFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Assuming that both an up spin (ρ↑) and a down spin (ρ↓) contribute to the current in parallel, and also that ρ↑ ≫ ρ↓, we may then express ρFM = (1/ρ↑ + 1/ρ↓)−1 ∼ ρ↓, which is a relatively low value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In the duu state, since the up and down spins ad- mix as the spin wave propagates, both ρ↑ and ρ↓ approach an averaged value of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' As a result, the resistivity in the duu state ρduu has a significant contribution from ρ↑ and is thus larger than ρFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 5(b) shows the Hall resistivity ρyx(H) measured at T < Tinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' At T = 2 K, there are two plateaus in ρyx(H), both of which correlate with the magnetization plateaus observed in the duu and FM states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' By fitting each plateau to a linear line, we extracted the anomalous part of ρyx(H) in the duu state (ρA,duu yx ) and FM state (ρA,FM yx ) using the y-intercept of their respective fitting line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' ρA,duu yx was extracted only for T ≤ Tcom since beyond that temperature there is no duu state, but the spins are still polarized at high fields and high temperatures so ρA,FM yx was calculated up to T ∼ Tinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' From the information extracted from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 5(a,b), we calculated the anomalous Hall conductivity (AHC) in the duu state (σA,duu xy ) and the FM state (σA,FM xy ) at each temperature using ρ0 ≡ ρxx(H = 0) and ρA yx as σA xy = ρA yx (ρA yx)2+ρ2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The results are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 5(c) for seven different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Each sample is uniquely represented by a specific color and symbol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' for example, the data of the sample T7 is plotted with black circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The solid line is used for σA,duu xy and the dashed line is for σA,FM xy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' At first glance, the data are all over the place and it seems difficult to draw a clear conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' However, since the resistivity is calculated from the resistance that depends on the geometric factors of each sample, the uncertainty in these sample-dependent factors may have contributed to the “randomness” of the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' To eliminate the trivial effect of geometric factors and extract the intrinsic properties of NdAlGe, we divided both σA,duu xy and σA,FM xy of each sample by its own σA,FM xy measured at 2 K (σA,FM xy (2K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Assum- ing ρxx ≫ ρyx [52], we show that the normalized AHC σA xy(T)/σA,FM xy (2K) is free of geometric factors (the super- scripts are omitted below for simplicity): σxy(T)/σxy(2K) = ρyx(T ) ρ2yx(T )+ρ2xx(T ) ρyx(2K) ρ2 yx(2K)+ρ2 xx(2K) ≃ ρyx(T ) ρ2xx(T ) ρyx(2K) ρ2 xx(2K) = ρyx(T) ρyx(2K) ρ2 xx(2K) ρ2xx(T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Since geometric factors do not depend on T, the geomet- 9 ric factor of ρyx (involving sample thickness) and ρxx (sample length, width, and thickness) are all eliminated in normalized AHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We plotted the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 5(d) and found interesting characteristics of σA,duu xy and σA,FM xy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 5(d), for σA,duu xy , the normalized AHC of different samples all collapsed onto a single curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In addition, as T decreases, the normalized σA,duu xy scales with the magnetization of the duu state (Mduu, light gray stripe) such that it saturates at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The convergence of the data from different samples, the scaling between σA,duu xy and Mduu, and the saturation of AHC at low T are strong evidence for an intrinsic AHE [8, 20, 53–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' However, in sharp contrast to σA,duu xy , the normalized σA,FM xy of different samples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 5(d) do not collapse but diverge into several curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Indeed, as T decreases, the normalized AHC does not follow the magnetization of the FM state (MFM, dark gray stripe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Since the NdAlGe crystals grown by flux method are prone to Ge vacancies, we expect that extrinsic disorders vary in each crystal and govern the variance in σA,FM xy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' When the different disorder levels among all samples are taken into account, the convergence of σA,duu xy among them becomes quite nontrivial, and strongly suggests a robust intrinsic contribution to the AHE due to Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The clear distinction between σA,duu xy and σA,FM xy marks two regimes of AHE in NdAlGe: an intrinsic AHE in the duu state and an extrinsic AHE in the FM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' DISCUSSION To better understand and interpret the magnetism and AHE in NdAlGe, we calculate band structure, Fermi surface, Weyl nodes, and AHC due to Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 6(a) shows the Brillouin zone and high-symmetry k- points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' a k-path along these high-symmetry k-points was selected to plot the band structure of NdAlGe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' At the first glance, the band structure of NdAlGe does not look much different from that of NdAlSi [26], but the similarities and differences are more visible when we look at the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 6(c), we can see the butterfly-shaped hole pockets along the Γ − X k-path, similar to the ones in NdAlSi near Q = (± 1 3, ± 1 3, l) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' These pockets fulfill the nesting condition for the incom- mensurate magnetic order to appear (see kAFM1 vector in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 6(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Besides, when looking at both Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 6(c) and (e) together, we find that the nesting butterfly-shaped hole pockets are also Weyl-like and Weyl nodes near dif- ferent pockets are separated by the nesting wave vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The inter-node scatterings between these Weyl nodes can provide the Weyl-mediated RKKY interactions and ac- count for the chiral component in the duu ferrimagnetic order (helical magnetism) [14, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The similarities be- tween NdAlSi and NdAlGe in the nesting Fermi pockets and the distribution of Weyl nodes provide a reasonable explanation for their similar magnetic orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' On the other hand, there are differences in the Fermi 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 7 Energy(eV) 400 -200 0 200 400 G S N S Z G X (a) (b) (c) (d) (e) (f) 𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (102 Ω-1cm-1) Energy (eV) 0 2 4 4 2 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 7 kAFM1 kAFM1 h+ e- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (a) Brillouin zone of NdAlGe and high-symmetry k-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (b) Band structure of NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The dashed line marks the Fermi level calculated by DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (c) Fermi surfaces of NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The blue pockets are electron pockets, while the red ones represent hole pockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (d) Side view of the Fermi surfaces of NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (e) The distribution of 56 Weyl nodes in the Brillouin zone found for the FM state of NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (f) Anomalous Hall conductivity (AHC), calculated for different energy relative to the Fermi level (marked by the dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' surfaces of NdAlSi and NdAlGe that distinguish their transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The most pronounced difference lies in the diminished electron pockets in NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In NdAlSi, in addition to the elongated electron pockets along the Z − Σ1 k-path at high kz, an octagon-like network of electron pockets that extend to lower kz and connect the elongated pockets is also present [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' However, in NdAlGe, this network of electron pockets is diminished and only the elongated electron pockets at high kz remain (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 6(c,d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Without this network, not only the number of electron carriers are reduced, but also the momentum dispersion of electrons is limited to a narrower range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' both factors may explain the dominant role of hole carriers in NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 6(f), we report the AHC contributed by intrinsic Berry curvature at different energies [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' At the Fermi level determined by our DFT calculations (indicated by the dashed line), σA xy ≃ 200 Ω−1cm−1 agrees with both the sign and the order of magnitude of σA,duu xy , which is ≃ 400 Ω−1cm−1 for sample T7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Although the calculation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 6(f) is done in the FM state, we expect it to re- flect the AHC in the duu state because of the similarity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 ev 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='0 E-Ef 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 N> Z XZ N X y T 30 xy yz 0 Zx10 in the net moment along z in both states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' For a more quantitative comparison, additional scaling analysis is required to determine the intrinsic AHC in σA,duu xy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We tried to perform the scaling analysis proposed by Tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' [58];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' although the data points do follow the scaling (σA,duu xy ∝ σ2 xx), the extracted intrinsic AHC seems unrea- sonably large, likely due to the narrow temperature range of the fitting, which is limited by Tcom [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' However, we argue that σA,duu xy should be dominated by intrinsic Berry curvature because 1) the collapse of data taken from samples of different disorders [20, 58], 2) the linear dependence of σA,duu xy on Mduu [60], 3) the saturation of AHC at low temperatures [54, 56], 4) the conductivity σxx falls in the regime where intrinsic AHE usually domi- nates [53, 54], and 5) the reasonable agreement between σA,duu xy and σxy calculated by DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Previously, the transition from intrinsic to extrinsic AHE in RAlX family was only observed among materi- als of different chemical compositions, and it was mainly driven by enhanced disorders [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Here, we argue that spin fluctuations may play a key role in such a transi- tion in NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In ferromagnets, it has been proposed that carriers scattering in a fluctuating spin background may lead to a chirality-driven AHE [60, 61], and a devi- ation of AHC from its scaling with M was shown to be a manifestation of such behavior in experiments [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In NdAlGe, σA,FM xy also deviates from MFM as T increases (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 5(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Besides, the magnetic fluctuations in the FM state seem to be strong, as suggested by the slow and smooth transition from the duu to the FM state, instead of the sharp and steep one in NdAlSi [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' As a result, we infer that the enhanced spin fluctuations as the magnetic field increases, which possibly intensified with disorders, may be the key factor driving the transition from intrinsic to extrinsic AHE in NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' A complete description of σA,FM xy , however, could be quite complicated and would require a combination of inherent Berry cur- vature from band structure, chirality-driven AHE due to spin fluctuations, and extrinsic disorders through skew scatterings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' CONCLUSION In conclusion, we report incommensurate magnetism in NdAlGe that onsets at Tinc = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8(2) K and consists of an Ising modulated SDW with a small helical chiral spin cant- ing of 3(1)°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The spin system transitions into a commen- surate ferrimagnetic state below Tcom = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1(1) K where the spins form a duu Ising spin structure while keeping the helical spin canting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Similar to NdAlSi, we found that the periodicity of the incommensurate SDW of NdAlGe matches the nesting wave vector between the two of its topologically non-trivial Fermi pockets, which confirms the possibility that Weyl-mediated RKKY interactions could also drive the collective magnetism of NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In contrast to NdAlSi, however, NdAlGe has a higher level of disorders, which has a minor effect on its magnetic properties but greatly modifies the transport ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Ef- fects of disorders in NdAlGe are manifested through the anisotropic ferrimagnetic domains of finite size as well as broad features in the temperature dependence of its magnetic heat capacity, magnetic order parameters, and electrical resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In terms of transport, the carrier concentration, mobility, and AHE of NdAlGe are all dras- tically different from those of NdAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In particular, we characterized an intrinsic as well as an extrinsic AHE regime in NdAlGe that are both absent in the Si analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In NdAlGe, we argued that the intrinsic AHE results mainly from its intrinsic Berry curvature, while the ex- trinsic AHE is tied to disorders and spin fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The lack of AHE in NdAlSi may be due to differences in the strength of spin-orbit coupling between Ge 4p and Si 3p electrons and an interplay between different mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Our work thus suggests that Weyl-mediated magnetism is a robust feature of non-centrosymmetric Weyl semimetals NdAlX, while the transport properties including AHE in Weyl semimetals can be strongly impacted by extrinsic effects despite the presence of prominent Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' ACKNOWLEDGMENTS H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' thanks Chunli Huang, Hiroaki Ishizuka, Christopher Eckberg, Allan MacDonald, Inti Sodemann, Yaroslav Tserkovnyak, and Collin Broholm for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This material is based upon work supported by the Air Force Office of Scientific Research under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' FA2386-21-1-4059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The work at TIFR Mumbai was supported by the Department of Atomic Energy of the government of India under Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 12- R&D-TFR- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='10-0100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We acknowledge the support of the National In- stitute of Standards and Technology, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Department of Commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The work at Northeastern University was sup- ported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' DE-SC0022216 and benefited from Northeastern University’s Advanced Scientific Computation Center and the Discovery Cluster and the National Energy Research Scientific Computing Center through DOE Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' DE-AC02-05CH11231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The identification of any commercial product or trade name does not imply endorsement or recommendation by the National Institute of Standards and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' A portion of this research used resources at the High Flux Isotope Reactor, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Appendix A: Crystal structure refinement of NdAlGe In addition to the powder XRD refinement shown in the main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 1 (NdAlGe made by 10 Al recipe with a refined ratio 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='01:1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='97), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 7 shows the refinement of NdAlSi made by 10 Al recipe and NdAlGe made by a 15 Al recipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The refined ratio for NdAlSi is essentially 1:1:1, 11 �� �� �� �� ��� �� � � � Intensity ��� � ������� 1G��6� I 41 m d � � ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='� ����� ������ ����� � ������ �� �� �� �� ��� 2Θ �G������� �� � � � Intensity ��� � ������� 1G���� ��� ��� I 41 m d ����� ������ ����� � ������ (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Powder XRD refinement of (a) NdAlSi, and (b) NdAlGe samples made with additional Al flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' and for NdAlGe (15 Al) is Nd : Al : Ge = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='90 : 1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We note that it is known from neutron scattering that the stoichiometry of NdAlSi single crystals is not exactly 1:1:1 [26], and we interpret our powder XRD refinement results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 7 as follows: relatively speaking, NdAlGe is more non-stoichiometric compared to NdAlSi, and using more Al flux to grow NdAlGe single crystals may result in a higher deficiency in the Nd and Ge sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (a) (b) T = 2 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (a) ρxx(T), and (b) ρyx(H) of NdAlGe (blue dashed lines, left y-axis) and NdAlSi (red solid line, right y-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The high-field part of the data of both materials is fitted to a linear expression (black line in panel (b)) to extract single-band carrier concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (a) (b) 2 K 3 K 4 K 5 K 6 K 7 K 2 K 7 K NdAlGe NdAlGe FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (a) M(H) data of NdAlGe recorded at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The data at each temperature are vertically shifted away from each other by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='2 µB/Nd for visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (b) The same data as in panel (a), but not shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Appendix B: Comparison between the transport properties of NdAlSi and NdAlGe Figure 8 compares ρxx(T) and ρyx(H) of NdAlGe and NdAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 8(a), it can be seen that at base temperature T = 2 K, ρxx of NdAlGe is much closer to its room-temperature value compared to that of NdAlSi, hence the higher RRR reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Figure 8(b) shows typical ρyx(H) curves for both materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' For NdAlGe, two plateaus corresponding to the duu and FM magnetic phases can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' For NdAlSi, however, although there are also two steps in its magnetization just like NdAlGe [26], its ρyx(H) curve is smooth for the first transition and only shows a small discontinuity at the transition field to the FM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The ρyx(H) curve is also mildly nonlinear overall such that it is difficult to argue an AHE in NdAlSi (see also Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We obtained the single-band carrier concentrations by fitting a linear line to the high-field part of ρyx(H) curves for both materials, and used them to calculate their respective single-band mobility from ρxx at 2 K (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Appendix C: M(H) of NdAlGe at different temperatures Figure 9 shows the details of M(H) data of NdAlGe at different temperatures below Tcom and Tinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The plateaus in the duu and FM states are evident and correspond to the plateaus in ρyx(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' At high magnetic field, the magnetization converges to the saturated value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The decent saturation of magnetization even at T = 7 K allows us to extract ρA, FM yx from the high-field plateaus in ρyx(H) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We note that the transition field from duu to FM state is not always the same and varies among samples, while the plateaus are persistent before and after the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 12 (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8 K 3 K 4 K 5 K 6 K 7 K NdAlSi NdAlSi 8 K 7 K 8 K 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8 K 3 K4 K 5 K 6 K 𝜎!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (Ω-1cm-1) 𝐸 − 𝐸" (eV) Γ Σ N Σ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Z Σ X (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (a) ρyx(H) data of NdAlSi recorded at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (b) M(H) data of NdAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' (c) Left panel: Band structure of NdAlSi in FM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Right panel: Anomalous Hall conductivity of NdAlSi, calculated by DFT considering intrinsic Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Appendix D: ρyx(H) and M(H) of NdAlSi at different temperatures Figure 10(a) and (b) compare ρyx(H) and M(H) data of NdAlSi side by side to reveal the absence of Hall resis- tivity plateaus despite clear plateaus in M(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Similar to NdAlGe, M(H) of NdAlSi has a low-field transition to duu state and a high-field transition to FM state, each of which results in a plateau in M(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In the Hall data, however, near H = 0 T, ρyx(H) is smooth and featureless;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' no feature can be associated with the transition to duu state in ρyx(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' At the transition to FM state, there is a concomitant jump in ρyx(H) as shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 10(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' It is tempting to subtract a smooth back- ground from ρyx(H) and interpret such a jump as AHE, but this is not feasible here since the ρyx data in the FM state (H > 7 T) below Tcom actually lie on top of the ρyx data above Tcom in the same field range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We note that M significantly drops (more than 30%) as T changes from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8 K to 8 K, so if there is a finite AHE, the Hall data recorded at these two temperatures should be quite different from each other due to the proportionality between M and ρA yx [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' As a result, we conclude that no Hall plateaus were observed in NdAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The absence of AHE in NdAlSi is intresting because it is in conflict with the AHC calculated by DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We performed AHC calculations for NdAlSi in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 10(c) by considering the intrinsic Berry curvature, similar to the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 6(f) for NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Near EF , there is a persistent AHC (σxy ∼ 500 Ω−1cm−1), in contrast to the absence of Hall plateaus in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 10(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' It would require further theoretical investigation to understand this discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Appendix E: Details on the spin structure refinement of NdAlGe We did magnetic refinement of our neutron diffraction data to determine the antiferromagnetic (AFM) spin struc- ture component of the commensurate phase of NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' To do so, rocking scans at 32 symmetrically nonequivalent Bragg positions were collected at T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 K within the manifold of Bragg peaks Q+ = G ± ( 1 3 + δ, 1 3 + δ, 0) and Q− = G ± ( 2 3 + δ, 2 3 + δ, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Here G refers to all nuclear allowed Bragg peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We performed representational analysis of the NdAlGe commensurate magnetism using SARAh refine [62] and found six possible basis vectors divided into two different irreps (Γ1 and Γ2) [63], whose real parts are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 11(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The two primitive Nd ions located at r1=(0,0,0) and r2=(1/2,0,1/4) within the unit cell have spins anti-parallel to each other for spin structures described by ⃗ψ1+ ⃗ψ2, ⃗ψ4- ⃗ψ5, and ⃗ψ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' These anti-parallel spin structures lead to strong Q− peaks and no intensity at Q+ peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' On the other hand, spin struc- tures described by ⃗ψ1- ⃗ψ2, ⃗ψ4+ ⃗ψ5, or ⃗ψ6 have parallel Nd spins at r1 and r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' This situation leads to strong Q+ peaks and no intensity at Q− peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 3(e) of the main text, we observed intensities at Q− positions that are two order of magnitude greater than at Q+ so the spin structure is dominantly anti-parallel and was re- fined to an Ising anisotropy ( ⃗ψ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We also detected weak intensity at Q− positions so there’s also a weak parallel spin component, which originates from an in-plane spin component µxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We did a magnetic refinement against both the Γ1 and Γ2 manifold and found our refined χ2 is ∼100 times smaller for the Γ1 manifold so we concluded that Γ1 is the appropriate irrep for NdAlGe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The final refinement is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 11(c) where the best solution was obtained with ⃗ψ1=- ⃗ψ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='14(2)µB and ⃗ψ3=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='8(4)µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 11(d) shows the χ2 dependence on both the angle direction of the spin canting within the ab plane (θxy), and its magnitude (µxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' In this plot, θxy = 0 is defined to be along the ordering vector direction [1,1,0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' 11(d), we deduced an in-plane spin canting of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='14(2)µB/Nd oriented 90(20)° away from the ordering vector direction, which produces an helical spin canting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' We determined the spin polarization of the ferromag- netic (FM) k = (0, 0, 0) magnetic structure of NdAlSi by collecting rocking scans at 24 symmetrically non- equivalent k = (0, 0, 0) Bragg positions covering the (H, H, L) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The nuclear and magnetic contributions to the Bragg diffraction were distinguished by collecting rocking scans within both the paramagnetic phase at 10 K and in the commensurate phase at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' Symmetry anal- ysis reveals three possible irreducible representations (ir- reps) to describe the k = (0, 0, 0) magnetic structure [63]: Γ1 and Γ3 that respectively correspond to ferromagnetic and antiferromagnetic structures where the spins are ori- ented along the c axis, and Γ5 that describes structures where the spins lie in the ab plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content=' The Ising ferromag- netic Γ1 is the only irrep that matches the symmetry 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='5 xy yz 0 Zx 500 0 500 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQfGQye/content/2301.04893v1.pdf'} +page_content='200013 ' 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amplifier, and we use non-oversampled +signals as input to PA for tractable localization performance +analysis4. Note that the PA affects the time domain signals and +each antenna at the Tx has a separate PA, and the PA model +in (16) does not consider the out-of-band emissions (only the +in-band distortion). For simplicity, the models are the same +for different PAs and hPA(ˇxt) returns the time domain signal +vector (by operating point-wise on each of the elements) with +PA nonlinearity introduced. +4) Array Calibration Error: The AGE and ADE are con- +sidered in the array calibration error. We define the complex +excitation coefficient of the n-th antenna at direction ϕ as [45] +bn(ϕ) = (1 + δa)ejδp, +(17) +where δa ∈ N(0, σ2 +AA), and δp ∈ N(0, σ2 +AP) are the relative +amplitude error and phase error, respectively. Regarding the +displacement error, we assume the n-th antenna position has a +displacement on the 2D plane of the local coordinate system +as +˜zn = zn + [0, δn,y, δn,z]⊤, +(18) +with dn ∈ R3 is the ideal position of the nth antenna in +the local coordinate system, δn,y, δn,z ∈ N(0, σ2 +ADE) are the +displacement error. The steering vector is then modified as +a(ϕ) → b(ϕ) ⊙ ej 2π +λ ˜Z⊤t, +(19) +where ˜Z = [˜z1, . . . , ˜zN] contains the geometry information +of all the antennas. The array calibration error is fixed for a +certain array and varies across different devices. +5) In-phase and quadrature imbalance: IQI operates on +the time domain signal and the transmitted symbol vector is +modified as [27], [46] +xg → F(αUFHxg + βUFHx∗ +g) = αUxg + βUx∗ +g, +(20) +where the FFT matrix F and IFFT matrix FH switch be- +tween time and frequency domain, αU = +1 +2 + 1 +2mUejψU, +βU = 1 +2 − 1 +2mUejψU with mU and ψU as the amplitude and +phase imbalance parameters at the UE side. We assume that +the IQI is compensated in the system, leading to a residual +impairment and the imbalance parameters can be modeled as +mU ∼ N(1, σ2 +IA) and φU ∼ N(0, σ2 +IP). Similarly, the IQI at +the receiving BS can be expressed as +yg → αByg + βBy∗ +g. +(21) +More accurate frequency-dependent IQI models can be found +in [47], [48], which is beyond the scope of this work. +4In order to fully characterize the effect of PAN, an oversampled model is +needed, which also captures the intersymbol interference introduced by the +nonlinearity, in addition to the symbol distortion (see (25) in [44]). +D. Hardware-impaired Model +Considering all types of HWIs described in Sec. II-C +and substituting (11)–(21) into (10), the observation can be +rewritten in the frequency domain. +1) Transmit Signal under HWI: The precoded transmitted +signal across subcarriers and antennas is modified from Xg = +xgv⊤ +g ∈ CK×NU to +ˇXg = FhPA(EUΞU(αUFHxg + βUFHx∗ +g)v⊤ +g +� +�� +� +precoded time domain signal before PA +). +(22) +2) Channel under HWI: The channel is modified from +Hk = αdk(τ)a(ϕB)a⊤(ϕU) ∈ CNB×NU in (4) to +ˇH = αdk(τ)CB(bB(ϕB) ⊙ ej 2π +λ ˜Z⊤ +B tB(ϕB) +� +�� +� +steering vector ˜aB(ϕB) +) +× (bU(ϕU) ⊙ ej 2π +λ ˜Z⊤ +U tU(ϕU) +� +�� +� +steering vector ˜aU(ϕU) +)C⊤ +U . +(23) +3) Received Signal under HWI: The received signal is +modified from yg ∈ CK×1 to (24). +E. Summary of the Models +To summarize, we have defined a MM in (1) without consid- +ering the HWI, which will be used for algorithm development. +With HWIs introduced, the impaired model defined in (24) will +be used as the TM. In the following section, we will evaluate +the impact of using the MM to process data generated by TM +on localization performance. For the sake of convenience in +performance analysis, we use µg(η) and ¯µg(η) to denote the +noise-free observation of (1) and (24), respectively. +III. LOCALIZATION ALGORITHM +Based on the models described above, a two-stage local- +ization5 problem can be formulated such that the channel +parameter vectors ˆηch = [η⊤ +1 , . . . , η⊤ +L]⊤ are firstly estimated +based on the observation vector ˆy1, . . . , ˆyL from all the BSs, +and then the stage vector ˆs is determined from ˆηch. +A. Mismatched Maximum Likelihood Estimator +The maximum likelihood estimation (MLE) can be em- +ployed when the observation y is generated from the same +model used by the algorithm. If y ∼ fTM(y|¯η), the MLE of +the UE position and channel gain is +ˆηMLE = arg max +¯η +ln fTM(y|¯η), +(25) +where ln fTM(y|¯η) is the log-likelihood of the TM. However, if +y ∼ fTM(y|¯η), but the estimator uses fMM(y|η) ̸= fTM(y|¯η), +the mismatched maximum likelihood estimation (MMLE) is +given by +ˆηMMLE = arg max +η +ln fMM(y|η). +(26) +More specifically, equation (26) formulates the MMLE for +channel parameters extraction, which can also be implemented +5In contrast, the direct localization estimates the state vector s from the +observed signal vector y directly. Considering the high complexity involved, +we adopt two-stage localization in this work. + +6 +ˇyg = F(αB(EB,gΞB,gFH( ˇXg ˇH⊤wg ⊙ d(τ)))) + βB(EB,gΞB,gFH( ˇXg ˇH⊤wg ⊙ d(τ)))∗) + ng. +(24) +in position and orientation estimation with known or approx- +imated likelihood function. A practical approach is to use the +gradient descent method with an initial point, which will be +detailed in the following subsections. +B. Channel Parameters Estimation +The channel parameters estimation will be performed with a +coarse estimation using ESPRIT, which provides a good initial +point for a refined estimation using (26). +1) Coarse Estimation using ESPRIT: We aim to obtain an +initial estimate of the channel parameters with a low com- +plexity, which can be solved using tensor-based beamspace +ESPRIT6 algorithm [13]. To implement the beamspace ES- +PRIT algorithm, we reformulate a beamspace channel matrix +H(b) based on the signal model in (1) as +H(b) +k += αdk(τ)WHaB(ϕB)a⊤ +U (ϕU)V +(27) +where W = T1⊗T2 ∈ CN1N2×M1M2 and V = (T3⊗T4)∗ ∈ +CN3N4×M3M4 are the combining matrix and precoder matrix +and the total number of transmissions G = M1M2M3M4. +Since the first row of the antenna position matrix ˜Z is all +zeros (see Sec. II-A and equation (3)), we can express the +steering vector in (3) as +aB(ϕB) = a(M1)(ω1) ⊗ a(M2)(ω2), +(28) +with +ω1 = π sin(φB) cos(θB), +ω2 = π sin(θB), +(29) +a(M1) +B +(ω1) = ej 2πfc sin(φB) cos(θB) +c +˜zB,2 = ej 2 +λc ω1˜zB,2, +(30) +a(M2) +B +(ω2) = ej 2πfc sin(θB) +c +˜zB,3 = ej 2 +λc ω2˜zB,3. +(31) +Here, ˜z⊤ +B,2 ∈ C1×NB and ˜z⊤ +B,3 ∈ C1×NB are the second and +third row of the matrix ˜Z, respectively. The combining matrix +can then be defined in terms of a grid of the spatial frequencies +¯ω1 = [¯ω1,1, . . . , ¯ω1,M1] and ¯ω2 = [¯ω2,1, . . . , ¯ω2,M2] as +T1 = [a(N1)(¯ω1,1), . . . , a(N1)(¯ω1,M1)]⊤ ∈ CN1×M1, +(32) +T2 = [a(N2)(¯ω2,1), . . . , a(N2)(¯ω2,M2)]⊤ ∈ CN2×M2, +(33) +where ¯ω1,m and ¯ω2,m are decided by beamforming directions +(detailed in Sec. V). A similar reasoning applies to the steering +vectors a(M3) +U +(ω3) and a(M4) +U +(ω4) at UE to define T3 and T4, +with +ω3 = π sin(φU) cos(θU), +ω4 = π sin(θU). +(34) +We further define b(Mn)(ωn) = TH +naNn(ωn) ∈ CMn for +n ∈ {1, 2, 3, 4} and b(M5)(ω5) = d(τ) (ω5 = 2π∆fτ), and +the beamspace channel matrix in (27) can be represented by +a tensor H ∈ CM1×M2×···×M5 as [14] +H(b) = αb(M1)(ω1) ◦ . . . ◦ b(M5)(ω5). +(35) +In practice, the estimated beamspace channel matrix can +be estimated with known pilot signals as vec( ˆH(b) +k ) += +[ˆy1,k/x1,k, . . . , ˆyG,k/xG,k]⊤. By rearranging the estimated +6While this work considers only the LOS channel, the ESPRIT also works +for the scenarios with NLOS paths. +channel into a tensor ˆH +(b) shown in (35), the beamspace +tensor-based ESPRIT method can then be used to estimate ω1 +to ω5 and obtain the AOA, AOD, and delay accordingly [13], +[14]. +2) Fine Estimation using MMLE: From ESPRIT, we can +obtain an initial estimate of the channel parameters ˆη0. The +refinement of the initial estimate can be formulated as an +optimization problem, based on (26), as +ˆη = arg min +η ∥y − µ(η)∥2. +(36) +Since α appears linearly in the noise-free observation µ, we +further define γ(η) = µ(c)/α with c = [ϕ⊤ +B , ϕ⊤ +U , τ]⊤. By +setting ∂∥y − µ(η)∥2/∂α = 0, we can have +ˆc = arg min +c ∥y − γH(c)y +∥γH(c)∥2 γ(c)∥2. +(37) +In this way, the nuisance parameters can be removed, which +reduces the dimension of the unknown parameters. +C. Localization Algorithm +1) Coarse Estimation: Given the estimated geometric pa- +rameter vector cl (1 ≤ l ≤ L) for all the channels, the +least squares solution for coarse estimation of position and +orientation as [49] +ˆRU,LS = +� +UVT, +if det(UVT) = 1, +UJVT, +if det(UVT) = −1, +(38) +[ˆpU,LS, ˆBU,LS]⊤ = (Q⊤ +3 Q3)−1Q⊤ +3 q, +(39) +where J = diag([1, 1, −1]), U and V are the unitary basis +matrices of the singular value decomposition of the matrix +Q1Q⊤ +2 , together with Q3, q are given by [49] +Q1 = −[RB,1t(ˆϕB,1), . . . , RB,Lt(ˆϕB,L)], +(40) +Q2 = [t(ˆϕU,1), . . . , t(ˆϕU,L)], +(41) +Q3 = +� +�� +I3 +RB,1t(ˆϕB,1) +... +... +I3 +RB,Lt(ˆϕB,L) +� +�� , +(42) +q = +� +�� +p(1) +B ++ RB,1ˆτ1t(ˆϕB,1) +... +pB,L + RB,LˆτLt(ˆϕB,L)]⊤ +� +�� . +(43) +Different from the algorithm in [49], the estimator for position +and clock offset in (39) does not require the orientation of the +UE RU, which is still sufficient as a coarse estimate, as will +be shown in the simulation section. +2) MMLE: Once the initial position and orientation results +are obtained, joint position and orientation estimation using +MMLE can be formulated as +ˆs = arg min +s +L +� +l=1 +(cl(s) − ˆcl)⊤Σ−1 +cl (cl(s) − ˆcl), +(44) +which can be solved using the manifold optimization toolbox +Manopt [50]. Note that the covariance matrix may not be +accurately obtained in practice. We formulate localization as +an MMLE problem with two purposes: (a) to evaluate the + +7 +performance improvement with known covariance matrices +compared to the coarse estimation; (b) to validate the derived +bound, which will be detailed in Sec. IV. +IV. LOWER BOUND ANALYSIS +In the next, we derive the CRB for MM, as well as the +MCRB for the mismatched estimator in (26). +A. CRB Analysis for the Mismatched Model +Based on the defined channel parameter vector η and state +vector s, the signal model in (1) and y ∼ fMM(y|η), the +channel estimation CRB of the MM for the lth channel can +be obtained as I(ηl)−1 ∈ R7×7 with [51] +I(ηl) = 2 +σ2n +G +� +g=1 +K +� +k=1 +Re +��∂µg,k +∂ηl +�H �∂µg,k +∂ηl +�� +. +(45) +Here, Re{·} extracts the real part of a complex variable. +Consequently, the FIM of all the channel parameters ηch can +be formulated as +I(ηch) = blkdiag(I(η1), . . . , I(ηL)). +(46) +where blkdiag(·) returns the block diagonal matrix created by +aligning the input matrices. The FIM of the state vector I(s) ∈ +R13×13 can then be formulated as +I(s) = M(M⊤ J⊤ +S I(cch)JS M)−1M⊤, +(47) +where I(cch) +∈ +R5L×5L is the EFIM of non-nuisance +parameters cch obtained from I(ηch), JS ≜ +∂cch +∂s +is the +Jacobian matrix using a denominator-layout notation, M = +blkdiag(I4×4, ¯M) with ¯M as [9] +¯M = +1 +√ +2 +� +� +−r3 +03×1 +r2 +03×1 +−r3 +−r1 +r1 +r2 +03×1 +� +� , +(48) +where r1, r2, and r3 are the first, second, and third columns +of the UE rotation matrix RU. +Based on I(η) in (45), we can define the AOD error bound +(ADEB), AOA error bound (AAEB), and delay error bound +(DEB) of the link between the UE and the lth BS) as +AAEB = +� +tr([I(ηl)−1]1:2,1:2), +(49) +ADEB = +� +tr([I(ηl)−1]3:4,3:4), +(50) +DEB = +� +([I(ηl)−1]5,5). +(51) +Similarly, based on I(s), we can define the position error +bound (PEB), clock offset error bound (CEB) and orientation +error bound (OEB) as +PEB = +� +tr([I(s)−1]1:3,1:3), +(52) +CEB = +� +([I(s)−1]4,4), +(53) +OEB = +� +tr([I(s)−1]5:13,5:13). +(54) +The bounds from (49)–(54) will be used to evaluate the +channel estimation and localization performance. In the next +subsections, we will first formulate the MCRB for channel +estimation, and then the mismatched lower bound for position +and orientation estimation will be derived. +B. Misspecified CRB of Channel Parameters +For a given channel model, the model is said to be mis- +matched or misspecified when y ∼ fTM(y|η), while the +estimation is based on the assumption that y ∼ fMM(y|η)), +where fTM(y|η) ̸= fMM(y|η). +The lower bound (LB) of using a mismatched estimator can +be obtained as [35] +LB(¯η, η0) = A−1 +η0 Bη0A−1 +η0 +� +�� +� +=MCRB(η0) ++ (¯η − η0)(¯η − η0)⊤ +� +�� +� +=Bias(η0) +, +(55) +where ¯η is the true channel parameter vector, η0 is the pseudo- +true parameter vector (which will be introduced soon), and +Aη0, Bη0 are two possible generalizations of the FIMs. The +LB is a bound in the sense that +E{(ˆηMMLE − ¯η)(ˆηMMLE − ¯η)⊤} ⪰ LB(¯η, η0), +(56) +where the expectation is with respect to fTM(y|η). What re- +mains is the formal definition and computation of the pseudo- +true parameter η0 and Aη0, Bη0. +1) Pseudo-true Parameter: Assume the probability density +function (PDF) of the TM, where the observation data come +from, is fTM(y|¯η), where y is the received signals and ¯η ∈ R7 +(7 unknowns for this case) is the vector containing all the +channel parameters. Similarly, the PDF of the MM for the +received signal y can be noted as fMM(y, η). The pseudo-true +parameter vector is defined as the point that minimizes the +Kullback-Leibler divergence between fTM(y|¯η) and fMM(y|η) +as +η0 = arg min +η DKL(fTM(y|¯η)∥fMM(y|η)). +(57) +We define ϵ(η) ≜ ¯µ(¯η)−µ(η), and the pseudo-true parameter +can be obtained as [36] +η0 = arg min +η ∥ϵ(η)∥2 = arg min +η ∥¯µ(¯η) − µ(η)∥2. +(58) +Hence, η0 can be found by solving (36) with the observation +y = +¯µ(¯η), which can be accomplished using the same +algorithm in Sec. III, initialized with the true value ¯η. +2) MCRB Component Matrices: The matrices Aη0 and +Bη0 can be obtained based on the pseudo-true parameter +vector η0 as [36] +[Aη0]i,j = +ˆ ∂2lnfMM(y|η) +∂ηi∂ηj +fTM(y|¯η)dy +���� +η=η0 += +2 +σ2n +Re +� +∂2µ(η) +∂ηi∂ηj +ϵ(η) − ∂µ(η) +∂ηj +�∂µ(η) +∂ηi +�H������ +η=η0 +(59) +and +[Bη0]i,j = +ˆ ∂lnfMM(y|η) +∂ηi +∂lnfMM(y|η) +∂ηj +fTM(y|¯η)dy +���� +η=η0 += 4 +σ4n +Re +�∂2µ(η) +∂ηi +ϵ(η) +� +Re +�∂2µ(η) +∂ηj +ϵ(η) +� ++ 2 +σ2n +Re +� +∂µ(η) +∂ηj +�∂µ(η) +∂ηi +�H������ +η=η0 +. +(60) +C. Absolute Lower Bound (ALB) for Localization +Another way to interpret the LB specified in (55) is that the +estimated channel parameter vector from an efficient estimator + +8 +follows a nonzero-mean multi-variable Gaussian distribution +as +ˆηl ∼ N(η0,l, A−1 +η0,lBη0,lA−1 +η0,l), +(61) +while the assumed distribution of the MMLE is +ˆηl ∼ N(ηl(¯s), I(ηl)−1), +(62) +where ¯s is the true state of the UE. As a result, the position and +orientation estimation (from the channel parameter vectors of +all the paths) of the two-stage localization problem is another +mismatched problem and the bound follows as +LB(¯s, s0) = MCRB(s0) + (¯s − s0)(¯s − s0)⊤ +� +�� +� +Absolute lower bound (ALB) +. +(63) +Similar to (55), ¯s is the true state parameter vector, s0 is the +pseudo-true state parameter vector. +It is possible to derive the localization LB constrained +MCRB [52]; however, considering the high complexity when +involving 3D orientation estimation, we will focus on the +bias term, defined as the absolute lower bound (ALB) of the +localization performance as ALB = (¯s − s0)(¯s − s0)⊤, which +can sufficiently evaluate the effect of HWIs on localization +as will be shown in Sec. V-C2 Following a similar derivation +in (58). The pseudo-true parameters for state vector s can be +obtained as +s0 = arg min +¯s +� +l +(η0,l − ηl(¯s))⊤I(ηl)(η0,l − ηl(¯s)), (64) +where η0,l += +arg minη ∥¯µ(¯ηl) − µ(ηl)∥2 is the biased +mapping obtained by calculating the pseudo-true parameters +of the lth channel from (58), and I(ηl) is the inverse of the +covariance matrix that can be obtained from (45). +D. Summary of Different Bounds +In this section, we introduced different types of lower +bounds. For channel geometric parameters, the CRB and LB +are derived for AOA, AOD, and delay estimations. For state +parameters, the CRB and ALB are derived for the position, +orientation, and clock offset estimations. All types of the lower +bounds are summarized in Table I, which will be used in +Sec. V Numerical Results. +TABLE I +SUMMARY OF DIFFERENT LOWER BOUNDS +Types +Parameters +Remarks +AOA +AOD +Delay +Channel Parameters +CRB +AAEB +ADEB +DEB +(49)-(51) +LB +AALB +ADLB +DLB +(55) +Position +Orientation +Clock Offset +State Parameters +CRB +PEB +OEB +CEB +(52)-(54) +ALB +PALB +OALB +CALB +(63) +V. NUMERICAL RESULTS +A. Default Parameters +We consider a 3D MIMO uplink scenario with one UE +and two BSs, and the simulation parameters7 can be found +7The PA parameters are estimated from the measurements of the RF +WebLab, which can be remotely accessed at www.dpdcompetition.com. Part +of the parameters come from the Hexa-X Deliverable 3.1. +in Table II. We utilize 10% of the total number of subcarriers +Kcom for localization, resulting in K = 100 subcarriers as +pilot signals. The amplitude of the channel gain is calculated +as ρ = +λ +4πcτ . The selection of these parameters is to show +the performance of the estimator in comparison to the derived +bound. The analysis of each HWI type is also discussed in the +simulation results. +Regarding the evaluation of communication performance, +only the first BS is considered, and 16-QAM modulation +is adopted. Different from localization, where BS-UE beam +sweeping is needed, we evaluate the effect on communication +with fixed precoder and combiner vectors across different +transmissions. By considering all HWIs, we assume the chan- +nel can be perfectly estimated (with a sufficient number of +pilots) as ˆH = ˇH = ˆaBˆaU with ˆaB = √αCB˜aB(ϕB) and +ˆaU = √α˜aU(ϕU)C⊤ +U from (23). In order to maximize the SNR +with the amplitude constraints of the precoder and combiner, +we choose w and v respectively as the conjugate of ˆaB and +ˆaU with each of the elements normalized to a unit amplitude. +For each realization, 20 OFDM symbols are sent with data +drawn randomly from 16-QAM, and SER is used to evaluate +the effect of HWIs on communication. +For localization, the pilot signal xg,k is chosen with random +phase and a constant amplitude |xg,k|2 = P/NU. To assist the +beamspace ESPRIT algorithm, we set the number of sweeping +beams as M1 = 4, M2 = 4, M3 = 3, M4 = 3 with +a total number of transmission G = 144. For a specific +spatial frequency vector ¯ωn (n ∈ {1, 2, 3, 4}), we assume +the sweeping range as (Mn − 1)∆ω centered at the location +prior ˚ωn = ωn + δω, where ωn is defined in (29), (34), +and δω is the error). More specifically, we choose ¯ωn,m = +ωn + δω + 2m−Mn−1 +2 +∆ω, with ∆ω = 0.15 and δω = 0.05 in +the simulation. The sweeping priority is set to ‘BS-first’ by +default, which means that the UE can change its precoder +vector when the BS finishes the M1M2 += 16 different +sweeping beams. Different error bounds (i.e., CRBs, LBs, +ALBs from Table I) are utilized as localization performance +metrics. +B. The Effect of HWIs on Communication +1) The Effect of HWIs on SER: We approximate the effect +of HWIs on communication as the random noise and evaluate +the effect on SER based on numerical and analytical results8. +Considering that the effects of some HWIs depend on the +amplitude of the symbol (e.g., PAN), we also obtain the +minimum and maximum noise levels across different symbols +to evaluate the lower bound and upper bound of the SER. The +SERs of 16-QAM with different transmit power for different +HWI coefficients are visualized in Fig. 2, where the black +solid curve is the benchmark SER without HWIs. By default, +cHWI = 1, and the HWI level is the same as the parameters +in Table II. A value of cHWI = 10 indicates that the standard +deviations (e.g., σPN, σCFO) of all the impairments (except for +8The SER of M-QAM can be calculated as SERM += +1 − (1 − +2 +√ +M−1 +√ +M +Q( +� +3SNR +M−1 ))2 [53, (6,23)], where Q(·) is the Q-function and SNR +is effective SNR considering both approximated HWI noise and background +noise. + +9 +TABLE II +DEFAULT SIMULATION PARAMETERS +Parameters +True Model +Mismatched Model +BS +p1 +B = [0, 0, 3]⊤, p2 +B = [0, 5, 3]⊤ +BS Orientations +o1 +B = [0◦, 15◦, 0◦]⊤, o2 +B = [−30◦, 15◦, 0◦]⊤ +BS Antennas +N 1 +B = N 2 +B = 8 × 8 +UE Position +pU = [4, 2, 1.5]⊤ +UE Orientation +oU = [180◦, 0◦, 0◦]⊤ +UE Antennas +NU = 4 × 4 +RFC at BS/UE +1 +Carrier Frequency +fc = 140 GHz +Bandwidth +W = 1 GHz +Transmissions +G = 4 × 4 × 3 × 3 = 144 +Subcarriers (Total) +Kcom = 1040 (∆f = 960 kHz) +Subcarriers (Pilots) +K = 100 +Length of the CP +Kcp = 7 +Load Impedance +R = 50 Ω +Noise PSD +N0 = −173.855 dBm/Hz +Noise Figure +10 dB +Phase Noise +σPN = 2.5◦ +σPN = 0◦ +Carrier Freq. Offset +σCFO = 5e−4 (0.036 ppm) +σCFO = 0 +Mutual Coupling +σMC = 0.002 +σMC = 0 +β1 = 0.9798+0.0286j +Power Amplifier +β2 = 0.0122-0.0043j +n/a +β3 = −0.0007+0.0001j +PA Clipping Voltage +xclip = 1 V +n/a +Array Gain Error +σGA = σGP = 0.002 +σRA = σRP = 0 +Antenna Disp. Error +σAD = 5 um (2.3e−3λ) +σAD = 0 +IQ Imbalance +σIA = σIP = 0.02 +σIA = σIP = 0 +PAN) are multiplied by 10. We can see from the figure that the +analytical SERs with approximated noise levels (red, blue, and +green markers) are close to the numerical SERs (solid red, blue +and green curves), and both are within the lower and upper +bounds (shaded areas). We can also see from Fig. 2 that the +selected impairment level (i.e., cHWI = 1) has limited effects +on communication. However, we will show the localization +performance will be affected by the same level of HWIs in +Sec. V-C. +−10 +−5 +0 +5 +10 +15 +10−7 +10−5 +10−3 +10−1 +P [dBm] +SER (16-QAM) +Anal. without HWI +Numer. HWI (cHWI = 0.1) +Anal.-Approx. HWI (cHWI = 0.1) +Numer. HWI (cHWI = 1) +Anal.-Approx. HWI (cHWI = 1) +Numer. HWI (cHWI = 2) +Anal.-Approx. HWI (cHWI = 2) +Fig. 2. +The effect of different HWI levels on SER. Numerical results for +100 realizations and analytical results calculated with approximated equivalent +HWI noise. The boundaries of the shadow areas indicate the upper and lower +bounds for SER. +2) The Effect of Individual HWIs on SER: We are also +interested in the effect of individual HWIs on communication. +By considering PN, CFO, PAN, and IQI one by one, the +SERs under HWI are shown in Fig. 3. Benchmarked by +−10 +−5 +0 +5 +10 +15 +10−7 +10−5 +10−3 +10−1 +P [dBm] +SER (16QAM) +PN +PAN +CFO +IQI +MC+AGE+ADE +Without HWI +−10 +−5 +0 +5 +10 +15 +10−7 +10−5 +10−3 +10−1 +P [dBm] +SER (16QAM) +PN +PAN +CFO +IQI +MC+AGE+ADE +Without HWI +Fig. 3. The effect of individual HWIs on SER using approximated equivalent +HWI noise. Under current simulation parameters, the PN, PAN, CFO and IQI +increase the SER, whereas the MC, AGE and ADE have negligible effects on +communication. +the solid black curve without HWIs, these factors degrade +SERs. We also performed simulations by including MC, AGE, +ADE at the same time, as shown in the dashed curve with +cross markers, and found their effects on communication are +negligible under the current simulation setup. +3) Insights into the Impact of HWI on Communication: To +gain further insight into the effect of HWI on communication, +we separate the overall system noise into equivalent HWI +noise and background noise. We can see from Fig. 4 that +the equivalent HWI noise is determined by the HWI level +and has an approximately linear relationship with the transmit +power (when working within the linear region of the PA). In +addition to the fixed background noise, the overall equivalent +noise level keeps increasing and is dominated by the HWIs at +high transmit power. +−10 +−5 +0 +5 +10 +15 +−110 +−100 +−90 +−80 +−70 +−60 +P [dBm] +Equivalent Noise Level [dBm] +Overall Noise (cHWI = 2) +HWI Noise (cHWI = 2) +Overall Noise (cHWI = 1) +HWI Noise (cHWI = 1) +Overall Noise (cHWI = 0.1) +HWI Noise (cHWI = 0.1) +Background Noise +Fig. 4. Visualization of overall system noise, equivalent HWI, and background +noise with different transmit power P. The background noise has a large effect +on communication in low transmit power, whereas the HWIs contribute more +in high transmit power. +C. The Effect of HWIs on Localization +Before analyzing the HWIs in detail, we first establish the +validity of the derived bounds by comparing them against the +performance of practical algorithms. +1) Channel Estimation Results: For convenient analysis, we +adopt one specific realization of the HWIs for the system. The +results of channel parameters estimation using ESPRIT (circle, + +10 +−10 +0 +10 +20 +30 +40 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +P [dBm] +AOA [◦] / AOD [◦] / Delay [m] +AOA-ESPRIT +AOD-ESPRIT +Delay-ESPRIT +AOA-MMLE +AOD-MMLE +Delay-MMLE +AAEB +ADEB +DEB +AALB +ADLB +DLB +Fig. 5. Comparison between channel parameters estimation results (ESPRIT +and MMLE) and different lower bounds (CRB of the MM and the LB of the +mismatched estimator) in terms of AOA, AOD and delay. Due to the HWIs, +the performance starts to saturate when the transmit power exceeds 30 dBm. +Although the performance of the coarse estimation using ESPRIT (using a +mismatched model) may not attain the theoretical bounds (especially for delay +estimation), the refined results using MMLE can reach the LB (solid curves +align well with the cross-marked dotted curve). +−10 +0 +10 +20 +30 +40 +10−4 +10−3 +10−2 +10−1 +100 +P [dBm] +Pos [m] / Ori [◦] / Clock [m] +POS-LS +ORI-LS +Clock-LS +POS-MMLE +ORI-MMLE +Clock-MMLE +PEB +OEB +CEB +PALB +OALB +CALB +Fig. 6. +Comparison between localization results (position, orientation, and +clock offset estimation) and different lower bounds (CRB of the MM and +the LB of the mismatched estimator). We noticed that the LS estimators are +sufficient for this 2-BS scenario, and the refined results using MMLE attain +the ALBs. +square, and diamond markers) and MMLE (solid curves) are +shown in Fig. 5. The estimators are benchmarked by the CRBs +of the ideal/mismatched model (CRB-MM, dashed curves) and +the LB using a mismatched model (dotted curves with cross +markers). Note that the average transmit power P is calculated +without considering the nonlinearity of the power amplifier +(calculated before the PA). When the transmit power P is low, +the LB is determined by the MCRB (since the bias part is con- +stant, see (55)) and has a similar performance as CRBs. This +indicates that in low transmit power, the mismatched model +will not significantly affect the performance, as the expected +accuracy is low and limited by the noise. With the increase of +transmit power, the contribution of MCRB decreases due to an +increased SNR, and eventually, the mismatched localization is +lower bounded by the absolute lower bound (ALB) (bias part +in (55)). This indicates that the localization performance can +no longer be improved by increasing transmit power, which +cannot be ignored in scenarios requiring high-accuracy local- +ization performance9. Regarding the estimators, the ESPRIT +(using a mismatched model) provides low-complexity results +with limited performance in delay estimation. However, the +refined results using MMLE can reach the LB (solid curves +align well with the dotted curve). +2) Localization Results: Based on the estimated channel +parameters, we are able to estimate the UE position and +orientation. Similar to the channel estimation results, two +estimators (LS and MMLE) and two bounds (CRB and LB) +are evaluated. The results for localization are shown in Fig. 6. +From the figure, we can see that at low transmit powers, the +LB and CRBs coincide, implying that the HWIs are not the +main source of error. At higher transmit powers (10 dBm for +OEB, and 20 dBm for PEB), LB deviates from the CRBs, and +the positioning performance is thus more severely affected by +HWIs. The MMLE in high SNR is close to the ALB, indicating +the validity of the MCRB analysis. +0 +5 +10 +15 +20 +25 +30 +35 +40 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +P [dBm] +AALB (Average) +AALB (Multi) +AAEB +ADLB (Average) +ADLB (Multi) +ADEB +DLB (Average) +DLB (Multi) +DEB +(a) PN +0 +5 +10 +15 +20 +25 +30 +35 +40 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +P [dBm] +(b) CFO +0 +5 +10 +15 +20 +25 +30 +35 +40 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +P [dBm] +(c) MC +0 +5 +10 +15 +20 +25 +30 +35 +40 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +P [dBm] +(d) AGE +0 +5 +10 +15 +20 +25 +30 +35 +40 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +P [dBm] +(e) ADE +0 +5 +10 +15 +20 +25 +30 +35 +40 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +P [dBm] +(f) IQI +Fig. 7. +LBs of channel parameter estimation under different types of +impairment with multiple realizations: (a) Phase noise, (b) Carrier frequency +offset, (c) Mutual coupling, (d) Array gain error, (e) Antenna displacement +error, (f) IQ-imbalance. +Now that the validity of the bounds has been established, +we rely solely on the bounds to evaluate the effect of HWIs +on localization. First, the impairments are studied individually, +then the impact of the waveform type is evaluated, and finally, +the impairment levels are varied. +9Note that the analysis here is under the same level of residual noise (e.g., +PN, CFO, IQI). In practice, the impairment levels depend on specific HWI +calibration algorithms and transmit power. + +11 +3) The Effect of Individual Impairments: To understand the +effect of different types of HWIs, we study the LB for AOA, +AOD, and delay estimation by considering one type of HWIs +at a time. The results are shown in Fig. 7 for (a) PN, (b) +CFO, (c) MC, (d) AGE, (e) ADE and (f) IQI. The effect of +PA will be separately discussed in Sec. V-C4. Considering we +define the HWIs as random variables with a fixed impairment +level as shown in Table II, we perform multiple hardware +realizations with a fixed pilot signal and plot all the resultant +LBs in the shaded regions. We can see that different types of +the HWIs affect angle and delay estimation differently. The +PN, CFO, and IQI introduce noise on the symbols across +different subcarriers and hence affect delay estimation10. Since +the phase change introduced by CFO affects the phase changes +across beams, angle estimation will also be affected. Instead of +affecting the phase changes between different subcarriers, the +MC, AGE, and ADE distort the steering vectors and therefore +have a more significant effect on the angle estimation. For all +the HWIs, the negative effect on the performance occurs when +the transmit power is high. +One special observation is that the effect of CFO on the +AOA is less pronounced than on AOD in Fig. 7 (b). This is +because the sweeping strategy is ‘BS-first’. For a system with +analog arrays, the estimation of AOA/AOD relies on phase +shifts across consecutive beams over time, meaning the angle +cannot be estimated from a single receive beam, like in a +digital array. If the BS sweeps across different beams while +the UE is using a fixed beam, the AOA can be estimated +in one BS sweep, and the effect of CFO will be minor. +However, the AOD estimation requires multiple BS sweeps, +which increases the effect of CFO. To verify the explanation, +we further changed the sweeping strategy from ‘BS-first’ to +‘UE-first,’ and the results with different array sizes can be +found in Fig. 8. We can see that the AOA is less affected if +the sweeping is ‘BS-first’ (blue curves in (a)) as shown in (12). +Similarly, the AODs are less affected if the sweeping is ‘UE- +first’ (dashed red curves in (b)) with a large UE array size. +However, when the array size is small, sweeping order will +have less impact (i.e., the gaps are small between the dashed +curves in (a) and the solid curves in (b)). +4) The Effect of PA with Different Pilot Signals: High peak- +to-average-power ratio (PAPR) is one of the critical issues in +implementing the OFDM signals, and a promising alternative +is to use DFT-S-OFDM [54]. When increasing the transmit +power, the PAN is more likely to happen, as can be seen +in Fig. 9 (a). The delay estimation suffers more from the +nonlinear distortion because of the clipping of transmit signal, +which distorts the uniformity of phase changes across the +subcarriers. The effect on angle estimation is less pronounced +(at the same level of transmit power) since different antenna +elements experience similar distortions with identical PAs +adopted in this work. We compare using the random OFDM +symbols and the FFT version of the benchmark symbols (a +special case of DFT-S-OFDM by choosing an identity mapping +matrix [54]), and the results are shown in Fig. 9. Due to the +10If multiple RFCs or several local oscillators are adopted in the array, PN +may have a larger effect on angle estimation. +0 +10 +20 +30 +40 +50 +10−3 +10−2 +10−1 +100 +P [dBm] +Angle Error [◦] +BS 8x8, UE 4x4, BS first +BS 8x8, UE 4x4, UE first +BS 4x4, UE 8x8, BS first +BS 4x4, UE 8x8, UE first +(a) AALB (average) +0 +10 +20 +30 +40 +50 +10−3 +10−2 +10−1 +100 +P [dBm] +Angle Error [◦] +(b) ADLB (average) +Fig. 8. The effect of CFO on channel geometrical parameters with different +sweeping strategies. The ‘BS first’ strategy (blue curves) works better for +AOA estimation, while the ‘UE first’ strategy (red curves) works better for +AOD estimation. +20 +25 +30 +35 +40 +45 +50 +55 +60 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +P [dBm] +AALB (Average) +AALB (Multi) +AAEB +ADLB (Average) +ADLB (Multi) +ADEB +DLB (Average) +DLB (Multi) +DEB +(a) OFDM +20 +25 +30 +35 +40 +45 +50 +55 +60 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +P [dBm] +(b) DFT-S-OFDM +Fig. 9. The effect of PA on channel parameters estimation using (a) OFDM, +and (b) DFT-S-OFDM. +reduced PAPR by DFT-S-OFDM, the localization performance +can be improved, as shown in the right figure. +5) Evaluation of HWIs with Different Impairment Levels: +We further evaluate the position and orientation ALBs with +different levels of HWIs by defining a HWI coefficient cHWI. +With different value of cHWI, the position ALB and orientation +ALB by considering all the HWIs, and individual HWIs, are +shown in Fig. 10 (a) and (b). All the results indicate the 75th +percentile of the total 100 realizations. We notice that the effect +of PN, MC, AR, AG, and IQI on the localization increases +approximately in a linear trend with impairment level. The +CFO has a larger effect in high impairment level as the error +residue accumulates over time. Based on Fig. 10, we can +quantize the contribution of individual HWIs (e.g., if the ALBs +are much smaller than current CRB, the negative contribution +of HWI on localization is negligible). In addition, it can also +identify dominant impairment factors for further compensation +(e.g., ADE is one of the dominant factors under current system +parameters). + +12 +−1 +−0.5 +0 +0.5 +1 +10−6 +10−3 +100 +10log(cHWI) +PALB [m] +ALL +PN +CFO +MC +AGE +ADE +IQI +(a) PALB +−1 +−0.5 +0 +0.5 +1 +10−6 +10−3 +100 +10log(cHWI) +OALB +ALL +PN +CFO +MC +AGE +ADE +IQI +(b) OALB +Fig. 10. An example of ALB with different levels of impairments: (a) PALB, +(b) OALB. The ALBs of the position and orientation affected by the HWIs +increase with cHWI (reflecting the impairment level). +D. Summary +From the simulation, we found that the HWIs affect both +localization and communication, especially at high transmit +power. The equivalent noise is mainly contributed by HWIs +for communication, and the localization performance will +saturate due to model mismatch. However, different types of +HWIs affect localization and communication differently. The +effect of the individual impairment on angle/delay estimation +and communication (i.e., SER) is summarized in Table III, +with two levels of impacts H/L to denote High/Low. Note +that in this uplink scenario, the position estimation is mainly +affected by AOA and TOA information, while the orientation +estimation is mainly affected by AOD. +As for the angle estimation for localization, the performance +is strongly affected by CFO, MC, AGE, and ADE. When +talking about the TOA, it is mainly affected by PN, CFO +and IQI. Since communication does not exploit the phase +relationship between antennas (e.g., no sweeping is needed +once the communication link is established), SER will be +affected by the same factors as delay estimation, which are +verified in Fig. 7. It should be noted that the effect of CFO on +AOA and AOD estimation depends on the sweeping order and +number of transmissions, while the effect of PA depends on +the transmit power and the nonlinear region of the amplifier. +VI. CONCLUSION +As the requirements on localization and communication +performance are more stringent to support new applications, +HWIs become a prominent factor affecting the performance +in 6G systems. We have modeled different types of HWIs and +utilized the MCRB to evaluate the localization error caused +by model-mismatch. The effects of HWIs on angle/delay +and position/orientation estimation are evaluated. We found +that PN and IQI have a stronger effect on delay estimation, +while MC, AGE, and ADE have a more significant effect +TABLE III +SUMMARY OF THE EFFECTS OF HWIS ON LOCALIZATION AND +COMMUNICATION +Type of HWI +AOD +AOA +TOA +SER +Phase Noise +L +L +H +H +Carrier Frequency Offset +H∗ +H∗ +H +H +Mutual Coupling +H +H +L +L +Power Amplifier Nonlinearity +H∗ +H∗ +H∗ +H∗ +Array Gain Error +H +H +L +L +Antenna Displacement Error +H +H +L +L +IQ Imbalance +L +L +H +H +∗The effect of CFO on angle estimations depends on the sweeping order and number +of transmissions. The effect of PAN depends on the transmit power and the nonlinear +region of the amplifier. +on angle estimation. The CFO and PAN affect both angle +and delay, where the former one depends on the sweeping +strategy and number of transmissions, and the latter factor +is determined by the transmit power (or amplitude) of the +signals. Furthermore, we evaluated the effect of individual +HWIs on the communication performance in terms of SER. +The dominant impairments that degrade SER (i.e., PN, CFO, +PA, and IQI) are in good agreement with the factors that affect +delay estimation. +In summary, the localization and communication perfor- +mance that improves with transmit power in an ideal model +will saturate due to the effect of HWIs. To fully realize the +potential of 6G joint localization and communication system, +a dedicated pilot signal design and algorithms for estimating +and mitigating HWI are needed. Further works can consider +the effect of HWIs in multipath and reconfigurable intelligent +surface-aided scenarios, as well as learning-based methods for +mismatch mitigation. +REFERENCES +[1] H. Chen et al., “A tutorial on terahertz-band localization for 6G +communication systems,” IEEE Communications Surveys & Tutorials, +May. 2022. +[2] H. 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Berardinelli, “Generalized DFT-s-OFDM waveforms without cyclic +prefix,” IEEE Access, vol. 6, pp. 4677–4689, Dec. 2017. + diff --git a/F9AzT4oBgHgl3EQfHPuf/content/tmp_files/load_file.txt b/F9AzT4oBgHgl3EQfHPuf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..68ec04f009c62465bf3abbaf6a44b0fc62215961 --- /dev/null +++ b/F9AzT4oBgHgl3EQfHPuf/content/tmp_files/load_file.txt @@ -0,0 +1,1233 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf,len=1232 +page_content='1 Modeling and Analysis of 6G Joint Localization and Communication under Hardware Impairments Hui Chen, Member, IEEE, Musa Furkan Keskin, Member, IEEE, Sina Rezaei Aghdam, Member, IEEE, Hyowon Kim, Member, IEEE, Simon Lindberg, Member, IEEE, Andreas Wolfgang, Member, IEEE, Traian E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Abrudan, Member, IEEE, Thomas Eriksson, Senior Member, IEEE, and Henk Wymeersch, Senior Member, IEEE Abstract—Localization (position and orientation estimation) is envisioned as a key enabler to satisfy the requirements of communication and context-aware services in the sixth generation (6G) communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' User localization can be achieved based on delay and angle estimation using uplink or downlink pilot signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' However, hardware impairments (HWIs) distort the signals at both the transmitter and receiver sides and thus affect the localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' While this impact can be ignored at lower frequencies where HWIs are less severe, and the localization requirements are not stringent, modeling and analysis efforts are needed for high-frequency 6G bands (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', sub-THz) to assess degradation in localization accuracy due to HWIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In this work, we model various types of impairments for a sub- THz multiple-input-multiple-output communication system and conduct a misspecified Cram´er-Rao bound analysis to evaluate HWI-induced performance losses in terms of angle/delay estima- tion and the resulting 3D position/orientation estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Complementary to the localization analysis, we also investigate the effect of individual and overall HWIs on communication in terms of symbol error rate (SER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Our extensive simulation results demonstrate that each type of HWI leads to a different level of degradation in angle and delay estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The prominent factors on delay estimation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', phase noise and carrier frequency offset) will have a dominant negative effect on SER, while the impairments affecting only the angle estimation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', mutual coupling and antenna displacement) induce slight degradation in SER performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Index Terms—Localization, 6G, hardware impairment, THz communications, CRB, MCRB, MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' INTRODUCTION Localization refers to the process of estimating the position and orientation of a connected device or user equipment (UE), which is expected to have a tight interaction with communication in future wireless systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Localization can benefit from a large array dimension and wide bandwidth of high-frequency signals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', mmWave and sub-THz) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In return, the position and orientation information can im- prove spatial efficiency and optimize resource allocation for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Chen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Keskin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Aghdam, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Kim, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Eriksson and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Wymeer- sch are with the Department of Electrical Engineering, Chalmers University of Technology, 412 58 Gothenburg, Sweden (email: hui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='chen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' furkan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' sinar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' hyowon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' thomase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' henkw@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='se).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Lindberg and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Wolfgang are with Qamcom Research & Technology, Gothenburg, Sweden (email: simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='lindberg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' andreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='wolfgang@qamcom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='se).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Abrudan is with Nokia Bell Labs, Finland (email: traian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='abrudan@nokia-bell-labs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' This work was supported, in part, by the European Commission through the H2020 project Hexa-X (Grant Agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 101015956) and by the MSCA-IF grant 888913 (OTFS-RADCOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' communication [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' As a result, high-accuracy context-aware applications such as the tactile Internet, augmented reality, and smart cities will be supported in next-generation wireless networks [4]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In global navigation satellite systems (GNSSs) and tra- ditional cellular networks, range-based algorithms, such as trilateration, are usually applied for estimating position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' When moving to higher carrier frequencies, more antennas can be packed in a single array due to shorter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' As a consequence, in addition to delay estimation, angle-of-arrival (AOA) and angle-of-departure (AOD) information can be exploited for localization, and a variety of new localization techniques have recently emerged in the fifth/sixth generation (5G/6G) systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', [7]–[10], considering localization with minimal infrastructure requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Multipath components (MPCs), which are usually considered as destructive signals, can be resolved in the emerging wireless systems, thereby enabling single-base station (BS) positioning and mapping [7] as well as simultaneous localization and mapping (SLAM) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' When the UE is equipped with an antenna array, orientation estimation is also possible [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In Doppler-assisted localiza- tion, although new unknowns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', velocity) are introduced, localization performance can be improved because mobility forms a virtual array with a large aperture compared to the stationary scenarios [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Most localization works rely on idealized models of the received signals as a function of the channel parameters (angles, delays, Dopplers) induced by the propagation environment, based on the assumption of deter- ministic and sparse channels in high-frequency systems [1], [11]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' However, in sub-THz bands for 6G communica- tions, pilot signals can be distorted due to the presence of hardware impairments (HWIs) such as phase noise (PN), carrier frequency offset (CFO), mutual coupling (MC), power amplifier nonlinearity (PAN), array gain error (AGE), antenna displacement error (ADE), in-phase and quadrature imbalance (IQI), etc [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Consequently, when algorithm derivation is based on a mismatched model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', without considering the HWIs in the channel model), the localization performance is unavoidably affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The effect of HWIs on communication have been stud- ied extensively in the literature [16]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In [16], differ- ent types of impairments have been accurately modeled and the effects on a multiple-input-multiple-output (MIMO)- orthogonal frequency-division multiplexing (OFDM) system are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In [17], an aggregate statistical HWI model con- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='01042v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='SP] 3 Jan 2023 2 sidering PAN, local oscillators with PN, and finite-resolution analog to digital converters (ADCs) is derived and validated with numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The residual additive transceiver hardware impairments, caused by direct current offset, MC, IQI and quantization noise, are discussed in [18], with the de- rived spectral efficiency to quantify the degradation caused by the HWIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In addition to modeling and analysis of the HWIs, research has also been conducted on impairment mitigation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' By incorporating signal distortions caused by hard- ware impairments, beamforming optimization is performed to maximize the received SNR at the destination [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' A channel estimation algorithm is designed by taking into account the transceiver impairments in [21], showing a better bit error rate and normalized mean-squared-error performance than the conventional algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Contrary to model-based solutions, channel estimation under HWI can also be formulated as a deep learning problem [20], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Nevertheless, these works focus only on communication performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Research on localization and sensing (here, sensing includes detection, angle, and delay estimation, as well as tracking of passive targets) considering HWIs is recently drawing attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The effect of PN on automotive radar [23]–[25], MC on AOA estimation [26], IQI on mmWave localization [27], and PAN on joint radar-communication systems [28] have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' However, these works only consider one or two types of impairments and cannot provide a thorough analysis in real scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In [29], [30], the impairments are modeled as additional Gaussian noise, with the variance determined by an ad hoc HWI factor, from which the error bounds for 3D localization are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' However, this approach fails to capture the contribution of each individual HWI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In [31], which forms the basis of the current paper, a simplified syn- chronized single-input-multiple-output (SIMO) uplink system is considered for 2D positioning, and the results show that dif- ferent types of impairments affect angle and delay estimation in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Nevertheless, the perfect synchronization assumption is impractical, and the impairments such as array calibration error and IQI are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Besides analyzing the effect of HWIs on localization or communication alone, more recent works consider the HWIs in joint localization and communication systems and use learning-based methods to mitigate the performance degradation [32], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Nevertheless, only a limited number of impairment types are discussed (MC and ADE in [32], IQI and DC offset in [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In addition, no theoretical analysis is performed in these works, and the relative importance of each HWI on communication compared to localization is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Hence, there is a need for a more systematic study that evaluates the effect of different types of HWI on both communication and localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In this paper, we investigate the problem of estimating the 3D position and 3D orientation of a multiple-antenna UE using several multiple-antenna BSs in a realistic uplink scenario for a sub-THz communications system under a wide variety of HWIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Specifically, we consider an OFDM-based system by rigorously modeling the impact of various HWIs on the received observations, and assume that the correspond- ing channel estimation and localization algorithms have no knowledge about these HWIs, resulting in degradation of lo- calization and communication performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The misspecified Cram´er-Rao bound (MCRB) [34]–[36] is employed to quantify the estimation performance loss due to model mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In addition, the effect of HWI on communication is evaluated numerically in terms of symbol error rate (SER) based on the developed model for a hardware-impaired channel under the same HWI levels, which allows a fair comparison of the impact of HWI on communication and localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The contributions are summarized as follows: Channel model with multiple HWIs: Based on the ideal MIMO model (mismatched model (MM)) with perfect hardware, we develop a more general channel model for the considered sub-THz system (true model (TM)) that can accommodate a variety of HWI types (including PN, CFO, MC, PAN, AGE, ADE, and IQI) in a 3D environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' To the best of the authors’ knowledge, this is the first study to derive a comprehensive and realistic signal model for localization and communications that provides explicit modeling of major HWIs that are likely to affect 6G communication systems at high-frequency operation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', mmWave and sub-THz bands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Analytical performance prediction of channel param- eter estimation and localization under HWIs: We leverage MCRB analysis to evaluate the effect of indi- vidual and combined HWIs on the estimation of channel parameters (AOD, AOA and delay estimation) and on the corresponding localization performance (3D position and 3D orientation estimation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' More specifically, the bounds provide the best performance of estimators using a MM to process the TM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Performance evaluation and comparison with commu- nication: Extensive simulations are performed to verify the performance analysis of the effect of HWI on lo- calization and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For communication, we approximate the HWIs as additive noise and evaluate the effect of individual and aggregated HWIs on communica- tion performance in terms of SER using a 16-quadrature amplitude modulation (QAM) modulation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In addition, the effect of different HWIs on localization and communication is evaluated with dominant factors identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We notice that the dominant factors that affect delay estimation will also affect communication, whereas the impairments that only affect AOA, AOD have a limited impact on communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Section II reviews the system models with and without HWIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Section III describes the channel estimation and localization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Theoretical performance analysis is carried out in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Next, the simulation results are presented in Section V, followed by the concluding remarks in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Notations and Symbols: Italic letters denote scalars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' a), bold lower-case letters denote vectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' a), and bold upper- case letters denote matrices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (·)⊤, (·)H, (·)−1, tr(·), and ∥·∥ represent the transpose, Hermitian transpose, inverse, trace, and ℓ-2 norm operations, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' A⊙B, A⊗B, a◦b are the Hadamard product, Kronecker product, and outer product, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' [·, ·, · · · , ·]⊤ denotes a column vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 3 IFFT D/A mix LO MC + AGE + ADE PN + CFO mix LO A/D FFT LNA LNA LNA LNA PN + CFO PA PA PA PA IQI MC + AGE + ADE PAN IQI UE xg lth BS yg Channel Hl Estimated UE state: s = [p⊤ U , BU, vec(RU)⊤]⊤ η1 · · ηL Channel parameter extraction: ηl = [φB, θB, φU, θU, τ, ρ, ξ]⊤ Estimated channel: ˆHl Received symbol: ˆyl = [y⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , y⊤ g ]⊤ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Block diagram of the hardware impairments considered at transmitter and receiver (highlighted in shaded regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' When the localization algorithm does not have perfect knowledge of the generative model, it operates under model mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' PN (phase noise), CFO (carrier frequency offset), MC (mutual coupling), PAN (power amplifier non-linearity), AGE (array gain error), ADE (antenna displacement error), and IQI (in-phase and quadrature imbalance) are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' tr(·) returns the trace of a matrix, [·]i,j is the element in the ith row, jth column of a matrix, and [·]a:b,c:d is the submatrix constructed from the ath to the bth row, and the cth to dth column of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' SYSTEM MODEL In this section, we start with a MIMO channel model (HWI- free model) and then describe the model considering the HWI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Geometric Model The block diagram of considered HWIs and localization procedures are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' An uplink MIMO system consisting of a UE and L BSs is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The BSs and UE are equipped with an uniform planar array (UPA) (antennas lie on the local YZ plane) driven by a single radio-frequency chain (RFC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The number of antenna elements at the l-th BS and the UE arrays is denoted as NB,l = NB,l,z × NB,l,y and NU = NU,z × NU,y where Nz and Ny are the number of antennas on the Z and Y axes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The BSs are perfectly synchronized while a clock offset BU exists between the UE and the BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We denote the array center and orientation of the l-th BS as pB,l ∈ R3 and oB,l ∈ R3 in the global coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Similarly, the position and orientation of the UE can be denoted as pU, oU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Since the orientation represented by an Euler angle vector is not unique, we use rotation matrices RB,l ∈ SO(3) and RU ∈ SO(3) in orientation estimation (more details can be found in [1], [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In localization, channel estimations are performed at each BS, and all estimates are combined to find the UE state parameter vector s = [p⊤ U , BU, vec(RU)⊤]⊤ ∈ R13, containing the UE position pU, clock offset BU, and rotation matrix RU, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Hardware Impairment-free Model Considering the transmitted OFDM symbol1 at the g-th transmission and k-th subcarrier, xg,k with an average transmit 1For positioning, constant modulus pilots are typically used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For commu- nication, different modulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', 16-QAM) can be adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' power E{|xg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='k|2} = P/NU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' its observation at a specific BS (the index l is omitted for convenience) can be formulated as yg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='k = w⊤ g Hkvgxg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='k + ng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (1) where wg ∈ CN is the combiner at the BS for the g-th transmission and vg ∈ CN is the precoder at the UE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' both with unit amplitude entries,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' ng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='k ∈ CN(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' wH g wgσ2 n) is the noise vector with each entry following a complex normal distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' with σ2 n = N0W (N0 is the noise power spectral density (PSD) and W = K∆f is the total bandwidth with K subcarriers and subcarrier spacing ∆f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We assume Hk re- mains the same during G transmissions (within the coherence time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The channel matrix at subcarrier k is given by Hk = αdk(τ)aB(ϕB)a⊤ U (ϕU) � �� � LOS path (2) + P � p=1 αpdp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='k(τp)aB(ϕB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='p)a⊤ U (ϕU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='p) � �� � NLOS paths ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' where for the LOS path,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' α = ρe−jξ is the complex channel gain assumed to be identical for different subcarriers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' dk(τ) = e−j2πk∆f τ (∆f is the subcarrier spacing) as a function of the path delay τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' while aB(ϕB) and aU(ϕU) are the receiver and transmitter steering vectors as a function of the AOA ϕB = [φB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' θB]⊤ (azimuth angle φB and elevation angle θB),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' and AOD φU = [φU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' θU]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' A steering vector a(ϕ) of an N- element array is a function of the direction of the (incoming or outgoing) signal and the locations of the antenna elements, which can be expressed as [1] a(ϕ) = ej 2πfc c Z⊤t(ϕ), (3) where we apply the exp operator element-wise, Z ∈ R3×N is the matrix containing the position of the N antennas in the local coordinate system (all zeros in the first row of Z) and t(ϕ) = [cos(θ) cos(φ), cos(θ) sin(φ), sin(θ)]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For the NLOS paths, each path can correspond to single or multi-bounce reflections, or diffuse scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Hence, the NLOS path will not be utilized for the positioning of the UE in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We further make the assumption that the LOS path is resolvable with respect to the NLOS paths (though the NLOS paths may 4 be mutually unresolved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' This is a reasonable assumption2 for 6G systems, due to large bandwidth and a large number of antennas [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Without significant loss of generality, the channel matrix for the kth subcarrier can thus be simplified as Hk = αdk(τ)aB(ϕB)a⊤ U (ϕU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (4) Correspondingly, the channel geometric parameter vector of the line-of-sight (LOS) path between a BS and the UE is defined as ηch = [η⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , η⊤ L]⊤ with ηl = [ϕ⊤ B,l, ϕ⊤ U,l, τl, ρl, ξl]⊤ ∈ R7 for the lth BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For later analysis, we define a vector by removing all the nuisance parame- ters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', complex channel gain for each path) as cch = [c⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , c⊤ L]⊤ with cl = [ϕ⊤ B,l, ϕ⊤ U,l, τl]⊤ ∈ R5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The geo- metric relationships between the channel parameters vector c and the state parameters s can be expressed as ϕB = �φB θB � = �arctan 2(tB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' tB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='1) arcsin(tB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='3) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (5) ϕU = � φU θU � = � arctan 2(tU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' tU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='1) arcsin(tU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='3) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (6) τ = ∥pU − pB∥ c + BU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (7) where c is the speed of light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' tB = [tB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' tB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' tB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='3]⊤ and tU = [tU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' tU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' tU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='3]⊤ are the direction vectors in the local coordinate system that can be expressed using global direction vectors and rotation matrices as tB = R−1 B pU − pB ∥pU − pB∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (8) tU = R−1 U pB − pU ∥pB − pU∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (9) Finally, by concatenating all the received symbols into a column, we obtain the received symbol block y ∈ RGK as y = [y⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , y⊤ g , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , y⊤ G]⊤, where yg = [yg,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , yg,K]⊤ can be expressed as yg = α(w⊤ g a(ϕB)a⊤(ϕU)vg)d(τ) ⊙ xg + ng, (10) in which d(τ) = [d1(τ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , dK(τ)]⊤, xg = [xg,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , xg,K]⊤, and ng = [ng,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , ng,K]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Hardware Impairments In this work, several types of HWIs are considered as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We study the effects of MC, PAN, AGE, ADE, PN, CFO, and IQI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Note that the impairments such as PN, CFO, MC, AGE, ADE and IQI exist both at the transmitter and the receiver, while the PAN appears only at the transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The HWIs are usually compensated offline during calibration or online with dedicated signals and routines, depending on whether the impairment is static or time-variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Both the offline and the online methods will have residual errors, which can be modeled as random perturbations around the nominal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' This work focus on the impact of these residual errors after calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For online methods, these random realizations correspond to different times for a specific device, 2For example, with a bandwidth of 1 GHz and 8 × 8 BS arrays, a delay resolution of 30 cm and an angle resolution of 22 degrees is achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Unless the UE is very close to a reflector, multipath can be resolved in the combined range-angle domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' while for offline methods, these random realizations should be interpreted as corresponding to an ensemble of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The imperfections of ADC, digital to analog converter (DAC), low-noise amplifier and mixer are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 1) Phase Noise and Carrier Frequency Offset: Imper- fect local oscillators (LOs) in the up-conversion and down- conversion processes add PN to the carrier wave phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In ad- dition, when the down-converting LO in the receiver does not perfectly synchronize with the received signal’s carrier [37], CFO occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In general, both PN and CFO are estimated and compensated by the receiver [38], so we only consider the residual PN and residual CFO at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' With PN and CFO, the observation, yg,k, is modified as in [39] yg,k → f ⊤ k EgΞgFHyg, (11) Eg = ej 2πϵgKtot K diag([1, ej 2πϵ K , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , ej 2π(K−1)ϵ K ]), (12) Ξg = diag([ejνg,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , ejνg,K]), (13) where yg is the received signals of the ideal model without PN or CFO (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', from (1)), F = [f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , fK] is the FFT matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The CFO matrix Eg considers both inter-OFDM symbol phase changes as well as inter-carrier interference [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' More specifically, Ktot = K + Kcp with Kcp as the length of the cyclic prefix, and ϵ is the residual CFO with ϵ ∼ N(0, σ2 CFO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Ξg is the residual3 PN matrix with νg,k ∼ N(0, σ2 PN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In (11), the vector yg is converted to the time domain by FHyg, where the successive PN samples, as well as the CFO, are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Finally, f ⊤ k extracts the k-th subcarrier after applying an FFT to EgΞgFHyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Note that the residual CFO ϵ is fixed for each realization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', one localization measurement with G transmission), while the PN νg,k is different for all the subcarriers and OFDM symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 2) Mutual Coupling: MC refers to the electromagnetic interaction between the antenna elements in an array [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For a UPA, we adopt the MC model as in [43] by assuming the antenna is only affected by the coupling of the surrounding elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' As a result, the MC matrix can be expressed as C = � ������ C1 C2 0 · · 0 C2 C1 0 · · 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 0 · · C2 C1 C2 0 · · 0 C2 C1 � ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (14) Here, C ∈ CNzNy×NzNy is the MC matrix with sub-matrices C1 = Toeplitz([1, cx, 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , 0]) ∈ CNy×Ny and C2 = Toeplitz([cx, cxy, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , 0]) ∈ CNy×Ny [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For convenience, we use one variable σMC to denote the severity of the MC such that cx ∼ CN(0, σ2 MC) and cxy ∼ CN(0, σ2 MC/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The MC leads to the following substitution of the channel matrix Hk → CBHkC⊤ U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (15) 3) Power Amplifier Nonlinearity: For the PA nonlinearity, we consider a Q-th order memoryless polynomial nonlinear 3Note that νg,k and ϵ represent residual PN and CFO that remains after the carrier synchronization process processing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', [41], [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Hence, νg,k is assumed to be independent across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 5 model with a clipping point xclip ∈ R as [16] hPA(ˇxt) = ��Q−1 q=0 βq+1ˇx|ˇx|q |ˇx| ≤ xclip, �Q−1 q=0 βq+1 ˇx |ˇx||xclip|q+1 |ˇx| > xclip, (16) where ˇxt = xt/R denotes the voltage of the transmitted time- domain signal (R is the load impedance in Ohms) in the time domain and β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , βQ are complex-valued parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We assume that (16) models both the effect of digital pre- distortion and power amplifier, and we use non-oversampled signals as input to PA for tractable localization performance analysis4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Note that the PA affects the time domain signals and each antenna at the Tx has a separate PA, and the PA model in (16) does not consider the out-of-band emissions (only the in-band distortion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For simplicity, the models are the same for different PAs and hPA(ˇxt) returns the time domain signal vector (by operating point-wise on each of the elements) with PA nonlinearity introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 4) Array Calibration Error: The AGE and ADE are con- sidered in the array calibration error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We define the complex excitation coefficient of the n-th antenna at direction ϕ as [45] bn(ϕ) = (1 + δa)ejδp, (17) where δa ∈ N(0, σ2 AA), and δp ∈ N(0, σ2 AP) are the relative amplitude error and phase error, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Regarding the displacement error, we assume the n-th antenna position has a displacement on the 2D plane of the local coordinate system as ˜zn = zn + [0, δn,y, δn,z]⊤, (18) with dn ∈ R3 is the ideal position of the nth antenna in the local coordinate system, δn,y, δn,z ∈ N(0, σ2 ADE) are the displacement error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The steering vector is then modified as a(ϕ) → b(ϕ) ⊙ ej 2π λ ˜Z⊤t, (19) where ˜Z = [˜z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , ˜zN] contains the geometry information of all the antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The array calibration error is fixed for a certain array and varies across different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 5) In-phase and quadrature imbalance: IQI operates on the time domain signal and the transmitted symbol vector is modified as [27], [46] xg → F(αUFHxg + βUFHx∗ g) = αUxg + βUx∗ g, (20) where the FFT matrix F and IFFT matrix FH switch be- tween time and frequency domain, αU = 1 2 + 1 2mUejψU, βU = 1 2 − 1 2mUejψU with mU and ψU as the amplitude and phase imbalance parameters at the UE side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We assume that the IQI is compensated in the system, leading to a residual impairment and the imbalance parameters can be modeled as mU ∼ N(1, σ2 IA) and φU ∼ N(0, σ2 IP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Similarly, the IQI at the receiving BS can be expressed as yg → αByg + βBy∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (21) More accurate frequency-dependent IQI models can be found in [47], [48], which is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 4In order to fully characterize the effect of PAN, an oversampled model is needed, which also captures the intersymbol interference introduced by the nonlinearity, in addition to the symbol distortion (see (25) in [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Hardware-impaired Model Considering all types of HWIs described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' II-C and substituting (11)–(21) into (10), the observation can be rewritten in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 1) Transmit Signal under HWI: The precoded transmitted signal across subcarriers and antennas is modified from Xg = xgv⊤ g ∈ CK×NU to ˇXg = FhPA(EUΞU(αUFHxg + βUFHx∗ g)v⊤ g � �� � precoded time domain signal before PA ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (22) 2) Channel under HWI: The channel is modified from Hk = αdk(τ)a(ϕB)a⊤(ϕU) ∈ CNB×NU in (4) to ˇH = αdk(τ)CB(bB(ϕB) ⊙ ej 2π λ ˜Z⊤ B tB(ϕB) � �� � steering vector ˜aB(ϕB) ) × (bU(ϕU) ⊙ ej 2π λ ˜Z⊤ U tU(ϕU) � �� � steering vector ˜aU(ϕU) )C⊤ U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (23) 3) Received Signal under HWI: The received signal is modified from yg ∈ CK×1 to (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Summary of the Models To summarize, we have defined a MM in (1) without consid- ering the HWI, which will be used for algorithm development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' With HWIs introduced, the impaired model defined in (24) will be used as the TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In the following section, we will evaluate the impact of using the MM to process data generated by TM on localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For the sake of convenience in performance analysis, we use µg(η) and ¯µg(η) to denote the noise-free observation of (1) and (24), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' LOCALIZATION ALGORITHM Based on the models described above, a two-stage local- ization5 problem can be formulated such that the channel parameter vectors ˆηch = [η⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , η⊤ L]⊤ are firstly estimated based on the observation vector ˆy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , ˆyL from all the BSs, and then the stage vector ˆs is determined from ˆηch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Mismatched Maximum Likelihood Estimator The maximum likelihood estimation (MLE) can be em- ployed when the observation y is generated from the same model used by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' If y ∼ fTM(y|¯η), the MLE of the UE position and channel gain is ˆηMLE = arg max ¯η ln fTM(y|¯η), (25) where ln fTM(y|¯η) is the log-likelihood of the TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' However, if y ∼ fTM(y|¯η), but the estimator uses fMM(y|η) ̸= fTM(y|¯η), the mismatched maximum likelihood estimation (MMLE) is given by ˆηMMLE = arg max η ln fMM(y|η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (26) More specifically, equation (26) formulates the MMLE for channel parameters extraction, which can also be implemented 5In contrast, the direct localization estimates the state vector s from the observed signal vector y directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Considering the high complexity involved, we adopt two-stage localization in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 6 ˇyg = F(αB(EB,gΞB,gFH( ˇXg ˇH⊤wg ⊙ d(τ)))) + βB(EB,gΞB,gFH( ˇXg ˇH⊤wg ⊙ d(τ)))∗) + ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (24) in position and orientation estimation with known or approx- imated likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' A practical approach is to use the gradient descent method with an initial point, which will be detailed in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Channel Parameters Estimation The channel parameters estimation will be performed with a coarse estimation using ESPRIT, which provides a good initial point for a refined estimation using (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 1) Coarse Estimation using ESPRIT: We aim to obtain an initial estimate of the channel parameters with a low com- plexity, which can be solved using tensor-based beamspace ESPRIT6 algorithm [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' To implement the beamspace ES- PRIT algorithm, we reformulate a beamspace channel matrix H(b) based on the signal model in (1) as H(b) k = αdk(τ)WHaB(ϕB)a⊤ U (ϕU)V (27) where W = T1⊗T2 ∈ CN1N2×M1M2 and V = (T3⊗T4)∗ ∈ CN3N4×M3M4 are the combining matrix and precoder matrix and the total number of transmissions G = M1M2M3M4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Since the first row of the antenna position matrix ˜Z is all zeros (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' II-A and equation (3)), we can express the steering vector in (3) as aB(ϕB) = a(M1)(ω1) ⊗ a(M2)(ω2), (28) with ω1 = π sin(φB) cos(θB), ω2 = π sin(θB), (29) a(M1) B (ω1) = ej 2πfc sin(φB) cos(θB) c ˜zB,2 = ej 2 λc ω1˜zB,2, (30) a(M2) B (ω2) = ej 2πfc sin(θB) c ˜zB,3 = ej 2 λc ω2˜zB,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (31) Here, ˜z⊤ B,2 ∈ C1×NB and ˜z⊤ B,3 ∈ C1×NB are the second and third row of the matrix ˜Z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The combining matrix can then be defined in terms of a grid of the spatial frequencies ¯ω1 = [¯ω1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , ¯ω1,M1] and ¯ω2 = [¯ω2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , ¯ω2,M2] as T1 = [a(N1)(¯ω1,1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , a(N1)(¯ω1,M1)]⊤ ∈ CN1×M1, (32) T2 = [a(N2)(¯ω2,1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , a(N2)(¯ω2,M2)]⊤ ∈ CN2×M2, (33) where ¯ω1,m and ¯ω2,m are decided by beamforming directions (detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' A similar reasoning applies to the steering vectors a(M3) U (ω3) and a(M4) U (ω4) at UE to define T3 and T4, with ω3 = π sin(φU) cos(θU), ω4 = π sin(θU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (34) We further define b(Mn)(ωn) = TH naNn(ωn) ∈ CMn for n ∈ {1, 2, 3, 4} and b(M5)(ω5) = d(τ) (ω5 = 2π∆fτ), and the beamspace channel matrix in (27) can be represented by a tensor H ∈ CM1×M2×···×M5 as [14] H(b) = αb(M1)(ω1) ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' ◦ b(M5)(ω5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (35) In practice, the estimated beamspace channel matrix can be estimated with known pilot signals as vec( ˆH(b) k ) = [ˆy1,k/x1,k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , ˆyG,k/xG,k]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' By rearranging the estimated 6While this work considers only the LOS channel, the ESPRIT also works for the scenarios with NLOS paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' channel into a tensor ˆH (b) shown in (35), the beamspace tensor-based ESPRIT method can then be used to estimate ω1 to ω5 and obtain the AOA, AOD, and delay accordingly [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 2) Fine Estimation using MMLE: From ESPRIT, we can obtain an initial estimate of the channel parameters ˆη0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The refinement of the initial estimate can be formulated as an optimization problem, based on (26), as ˆη = arg min η ∥y − µ(η)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (36) Since α appears linearly in the noise-free observation µ, we further define γ(η) = µ(c)/α with c = [ϕ⊤ B , ϕ⊤ U , τ]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' By setting ∂∥y − µ(η)∥2/∂α = 0, we can have ˆc = arg min c ∥y − γH(c)y ∥γH(c)∥2 γ(c)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (37) In this way, the nuisance parameters can be removed, which reduces the dimension of the unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Localization Algorithm 1) Coarse Estimation: Given the estimated geometric pa- rameter vector cl (1 ≤ l ≤ L) for all the channels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' the least squares solution for coarse estimation of position and orientation as [49] ˆRU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='LS = � UVT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' if det(UVT) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' UJVT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' if det(UVT) = −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (38) [ˆpU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='LS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' ˆBU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='LS]⊤ = (Q⊤ 3 Q3)−1Q⊤ 3 q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (39) where J = diag([1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' −1]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' U and V are the unitary basis matrices of the singular value decomposition of the matrix Q1Q⊤ 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' together with Q3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' q are given by [49] Q1 = −[RB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='1t(ˆϕB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , RB,Lt(ˆϕB,L)], (40) Q2 = [t(ˆϕU,1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , t(ˆϕU,L)], (41) Q3 = � �� I3 RB,1t(ˆϕB,1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' I3 RB,Lt(ˆϕB,L) � �� , (42) q = � �� p(1) B + RB,1ˆτ1t(ˆϕB,1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' pB,L + RB,LˆτLt(ˆϕB,L)]⊤ � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (43) Different from the algorithm in [49], the estimator for position and clock offset in (39) does not require the orientation of the UE RU, which is still sufficient as a coarse estimate, as will be shown in the simulation section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 2) MMLE: Once the initial position and orientation results are obtained, joint position and orientation estimation using MMLE can be formulated as ˆs = arg min s L � l=1 (cl(s) − ˆcl)⊤Σ−1 cl (cl(s) − ˆcl), (44) which can be solved using the manifold optimization toolbox Manopt [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Note that the covariance matrix may not be accurately obtained in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We formulate localization as an MMLE problem with two purposes: (a) to evaluate the 7 performance improvement with known covariance matrices compared to the coarse estimation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (b) to validate the derived bound, which will be detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' LOWER BOUND ANALYSIS In the next, we derive the CRB for MM, as well as the MCRB for the mismatched estimator in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' CRB Analysis for the Mismatched Model Based on the defined channel parameter vector η and state vector s, the signal model in (1) and y ∼ fMM(y|η), the channel estimation CRB of the MM for the lth channel can be obtained as I(ηl)−1 ∈ R7×7 with [51] I(ηl) = 2 σ2n G � g=1 K � k=1 Re ��∂µg,k ∂ηl �H �∂µg,k ∂ηl �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (45) Here, Re{·} extracts the real part of a complex variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Consequently, the FIM of all the channel parameters ηch can be formulated as I(ηch) = blkdiag(I(η1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' , I(ηL)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (46) where blkdiag(·) returns the block diagonal matrix created by aligning the input matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The FIM of the state vector I(s) ∈ R13×13 can then be formulated as I(s) = M(M⊤ J⊤ S I(cch)JS M)−1M⊤, (47) where I(cch) ∈ R5L×5L is the EFIM of non-nuisance parameters cch obtained from I(ηch), JS ≜ ∂cch ∂s is the Jacobian matrix using a denominator-layout notation, M = blkdiag(I4×4, ¯M) with ¯M as [9] ¯M = 1 √ 2 � � −r3 03×1 r2 03×1 −r3 −r1 r1 r2 03×1 � � , (48) where r1, r2, and r3 are the first, second, and third columns of the UE rotation matrix RU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Based on I(η) in (45), we can define the AOD error bound (ADEB), AOA error bound (AAEB), and delay error bound (DEB) of the link between the UE and the lth BS) as AAEB = � tr([I(ηl)−1]1:2,1:2), (49) ADEB = � tr([I(ηl)−1]3:4,3:4), (50) DEB = � ([I(ηl)−1]5,5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (51) Similarly, based on I(s), we can define the position error bound (PEB), clock offset error bound (CEB) and orientation error bound (OEB) as PEB = � tr([I(s)−1]1:3,1:3), (52) CEB = � ([I(s)−1]4,4), (53) OEB = � tr([I(s)−1]5:13,5:13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (54) The bounds from (49)–(54) will be used to evaluate the channel estimation and localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In the next subsections, we will first formulate the MCRB for channel estimation, and then the mismatched lower bound for position and orientation estimation will be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Misspecified CRB of Channel Parameters For a given channel model, the model is said to be mis- matched or misspecified when y ∼ fTM(y|η), while the estimation is based on the assumption that y ∼ fMM(y|η)), where fTM(y|η) ̸= fMM(y|η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The lower bound (LB) of using a mismatched estimator can be obtained as [35] LB(¯η, η0) = A−1 η0 Bη0A−1 η0 � �� � =MCRB(η0) + (¯η − η0)(¯η − η0)⊤ � �� � =Bias(η0) , (55) where ¯η is the true channel parameter vector, η0 is the pseudo- true parameter vector (which will be introduced soon), and Aη0, Bη0 are two possible generalizations of the FIMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The LB is a bound in the sense that E{(ˆηMMLE − ¯η)(ˆηMMLE − ¯η)⊤} ⪰ LB(¯η, η0), (56) where the expectation is with respect to fTM(y|η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' What re- mains is the formal definition and computation of the pseudo- true parameter η0 and Aη0, Bη0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 1) Pseudo-true Parameter: Assume the probability density function (PDF) of the TM, where the observation data come from, is fTM(y|¯η), where y is the received signals and ¯η ∈ R7 (7 unknowns for this case) is the vector containing all the channel parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Similarly, the PDF of the MM for the received signal y can be noted as fMM(y, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The pseudo-true parameter vector is defined as the point that minimizes the Kullback-Leibler divergence between fTM(y|¯η) and fMM(y|η) as η0 = arg min η DKL(fTM(y|¯η)∥fMM(y|η)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (57) We define ϵ(η) ≜ ¯µ(¯η)−µ(η), and the pseudo-true parameter can be obtained as [36] η0 = arg min η ∥ϵ(η)∥2 = arg min η ∥¯µ(¯η) − µ(η)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (58) Hence, η0 can be found by solving (36) with the observation y = ¯µ(¯η), which can be accomplished using the same algorithm in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' III, initialized with the true value ¯η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 2) MCRB Component Matrices: The matrices Aη0 and Bη0 can be obtained based on the pseudo-true parameter vector η0 as [36] [Aη0]i,j = ˆ ∂2lnfMM(y|η) ∂ηi∂ηj fTM(y|¯η)dy ���� η=η0 = 2 σ2n Re � ∂2µ(η) ∂ηi∂ηj ϵ(η) − ∂µ(η) ∂ηj �∂µ(η) ∂ηi �H������ η=η0 (59) and [Bη0]i,j = ˆ ∂lnfMM(y|η) ∂ηi ∂lnfMM(y|η) ∂ηj fTM(y|¯η)dy ���� η=η0 = 4 σ4n Re �∂2µ(η) ∂ηi ϵ(η) � Re �∂2µ(η) ∂ηj ϵ(η) � + 2 σ2n Re � ∂µ(η) ∂ηj �∂µ(η) ∂ηi �H������ η=η0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (60) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Absolute Lower Bound (ALB) for Localization Another way to interpret the LB specified in (55) is that the estimated channel parameter vector from an efficient estimator 8 follows a nonzero-mean multi-variable Gaussian distribution as ˆηl ∼ N(η0,l, A−1 η0,lBη0,lA−1 η0,l), (61) while the assumed distribution of the MMLE is ˆηl ∼ N(ηl(¯s), I(ηl)−1), (62) where ¯s is the true state of the UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' As a result, the position and orientation estimation (from the channel parameter vectors of all the paths) of the two-stage localization problem is another mismatched problem and the bound follows as LB(¯s, s0) = MCRB(s0) + (¯s − s0)(¯s − s0)⊤ � �� � Absolute lower bound (ALB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' (63) Similar to (55), ¯s is the true state parameter vector, s0 is the pseudo-true state parameter vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' It is possible to derive the localization LB constrained MCRB [52];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' however, considering the high complexity when involving 3D orientation estimation, we will focus on the bias term, defined as the absolute lower bound (ALB) of the localization performance as ALB = (¯s − s0)(¯s − s0)⊤, which can sufficiently evaluate the effect of HWIs on localization as will be shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' V-C2 Following a similar derivation in (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The pseudo-true parameters for state vector s can be obtained as s0 = arg min ¯s � l (η0,l − ηl(¯s))⊤I(ηl)(η0,l − ηl(¯s)), (64) where η0,l = arg minη ∥¯µ(¯ηl) − µ(ηl)∥2 is the biased mapping obtained by calculating the pseudo-true parameters of the lth channel from (58), and I(ηl) is the inverse of the covariance matrix that can be obtained from (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Summary of Different Bounds In this section, we introduced different types of lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For channel geometric parameters, the CRB and LB are derived for AOA, AOD, and delay estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For state parameters, the CRB and ALB are derived for the position, orientation, and clock offset estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' All types of the lower bounds are summarized in Table I, which will be used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' V Numerical Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' TABLE I SUMMARY OF DIFFERENT LOWER BOUNDS Types Parameters Remarks AOA AOD Delay Channel Parameters CRB AAEB ADEB DEB (49)-(51) LB AALB ADLB DLB (55) Position Orientation Clock Offset State Parameters CRB PEB OEB CEB (52)-(54) ALB PALB OALB CALB (63) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' NUMERICAL RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Default Parameters We consider a 3D MIMO uplink scenario with one UE and two BSs, and the simulation parameters7 can be found 7The PA parameters are estimated from the measurements of the RF WebLab, which can be remotely accessed at www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='dpdcompetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Part of the parameters come from the Hexa-X Deliverable 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We utilize 10% of the total number of subcarriers Kcom for localization, resulting in K = 100 subcarriers as pilot signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The amplitude of the channel gain is calculated as ρ = λ 4πcτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The selection of these parameters is to show the performance of the estimator in comparison to the derived bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The analysis of each HWI type is also discussed in the simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Regarding the evaluation of communication performance, only the first BS is considered, and 16-QAM modulation is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Different from localization, where BS-UE beam sweeping is needed, we evaluate the effect on communication with fixed precoder and combiner vectors across different transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' By considering all HWIs, we assume the chan- nel can be perfectly estimated (with a sufficient number of pilots) as ˆH = ˇH = ˆaBˆaU with ˆaB = √αCB˜aB(ϕB) and ˆaU = √α˜aU(ϕU)C⊤ U from (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In order to maximize the SNR with the amplitude constraints of the precoder and combiner, we choose w and v respectively as the conjugate of ˆaB and ˆaU with each of the elements normalized to a unit amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For each realization, 20 OFDM symbols are sent with data drawn randomly from 16-QAM, and SER is used to evaluate the effect of HWIs on communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For localization, the pilot signal xg,k is chosen with random phase and a constant amplitude |xg,k|2 = P/NU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' To assist the beamspace ESPRIT algorithm, we set the number of sweeping beams as M1 = 4, M2 = 4, M3 = 3, M4 = 3 with a total number of transmission G = 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For a specific spatial frequency vector ¯ωn (n ∈ {1, 2, 3, 4}), we assume the sweeping range as (Mn − 1)∆ω centered at the location prior ˚ωn = ωn + δω, where ωn is defined in (29), (34), and δω is the error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' More specifically, we choose ¯ωn,m = ωn + δω + 2m−Mn−1 2 ∆ω, with ∆ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='15 and δω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='05 in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The sweeping priority is set to ‘BS-first’ by default, which means that the UE can change its precoder vector when the BS finishes the M1M2 = 16 different sweeping beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Different error bounds (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', CRBs, LBs, ALBs from Table I) are utilized as localization performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The Effect of HWIs on Communication 1) The Effect of HWIs on SER: We approximate the effect of HWIs on communication as the random noise and evaluate the effect on SER based on numerical and analytical results8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Considering that the effects of some HWIs depend on the amplitude of the symbol (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', PAN), we also obtain the minimum and maximum noise levels across different symbols to evaluate the lower bound and upper bound of the SER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The SERs of 16-QAM with different transmit power for different HWI coefficients are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 2, where the black solid curve is the benchmark SER without HWIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' By default, cHWI = 1, and the HWI level is the same as the parameters in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' A value of cHWI = 10 indicates that the standard deviations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', σPN, σCFO) of all the impairments (except for 8The SER of M-QAM can be calculated as SERM = 1 − (1 − 2 √ M−1 √ M Q( � 3SNR M−1 ))2 [53, (6,23)], where Q(·) is the Q-function and SNR is effective SNR considering both approximated HWI noise and background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 9 TABLE II DEFAULT SIMULATION PARAMETERS Parameters True Model Mismatched Model BS p1 B = [0, 0, 3]⊤, p2 B = [0, 5, 3]⊤ BS Orientations o1 B = [0◦, 15◦, 0◦]⊤, o2 B = [−30◦, 15◦, 0◦]⊤ BS Antennas N 1 B = N 2 B = 8 × 8 UE Position pU = [4, 2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='5]⊤ UE Orientation oU = [180◦, 0◦, 0◦]⊤ UE Antennas NU = 4 × 4 RFC at BS/UE 1 Carrier Frequency fc = 140 GHz Bandwidth W = 1 GHz Transmissions G = 4 × 4 × 3 × 3 = 144 Subcarriers (Total) Kcom = 1040 (∆f = 960 kHz) Subcarriers (Pilots) K = 100 Length of the CP Kcp = 7 Load Impedance R = 50 Ω Noise PSD N0 = −173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='855 dBm/Hz Noise Figure 10 dB Phase Noise σPN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='5◦ σPN = 0◦ Carrier Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Offset σCFO = 5e−4 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='036 ppm) σCFO = 0 Mutual Coupling σMC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='002 σMC = 0 β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='9798+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='0286j Power Amplifier β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='0122-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='0043j n/a β3 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='0007+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='0001j PA Clipping Voltage xclip = 1 V n/a Array Gain Error σGA = σGP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='002 σRA = σRP = 0 Antenna Disp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Error σAD = 5 um (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='3e−3λ) σAD = 0 IQ Imbalance σIA = σIP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='02 σIA = σIP = 0 PAN) are multiplied by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We can see from the figure that the analytical SERs with approximated noise levels (red, blue, and green markers) are close to the numerical SERs (solid red, blue and green curves), and both are within the lower and upper bounds (shaded areas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We can also see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 2 that the selected impairment level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', cHWI = 1) has limited effects on communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' However, we will show the localization performance will be affected by the same level of HWIs in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' V-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' −10 −5 0 5 10 15 10−7 10−5 10−3 10−1 P [dBm] SER (16-QAM) Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' without HWI Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' HWI (cHWI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='1) Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='-Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' HWI (cHWI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='1) Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' HWI (cHWI = 1) Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='-Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' HWI (cHWI = 1) Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' HWI (cHWI = 2) Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='-Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' HWI (cHWI = 2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The effect of different HWI levels on SER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Numerical results for 100 realizations and analytical results calculated with approximated equivalent HWI noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The boundaries of the shadow areas indicate the upper and lower bounds for SER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 2) The Effect of Individual HWIs on SER: We are also interested in the effect of individual HWIs on communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' By considering PN, CFO, PAN, and IQI one by one, the SERs under HWI are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Benchmarked by −10 −5 0 5 10 15 10−7 10−5 10−3 10−1 P [dBm] SER (16QAM) PN PAN CFO IQI MC+AGE+ADE Without HWI −10 −5 0 5 10 15 10−7 10−5 10−3 10−1 P [dBm] SER (16QAM) PN PAN CFO IQI MC+AGE+ADE Without HWI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The effect of individual HWIs on SER using approximated equivalent HWI noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Under current simulation parameters, the PN, PAN, CFO and IQI increase the SER, whereas the MC, AGE and ADE have negligible effects on communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' the solid black curve without HWIs, these factors degrade SERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We also performed simulations by including MC, AGE, ADE at the same time, as shown in the dashed curve with cross markers, and found their effects on communication are negligible under the current simulation setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 3) Insights into the Impact of HWI on Communication: To gain further insight into the effect of HWI on communication, we separate the overall system noise into equivalent HWI noise and background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 4 that the equivalent HWI noise is determined by the HWI level and has an approximately linear relationship with the transmit power (when working within the linear region of the PA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In addition to the fixed background noise, the overall equivalent noise level keeps increasing and is dominated by the HWIs at high transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' −10 −5 0 5 10 15 −110 −100 −90 −80 −70 −60 P [dBm] Equivalent Noise Level [dBm] Overall Noise (cHWI = 2) HWI Noise (cHWI = 2) Overall Noise (cHWI = 1) HWI Noise (cHWI = 1) Overall Noise (cHWI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='1) HWI Noise (cHWI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='1) Background Noise Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Visualization of overall system noise, equivalent HWI, and background noise with different transmit power P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The background noise has a large effect on communication in low transmit power, whereas the HWIs contribute more in high transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The Effect of HWIs on Localization Before analyzing the HWIs in detail, we first establish the validity of the derived bounds by comparing them against the performance of practical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 1) Channel Estimation Results: For convenient analysis, we adopt one specific realization of the HWIs for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The results of channel parameters estimation using ESPRIT (circle, 10 −10 0 10 20 30 40 10−5 10−4 10−3 10−2 10−1 100 101 P [dBm] AOA [◦] / AOD [◦] / Delay [m] AOA-ESPRIT AOD-ESPRIT Delay-ESPRIT AOA-MMLE AOD-MMLE Delay-MMLE AAEB ADEB DEB AALB ADLB DLB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Comparison between channel parameters estimation results (ESPRIT and MMLE) and different lower bounds (CRB of the MM and the LB of the mismatched estimator) in terms of AOA, AOD and delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Due to the HWIs, the performance starts to saturate when the transmit power exceeds 30 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Although the performance of the coarse estimation using ESPRIT (using a mismatched model) may not attain the theoretical bounds (especially for delay estimation), the refined results using MMLE can reach the LB (solid curves align well with the cross-marked dotted curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' −10 0 10 20 30 40 10−4 10−3 10−2 10−1 100 P [dBm] Pos [m] / Ori [◦] / Clock [m] POS-LS ORI-LS Clock-LS POS-MMLE ORI-MMLE Clock-MMLE PEB OEB CEB PALB OALB CALB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Comparison between localization results (position, orientation, and clock offset estimation) and different lower bounds (CRB of the MM and the LB of the mismatched estimator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We noticed that the LS estimators are sufficient for this 2-BS scenario, and the refined results using MMLE attain the ALBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' square, and diamond markers) and MMLE (solid curves) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The estimators are benchmarked by the CRBs of the ideal/mismatched model (CRB-MM, dashed curves) and the LB using a mismatched model (dotted curves with cross markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Note that the average transmit power P is calculated without considering the nonlinearity of the power amplifier (calculated before the PA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' When the transmit power P is low, the LB is determined by the MCRB (since the bias part is con- stant, see (55)) and has a similar performance as CRBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' This indicates that in low transmit power, the mismatched model will not significantly affect the performance, as the expected accuracy is low and limited by the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' With the increase of transmit power, the contribution of MCRB decreases due to an increased SNR, and eventually, the mismatched localization is lower bounded by the absolute lower bound (ALB) (bias part in (55)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' This indicates that the localization performance can no longer be improved by increasing transmit power, which cannot be ignored in scenarios requiring high-accuracy local- ization performance9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Regarding the estimators, the ESPRIT (using a mismatched model) provides low-complexity results with limited performance in delay estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' However, the refined results using MMLE can reach the LB (solid curves align well with the dotted curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 2) Localization Results: Based on the estimated channel parameters, we are able to estimate the UE position and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Similar to the channel estimation results, two estimators (LS and MMLE) and two bounds (CRB and LB) are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The results for localization are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' From the figure, we can see that at low transmit powers, the LB and CRBs coincide, implying that the HWIs are not the main source of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' At higher transmit powers (10 dBm for OEB, and 20 dBm for PEB), LB deviates from the CRBs, and the positioning performance is thus more severely affected by HWIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The MMLE in high SNR is close to the ALB, indicating the validity of the MCRB analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='P [dBm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='AALB (Average) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='AALB (Multi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='AAEB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='ADLB (Average) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='ADLB (Multi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='ADEB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='DLB (Average) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='DLB (Multi) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='DEB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='(a) PN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='P [dBm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='(b) CFO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='P [dBm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='(c) MC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='P [dBm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='(d) AGE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='P [dBm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='(e) ADE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='P [dBm] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='(f) IQI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' LBs of channel parameter estimation under different types of impairment with multiple realizations: (a) Phase noise, (b) Carrier frequency offset, (c) Mutual coupling, (d) Array gain error, (e) Antenna displacement error, (f) IQ-imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Now that the validity of the bounds has been established, we rely solely on the bounds to evaluate the effect of HWIs on localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' First, the impairments are studied individually, then the impact of the waveform type is evaluated, and finally, the impairment levels are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 9Note that the analysis here is under the same level of residual noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', PN, CFO, IQI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In practice, the impairment levels depend on specific HWI calibration algorithms and transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 11 3) The Effect of Individual Impairments: To understand the effect of different types of HWIs, we study the LB for AOA, AOD, and delay estimation by considering one type of HWIs at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 7 for (a) PN, (b) CFO, (c) MC, (d) AGE, (e) ADE and (f) IQI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The effect of PA will be separately discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' V-C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Considering we define the HWIs as random variables with a fixed impairment level as shown in Table II, we perform multiple hardware realizations with a fixed pilot signal and plot all the resultant LBs in the shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We can see that different types of the HWIs affect angle and delay estimation differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The PN, CFO, and IQI introduce noise on the symbols across different subcarriers and hence affect delay estimation10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Since the phase change introduced by CFO affects the phase changes across beams, angle estimation will also be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Instead of affecting the phase changes between different subcarriers, the MC, AGE, and ADE distort the steering vectors and therefore have a more significant effect on the angle estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For all the HWIs, the negative effect on the performance occurs when the transmit power is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' One special observation is that the effect of CFO on the AOA is less pronounced than on AOD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 7 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' This is because the sweeping strategy is ‘BS-first’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' For a system with analog arrays, the estimation of AOA/AOD relies on phase shifts across consecutive beams over time, meaning the angle cannot be estimated from a single receive beam, like in a digital array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' If the BS sweeps across different beams while the UE is using a fixed beam, the AOA can be estimated in one BS sweep, and the effect of CFO will be minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' However, the AOD estimation requires multiple BS sweeps, which increases the effect of CFO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' To verify the explanation, we further changed the sweeping strategy from ‘BS-first’ to ‘UE-first,’ and the results with different array sizes can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We can see that the AOA is less affected if the sweeping is ‘BS-first’ (blue curves in (a)) as shown in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Similarly, the AODs are less affected if the sweeping is ‘UE- first’ (dashed red curves in (b)) with a large UE array size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' However, when the array size is small, sweeping order will have less impact (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', the gaps are small between the dashed curves in (a) and the solid curves in (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 4) The Effect of PA with Different Pilot Signals: High peak- to-average-power ratio (PAPR) is one of the critical issues in implementing the OFDM signals, and a promising alternative is to use DFT-S-OFDM [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' When increasing the transmit power, the PAN is more likely to happen, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 9 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The delay estimation suffers more from the nonlinear distortion because of the clipping of transmit signal, which distorts the uniformity of phase changes across the subcarriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The effect on angle estimation is less pronounced (at the same level of transmit power) since different antenna elements experience similar distortions with identical PAs adopted in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We compare using the random OFDM symbols and the FFT version of the benchmark symbols (a special case of DFT-S-OFDM by choosing an identity mapping matrix [54]), and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Due to the 10If multiple RFCs or several local oscillators are adopted in the array, PN may have a larger effect on angle estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 0 10 20 30 40 50 10−3 10−2 10−1 100 P [dBm] Angle Error [◦] BS 8x8, UE 4x4, BS first BS 8x8, UE 4x4, UE first BS 4x4, UE 8x8, BS first BS 4x4, UE 8x8, UE first (a) AALB (average) 0 10 20 30 40 50 10−3 10−2 10−1 100 P [dBm] Angle Error [◦] (b) ADLB (average) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The effect of CFO on channel geometrical parameters with different sweeping strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The ‘BS first’ strategy (blue curves) works better for AOA estimation, while the ‘UE first’ strategy (red curves) works better for AOD estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 20 25 30 35 40 45 50 55 60 10−6 10−5 10−4 10−3 10−2 10−1 100 P [dBm] AALB (Average) AALB (Multi) AAEB ADLB (Average) ADLB (Multi) ADEB DLB (Average) DLB (Multi) DEB (a) OFDM 20 25 30 35 40 45 50 55 60 10−6 10−5 10−4 10−3 10−2 10−1 100 P [dBm] (b) DFT-S-OFDM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The effect of PA on channel parameters estimation using (a) OFDM, and (b) DFT-S-OFDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' reduced PAPR by DFT-S-OFDM, the localization performance can be improved, as shown in the right figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 5) Evaluation of HWIs with Different Impairment Levels: We further evaluate the position and orientation ALBs with different levels of HWIs by defining a HWI coefficient cHWI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' With different value of cHWI, the position ALB and orientation ALB by considering all the HWIs, and individual HWIs, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 10 (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' All the results indicate the 75th percentile of the total 100 realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We notice that the effect of PN, MC, AR, AG, and IQI on the localization increases approximately in a linear trend with impairment level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The CFO has a larger effect in high impairment level as the error residue accumulates over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 10, we can quantize the contribution of individual HWIs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', if the ALBs are much smaller than current CRB, the negative contribution of HWI on localization is negligible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In addition, it can also identify dominant impairment factors for further compensation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', ADE is one of the dominant factors under current system parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 12 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='5 1 10−6 10−3 100 10log(cHWI) PALB [m] ALL PN CFO MC AGE ADE IQI (a) PALB −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='5 1 10−6 10−3 100 10log(cHWI) OALB ALL PN CFO MC AGE ADE IQI (b) OALB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' An example of ALB with different levels of impairments: (a) PALB, (b) OALB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The ALBs of the position and orientation affected by the HWIs increase with cHWI (reflecting the impairment level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Summary From the simulation, we found that the HWIs affect both localization and communication, especially at high transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The equivalent noise is mainly contributed by HWIs for communication, and the localization performance will saturate due to model mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' However, different types of HWIs affect localization and communication differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The effect of the individual impairment on angle/delay estimation and communication (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', SER) is summarized in Table III, with two levels of impacts H/L to denote High/Low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Note that in this uplink scenario, the position estimation is mainly affected by AOA and TOA information, while the orientation estimation is mainly affected by AOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' As for the angle estimation for localization, the performance is strongly affected by CFO, MC, AGE, and ADE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' When talking about the TOA, it is mainly affected by PN, CFO and IQI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Since communication does not exploit the phase relationship between antennas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', no sweeping is needed once the communication link is established), SER will be affected by the same factors as delay estimation, which are verified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' It should be noted that the effect of CFO on AOA and AOD estimation depends on the sweeping order and number of transmissions, while the effect of PA depends on the transmit power and the nonlinear region of the amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' CONCLUSION As the requirements on localization and communication performance are more stringent to support new applications, HWIs become a prominent factor affecting the performance in 6G systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We have modeled different types of HWIs and utilized the MCRB to evaluate the localization error caused by model-mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The effects of HWIs on angle/delay and position/orientation estimation are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' We found that PN and IQI have a stronger effect on delay estimation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' while MC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' AGE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' and ADE have a more significant effect TABLE III SUMMARY OF THE EFFECTS OF HWIS ON LOCALIZATION AND COMMUNICATION Type of HWI AOD AOA TOA SER Phase Noise L L H H Carrier Frequency Offset H∗ H∗ H H Mutual Coupling H H L L Power Amplifier Nonlinearity H∗ H∗ H∗ H∗ Array Gain Error H H L L Antenna Displacement Error H H L L IQ Imbalance L L H H ∗The effect of CFO on angle estimations depends on the sweeping order and number of transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The effect of PAN depends on the transmit power and the nonlinear region of the amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' on angle estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The CFO and PAN affect both angle and delay, where the former one depends on the sweeping strategy and number of transmissions, and the latter factor is determined by the transmit power (or amplitude) of the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Furthermore, we evaluated the effect of individual HWIs on the communication performance in terms of SER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' The dominant impairments that degrade SER (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=', PN, CFO, PA, and IQI) are in good agreement with the factors that affect delay estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' In summary, the localization and communication perfor- mance that improves with transmit power in an ideal model will saturate due to the effect of HWIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' To fully realize the potential of 6G joint localization and communication system, a dedicated pilot signal design and algorithms for estimating and mitigating HWI are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfHPuf/content/2301.01042v1.pdf'} +page_content=' Further works can consider the effect of HWIs in multipath and 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a/FNAzT4oBgHgl3EQfw_7Q/content/tmp_files/2301.01732v1.pdf.txt b/FNAzT4oBgHgl3EQfw_7Q/content/tmp_files/2301.01732v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cd0ef46c07068ab21e370d395f6181e7d85bab49 --- /dev/null +++ b/FNAzT4oBgHgl3EQfw_7Q/content/tmp_files/2301.01732v1.pdf.txt @@ -0,0 +1,901 @@ +arXiv:2301.01732v1 [eess.IV] 4 Jan 2023 +IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS, VOL. XX, NO. XX, SUBMITTED FEB. 2023 +1 +UNAEN: Unsupervised Abnomality Extraction +Network for MRI Motion Artifact Reduction +Yusheng Zhou, Hao Li, Jianan Liu, Zhengmin Kong, Tao Huang, Euijoon Ah, and Zhihan Lv +Abstract—Motion artifact reduction is one of the most +concerned problems in magnetic resonance imaging. As +a promising solution, deep learning-based methods have +been widely investigated for artifact reduction tasks in +MRI. As a retrospective processing method, neural network +does not cost additional acquisition time or require new +acquisition equipment, and seems to work better than tra- +ditional artifact reduction methods. In the previous study, +training such models require the paired motion-corrupted +and motion-free MR images. However, it is extremely tough +or even impossible to obtain these images in reality be- +cause patients have difficulty in maintaining the same state +during two image acquisition, which makes the training +in a supervised manner impractical. In this work, we pro- +posed a new unsupervised abnomality extraction network +(UNAEN) to alleviate this problem. Our network realizes the +transition from artifact domain to motion-free domain by +processing the abnormal information introduced by artifact +in unpaired MR images. Different from directly generating +artifact reduction results from motion-corrupted MR im- +ages, we adopted the strategy of abnomality extraction to +indirectly correct the impact of artifact in MR images by +learning the deep features. Experimental results show that +our method is superior to state-of-the-art networks and can +potentially be applied in real clinical settings. +Index Terms— Magnetic Resonance Imaging, Motion Ar- +tifact Reduction, Unsupervised Learning. +I. INTRODUCTION +M +AGNETIC resonance imaging (MRI) is a non-invasive +medical imaging technique used in the diagnosis of +various diseases without radiation exposure. However, due to +the long acquisition time, MRI is sensitive to the patient’s +movement [1], and incorrect K-space signal filling cause +blurring or ghosting artifacts, which in turn affects the patient’s +diagnosis. To solve motion-related problems, researchers have +proposed a variety of methods to prevent movement or correct +artifacts [2]–[6]. An effective method is to introduce new +equipment to accelerate the acquisition and compensate or +Yusheng Zhou and Hao Li contribute equally to the work and are co- +first authors. +Zhengmin Kong is the corresponding author. +Yusheng Zhou and Zhengmin Kong are with School of Electrical +Engineering and Automation , Wuhan University, China. +Hao Li is with the Department of Neuroradiology, University Hospital +Heidelberg, Heidelberg, Germany. +Jianan Liu is with Vitalent Consulting, Gothenburg, Sweden. (Email: +jianan.liu@vitalent.se) +Tao Huang and Euijoon Ahn are with the College of Science +and Engineering, James Cook University, Cairns, Australia. (Email: +tao.huang1@jcu.edu.au; euijoon.ahn@jcu.edu.au) +Zhihan Lv is with the Department of Game Design, Faculty of Arts, +Uppsala University, Sweden (Email: lvzhihan@gmail.com) +reacquire the K-space data partially in a prospective manner. +Although it can significantly prevent the appearance of motion +artifacts, it has not been widely applied due to the expensive +cost. Therefore, compared with high-cost prospective meth- +ods, retrospective artifact removal is still the main research +direction at present. +In recent years, artifact reduction techniques based on su- +pervision and deep learning have been proposed to address the +artifact problem in MRI [7]–[9]. It does not increase scanning +time and requires no additional acquisition equipment. A large +number of training samples are used to train neural networks. +Motion-free MR images is used as the correction guide to re- +duce artifacts in paired motion-corrupted MR images, showing +better performance over traditional methods in several studies. +However, the acquisition of paired MR images is extremely +tough or even impossible due to the difficulty in maintaining +the same state of the patients during the two image acquisition. +Image misalignment caused by state deviation is mistakenly +considered as a type of artifact, and then descends the artifact +reduction ability of the model, restricting the use of these +method in real clinical practice. +It is necessary to develop training methods that are appli- +cable when no paired MR images are available [10], [11], +and the successful popularization of unsupervised learning in +various tasks in the field of computer vision [12]–[16] gives us +a possible way to solve above problems. As another branch of +deep learning, unsupervised learning can find hidden patterns +or features from data without requiring feedback information +such as labels or categories, and does not over-rely on prior +knowledge of dataset. In particular, several recent models +based on unsupervised learning have shown promising results +without paired training samples, such as ISCL [17] for image +denoising task proposed by Lee et al., ADN [18] for computed +tomography (CT) metal artifact reduction task proposed by +Liao et al. and CycleGAN [19] proposed by Zhu et al. for +realizing images style transfer. However, although these tasks +are similar to motion artifact reduction, it does not mean that +the former models can be directly applied to the latter task. +As a common basis of the methods mentioned above, +generative adversarial network (GAN) [12] is one of the +most attractive technologies at present and one of the most +promising methods to handle the distribution of complex data. +Originally designed to generate data that doesn’t exist in the +real world, GAN comes in many variations for different tasks +[19]–[22]. Especially in the field of image generation, includ- +ing unconditional generation [12], [21], conditional generation +[20], [22] and image-to-image translation [19], etc., GAN’s + +LOGO2 +IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS, VOL. XX, NO. XX, SUBMITTED FEB. 2023 +studies have accumulated a solid fundamental of knowledge. +In order to avoid the unavailablility of paired MR images, we +proposed an unsupervised MRI artifact reduction framework +inspired by GAN, which trains the network by using unpaired +motion-free MR images and motion-corrupted MR images. +The contributions of this work are summarized as follows: +• We proposed an unsupervised abnomality extraction +network (UNAEN) to extract artifact residual maps by +learning the deep feature differences between unpaired +motion-free images and motion-corrupted images, indi- +rectly achieving the reduction of motion artifacts in MR +images. +• Different from the existing domain transfer methods in +the literature, UNAEN aimed to extract the abnormal +information in the image that causes the deep features +difference, and eliminated these abnormal information +to make the motion-corrupted close to the motion- +free distribution, improving the model’s representation +learning ability of artifact. +• Experimental results showed that compared with some +unsupervised models, the proposed model got higher +evaluation metrics and generated image with superior +quality. +II. RELATED WORK +A. Conventional Artifact Reduction +The most straightforward method to address the problem +of motion artifacts in MRI is to restrain the patients’ motions +by means of sedation or breath-holding during K-space data +acquisition [2]. However, patients cannot control physiological +involuntary movements such as cerebrospinal fluid pulsation +or intestinal peristalsis. In order to reduce the burden on +patients, some fast acquisition strategies have been proposed. +Compressed sensing [3] is an acquisition and reconstruction +technique based on signal sparsity, and its application to +K-space undersampling can shorten the scan time. Parallel +imaging [4] technique uses multiple coils with different sensi- +tivities to collect data during MR scanning to reduce the phase +encodings and thus the scan time. Although these methods to +accelerate the acquisition of K-space data can suppress motion +artifacts to a certain extent, they do not fundamentally solve +the problem. +Traditional artifact reduction methods include prospective +methods and retrospective methods. Prospective motion arti- +fact correction [5], [6] can compensate or reacquire K-space +partially during acquisition, which has great potential. But +because of requiring additional expensive hardware, it have +not been widely used in the clinic. Unlike the prospective +methods, the retrospective methods have no additional equip- +ment requirements. Retrospective motion artifact correction +[23]–[25] can estimate motions without obtaining information. +But these algorithms are computationally limited due to the +complexity and unpredictability of patients’ motions. Overall, +the traditional algorithms mentioned above all have some +shortcomings when dealing with the motion artifacts. +B. Deep Artifact Reduction +With the great success of deep learning in the field of +computer vision, some researchers have proposed retrospective +artifact reduction schemes based on deep learning (especially +convolutional neural network, CNN). The CNN model can be +trained with motion-corrupted images as input and the same +individual’s motion-free images as ground truth. As one of the +first studies for motion correction using deep learning, Johnson +et al. reconstructed the motion-corrected MR image from the +vector of motion-deformed k-space by the deep neural network +(DNN) [8]. Han et al. proposed a denoising algorithm based on +U-net to remove the streak artifacts induced in images obtained +via radial acquisition [7]. And Sommer et al. applied a fully +convolutional neural networks to extracted motion artifact- +only image, which subtracts the motion-clean image from +the motion-corrupted image, resulting in less deformation [9]. +However, in most cases it is difficult or impossible to obtain +paired MRI dataset to train neural networks. Although several +algorithms on motion simulation have been proposed to solve +this problem, these algorithms only consider simple and fixed +motion patterns to corrupt MR images from the image domain +[26] or K-space [27], [28]. In fact, the motion of patients is +more random and unpredictable. Models trained on datasets +generated by simulation artifacts perform poorly in practical +applications. +C. Unsupervised Image-to-Image Translation +Artifact reduction can be regarded as a task of image-to- +image translation. In recent years, some training strategies +based on unpaired images have attracted much attention. Deep +Image Prior (DIP) [29] demonstrated the feasibility of hand- +crafted prior generated by a randomly initialized network for +image denoising task. However, the disadvantage is that a large +amount of resources are consumed for iterative computation +for each image. Noise2Noise (N2N) [30] and Noise2Void +(N2V) [31] only used noisy images to train a CNN denoiser. +Although satisfactory denoising effect can be achieved without +noisy-clean image pairs, it is also necessary to know the +distribution of pixel-independent noise in order to choose +the applicable loss functions. Recently, generative adversarial +network (GAN) [12] had shown great potential in image gen- +eration and representation learning. The GCBD [32] proposed +by Chen et al. illustrated that GAN can train to estimate the +noise distribution of the noisy images. UIDnet [33] applied +a conditional GAN (cGAN) [22] to generate clean-pseudo +noisy pairs for training a denoising network. CycleGAN [19] +is a cyclic symmetric network consisted of two generators and +two discriminators, which is mainly used for domain adaption. +ISCL [17] added a noise extractor on the basis of CycleGAN +for cooperative learning with the generators. By combining +generative model and disentanglement network, ADN [18] +constructed multiple encoders and decoders to separate the +contents and artifacts in the CT images and get comparable +results with supervised learning. + +AUTHOR et al.: PREPARATION OF PAPERS FOR IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS (FEBRUARY 2023) +3 +III. PROPOSED METHOD +In this work, an unsupervised de-motion artifact model +named Unsupervised Abnomality Extraction Network (UN- +AEN) which uses the unpaired MR images to train, is proposed +as shown in Fig.1. In order to promote the representation +learning ability of motion artifact, an artifact extractor was +designed to intercept the artifact residual maps from the +motion-corrupted MR images, instead of using the generator to +directly generate the motion correction result. Compared with +general GAN, the mapping function between artifact domain +and motion-free domain could be obtained more easily. In +addition, we used an artifact reconstructor to restore the orig- +inal input from the motion artifact-reduced images to prevent +the artifact extractor from mismapping. In the experiment, +we compared the performance of UNEAN with some state- +of-the-art models such as CycleGAN, ISCL, UIDnet. The +experimental results show that our proposed model can achieve +better artifact reduction effect. +A. Network Architecture +Specifically, the UNAEN framework contains two modules: +forward module for artifact reduction and backward module +for artifact reconstruction. The forward module comes with an +artifact extractor Ge for learning the artifact distribution in the +motion-corrupted MR images. There is an artifact reconstruc- +tor Gr in the backward module that restores the corresponding +original input from the output generated by the forward mod- +ule. We take the unpaired images {(xa, y)|xa ∈ Xa, y ∈ Y } +as training samples, where Xa and Y represent the motion- +corrupted MRI set and motion-free MRI set, respectively. The +Ge and Gr are both generators of UNAEN. To train generators, +we employed Df and Db as discriminators in the forward and +backward modules to distinguish between a real sample and a +fake sample. +The workflow of UNAEN is shown as the arrows in the +Fig.1. We took the motion-corrupted MR image xa as input +fed into Ge to extract the artifact residual map Ge(xa), which +affects the texture information of MRI. The forward module +will generate the corresponding artifact-reduced image x by +subtracting Ge(xa) from xa: +x = xa − Ge(xa), +(1) +To enable the forward module to translate an instance xa +into a counterpart x rather than any instance, we introduced +the backward module. The main target of Gr is to translate +back the x into the original xa. So Gr is used to restore the +generated x and output the restored artifact-corrupted image +xa: +xa = Gr(x), +(2) +There is a cycle consistency between xa and xa and they +are expected to be identical. Since x and y are unpaired and +only have similar content, a forward discriminator Df should +be applied to distinguish between the generated image x and +real motion-free image y. To promote the reconstruction ability +of xa, we train a backward discriminator Db to distinguish +between the original input xa and restored artifact-corrupted +result xa. +During the training step, we train the generators and dis- +criminators alternately. The generators aim to generate samples +that are closed to real data while discriminators try not to be +deceived by the output of generators. During the inference +step, only the trained Ge are required. We can obtain the +motion artifact-reduced images as long as we subtract the +artifact residual maps extracted by the Ge from corresponding +motion-corrupted inputs. More details about generators and +discriminators will be discussed in the following subsection. +B. Loss Functions +In our experiments, we employed three types of loss +functions which are the L1 loss, SSIM loss [34], [35] and +adversarial loss: +L1(x, y) = 1 +N +N +� +i=1 +|x − y| +(3) +LSSIM(x, y) = 1 +N +N +� +i=1 +��1 − SSIM(x, y)2�� +(4) +Ladv(x, D) = 1 +N +N +� +i=1 +� +(D(x) − 1)2 +(5) +where D represents the Df or Db. SSIM (Structural Similarity +Index Measure) is an indicator to quantify the similarity +between two digital images. See Eq.(10) for specific formula. +In addition, we use the least square loss [36] as the adversarial +loss in our model instead of the negative log likelihood [12] +for stabilizing the training procedure. +To train Ge, we use a discriminator Df which aims to +classify the motion artifact-reduced output x as a motion-free +image. The adversarial loss function LGe as follow: +LGe adv(x, Df) = 1 +N +N +� +i=1 +� +(Df(x) − 1)2 +(6) +To train Gr, we use a discriminator Db which aims to +classify the restored artifact-corrupted result xa as the orig- +inal motion-corrupted image. The following adversarial loss +function is used to train the Gr: +LGr adv(xa, Db) = 1 +N +N +� +i=1 +� +(Db(xa) − 1)2 +(7) +Moreover, we adopt the cycle consistency loss to restrain +the restoration of xa. It is calculated as a weighted sum of +L1 loss and SSIM loss between the input and reconstruction +images: +LGr cyc(xa, xa) = L1(xa, xa)+λSSIM ∗LSSIM(xa, xa) (8) +where λSSIM is the weight of SSIM loss. We set λSSIM = +0.5 in our experiments. +So, the final objective function that optimizes the Ge and +Gr networks can be represented as: +LG = λGe adv ∗ LGe adv + λGr adv ∗ LGr adv + LGr cyc (9) +where λGe adv and λGr adv are the weights of the adversarial +losses of Ge and Gr, respectively. We set λGe adv = 0.1 and +λGr adv = 0.1 in our experiments. + +4 +IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS, VOL. XX, NO. XX, SUBMITTED FEB. 2023 +Fig. 1. The architecture of UNAEN. It consists of two generators and two discriminators. The network is fed unpaired motion artifact-corrupted and +motion artifact-free images in training. Motion artifact reduced output can be obtained by subtracting the artifact residual map extracted by Ge from +motion-corrupted input, and Gr converts the output to original input. Df compared the output with motion artifact-free input to identify whether the +artifact removal is successful while Db is used to check whether Gr is restored successfully. +Fig. 2. +The detailed structures of generator and discriminator. The generator adopt the RCAN backbone with a depth of 5 residual groups (RG) +and a long skip connection, and the discriminator is a VGG network. +C. Motion Simulation +We referred to the paper [37] to simulate the motion in MR +images. The method of splicing lines from multiple K-space +was used to simulate the generation of real motion artifacts. +Firstly, a group of images was generated from the original +images by rotating them in specific directions and to specific +degrees. The severity can be managed by the frequency of +motion. Then the original image and the generated images +were transformed to K-space using FFT, and K-space segments +of the original image were replaced with segments from the +generated images’ K-spaces, according to a predefined pattern. +Finally, the damaged original K-space data is transferred back +to the image domain by iFFT to obtain the simulation motion- +corrupted MR image. +In the process of motion simulation, we used the echo +group (EG) as the minimum time period unit to obtain a +certain number of successive echoes, and the duration of +any action must be an integer multiple of EG. To simulate +the motion of patients’ head, we set the original images to +be rotated 5 degrees to the left and to the right in plane. +Specifically, we used the K-space segments of the rotated +images to periodically replace the K-space segments of the +original image from the center line to the edge line. + +Generator Architectures +Discriminator Architecture +RG +RG +RG +RG +Channel +RCAB +RCAB +RCAB +2D Conv +ReLU +Attention +RCAB +Leaky ReLU +Batch Norm +FC +Tanh + Element-wise sumDb +G +MotionArtifacts +Extracted Motion Artifacts +MotionArtifacts +Restored Artifacts +-corrupted Image xa +-reduced Image x +-corrupted Image xa +D +Motion Artifacts +-free Image yAUTHOR et al.: PREPARATION OF PAPERS FOR IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS (FEBRUARY 2023) +5 +IV. EXPERIMENTS +In this section, a brief description of the dataset is presented, +and implementation details, including the network architecture +and hyper-parameters, are introduced. Experimental results are +presented with analyses and discussions. +A. Dataset Description +In this study, the fastMRI brain dataset [38] is used to +evaluate the proposed method. It includes 6970 fully sampled +brain MRIs (3001 at 1.5T and 3969 at 3T) collected at NYU +Langone Health on Siemens scanners using T1-weighted, T2- +weighted, and FLAIR acquisitions. Some of the T1-weighted +acquisitions included admissions of contrast agents. The Brain +MRI DICOM set, which exhibits a wide variety of recon- +struction matrix sizes, were acquired with a larger diversity +of scanners, manners of acquisition, reconstruction methods, +and post-processing algorithms. See paper [38], [39] for more +details. +In our experiments, the slices with large background in brain +MRI dataset were firstly discarded. To reduce the influence of +external factors and MRI acquisition methods on the exper- +iment results, we randomly selected 5000 slices only from +the T1 weighted slices with 3T field strength, whose matrix +size is 320 x 320. All selected images were corrupted from +the K-space by using a certain motion simulation algorithm +mentioned above. Specifically, 1 EG contained 10 echos and +the movement interval TS was set to 3EG, 6EG and 9EG, +resulting in a K-space corrupted line ratio of 75%, 60% and +50%, respectively. Then the dataset was divided into training +set, validation set and test set. The unsupervised MRI de- +motion artifact method requires unpaired motion-free MR im- +ages and motion-corrupted MR images, so we further divided +the training set into two non-overlapping groups. One group +contains only motion-free images as learning target while the +other group contains only motion-corrupted images as input +to the model. The validation set were used to monitor the +networks’ performance during training and test set to evaluate +the networks after training. All of images were normalized to +0 to 1. To save computation resource, we cropped images into +128 x 128 patches. +B. Evaluation Metrics +In order to make a comprehensive comparison, we used +SSIM and PSNR as the basic evaluation metrics in our +experiments. +As mentioned in III-B, SSIM (Structural Similarity Index +Measure) can quantify the similarity of two images. It was +defined to compare the brightness, contrast, and structure +between the motion artifact-reduced output x and the ground +truth. The SSIM is never greater than 1 and a larger value +represents a better motion correction result. The specific +expression is as follow: +SSIM(X, Y ) = +(2µXµY + C1)(2σXY + C2) +(µ2 +X + µ2 +Y + C1)(σ2 +X + σ2 +Y + C2) +(10) +where µ and σ donate the mean and standard deviation of the +images, respectively (σ2 +XY donates the covariance of x and y). +C1 and C2 are constants. +The PSNR (Peak Signal-to-Noise Ratio) is one of the +widely employed image quality indicators, which represents +the ratio between the maximum possible signal value and the +interference noise value that affects the signal representation +accuracy. It is usually measured in decibels (db) and a higher +value indicates a lower distortion. PSNR can be calculated +according to the following formula: +PSNR = 10 log10 +MaxV alue2 +MSE +(11) +MSE = +1 +mn +m−1 +� +i=0 +n−1 +� +j=0 +[I(i, j) − K(i, j)]2 +(12) +where MaxV alue is the largest possible pixel value and +MSE calculates the mean square error of two images. It is +difficult for human eyes to perceive the difference when PSNR +exceeds 30. +C. Experiment Configurations +We constructed two generators (artifact extractor Ge and +artifact reconstructor Gr) and two discriminators to train +UNAEN. The detailed structure of all networks as shown in +the Fig.2. The backbone of generator was built by the Residual +Channel Attention Network (RCAN) [40], [41] with a depth +of 5 residual groups (RG) and a long skip connection. Each +residual group (RG) has 5 residual channel attention blocks +(RCAB) and a long skip connection. We set the number of +feature channels to 64 at each base block of the generator. For +the discriminator, we just used simple convolutional units to +build the network, each unit consists of a 3 x 3 convolutional +layer and a leaky rectified linear unit (leaky ReLU) activation +layer [42]. The size of feature map was reduced by half after +each two convolution. All but the first unit have a batch +normalization layer [43]. Similarly, we set the number of +feature channels to 64 in the first convolutional layer of the +discriminator and doubled after each two convolutional layer. +All of our experiments were implemented on a desktop +system with 64GB RAM and two NVIDIA GeForce RTX 2080 +Ti graphics cards and used torch 1.8.1 as the back end. Before +each epoch of training process, all the motion-free and motion- +corrupted image patches were shuffled. We trained our model +for 50 epochs using the ADAM optimizer with β1 = 0.9, β2 += 0.99 and set batch size to 4. In each batch, the motion-free +patches and motion-corrupted patches fed to the networks were +unpaired. The initial learning rate was set to 10-4 and droppd +by half every 10 epochs. The generators were trained twice +for every time the discriminators trained. +D. Artifact Reduction on fastMRI +As shown in the Table I, we compared the performance of +the proposed model with other baseline methods on fastMRI +brain datasets with varying degrees of artifacts severity. The +SSIMs and PSNRs of the motion artifact-corrupted images +revealed the severity difference of motion artifacts. We ob- +served that the proposed unsupervised model was significantly +superior to all comparison unsupervised methods, where the + +6 +IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS, VOL. XX, NO. XX, SUBMITTED FEB. 2023 +Fig. 3. +Comparison of the qualitative performance of UNAEN and other unsupervised models on the fastMRI brain dataset. There visualized the +artifact reduction results with varying degrees of artifact severity and corresponding error heat maps showing the difference between ground truth +and each result. +TABLE I +QUANTITATIVE COMPARISON WITH THE STATE-OF-THE-ART UNSUPERVISED NETWORKS FOR MRI MOTION ARTIFACT REDUCTION ON FASTMRI +BRAIN DATASET +Methods +TS=3EG +TS=6EG +TS=9EG +SSIM +PSNR +SSIM +PSNR +SSIM +PSNR +Before Reduction +0.7981 +26.6165 +0.8824 +30.4109 +0.9225 +33.4192 +UIDnet (AAAI 2020) [33] +0.8551 +27.1392 +0.9168 +30.4248 +0.9411 +32.5677 +CycleGAN (ICCV 2017) [19] +0.8714 +27.4449 +0.9261 +31.1473 +0.9559 +33.4017 +ISCL (IEEE TMI 2021) [17] +0.8958 +29.3085 +0.9410 +32.4944 +0.9585 +34.4717 +UNAEN (Ours) +0.9126 +30.5387 +0.9504 +33.5448 +0.9674 +35.9265 + +Ground Truth +Before Correction +UIDNet +CycleGAN +ISCL +UNAEN (Ours) +3 +SSIM / PSNR +0.7898 / 26.1342 +0.8620 / 27.3402 +0.8818 / 28.1277 +0.9024 /29.1901 +0.9306 / 31.0245 +人S +0.20 +Error Map +0.10 +0.00 +1 +9= +SSIM / PSNR +0.8516 / 29.0333 +0.9093 / 30.6505 +0.9159/30.9561 +0.9376 / 33.0140 +0.9530 / 35.3089 +TS +0.20 +Error Map +0.10 +0.00 +志 +老 +=9 +SSIM/ PSNR +0.8561/29.2947 +0.9139/29.9313 +0.9442/30.9974 +0.9504/31.5782 +0.9656 / 34.3280 +TS +0.20 +Error Map +0.10 +0.00AUTHOR et al.: PREPARATION OF PAPERS FOR IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS (FEBRUARY 2023) +7 +SSIM was higher than 0.0089 to 0.0575 and the PSNR was +higher than 1.0504 to 3.3995 dB according to experimental +results. +Fig.3 visualized the artifact reduction effects of different +model and showed the qualitative performance on three de- +grees of artifact severity by displaying the reduction results and +corresponding error heat maps comparing to ground truth. All +four unsupervised methods we compared (UIDnet, CycleGAN, +ISCL, and UNAEN) successfully reduced the motion artifact. +UIDnet seemed to have the weakest reduction ability and its +outputs still retained significant artifact traces in the marginal +region of the tissue. Similarly, CycleGAN generated blurry im- +ages even though it had a higher SSIM and PSNR than UIDnet. +ISCL had better artifact reduction performance and improved +image quality. However, evident errors on the boundaries of +distinct soft tissues were observed in the reduction results, +as shown in the error heat maps. On the contrary, UNEAN +achieved higher metrics values and minimized errors, and with +the increase of artifact severity, the performance gap with other +methods was larger. In summary, UNAEN outperformed other +compared models in terms of overall image quality and feature +details in the experiment of fastMRI brain dataset. +V. DISCUSSION AND CONCLUSION +In this paper, we proposed an improved GAN model to +get an artifact reduction network, which trained by unpaired +MR images in an unsupervised manner to circumvent the +difficulty of obtaining paired MR images. We conducted sev- +eral experiments on two different dataset to qualitatively and +quantitively prove the outstanding performance of proposed +model by compared to UIDnet, CycleGAN and ISCL. +Unlike other unsupervised networks, UIDnet trains a cGAN +[22] which adds artifacts to clean images in order to generate +paired images to train a de-artifacts network under supervision. +Due to its indirect training strategy, more errors will be caused +than other models, limiting the ability to remove artifacts and +resulting in the fewest SSIM and PSNR in the experiments. +The network error which represented as geometric uncertainty +in image detail, could result in inaccurate surgery or therapy +doses, indicating that the approach is less applicable in real +clinics. +As an unsupervised network for domain transfer tasks, +CycleGAN can transfer images between different styles. To +generate a tighter mapping space, two symmetric generators +are used to realize the conversion between motion-corrupted +and motion-free image domains. The special learning method +slightly promotes the artifact reduction effect while causes +the problem of calculation redundancy. However, most of the +time we just need the artifact removal function rather than +the reverse process, which would make training the model +more difficult. Consuming more computing resources is not +proportional to the improvement in evaluation metrics. +ISCL is a variation of CycleGAN that adds an additional +extractor and collaborates with generators to accomplish co- +operative learning. The generators are responsible for direct +conversion between image domains, while the extractor can +extract artifacts from artifact observations. The experimen- +tal results showed that cooperative learning can further im- +prove the SSIM and PSNR values, but has no effect on the +boundaries of soft tissues. Unlike ISCL, UNAEN has no +cooperative learning, no bidirectional cycle consistency, and +the abandonment of redundant training makes the model pay +more attention to the artifact removal process and promote +the representation ability of artifacts. Experimental results +demonstrated that our modifications could successfully extract +the artifact residual components of the images and suppress the +motion artifact with little impact on the image quality, which +significantly improved the metrics values and generated high +quality artifact reduction results. +Given the effectiveness of UNAEN for unpaired images, +we expect more applications to artifact reduction since ob- +taining paired images is commonly impractical. In the real +clinical settings, UNAEN, as a retrospective method, can +correct movements of patients to avoid the destruction of +textures caused by artifacts. It is critical when researchers or +medical staffs do not have access to the original data and +associated reconstruction algorithms. In addition, we did not +make assumptions about the nature of artifacts during the +construction of UNAEN architecture, which makes it possible +for the proposed model to be generalized in other artifact +reduction problems, such as deblurring and denoising. We will +further explore the possibility of realizing these extensions. +Despite the superior artifact reduction effect of UNAEN, +there are still limitations in this study. Firstly, we generated ar- +tifacts of brain MRI only through simple periodic motion, but +the movement of patients during K-space data acquisition may +be more complex and irregular in real scenes. The performance +of the proposed model trained with authentic motion-corrupted +and motion-free images remains to be investigated. Besides, +another limitation is that training the network is difficult, +e.g., finding optimal hyper-parameters, due to complex loss +functions and adversarial networks. For the selection of some +hyper-parameters, we directly gave the conclusions without +listing relevant comparative experimental results, because their +adjustments have limited impact on the overall performance of +the network. We payed more attention to the modification of +the model architecture, and the optimization of the details is +one of goals of our future work. +REFERENCES +[1] M. Zaitsev, J. Maclaren, and M. Herbst, “Motion artifacts in mri: A +complex problem with many partial solutions,” Journal of magnetic +resonance imaging : JMRI, vol. 42, no. 4, p. 887—901, October 2015. +[2] A. Stadler, W. Schima, A. Ba’ssalamah, J. Kettenbach, and E. Eisen- +huber, “Artifacts in body mr imaging: Their appearance and how to +eliminate them,” European radiology, vol. 17, pp. 1242–55, 06 2007. +[3] Z. Yang, C. 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Szegedy, “Batch normalization: Accelerating deep +network training by reducing internal covariate shift,” in International +conference on machine learning. +PMLR, 2015, pp. 448–456. + diff --git a/FNAzT4oBgHgl3EQfw_7Q/content/tmp_files/load_file.txt b/FNAzT4oBgHgl3EQfw_7Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c731248ceca5d6dc0e33b9c907b667922125966e --- /dev/null +++ b/FNAzT4oBgHgl3EQfw_7Q/content/tmp_files/load_file.txt @@ -0,0 +1,776 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf,len=775 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='01732v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='IV] 4 Jan 2023 IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' XX, SUBMITTED FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' 2023 1 UNAEN: Unsupervised Abnomality Extraction Network for MRI Motion Artifact Reduction Yusheng Zhou, Hao Li, Jianan Liu, Zhengmin Kong, Tao Huang, Euijoon Ah, and Zhihan Lv Abstract—Motion artifact reduction is one of the most concerned problems in magnetic resonance imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' As a promising solution, deep learning-based methods have been widely investigated for artifact reduction tasks in MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' As a retrospective processing method, neural network does not cost additional acquisition time or require new acquisition equipment, and seems to work better than tra- ditional artifact reduction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In the previous study, training such models require the paired motion-corrupted and motion-free MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' However, it is extremely tough or even impossible to obtain these images in reality be- cause patients have difficulty in maintaining the same state during two image acquisition, which makes the training in a supervised manner impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In this work, we pro- posed a new unsupervised abnomality extraction network (UNAEN) to alleviate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Our network realizes the transition from artifact domain to motion-free domain by processing the abnormal information introduced by artifact in unpaired MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Different from directly generating artifact reduction results from motion-corrupted MR im- ages, we adopted the strategy of abnomality extraction to indirectly correct the impact of artifact in MR images by learning the deep features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Experimental results show that our method is superior to state-of-the-art networks and can potentially be applied in real clinical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Index Terms— Magnetic Resonance Imaging, Motion Ar- tifact Reduction, Unsupervised Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' INTRODUCTION M AGNETIC resonance imaging (MRI) is a non-invasive medical imaging technique used in the diagnosis of various diseases without radiation exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' However, due to the long acquisition time, MRI is sensitive to the patient’s movement [1], and incorrect K-space signal filling cause blurring or ghosting artifacts, which in turn affects the patient’s diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' To solve motion-related problems, researchers have proposed a variety of methods to prevent movement or correct artifacts [2]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' An effective method is to introduce new equipment to accelerate the acquisition and compensate or Yusheng Zhou and Hao Li contribute equally to the work and are co- first authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Zhengmin Kong is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Yusheng Zhou and Zhengmin Kong are with School of Electrical Engineering and Automation , Wuhan University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Hao Li is with the Department of Neuroradiology, University Hospital Heidelberg, Heidelberg, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Jianan Liu is with Vitalent Consulting, Gothenburg, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' (Email: jianan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='liu@vitalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='se) Tao Huang and Euijoon Ahn are with the College of Science and Engineering, James Cook University, Cairns, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' (Email: tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='huang1@jcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' euijoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='ahn@jcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='au) Zhihan Lv is with the Department of Game Design, Faculty of Arts, Uppsala University, Sweden (Email: lvzhihan@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='com) reacquire the K-space data partially in a prospective manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Although it can significantly prevent the appearance of motion artifacts, it has not been widely applied due to the expensive cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Therefore, compared with high-cost prospective meth- ods, retrospective artifact removal is still the main research direction at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In recent years, artifact reduction techniques based on su- pervision and deep learning have been proposed to address the artifact problem in MRI [7]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' It does not increase scanning time and requires no additional acquisition equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' A large number of training samples are used to train neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Motion-free MR images is used as the correction guide to re- duce artifacts in paired motion-corrupted MR images, showing better performance over traditional methods in several studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' However, the acquisition of paired MR images is extremely tough or even impossible due to the difficulty in maintaining the same state of the patients during the two image acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Image misalignment caused by state deviation is mistakenly considered as a type of artifact, and then descends the artifact reduction ability of the model, restricting the use of these method in real clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' It is necessary to develop training methods that are appli- cable when no paired MR images are available [10], [11], and the successful popularization of unsupervised learning in various tasks in the field of computer vision [12]–[16] gives us a possible way to solve above problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' As another branch of deep learning, unsupervised learning can find hidden patterns or features from data without requiring feedback information such as labels or categories, and does not over-rely on prior knowledge of dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In particular, several recent models based on unsupervised learning have shown promising results without paired training samples, such as ISCL [17] for image denoising task proposed by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=', ADN [18] for computed tomography (CT) metal artifact reduction task proposed by Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' and CycleGAN [19] proposed by Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' for realizing images style transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' However, although these tasks are similar to motion artifact reduction, it does not mean that the former models can be directly applied to the latter task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' As a common basis of the methods mentioned above, generative adversarial network (GAN) [12] is one of the most attractive technologies at present and one of the most promising methods to handle the distribution of complex data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Originally designed to generate data that doesn’t exist in the real world, GAN comes in many variations for different tasks [19]–[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Especially in the field of image generation, includ- ing unconditional generation [12], [21], conditional generation [20], [22] and image-to-image translation [19], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=', GAN’s LOGO2 IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' XX, SUBMITTED FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' 2023 studies have accumulated a solid fundamental of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In order to avoid the unavailablility of paired MR images, we proposed an unsupervised MRI artifact reduction framework inspired by GAN, which trains the network by using unpaired motion-free MR images and motion-corrupted MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The contributions of this work are summarized as follows: We proposed an unsupervised abnomality extraction network (UNAEN) to extract artifact residual maps by learning the deep feature differences between unpaired motion-free images and motion-corrupted images, indi- rectly achieving the reduction of motion artifacts in MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Different from the existing domain transfer methods in the literature, UNAEN aimed to extract the abnormal information in the image that causes the deep features difference, and eliminated these abnormal information to make the motion-corrupted close to the motion- free distribution, improving the model’s representation learning ability of artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Experimental results showed that compared with some unsupervised models, the proposed model got higher evaluation metrics and generated image with superior quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Conventional Artifact Reduction The most straightforward method to address the problem of motion artifacts in MRI is to restrain the patients’ motions by means of sedation or breath-holding during K-space data acquisition [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' However, patients cannot control physiological involuntary movements such as cerebrospinal fluid pulsation or intestinal peristalsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In order to reduce the burden on patients, some fast acquisition strategies have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Compressed sensing [3] is an acquisition and reconstruction technique based on signal sparsity, and its application to K-space undersampling can shorten the scan time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Parallel imaging [4] technique uses multiple coils with different sensi- tivities to collect data during MR scanning to reduce the phase encodings and thus the scan time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Although these methods to accelerate the acquisition of K-space data can suppress motion artifacts to a certain extent, they do not fundamentally solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Traditional artifact reduction methods include prospective methods and retrospective methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Prospective motion arti- fact correction [5], [6] can compensate or reacquire K-space partially during acquisition, which has great potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' But because of requiring additional expensive hardware, it have not been widely used in the clinic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Unlike the prospective methods, the retrospective methods have no additional equip- ment requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Retrospective motion artifact correction [23]–[25] can estimate motions without obtaining information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' But these algorithms are computationally limited due to the complexity and unpredictability of patients’ motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Overall, the traditional algorithms mentioned above all have some shortcomings when dealing with the motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Deep Artifact Reduction With the great success of deep learning in the field of computer vision, some researchers have proposed retrospective artifact reduction schemes based on deep learning (especially convolutional neural network, CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The CNN model can be trained with motion-corrupted images as input and the same individual’s motion-free images as ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' As one of the first studies for motion correction using deep learning, Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' reconstructed the motion-corrected MR image from the vector of motion-deformed k-space by the deep neural network (DNN) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' proposed a denoising algorithm based on U-net to remove the streak artifacts induced in images obtained via radial acquisition [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' And Sommer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' applied a fully convolutional neural networks to extracted motion artifact- only image, which subtracts the motion-clean image from the motion-corrupted image, resulting in less deformation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' However, in most cases it is difficult or impossible to obtain paired MRI dataset to train neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Although several algorithms on motion simulation have been proposed to solve this problem, these algorithms only consider simple and fixed motion patterns to corrupt MR images from the image domain [26] or K-space [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In fact, the motion of patients is more random and unpredictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Models trained on datasets generated by simulation artifacts perform poorly in practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Unsupervised Image-to-Image Translation Artifact reduction can be regarded as a task of image-to- image translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In recent years, some training strategies based on unpaired images have attracted much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Deep Image Prior (DIP) [29] demonstrated the feasibility of hand- crafted prior generated by a randomly initialized network for image denoising task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' However, the disadvantage is that a large amount of resources are consumed for iterative computation for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Noise2Noise (N2N) [30] and Noise2Void (N2V) [31] only used noisy images to train a CNN denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Although satisfactory denoising effect can be achieved without noisy-clean image pairs, it is also necessary to know the distribution of pixel-independent noise in order to choose the applicable loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Recently, generative adversarial network (GAN) [12] had shown great potential in image gen- eration and representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The GCBD [32] proposed by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' illustrated that GAN can train to estimate the noise distribution of the noisy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' UIDnet [33] applied a conditional GAN (cGAN) [22] to generate clean-pseudo noisy pairs for training a denoising network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' CycleGAN [19] is a cyclic symmetric network consisted of two generators and two discriminators, which is mainly used for domain adaption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' ISCL [17] added a noise extractor on the basis of CycleGAN for cooperative learning with the generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' By combining generative model and disentanglement network, ADN [18] constructed multiple encoders and decoders to separate the contents and artifacts in the CT images and get comparable results with supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' AUTHOR et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' : PREPARATION OF PAPERS FOR IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS (FEBRUARY 2023) 3 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' PROPOSED METHOD In this work, an unsupervised de-motion artifact model named Unsupervised Abnomality Extraction Network (UN- AEN) which uses the unpaired MR images to train, is proposed as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In order to promote the representation learning ability of motion artifact, an artifact extractor was designed to intercept the artifact residual maps from the motion-corrupted MR images, instead of using the generator to directly generate the motion correction result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Compared with general GAN, the mapping function between artifact domain and motion-free domain could be obtained more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In addition, we used an artifact reconstructor to restore the orig- inal input from the motion artifact-reduced images to prevent the artifact extractor from mismapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In the experiment, we compared the performance of UNEAN with some state- of-the-art models such as CycleGAN, ISCL, UIDnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The experimental results show that our proposed model can achieve better artifact reduction effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Network Architecture Specifically, the UNAEN framework contains two modules: forward module for artifact reduction and backward module for artifact reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The forward module comes with an artifact extractor Ge for learning the artifact distribution in the motion-corrupted MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' There is an artifact reconstruc- tor Gr in the backward module that restores the corresponding original input from the output generated by the forward mod- ule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' We take the unpaired images {(xa, y)|xa ∈ Xa, y ∈ Y } as training samples, where Xa and Y represent the motion- corrupted MRI set and motion-free MRI set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The Ge and Gr are both generators of UNAEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' To train generators, we employed Df and Db as discriminators in the forward and backward modules to distinguish between a real sample and a fake sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The workflow of UNAEN is shown as the arrows in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' We took the motion-corrupted MR image xa as input fed into Ge to extract the artifact residual map Ge(xa), which affects the texture information of MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The forward module will generate the corresponding artifact-reduced image x by subtracting Ge(xa) from xa: x = xa − Ge(xa), (1) To enable the forward module to translate an instance xa into a counterpart x rather than any instance, we introduced the backward module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The main target of Gr is to translate back the x into the original xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' So Gr is used to restore the generated x and output the restored artifact-corrupted image xa: xa = Gr(x), (2) There is a cycle consistency between xa and xa and they are expected to be identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Since x and y are unpaired and only have similar content, a forward discriminator Df should be applied to distinguish between the generated image x and real motion-free image y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' To promote the reconstruction ability of xa, we train a backward discriminator Db to distinguish between the original input xa and restored artifact-corrupted result xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' During the training step, we train the generators and dis- criminators alternately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The generators aim to generate samples that are closed to real data while discriminators try not to be deceived by the output of generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' During the inference step, only the trained Ge are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' We can obtain the motion artifact-reduced images as long as we subtract the artifact residual maps extracted by the Ge from corresponding motion-corrupted inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' More details about generators and discriminators will be discussed in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Loss Functions In our experiments, we employed three types of loss functions which are the L1 loss, SSIM loss [34], [35] and adversarial loss: L1(x, y) = 1 N N � i=1 |x − y| (3) LSSIM(x, y) = 1 N N � i=1 ��1 − SSIM(x, y)2�� (4) Ladv(x, D) = 1 N N � i=1 � (D(x) − 1)2 (5) where D represents the Df or Db.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' SSIM (Structural Similarity Index Measure) is an indicator to quantify the similarity between two digital images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' (10) for specific formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In addition, we use the least square loss [36] as the adversarial loss in our model instead of the negative log likelihood [12] for stabilizing the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' To train Ge, we use a discriminator Df which aims to classify the motion artifact-reduced output x as a motion-free image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The adversarial loss function LGe as follow: LGe adv(x, Df) = 1 N N � i=1 � (Df(x) − 1)2 (6) To train Gr, we use a discriminator Db which aims to classify the restored artifact-corrupted result xa as the orig- inal motion-corrupted image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The following adversarial loss function is used to train the Gr: LGr adv(xa, Db) = 1 N N � i=1 � (Db(xa) − 1)2 (7) Moreover, we adopt the cycle consistency loss to restrain the restoration of xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' It is calculated as a weighted sum of L1 loss and SSIM loss between the input and reconstruction images: LGr cyc(xa, xa) = L1(xa, xa)+λSSIM ∗LSSIM(xa, xa) (8) where λSSIM is the weight of SSIM loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' We set λSSIM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='5 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' So, the final objective function that optimizes the Ge and Gr networks can be represented as: LG = λGe adv ∗ LGe adv + λGr adv ∗ LGr adv + LGr cyc (9) where λGe adv and λGr adv are the weights of the adversarial losses of Ge and Gr, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' We set λGe adv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='1 and λGr adv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='1 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' 4 IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' XX, SUBMITTED FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The architecture of UNAEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' It consists of two generators and two discriminators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The network is fed unpaired motion artifact-corrupted and motion artifact-free images in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Motion artifact reduced output can be obtained by subtracting the artifact residual map extracted by Ge from motion-corrupted input, and Gr converts the output to original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Df compared the output with motion artifact-free input to identify whether the artifact removal is successful while Db is used to check whether Gr is restored successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The detailed structures of generator and discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The generator adopt the RCAN backbone with a depth of 5 residual groups (RG) and a long skip connection, and the discriminator is a VGG network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Motion Simulation We referred to the paper [37] to simulate the motion in MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The method of splicing lines from multiple K-space was used to simulate the generation of real motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Firstly, a group of images was generated from the original images by rotating them in specific directions and to specific degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The severity can be managed by the frequency of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Then the original image and the generated images were transformed to K-space using FFT, and K-space segments of the original image were replaced with segments from the generated images’ K-spaces, according to a predefined pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Finally, the damaged original K-space data is transferred back to the image domain by iFFT to obtain the simulation motion- corrupted MR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In the process of motion simulation, we used the echo group (EG) as the minimum time period unit to obtain a certain number of successive echoes, and the duration of any action must be an integer multiple of EG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' To simulate the motion of patients’ head, we set the original images to be rotated 5 degrees to the left and to the right in plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Specifically, we used the K-space segments of the rotated images to periodically replace the K-space segments of the original image from the center line to the edge line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Generator Architectures Discriminator Architecture RG RG RG RG Channel RCAB RCAB RCAB 2D Conv ReLU Attention RCAB Leaky ReLU Batch Norm FC Tanh Element-wise sumDb G MotionArtifacts Extracted Motion Artifacts MotionArtifacts Restored Artifacts corrupted Image xa reduced Image x corrupted Image xa D Motion Artifacts free Image yAUTHOR et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' : PREPARATION OF PAPERS FOR IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS (FEBRUARY 2023) 5 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' EXPERIMENTS In this section, a brief description of the dataset is presented, and implementation details, including the network architecture and hyper-parameters, are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Experimental results are presented with analyses and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Dataset Description In this study, the fastMRI brain dataset [38] is used to evaluate the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' It includes 6970 fully sampled brain MRIs (3001 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='5T and 3969 at 3T) collected at NYU Langone Health on Siemens scanners using T1-weighted, T2- weighted, and FLAIR acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Some of the T1-weighted acquisitions included admissions of contrast agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The Brain MRI DICOM set, which exhibits a wide variety of recon- struction matrix sizes, were acquired with a larger diversity of scanners, manners of acquisition, reconstruction methods, and post-processing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' See paper [38], [39] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In our experiments, the slices with large background in brain MRI dataset were firstly discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' To reduce the influence of external factors and MRI acquisition methods on the exper- iment results, we randomly selected 5000 slices only from the T1 weighted slices with 3T field strength, whose matrix size is 320 x 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' All selected images were corrupted from the K-space by using a certain motion simulation algorithm mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Specifically, 1 EG contained 10 echos and the movement interval TS was set to 3EG, 6EG and 9EG, resulting in a K-space corrupted line ratio of 75%, 60% and 50%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Then the dataset was divided into training set, validation set and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The unsupervised MRI de- motion artifact method requires unpaired motion-free MR im- ages and motion-corrupted MR images, so we further divided the training set into two non-overlapping groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' One group contains only motion-free images as learning target while the other group contains only motion-corrupted images as input to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The validation set were used to monitor the networks’ performance during training and test set to evaluate the networks after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' All of images were normalized to 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' To save computation resource, we cropped images into 128 x 128 patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Evaluation Metrics In order to make a comprehensive comparison, we used SSIM and PSNR as the basic evaluation metrics in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' As mentioned in III-B, SSIM (Structural Similarity Index Measure) can quantify the similarity of two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' It was defined to compare the brightness, contrast, and structure between the motion artifact-reduced output x and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The SSIM is never greater than 1 and a larger value represents a better motion correction result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The specific expression is as follow: SSIM(X, Y ) = (2µXµY + C1)(2σXY + C2) (µ2 X + µ2 Y + C1)(σ2 X + σ2 Y + C2) (10) where µ and σ donate the mean and standard deviation of the images, respectively (σ2 XY donates the covariance of x and y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' C1 and C2 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The PSNR (Peak Signal-to-Noise Ratio) is one of the widely employed image quality indicators, which represents the ratio between the maximum possible signal value and the interference noise value that affects the signal representation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' It is usually measured in decibels (db) and a higher value indicates a lower distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' PSNR can be calculated according to the following formula: PSNR = 10 log10 MaxV alue2 MSE (11) MSE = 1 mn m−1 � i=0 n−1 � j=0 [I(i, j) − K(i, j)]2 (12) where MaxV alue is the largest possible pixel value and MSE calculates the mean square error of two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' It is difficult for human eyes to perceive the difference when PSNR exceeds 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Experiment Configurations We constructed two generators (artifact extractor Ge and artifact reconstructor Gr) and two discriminators to train UNAEN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The detailed structure of all networks as shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The backbone of generator was built by the Residual Channel Attention Network (RCAN) [40], [41] with a depth of 5 residual groups (RG) and a long skip connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Each residual group (RG) has 5 residual channel attention blocks (RCAB) and a long skip connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' We set the number of feature channels to 64 at each base block of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' For the discriminator, we just used simple convolutional units to build the network, each unit consists of a 3 x 3 convolutional layer and a leaky rectified linear unit (leaky ReLU) activation layer [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The size of feature map was reduced by half after each two convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' All but the first unit have a batch normalization layer [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Similarly, we set the number of feature channels to 64 in the first convolutional layer of the discriminator and doubled after each two convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' All of our experiments were implemented on a desktop system with 64GB RAM and two NVIDIA GeForce RTX 2080 Ti graphics cards and used torch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='1 as the back end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Before each epoch of training process, all the motion-free and motion- corrupted image patches were shuffled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' We trained our model for 50 epochs using the ADAM optimizer with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='9, β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='99 and set batch size to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In each batch, the motion-free patches and motion-corrupted patches fed to the networks were unpaired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The initial learning rate was set to 10-4 and droppd by half every 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The generators were trained twice for every time the discriminators trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Artifact Reduction on fastMRI As shown in the Table I, we compared the performance of the proposed model with other baseline methods on fastMRI brain datasets with varying degrees of artifacts severity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The SSIMs and PSNRs of the motion artifact-corrupted images revealed the severity difference of motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' We ob- served that the proposed unsupervised model was significantly superior to all comparison unsupervised methods, where the 6 IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' XX, SUBMITTED FEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Comparison of the qualitative performance of UNAEN and other unsupervised models on the fastMRI brain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' There visualized the artifact reduction results with varying degrees of artifact severity and corresponding error heat maps showing the difference between ground truth and each result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' TABLE I QUANTITATIVE COMPARISON WITH THE STATE-OF-THE-ART UNSUPERVISED NETWORKS FOR MRI MOTION ARTIFACT REDUCTION ON FASTMRI BRAIN DATASET Methods TS=3EG TS=6EG 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+page_content='20 Error Map 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='00AUTHOR et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' : PREPARATION OF PAPERS FOR IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS (FEBRUARY 2023) 7 SSIM was higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='0089 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='0575 and the PSNR was higher than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='0504 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='3995 dB according to experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='3 visualized the artifact reduction effects of different model and showed the qualitative performance on three de- grees of artifact severity by displaying the reduction results and corresponding error heat maps comparing to ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' All four unsupervised methods we compared (UIDnet, CycleGAN, ISCL, and UNAEN) successfully reduced the motion artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' UIDnet seemed to have the weakest reduction ability and its outputs still retained significant artifact traces in the marginal region of the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Similarly, CycleGAN generated blurry im- ages even though it had a higher SSIM and PSNR than UIDnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' ISCL had better artifact reduction performance and improved image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' However, evident errors on the boundaries of distinct soft tissues were observed in the reduction results, as shown in the error heat maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' On the contrary, UNEAN achieved higher metrics values and minimized errors, and with the increase of artifact severity, the performance gap with other methods was larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In summary, UNAEN outperformed other compared models in terms of overall image quality and feature details in the experiment of fastMRI brain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION In this paper, we proposed an improved GAN model to get an artifact reduction network, which trained by unpaired MR images in an unsupervised manner to circumvent the difficulty of obtaining paired MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' We conducted sev- eral experiments on two different dataset to qualitatively and quantitively prove the outstanding performance of proposed model by compared to UIDnet, CycleGAN and ISCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Unlike other unsupervised networks, UIDnet trains a cGAN [22] which adds artifacts to clean images in order to generate paired images to train a de-artifacts network under supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Due to its indirect training strategy, more errors will be caused than other models, limiting the ability to remove artifacts and resulting in the fewest SSIM and PSNR in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The network error which represented as geometric uncertainty in image detail, could result in inaccurate surgery or therapy doses, indicating that the approach is less applicable in real clinics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' As an unsupervised network for domain transfer tasks, CycleGAN can transfer images between different styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' To generate a tighter mapping space, two symmetric generators are used to realize the conversion between motion-corrupted and motion-free image domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The special learning method slightly promotes the artifact reduction effect while causes the problem of calculation redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' However, most of the time we just need the artifact removal function rather than the reverse process, which would make training the model more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Consuming more computing resources is not proportional to the improvement in evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' ISCL is a variation of CycleGAN that adds an additional extractor and collaborates with generators to accomplish co- operative learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The generators are responsible for direct conversion between image domains, while the extractor can extract artifacts from artifact observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The experimen- tal results showed that cooperative learning can further im- prove the SSIM and PSNR values, but has no effect on the boundaries of soft tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Unlike ISCL, UNAEN has no cooperative learning, no bidirectional cycle consistency, and the abandonment of redundant training makes the model pay more attention to the artifact removal process and promote the representation ability of artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Experimental results demonstrated that our modifications could successfully extract the artifact residual components of the images and suppress the motion artifact with little impact on the image quality, which significantly improved the metrics values and generated high quality artifact reduction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Given the effectiveness of UNAEN for unpaired images, we expect more applications to artifact reduction since ob- taining paired images is commonly impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In the real clinical settings, UNAEN, as a retrospective method, can correct movements of patients to avoid the destruction of textures caused by artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' It is critical when researchers or medical staffs do not have access to the original data and associated reconstruction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' In addition, we did not make assumptions about the nature of artifacts during the construction of UNAEN architecture, which makes it possible for the proposed model to be generalized in other artifact reduction problems, such as deblurring and denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' We will further explore the possibility of realizing these extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Despite the superior artifact reduction effect of UNAEN, there are still limitations in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Firstly, we generated ar- tifacts of brain MRI only through simple periodic motion, but the movement of patients during K-space data acquisition may be more complex and irregular in real scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' The performance of the proposed model trained with authentic motion-corrupted and motion-free images remains to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' Besides, another limitation is that training the network is difficult, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=', finding optimal hyper-parameters, due to complex loss functions and adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' For the selection of some hyper-parameters, we directly gave the conclusions without listing relevant comparative experimental results, because their adjustments have limited impact on the overall performance of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfw_7Q/content/2301.01732v1.pdf'} +page_content=' We payed more attention to the modification of the model architecture, and the optimization of the details is one of goals of our 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Exist- +ing works assume that transceivers of semantic communications +share the same KB. However, intelligent transceivers may suffer +from the communication burden or worry about privacy leakage +to exchange data in KBs. Besides, the transceivers may indepen- +dently learn from the environment and dynamically update their +KBs, leading to timely sharing of the KBs infeasible. All these +cause the mismatch between the KBs, which may result in a +semantic-level misunderstanding on the receiver side. To address +this issue, we propose a transceiver cooperative learning-assisted +semantic communication (TCL-SC) scheme against mismatched +KBs. In TCL-SC, the transceivers cooperatively train seman- +tic encoder and decoder neuron networks (NNs) of the same +structure based on their own KBs. They periodically share the +parameters of NNs. To reduce the communication overhead of +parameter sharing, parameter quantization is adopted. More- +over, we discuss the impacts of the number of communication +rounds on the performance of semantic communication systems. +Experiments on real-world data demonstrate that our proposed +TCL-SC can reduce the semantic-level misunderstanding on the +receiver side caused by the mismatch between the KBs, especially +at the low signal-to-noise (SNR) ratio regime. +I. INTRODUCTION +Due to tremendous breakthroughs in artificial intelligence +(AI) and powerful chipsets, many intelligent applications such +as Sari of the Apple company, and self-driving vehicles, have +sprung up. Semantic communications, as a communication +paradigm beyond transmitting bits [1], [2], aim to precisely +convey the meaning of messages, rather than accurately trans- +mitting each symbol. It can significantly reduce data traffic +and meanwhile well support the communication requirement +of intelligent agents. It has been recognized as a promising +technology to make wireless networks significantly more in- +telligent, energy-efficient, and sustainable. +Owing to advances in deep learning, in particular natural +language processing (NLP) and computer vision, digging the +semantic meaning of data for transmission becomes possible. +In recent years, semantic communication systems learned on +background knowledge bases (KBs) at transceivers have been +developed for delivering text [3]–[5], image [6], [7], speech +[8], as well as multimodal data [9]. In semantic communication +systems, the transmitter uses a semantic encoding module +Y. Wang and S. Guo are all with the School of Control Science and +Engineering, Shandong University, China, and also with Shandong Key +Laboratory of Wireless Communication Technologies, Shandong University, +China (e-mail: yh-wang@mail.sdu.edu.cn; shuaishuai guo@sdu.edu.cn). +to extract semantic information based on its own KB, and +the receiver uses a semantic decoding module to recover +the meaning of messages based on its own KB. To make +the transceivers have the same interpretation of the transmit- +ted semantic data, existing works, e.g., [3]–[9], assume the +transceivers have the same KBs. However, in practice, the KBs +of the transmitter and receiver may be the same initially, they +may become different due to the variations of the environment +and/or the strength of the device’s ability to acquire data. The +mismatch between the KBs of the transmitter and receiver can +cause misunderstanding on the receiver side. For instance, the +KB of the transmitter has knowledge about cars, ships, and +birds, while the KB of the receiver has knowledge about cars, +ships, and planes. When the transmitter transmits semantic +information about birds, the receiver cannot understand it +precisely. This calls for a way to address the problem of the +mismatch between the KBs and avoid misunderstanding at the +receiver. +The communication participant sharing their own data with +another participant to update the KB seems to be a viable ap- +proach. However, there are the following intractable challenges +when applying it to address the above problem: +• Massive communication overhead: In reality, the data +acquired by devices are mainly images and videos. Direct +sharing of these data to another communication partici- +pant may result in unaffordable communication overhead. +• Expensive training cost: In general, the storage capacity +and computing power of the device are limited. Shared +data, especially images and videos, undoubtedly increase +the storage cost and training cost of the device. +• Privacy concerns: Data shared during the communica- +tion may be illegally stolen by adversaries leading to +privacy leakage. Also out of privacy protection, some +participants are reluctant to share the acquired data with +another communication participant. +These challenges motivate us to propose a feasible approach +to address the mismatch between the KBs in semantic com- +munication systems. +In this article, a new semantic communication framework +is developed to combat the mismatched KBs of the trans- +mitter and receiver without causing excessive communication +overhead, training costs as well as privacy concerns. As indi- +cated in previous studies [10], only exchanging parameters of +neuron networks (NNs) without data exchange could achieve +a balance between data privacy protection and data sharing + +2 +Source Encoding +Channel Encoding +Wireless +Channel +Channel Decoding +Bits +01001 +01001 +Bits +Noise +JSCC Encoding +Wireless +Channel +JSCC Decoding +Semantic Decoding +Semantic Encoding +Knowledge Base of +Transmitter +Knowledge Base of +Receiver +Sharing Knowledge +Semantic Features +Semantic Features +Noise +Semantic Noise +Source Decoding +Semantic communication: Accurate transmission of the meaning of messages +Traditional communication: Bit-level error-free transmission +Fig. 1. A comparison between traditional communication system and deep learning-enabled semantic communication system. The upper region of the graph +corresponds to the traditional communication system, and the lower region corresponds to the semantic communication system. In the traditional communication +system, the source is encoded as bit sequences by source encoding and recovered by source decoding module at the receiver. In the semantic communication +system, semantic encoding module is fused to extract semantic features. In addition to being disturbed by physical noise, semantic communication is also +disturbed by semantic noise which is caused by the ambiguity of words, sentences or symbols between the transmitter and receiver. +computing. Inspired by these findings, we design a transceiver +cooperative learning-aided semantic communication (TCL- +SC) framework, in which the transceivers cooperatively train +the semantic encoder and decoder NNs of the same structure +based on their own KBs and periodically share the parame- +ters of NNs. Obviously, TCL-SC still needs a large amount +of communication overhead for exchanging the parameters +precisely. Inspired by previous work [11], quantization is +adopted to reduce communication overhead. Also, the impact +of the number of communication rounds on the performance of +semantic communications is investigated, trying to find out the +appropriate value that can reduce the communication overhead +without reducing the performance. +The main contributions of this article can be summarized +as follows. +• A TCL-SC scheme is proposed by using parameter shar- +ing to minimize the semantic loss caused by mismatched +KBs, which avoids excessive costs as data sharing needs. +• To reduce the communication overhead caused by pa- +rameter sharing, parameter quantization is adopted to +reduce the weights resolution. The effect of the number of +communication rounds on TCL-SC is investigated, and a +suitable value is chosen to achieve a compromise between +communication overhead and performance. +• The effectiveness of the proposed TCL-SC is validated +on text transmission tasks. Extensive experiments demon- +strate that, compared with baselines, the proposed TCL- +SC can well combat mismatched KBs and reduce the +misunderstanding on the receiver side, especially at the +low signal-to-noise (SNR) ratio. +In the remainder of the article, deep learning-enabled se- +mantic communications are first introduced, and the differ- +ence from traditional communication systems is emphasized. +Next, the TCL-aided semantic communication scheme against +mismatched KBs is illustrated. Then, comprehensive experi- +ments are conducted are concluded and future directions are +presented. Finally, conclusions are drawn. +II. DEEP LEARNING-ENABLED SEMANTIC +COMMUNICATION SYSTEM +Recent advances in deep learning have made substantial +progress in semantic communications, which has been dormant +for many years [1], [2]. As shown in Fig. 1, a DL-enabled +semantic communication system consists of a transmitter and +a receiver. The transmitter includes a semantic coding mod- +ule and a joint source and channel coding (JSCC) module, +where the semantic encoding module based on the KB of +the transmitter extracts semantic features from the raw data +input, and the JSCC encoding module maps extracted semantic +features to the channel input. As the opposite process, the +receiver is equipped with a JSCC decoding module and a +semantic decoding module. The JSCC decoding module de- +codes the channel output for mitigating the channel distortion +and attenuation, then the semantic decoding module based on +the KB of the receiver recovers it to the original messages. +According to the previous papers [3]–[8], for text sources, +the semantic encoding and semantic decoding modules can be +composed of long short-term memory (LSTM) [3] or Trans- +former [4] networks. For image sources, convolutional neural +networks [6], [7] can be used to extract and recover semantic +information. For speech sources, SE-ResNet networks [8] can +be used to construct the semantic encoding and semantic +decoding module. The JSCC encoding and decoding modules +can generally be constructed by fully connected layers. +Compared to traditional communication systems which pur- +sue bit-level error-free transmission, semantic communications + +3 +aim an accurate transmission of the meaning of messages. +The semantic encoding and decoding neuron networks (NNs) +play the role of extracting and recovering the meaning of +messages. For accuracy, they have to be trained on a KB. +The KB plays an important role in provisioning the accuracy +of message meaning. Existing works assume that transceivers +of semantic communications share the same KB. However, +intelligent transceivers may suffer from the communication +burden or worry about privacy leakage to exchange data +in KBs. Besides, the transceivers may independently learn +from the environment and dynamically update their KBs, +leading to timely sharing of the KBs infeasible. All these +cause the mismatch between the KBs, which may result in +a semantic-level misunderstanding on the receiver side. Thus, +it is important to develop a semantic communication scheme +to mitigate the impact of the mismatch. +III. TRANSCEIVER COOPERATIVE LEARNING-AIDED +SEMANTIC COMMUNICATION SYSTEM FOR TEXT +TRANSMISSION +In this section, a transceiver cooperative learning-aided +semantic communication system to address the mismatch +between KBs is illustrated. +A. Transceiver Cooperative Learning-aided Semantic Com- +munication +The proposed TCL-SC is shown in Fig. 2. The transceivers +cooperatively train the encoding and decoding NNs of the +same structure based on their own KBs (i.e. transceiver self- +learning) and periodically share the parameters of NNs (i.e. +transceiver cooperative learning). +In the transceiver self-learning phase, the input data S ∈ +ℜB×L as a dense vector goes through the semantic encod- +ing module, which extracts the semantic information X ∈ +ℜB×L×D, with B being the batch size, L representing the +input data size, and D stands for the output dimension of +semantic encoding module. The JSCC encoding module then +maps X to the channel input symbols Z ∈ ℜB×NL×2. The +encoded features in Z are transmitted to the receiver over +the wireless channel with noise, and the JSCC decoding +module decodes the noise-corrupted features ˆZ ∈ ℜB×NL×2 +to ˆX ∈ ℜB×L×D. Finally, the semantic decoding module +recovers ˆX to ˆS ∈ ℜB×L×D. The whole network learns in +an end-to-end manner. In the transceiver cooperative learning +phase, one transceiver (e.g. A) shares the network parameters +of the system model with the same architecture to the other +transceiver (e.g. B), and the network parameters of both +transceivers are aggregated on transceiver B. For simplicity, +the weight assigned to the model aggregation is set to be +propositional to the data sizes of the transceivers. Transceiver +B then sends the aggregated network parameters to transceiver +A, and the network parameters of both transceivers are up- +dated. The two phases alternate until the termination condition +is met. +Note that, TCL-SC1 could address the mismatch between +KBs of the transmitter and receiver skillfully, but it also +suffers from extra communication overhead. The communi- +cation overhead depends on the number of transmitted bits +as well as the communication rounds. Therefore to reduce +the communication overhead, one can either the number of +transmitted bits or that of the communication rounds. +B. Network Parameter Quantization +As aforementioned, a feasible way is to convert the weights +of the trained NNs from high-precision to low-precision. The +parameter compression can not only reduce the communica- +tion overhead of the parameter sharing but also can improve +the inference speed [11]. In [12], the authors proposed a +quantization approach that quantizes both weights and ac- +tivations from FP32 representation to INT8 representation. +The approach can reduce the model size without significantly +degrading the model performance. Thus, we use the approach +proposed in [12] for the parameter sharing in TCL-SC. This +not only reduces the communication overhead, but also makes +the proposed TCL-SC suitable for devices with limited re- +sources. +C. Impact of The Number of Communication Rounds +Intuitively, reducing the number of communication rounds +can also reduce the communication overhead caused by +sharing parameters of NNs. However, the performance of +the proposed TCL-SC scheme may severely degrade with +the number of communication rounds. This is because the +performance of semantic communication systems may fall +into local optimality as the number of communication rounds +decreases. Hence, we expect to find an appropriate value for +achieving a compromise between communication overhead +and performance. +The impact of the number of communication rounds on the +proposed TCL-SC performance is investigated in two cases. +• Case 1: The data perceived by transceiver A is different +that perceived by transceiver B, but the data size is +approximately equal, denoted as ∥A∥ ≈ ∥B∥. +• Case 2: The data perceived by transceiver A is different +that perceived by transceiver B, and the data size of +transceiver A is much larger than that of transceiver B, +denoted as ∥A∥ ≫ ∥B∥. +The impact of communication rounds is investigated by ex- +periments. In this article, text transmission experiments are +conducted. We select the dataset proceedings of the European +Parliament [13]. The dataset contains around 2 million sen- +tences and 50 million words. We pre-process the dataset and +select sentences consisting of 4 − 30 words from it. In case +1, we divide the dataset into four parts: 10% as initial public +data, 40% as private data of transceiver A, 40% as private +data of transceiver B, and 10% as test data. In case 2, 60% is +the private data of transceiver A, 20% is the private data of +1Federated learning typically involves a large number of mobile devices +that are connected to a central server. However, in the proposed TCL-SC, +only two transceivers are considered. + +4 +Transceiver A +Transceiver B +Transceiver +self-learning +Transceiver +cooperative +learning +Model aggregation: +Wireless +Channel +B×L +B×L×D +B×L×D +B×L×2N +B×NL×2 +B×L×2N +B×L×D +B×L +B×L×D +Semantic +Encoding +JSCC +Encoding +Embedding +Layer +Softmax +JSCC +Decoding +Semantic +Decoding +S +X +Z +Fig. 2. The framework of our proposed TCL-SC: (1) transceiver self-learning: the transceivers (e.g., A and B) train the semantic encoder and decoder of the +same structure based on their own KBs. In order to reduce the communication overhead in cooperative learning, network parameter quantization is introduced +to reduce the resolution of weights; (2) transceiver cooperative learning: the model parameters of transceivers A and B are aggregated on transceiver B. During +model aggregation, the weight (e.g., mA) assigned to the model is set according to the data size of the transceiver. +transceiver B, and the rest is unchanged. The proposed TCL- +SC scheme is trained over the additive white Gaussian noise +(AWGN) channels with a signal-to-noise ratio (SNR) of 15 dB +with a cross-entropy loss function. The total learning epochs +of the transceivers is 80, and the number of communication +rounds is set to 1, 4, 8, 10, 20, and 40, respectively. We adopt +bilingual evaluation understudy (BLEU) score [14] as the +performance metric. The BLEU score measures the similarity +between words, and its value ranges from 0 to 1. The larger the +value, the better the system performance. The BLEU scores +of the experiments with different numbers of communication +rounds are illustrated in Fig. 3. +It is observed that the addition of communication rounds +increases the performance initially in case 1. However, with +the increase of communication rounds to a certain number, +the performance is no longer significantly improved. In case +2, the BLEU score has shown roughly the same tendency. +Experiment results suggest that the number of communication +rounds equal to 8 could provide a reasonable performance +compromise. +IV. PERFORMANCE COMPARISON AND DISCUSSIONS +In this section, the effectiveness of the proposed TCL-SC is +verified by comparing it with other semantic communication +scheme and the traditional source coding and channel coding +approach over the AWGN channels. The text transmission is +chosen for the experiment. It is noteworthy the proposed TCL- +SC scheme is not limited to the source type and can be applied +to semantic communication systems with any source type. +A. Experimental Setup +Dataset: The proceedings of the European Parliament [13] +is adopted to demonstrate the performance comparison. We +pre-process the dataset and select sentences consisting of 4−30 +0 +5 +10 +15 +20 +25 +30 +35 +40 +Number of Communication Rounds +0.6 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +BLEU(1-grams) +AWGN Channel - SNR = 15dB +||A||>>||B|| +Fig. 3. Impact of the number of communication rounds on the performance. +Initially, the addition of communication rounds increases performance, but +after a point, the gains stagnate. +words from it. Further, we divide the dataset into four parts: +10% as initial public data, 60% as private data of transceiver +A, 20% as private data of transceiver B, and 10% as test data. +Network: For fair comparisons, the network structures of +the transmitter and receiver in TCL-SC are the same as that +of DeepSC [4], where the semantic encoding and decoding +modules for extracting and recovering semantic information +are composed of Transformer encoder and decoder. +Baselines: A DL-enabled semantic communication scheme +and a traditional communication scheme are chosen as base- +lines. +• DeepSC [4]: DeepSC is a deep learning-based semantic +communication system, which extracts semantic informa- +tion from texts via the Transformer encoder and then + +5 +maps semantic information to the channel input. The loss +function of DeepSC is the cross-entropy. +• Separate source-channel coding scheme: We consider +Huffman coding for source coding. As for channel cod- +ing, we use Reed-Solomon (RS) coding [15]. The 16 +quadrature amplitude modulation (16-QAM) is adopted. +It is noteworthy that the KB mismatch problem will +not affect the separate source-channel coding scheme, +because it aims to achieve perfect bit transmission. +Metrics: In addition to the BLEU score, we also adopt +sentence similarity [4] as performance metric. The sentence +similarity compares the difference between sentences. Similar +to the BLEU score, the sentence similarity ranges from [0, 1], +and the larger the value, the better the performance of the +communication system. +B. Experimental Results +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +SNR (dB) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +BLEU(1-grams) +DeepSC +Huffman+RS +TCL-SC +(a) +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +SNR (dB) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +BLEU(2-grams) +DeepSC +Huffman+RS +TCL-SC +(b) +Fig. 4. BLEU score versus SNR, where both the DeepSC and TCL-SC are +trained over the AWGN channels at an SNR of 15 dB. +With the same number of transmitted symbols, we test the +BLEU score under different SNR over the AWGN channels, as +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +SNR (dB) +0.6 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +Sentence Similarity +DeepSC +Huffman+RS +TCL-SC +Fig. 5. Sentence similarity versus SNR, where both the DeepSC and TCL-SC +are trained over the AWGN channels at an SNR of 15 dB. +shown in Fig. 4. BLEU(1-gram) and BLEU(2-grams) calcu- +lated the 1-gram difference and 2-grams difference between +the transmitted and recovered sentences, respectively. Take +the sentence “the cat is on the sofa” for example, 1-gram: +“the”,“is”,“on”,“the”, and “sofa”, 2-gram: “the cat”,“cat is”,“is +on”,“on the”, and “the sofa”. Furthermore, take 2-grams as an +example, if the recovered sentence contains the above phrases +and the number is the same, then the BLEU score is 1. +As can be seen from Fig. 4.(a), the BLEU(1-gram) score of +the traditional communication scheme is higher than that of +DeepSC and the proposed TCL-SC when the SNR is greater +than 12 dB. This is because channel impairments are relatively +small. However, in the low SNR regime, the BLEU(1-gram) +score of the proposed TCL-SC achieved is higher than that +of the two baselines. In particular, the performance of the +proposed TCL-SC is much better than that of the DeepSC in +both low and high SNR regimes. In Fig. 4.(b), BLEU(2-grams) +score obtained by each scheme decreased slightly but has +shown roughly the same tendency with BLEU(1-gram) score. +These comparisons demonstrate that the existing semantic +communication schemes cannot cope with the mismatched +KBs, and the proposed TCL-SC could reduce the misunder- +standing on the receiver side caused by the mismatched KBs. +Notice that, the BLEU score is still the pursuit of the +accuracy of the symbol in essence, and it does not measure the +performance of the communication system from the semantic +level. For example, “sofa” is recovered to “lounges”, and +the BLEU score will drop, but in fact, the semantics have +not changed. In order to more appropriately measure the +performance of semantic communication system, in Fig. 5, +we show the relationship between sentence similarity and the +SNR for each scheme over the AWGN channels. +It can be observed that, at the high SNR, the sentence +similarity obtained by the traditional method is higher than +that obtained by the proposed TCL-SC. The sentence similarity +obtained by DeepSC is much smaller than that obtained by the +proposed TCL-SC. At the low SNR, the sentence similarity +obtained by the proposed TCL-SC is higher than that of the + +6 +comparison schemes. These comparisons further indicate that +our proposed TCL-SC could well cope with the mismatched +KBs in semantic communication systems, especially at the low +SNR. +V. FUTURE DIRECTIONS +While the proposed TCL-SC has been demonstrated to be +effective in combating mismatched KBs, there are still some +issues worth further investigation. +Imperfect transmission of NN parameters: As this article +focuses on addressing the mismatch between the KBs, in the +phase of transceiver cooperative learning, the transmission of +the NN parameters is assumed to be perfect. In practice, the +channel for NN parameter transmission is typically imperfect. +The noisy model parameter may greatly affect the performance +of TCL-SC. Therefore, the impact of noisy parameter trans- +mission on TCL-SC needs further investigation. +Data importance-based weights setting: For simplicity, +the weight in TCL-SC assigned to the model aggregation is +set to be propositional to the data sizes of the transceivers. +The data importance that describes the contributions to model +updates has been overlooked. In the future, the importance of +data can be reflected in the weighted model aggregation. +TCL-SC for multi-user communications: In this article, +we only considered point-to-point semantic communication. +In practice, a single transceiver can usually communicate +with multiple transceivers (e.g., transceiver A↔transceiver B, +transceiver C↔transceiver A). In the multi-transceiver com- +munication scenario, it is much more complicated to adopt +TCL-SC against the mismatched KBs. Thus, more research +efforts, such as how to avoid the semantic-level interference +of multi-transceivers cooperative learning and how to optimize +computing and communication resources, can be done on +TCL-SC in multi-user communication scenarios in future +work. +VI. CONCLUSIONS +This article proposed a TCL-SC scheme, where the +transceivers periodically share the network parameters of the +semantic encoder and decoder instead of directly exchang- +ing data. In TCL-SC, the network parameter quantization is +adopted to reduce the weights resolution, thus reducing the +communication overhead caused by sharing the parameters +of NNs. Furthermore, we studied the effect of the number +of communication rounds on TCL-SC. Experiments of text +transmission have demonstrated our proposed TCL-SC could +well reduce the misunderstanding on the receiver side caused +by the mismatched KBs, especially at the low SNR ratio +regime. Future research directions regarding TCL-SC were +discussed. +REFERENCES +[1] Z. Qin, X. Tao, J. Lu, W. Tong, and G. Y. Li, “Semantic communications: +Principles and challenges,” arXiv preprint arXiv:2201.01389, 2021. +[2] S. Guo, Y. Wang, and P. Zhang, “Signal shaping for semantic commu- +nication systems with a few message candidates,” in Proc. IEEE VTC, +London & Beijing, Sep. 2022, pp. 1–5. +[3] N. Farsad, M. Rao, and A. Goldsmith, “Deep learning for joint source- +channel coding of text,” in Proc. IEEE int’l conf. Acoustics, Speech +Signal process (ICASSP), Calgary, Canada, 2018, pp. 2326–2330. +[4] H. Xie, Z. Qin, G. Y. Li, and B. H. Juang, “Deep learning enabled +semantic communication systems,” IEEE Trans. Signal Process., vol. 69, +pp. 2663–2675, Jun. 2020. +[5] P. Jiang, C.-K. Wen, S. Jin, and G. Y. Li, “Deep source-channel coding +for sentence semantic transmission with harq,” IEEE Trans. Commun., +vol. 70, no. 8, pp. 5225–5240, Aug. 2022. +[6] E. Bourtsoulatze, D. B. Kurka, and D. G¨und¨uz, “Deep joint source- +channel coding for wireless image transmission,” IEEE Trans. Cogn. +Commun. Netw., vol. 5, no. 3, pp. 567–579, May. 2019. +[7] J. Shao and J. Zhang, “Communication-computation +trade-off in +resource-constrained edge inference,” IEEE Commun. Mag., vol. 58, +no. 12, pp. 20–26, Dec. 2020. +[8] Z. Weng and Z. Qin, “Semantic communication systems for speech +transmission,” IEEE J. Sel. Areas Commun., vol. 39, no. 8, pp. 2434– +2444, Aug. 2021. +[9] G. Zhang, Q. Hu, Z. Qin, Y. Cai, G. Yu, X. Tao, and G. Y. Li, “A unified +multi-task semantic communication system for multimodal data,” arXiv +preprint arXiv:2209.07689, 2021. +[10] N. Shlezinger, M. Chen, Y. C. Eldar, H. V. Poor, and S. Cui, “Uveqfed: +Universal vector quantization for federated learning,” IEEE Trans. Signal +Process., vol. 69, pp. 500–514, Dec. 2020. +[11] H. Xie and Z. Qin, “A lite distributed semantic communication system +for internet of things,” IEEE J. Sel. Areas Commun., vol. 39, no. 1, pp. +142–153, Jan. 2020. +[12] B. Jacob, S. Kligys, B. Chen, M. Zhu, M. Tang, A. Howard, H. Adam, +and D. Kalenichenko, “Quantization and training of neural networks for +efficient integer-arithmetic-only inference,” in Proc. IEEE Conf. Comput. +Vis. Pattern Recognit., Salt Lake City, UT, USA, Jun. 2018, pp. 2704– +2713. +[13] K. Philipp, “Europarl: A parallel corpus for statistical machine transla- +tion,” in MT summit, 2005, pp. 79–86. +[14] K. Papineni, S. Roukos, T. Ward, and W.-J. Zhu, “Bleu: A method for +automatic evaluation of machine translation,” in Proc. Annu. Meeting +Assoc. Comput. Linguistics, Philadelphia, PA, USA, Jul. 2002, pp. 311– +318. +[15] I. S. Reed and G. Solomon, “Polynomial codes over certain finite fields,” +J. Soc. Ind. Appl. Math., vol. 8, no. 2, pp. 300–304, Jan. 1960. + diff --git a/ItE1T4oBgHgl3EQfYARl/content/tmp_files/load_file.txt b/ItE1T4oBgHgl3EQfYARl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd7b8d071c939061ac91b67a065ae71c23d4f714 --- /dev/null +++ b/ItE1T4oBgHgl3EQfYARl/content/tmp_files/load_file.txt @@ -0,0 +1,458 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf,len=457 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='03133v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='NI] 9 Jan 2023 1 Transceiver Cooperative Learning-aided Semantic Communications Against Mismatched Background Knowledge Bases Yanhu Wang, Student Member, IEEE, and Shuaishuai Guo, Senior Member, IEEE Abstract—Semantic communications learned on background knowledge bases (KBs) have been identified as a promising technology for communications between intelligent agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Exist- ing works assume that transceivers of semantic communications share the same KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' However, intelligent transceivers may suffer from the communication burden or worry about privacy leakage to exchange data in KBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Besides, the transceivers may indepen- dently learn from the environment and dynamically update their KBs, leading to timely sharing of the KBs infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' All these cause the mismatch between the KBs, which may result in a semantic-level misunderstanding on the receiver side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' To address this issue, we propose a transceiver cooperative learning-assisted semantic communication (TCL-SC) scheme against mismatched KBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In TCL-SC, the transceivers cooperatively train seman- tic encoder and decoder neuron networks (NNs) of the same structure based on their own KBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' They periodically share the parameters of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' To reduce the communication overhead of parameter sharing, parameter quantization is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' More- over, we discuss the impacts of the number of communication rounds on the performance of semantic communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Experiments on real-world data demonstrate that our proposed TCL-SC can reduce the semantic-level misunderstanding on the receiver side caused by the mismatch between the KBs, especially at the low signal-to-noise (SNR) ratio regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' INTRODUCTION Due to tremendous breakthroughs in artificial intelligence (AI) and powerful chipsets, many intelligent applications such as Sari of the Apple company, and self-driving vehicles, have sprung up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Semantic communications, as a communication paradigm beyond transmitting bits [1], [2], aim to precisely convey the meaning of messages, rather than accurately trans- mitting each symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' It can significantly reduce data traffic and meanwhile well support the communication requirement of intelligent agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' It has been recognized as a promising technology to make wireless networks significantly more in- telligent, energy-efficient, and sustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Owing to advances in deep learning, in particular natural language processing (NLP) and computer vision, digging the semantic meaning of data for transmission becomes possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In recent years, semantic communication systems learned on background knowledge bases (KBs) at transceivers have been developed for delivering text [3]–[5], image [6], [7], speech [8], as well as multimodal data [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In semantic communication systems, the transmitter uses a semantic encoding module Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Wang and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Guo are all with the School of Control Science and Engineering, Shandong University, China, and also with Shandong Key Laboratory of Wireless Communication Technologies, Shandong University, China (e-mail: yh-wang@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' shuaishuai guo@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' to extract semantic information based on its own KB, and the receiver uses a semantic decoding module to recover the meaning of messages based on its own KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' To make the transceivers have the same interpretation of the transmit- ted semantic data, existing works, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=', [3]–[9], assume the transceivers have the same KBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' However, in practice, the KBs of the transmitter and receiver may be the same initially, they may become different due to the variations of the environment and/or the strength of the device’s ability to acquire data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The mismatch between the KBs of the transmitter and receiver can cause misunderstanding on the receiver side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' For instance, the KB of the transmitter has knowledge about cars, ships, and birds, while the KB of the receiver has knowledge about cars, ships, and planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' When the transmitter transmits semantic information about birds, the receiver cannot understand it precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' This calls for a way to address the problem of the mismatch between the KBs and avoid misunderstanding at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The communication participant sharing their own data with another participant to update the KB seems to be a viable ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' However, there are the following intractable challenges when applying it to address the above problem: Massive communication overhead: In reality, the data acquired by devices are mainly images and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Direct sharing of these data to another communication partici- pant may result in unaffordable communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Expensive training cost: In general, the storage capacity and computing power of the device are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Shared data, especially images and videos, undoubtedly increase the storage cost and training cost of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Privacy concerns: Data shared during the communica- tion may be illegally stolen by adversaries leading to privacy leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Also out of privacy protection, some participants are reluctant to share the acquired data with another communication participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' These challenges motivate us to propose a feasible approach to address the mismatch between the KBs in semantic com- munication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In this article, a new semantic communication framework is developed to combat the mismatched KBs of the trans- mitter and receiver without causing excessive communication overhead, training costs as well as privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' As indi- cated in previous studies [10],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' only exchanging parameters of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='neuron networks (NNs) without data exchange could achieve ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Channel Decoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='01001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='01001 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Bits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Noise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='JSCC Encoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Wireless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Channel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='JSCC Decoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Semantic Decoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Semantic Encoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Knowledge Base of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Transmitter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Knowledge Base of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Receiver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Sharing Knowledge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Semantic Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Semantic Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Noise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Semantic Noise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Source Decoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Semantic communication: Accurate transmission of the meaning of messages ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Traditional communication: Bit-level error-free transmission ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' A comparison between traditional communication system and deep learning-enabled semantic communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The upper region of the graph corresponds to the traditional communication system, and the lower region corresponds to the semantic communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In the traditional communication system, the source is encoded as bit sequences by source encoding and recovered by source decoding module at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In the semantic communication system, semantic encoding module is fused to extract semantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In addition to being disturbed by physical noise, semantic communication is also disturbed by semantic noise which is caused by the ambiguity of words, sentences or symbols between the transmitter and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Inspired by these findings, we design a transceiver cooperative learning-aided semantic communication (TCL- SC) framework, in which the transceivers cooperatively train the semantic encoder and decoder NNs of the same structure based on their own KBs and periodically share the parame- ters of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Obviously, TCL-SC still needs a large amount of communication overhead for exchanging the parameters precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Inspired by previous work [11], quantization is adopted to reduce communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Also, the impact of the number of communication rounds on the performance of semantic communications is investigated, trying to find out the appropriate value that can reduce the communication overhead without reducing the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The main contributions of this article can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' A TCL-SC scheme is proposed by using parameter shar- ing to minimize the semantic loss caused by mismatched KBs, which avoids excessive costs as data sharing needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' To reduce the communication overhead caused by pa- rameter sharing, parameter quantization is adopted to reduce the weights resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The effect of the number of communication rounds on TCL-SC is investigated, and a suitable value is chosen to achieve a compromise between communication overhead and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The effectiveness of the proposed TCL-SC is validated on text transmission tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Extensive experiments demon- strate that, compared with baselines, the proposed TCL- SC can well combat mismatched KBs and reduce the misunderstanding on the receiver side, especially at the low signal-to-noise (SNR) ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In the remainder of the article, deep learning-enabled se- mantic communications are first introduced, and the differ- ence from traditional communication systems is emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Next, the TCL-aided semantic communication scheme against mismatched KBs is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Then, comprehensive experi- ments are conducted are concluded and future directions are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Finally, conclusions are drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' DEEP LEARNING-ENABLED SEMANTIC COMMUNICATION SYSTEM Recent advances in deep learning have made substantial progress in semantic communications, which has been dormant for many years [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 1, a DL-enabled semantic communication system consists of a transmitter and a receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The transmitter includes a semantic coding mod- ule and a joint source and channel coding (JSCC) module, where the semantic encoding module based on the KB of the transmitter extracts semantic features from the raw data input, and the JSCC encoding module maps extracted semantic features to the channel input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' As the opposite process, the receiver is equipped with a JSCC decoding module and a semantic decoding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The JSCC decoding module de- codes the channel output for mitigating the channel distortion and attenuation, then the semantic decoding module based on the KB of the receiver recovers it to the original messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' According to the previous papers [3]–[8], for text sources, the semantic encoding and semantic decoding modules can be composed of long short-term memory (LSTM) [3] or Trans- former [4] networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' For image sources, convolutional neural networks [6], [7] can be used to extract and recover semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' For speech sources, SE-ResNet networks [8] can be used to construct the semantic encoding and semantic decoding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The JSCC encoding and decoding modules can generally be constructed by fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Compared to traditional communication systems which pur- sue bit-level error-free transmission, semantic communications 3 aim an accurate transmission of the meaning of messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The semantic encoding and decoding neuron networks (NNs) play the role of extracting and recovering the meaning of messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' For accuracy, they have to be trained on a KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The KB plays an important role in provisioning the accuracy of message meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Existing works assume that transceivers of semantic communications share the same KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' However, intelligent transceivers may suffer from the communication burden or worry about privacy leakage to exchange data in KBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Besides, the transceivers may independently learn from the environment and dynamically update their KBs, leading to timely sharing of the KBs infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' All these cause the mismatch between the KBs, which may result in a semantic-level misunderstanding on the receiver side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Thus, it is important to develop a semantic communication scheme to mitigate the impact of the mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' TRANSCEIVER COOPERATIVE LEARNING-AIDED SEMANTIC COMMUNICATION SYSTEM FOR TEXT TRANSMISSION In this section, a transceiver cooperative learning-aided semantic communication system to address the mismatch between KBs is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Transceiver Cooperative Learning-aided Semantic Com- munication The proposed TCL-SC is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The transceivers cooperatively train the encoding and decoding NNs of the same structure based on their own KBs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' transceiver self- learning) and periodically share the parameters of NNs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' transceiver cooperative learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In the transceiver self-learning phase, the input data S ∈ ℜB×L as a dense vector goes through the semantic encod- ing module, which extracts the semantic information X ∈ ℜB×L×D, with B being the batch size, L representing the input data size, and D stands for the output dimension of semantic encoding module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The JSCC encoding module then maps X to the channel input symbols Z ∈ ℜB×NL×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The encoded features in Z are transmitted to the receiver over the wireless channel with noise, and the JSCC decoding module decodes the noise-corrupted features ˆZ ∈ ℜB×NL×2 to ˆX ∈ ℜB×L×D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Finally, the semantic decoding module recovers ˆX to ˆS ∈ ℜB×L×D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The whole network learns in an end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In the transceiver cooperative learning phase, one transceiver (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' A) shares the network parameters of the system model with the same architecture to the other transceiver (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' B), and the network parameters of both transceivers are aggregated on transceiver B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' For simplicity, the weight assigned to the model aggregation is set to be propositional to the data sizes of the transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Transceiver B then sends the aggregated network parameters to transceiver A, and the network parameters of both transceivers are up- dated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The two phases alternate until the termination condition is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Note that, TCL-SC1 could address the mismatch between KBs of the transmitter and receiver skillfully, but it also suffers from extra communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The communi- cation overhead depends on the number of transmitted bits as well as the communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Therefore to reduce the communication overhead, one can either the number of transmitted bits or that of the communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Network Parameter Quantization As aforementioned, a feasible way is to convert the weights of the trained NNs from high-precision to low-precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The parameter compression can not only reduce the communica- tion overhead of the parameter sharing but also can improve the inference speed [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In [12], the authors proposed a quantization approach that quantizes both weights and ac- tivations from FP32 representation to INT8 representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The approach can reduce the model size without significantly degrading the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Thus, we use the approach proposed in [12] for the parameter sharing in TCL-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' This not only reduces the communication overhead, but also makes the proposed TCL-SC suitable for devices with limited re- sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Impact of The Number of Communication Rounds Intuitively, reducing the number of communication rounds can also reduce the communication overhead caused by sharing parameters of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' However, the performance of the proposed TCL-SC scheme may severely degrade with the number of communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' This is because the performance of semantic communication systems may fall into local optimality as the number of communication rounds decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Hence, we expect to find an appropriate value for achieving a compromise between communication overhead and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The impact of the number of communication rounds on the proposed TCL-SC performance is investigated in two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Case 1: The data perceived by transceiver A is different that perceived by transceiver B, but the data size is approximately equal, denoted as ∥A∥ ≈ ∥B∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Case 2: The data perceived by transceiver A is different that perceived by transceiver B, and the data size of transceiver A is much larger than that of transceiver B, denoted as ∥A∥ ≫ ∥B∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The impact of communication rounds is investigated by ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In this article, text transmission experiments are conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' We select the dataset proceedings of the European Parliament [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The dataset contains around 2 million sen- tences and 50 million words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' We pre-process the dataset and select sentences consisting of 4 − 30 words from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In case 1, we divide the dataset into four parts: 10% as initial public data, 40% as private data of transceiver A, 40% as private data of transceiver B, and 10% as test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In case 2, 60% is the private data of transceiver A, 20% is the private data of 1Federated learning typically involves a large number of mobile devices that are connected to a central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' However, in the proposed TCL-SC, only two transceivers are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 4 Transceiver A Transceiver B Transceiver self-learning Transceiver cooperative learning Model aggregation: Wireless Channel B×L B×L×D B×L×D B×L×2N B×NL×2 B×L×2N B×L×D B×L B×L×D Semantic Encoding JSCC Encoding Embedding Layer Softmax JSCC Decoding Semantic Decoding S X Z Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The framework of our proposed TCL-SC: (1) transceiver self-learning: the transceivers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=', A and B) train the semantic encoder and decoder of the same structure based on their own KBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In order to reduce the communication overhead in cooperative learning, network parameter quantization is introduced to reduce the resolution of weights;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' (2) transceiver cooperative learning: the model parameters of transceivers A and B are aggregated on transceiver B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' During model aggregation, the weight (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=', mA) assigned to the model is set according to the data size of the transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' transceiver B, and the rest is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The proposed TCL- SC scheme is trained over the additive white Gaussian noise (AWGN) channels with a signal-to-noise ratio (SNR) of 15 dB with a cross-entropy loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The total learning epochs of the transceivers is 80, and the number of communication rounds is set to 1, 4, 8, 10, 20, and 40, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' We adopt bilingual evaluation understudy (BLEU) score [14] as the performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The BLEU score measures the similarity between words, and its value ranges from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The larger the value, the better the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The BLEU scores of the experiments with different numbers of communication rounds are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' It is observed that the addition of communication rounds increases the performance initially in case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' However, with the increase of communication rounds to a certain number, the performance is no longer significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In case 2, the BLEU score has shown roughly the same tendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Experiment results suggest that the number of communication rounds equal to 8 could provide a reasonable performance compromise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' PERFORMANCE COMPARISON AND DISCUSSIONS In this section, the effectiveness of the proposed TCL-SC is verified by comparing it with other semantic communication scheme and the traditional source coding and channel coding approach over the AWGN channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The text transmission is chosen for the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' It is noteworthy the proposed TCL- SC scheme is not limited to the source type and can be applied to semantic communication systems with any source type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Experimental Setup Dataset: The proceedings of the European Parliament [13] is adopted to demonstrate the performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' We pre-process the dataset and select sentences consisting of 4−30 0 5 10 15 20 25 30 35 40 Number of Communication Rounds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='95 BLEU(1-grams) AWGN Channel - SNR = 15dB ||A||>>||B|| Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Impact of the number of communication rounds on the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Initially, the addition of communication rounds increases performance, but after a point, the gains stagnate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' words from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Further, we divide the dataset into four parts: 10% as initial public data, 60% as private data of transceiver A, 20% as private data of transceiver B, and 10% as test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Network: For fair comparisons, the network structures of the transmitter and receiver in TCL-SC are the same as that of DeepSC [4], where the semantic encoding and decoding modules for extracting and recovering semantic information are composed of Transformer encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Baselines: A DL-enabled semantic communication scheme and a traditional communication scheme are chosen as base- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' DeepSC [4]: DeepSC is a deep learning-based semantic communication system, which extracts semantic informa- tion from texts via the Transformer encoder and then 5 maps semantic information to the channel input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The loss function of DeepSC is the cross-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Separate source-channel coding scheme: We consider Huffman coding for source coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' As for channel cod- ing, we use Reed-Solomon (RS) coding [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The 16 quadrature amplitude modulation (16-QAM) is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' It is noteworthy that the KB mismatch problem will not affect the separate source-channel coding scheme, because it aims to achieve perfect bit transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Metrics: In addition to the BLEU score, we also adopt sentence similarity [4] as performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The sentence similarity compares the difference between sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Similar to the BLEU score, the sentence similarity ranges from [0, 1], and the larger the value, the better the performance of the communication system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Experimental Results 0 2 4 6 8 10 12 14 16 18 SNR (dB) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='9 1 BLEU(1-grams) DeepSC Huffman+RS TCL-SC (a) 0 2 4 6 8 10 12 14 16 18 SNR (dB) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='9 1 BLEU(2-grams) DeepSC Huffman+RS TCL-SC (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' BLEU score versus SNR, where both the DeepSC and TCL-SC are trained over the AWGN channels at an SNR of 15 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' With the same number of transmitted symbols, we test the BLEU score under different SNR over the AWGN channels, as 0 2 4 6 8 10 12 14 16 18 SNR (dB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='95 1 Sentence Similarity DeepSC Huffman+RS TCL-SC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Sentence similarity versus SNR, where both the DeepSC and TCL-SC are trained over the AWGN channels at an SNR of 15 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' BLEU(1-gram) and BLEU(2-grams) calcu- lated the 1-gram difference and 2-grams difference between the transmitted and recovered sentences, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Take the sentence “the cat is on the sofa” for example, 1-gram: “the”,“is”,“on”,“the”, and “sofa”, 2-gram: “the cat”,“cat is”,“is on”,“on the”, and “the sofa”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Furthermore, take 2-grams as an example, if the recovered sentence contains the above phrases and the number is the same, then the BLEU score is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' (a), the BLEU(1-gram) score of the traditional communication scheme is higher than that of DeepSC and the proposed TCL-SC when the SNR is greater than 12 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' This is because channel impairments are relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' However, in the low SNR regime, the BLEU(1-gram) score of the proposed TCL-SC achieved is higher than that of the two baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In particular, the performance of the proposed TCL-SC is much better than that of the DeepSC in both low and high SNR regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' (b), BLEU(2-grams) score obtained by each scheme decreased slightly but has shown roughly the same tendency with BLEU(1-gram) score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' These comparisons demonstrate that the existing semantic communication schemes cannot cope with the mismatched KBs, and the proposed TCL-SC could reduce the misunder- standing on the receiver side caused by the mismatched KBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Notice that, the BLEU score is still the pursuit of the accuracy of the symbol in essence, and it does not measure the performance of the communication system from the semantic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' For example, “sofa” is recovered to “lounges”, and the BLEU score will drop, but in fact, the semantics have not changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In order to more appropriately measure the performance of semantic communication system, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' 5, we show the relationship between sentence similarity and the SNR for each scheme over the AWGN channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' It can be observed that, at the high SNR, the sentence similarity obtained by the traditional method is higher than that obtained by the proposed TCL-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The sentence similarity obtained by DeepSC is much smaller than that obtained by the proposed TCL-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' At the low SNR, the sentence similarity obtained by the proposed TCL-SC is higher than that of the 6 comparison schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' These comparisons further indicate that our proposed TCL-SC could well cope with the mismatched KBs in semantic communication systems, especially at the low SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' FUTURE DIRECTIONS While the proposed TCL-SC has been demonstrated to be effective in combating mismatched KBs, there are still some issues worth further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Imperfect transmission of NN parameters: As this article focuses on addressing the mismatch between the KBs, in the phase of transceiver cooperative learning, the transmission of the NN parameters is assumed to be perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In practice, the channel for NN parameter transmission is typically imperfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The noisy model parameter may greatly affect the performance of TCL-SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Therefore, the impact of noisy parameter trans- mission on TCL-SC needs further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Data importance-based weights setting: For simplicity, the weight in TCL-SC assigned to the model aggregation is set to be propositional to the data sizes of the transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' The data importance that describes the contributions to model updates has been overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In the future, the importance of data can be reflected in the weighted model aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' TCL-SC for multi-user communications: In this article, we only considered point-to-point semantic communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In practice, a single transceiver can usually communicate with multiple transceivers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=', transceiver A↔transceiver B, transceiver C↔transceiver A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In the multi-transceiver com- munication scenario, it is much more complicated to adopt TCL-SC against the mismatched KBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Thus, more research efforts, such as how to avoid the semantic-level interference of multi-transceivers cooperative learning and how to optimize computing and communication resources, can be done on TCL-SC in multi-user communication scenarios in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' CONCLUSIONS This article proposed a TCL-SC scheme, where the transceivers periodically share the network parameters of the semantic encoder and decoder instead of directly exchang- ing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' In TCL-SC, the network parameter quantization is adopted to reduce the weights resolution, thus reducing the communication overhead caused by sharing the parameters of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE1T4oBgHgl3EQfYARl/content/2301.03133v1.pdf'} +page_content=' Furthermore, we studied the effect of the number of communication rounds on TCL-SC.' metadata={'source': 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Its results end up in a paper’s “background” section or in standalone +articles, which include meta-analyses and systematic literature reviews. The required re- +search is aided through the use of online scientific publication databases and search engines, +such as Web of Science, Scopus, and Google Scholar. However, use of online databases suf- +fers from a lack of repeatability and transparency, as well as from technical restrictions. +Thankfully, open data, powerful personal computers, and open source software now make +it possible to run sophisticated publication studies on the desktop in a self-contained en- +vironment that peers can readily reproduce. Here we report a Python software package +and an associated command-line tool that can populate embedded relational databases with +slices from the complete set of Crossref publication metadata,1 ORCID author records,2 and +other open data sets, for in-depth processing through performant queries. We demonstrate +the software’s utility by analyzing the underlying dataset’s contents, by visulizing the evolu- +tion of publications in diverse scientific fields and relationships between them, by outlining +scientometric facts associated with COVID-19 research, and by replicating commonly-used +bibliometric measures of productivity and impact. +Research synthesis is becoming an increasingly important3 and popular scientific method. +By our own calculations about 437 thousand scientific studies published from 1846 — the year +of the first one we found4 — onward are based on the analysis of previously published primary +studies. (The Methods section provides details on how all figures appearing in this report where +obtained in a repeatable manner.) These studies are typically identified in their titles with terms +such as “systematic review”, “systematic literature review”, or “systematic mapping study” (sec- +ondary studies using methods that help make their findings unbiased and repeatable — 251 850 +titles); “secondary study”, “literature survey”, or “literature review” (a not necessarily systematic +study reviewing primary studies — 77 037 titles); “tertiary study” or “umbrella review”5 (a study +reviewing secondary studies — 4 039 titles); “meta-analysis” (a systematic secondary study em- +ploying statistical methods — 92 363 titles);6 as well as (systematic by definition) “scientometric” +(employing quantitative methods to study scientific research — 2 769 titles) and “bibliometric” +(using statistical methods to study written communications — 12 361 titles) studies.6 The number +of such studies published each year has risen considerably over the past two decades, particularly +so for systematic literature reviews (see extended Figure 4). A major data source for research syn- +thesis studies are online specialized and general purpose bibliographic and article databases,7 such +as ERIC, Google Scholar, Inspec, Scopus, and Web of Science. +1 +arXiv:2301.13312v1 [cs.DL] 30 Jan 2023 + +Performing systematic studies on published literature through the available online systems +can be problematic.8 First, their constantly updated contents, bubble effects,9 location- and license- +dependent results,10 and periodic changes to their internal workings compromise reproducibil- +ity.11,12 Even when the search strategy is well-documented to aid reproducibility, by following best +practice reporting guidance such as PRISMA,13 which is often not the case,14,15 it is difficult to +repeat a query to an online service, and obtain the same results as those that have been published.8 +An associated problem is the lack of transparency.16 Most online services work with proprietary +data collections and algorithms, making it difficult to understand and explain the obtained results. +As an example, Clarivate’s journal impact factor calculation depends on an opaque collection of +journals17 and list of “citable items”18 tagged so by the vendor. In addition the reproducibility of +such studies is hampered in the short term by the fees required for accessing some online services +and in the long term by the commercial survival of the corresponding companies.19 Also, service +access costs on their own can restrict institutions with limited funding from conducting systematic +literature studies. Finally, there are technical limitations. Some services lack a way to access them +programmatically (an application programming interface — API),8 forcing researchers to resort to +tricky and unreliable contortions, such as screen scrapping. Both APIs and offered query languages +are not standardized,20 and often restrict the allowed operations.8 In addition, the network-based +APIs suffer from corresponding latency21 and also often from rate and ceiling limits to the number +of allowed invocations.22 These restrictions make it difficult to run studies that require performing +a large number or sophisticated queries. +The outlined problems can be addressed thanks to sustained exponential advances in com- +puting power,23 drops in associated costs, and Open Science initiatives.24 The Alexandria3k system +presented here is an open-source software library and command-line tool that builds on these ad- +vances to allow the conduct of sophisticated systematic research of published literature, (e.g. liter- +ature reviews, meta-analyses, bibliometric and scientometric studies) in a transparent, repeatable, +reproducible, and efficient manner. Alexandria3k allows researchers to process on a personal com- +puter publications’ metadata (including citations) from most major international academic publish- +ers as well as corresponding author, funder, organization, and journal details. Specifically, Alexan- +dria3k works on data snapshots offered periodically by initiatives, such as Crossref (publication +metadata, journal names, funder names),1 ORCID (author details),2 ROR (research organization +registry),25 and others. Using Alexandria3k researchers can query and process that data through +SQL queries launched by means of command-line tool invocations or Python scripts. Researchers +can ensure the transparency, reproducibility, and exact repeatability of their methods by document- +ing or publishing the version of the data used and the employed commands.24 The primary data +sets can be stored and processed locally on a modern laptop, because they amount to a few hundred +gigabytes in their compressed format (157 GB for Crossref, 25 GB for ORCID data; the download- +ing of the Crossref data is facilitated by its availability through the BitTorrent protocol).26 The data +are decompressed in small chunks ensuring that both main and secondary memory requirements +are kept within the limits of what a typical personal computer can accommodate. (Keeping the data +decompressed or populating a relational or graph database with all of it would require more than +1.5 TB of storage space.) In addition, Alexandria3k offers facilities for running relational database +2 + +queries on data partitions, sampling records, and populating a relational database with a subset +of records or fields. All these facilities help tasks to execute in reasonable time. On a populated +and suitably indexed database with millions of records many queries finish in minutes. Queries or +database population tasks involving a full scan of the entire Crossref publication data set complete +in less than five hours. +Contents, structure, and use +In total, Alexandria3k offers relational query access to 2.6 billion records. These are organized in +a relational schema illustrated in supplemental figures 5–8. +Most records are publication metadata obtained from the Crossref Public Data File. These +contain publication details (DOI, title, abstract, date, venue, type, pages, volume, ...), a publica- +tion’s references to other publications (DOI, title, author, page, ISSN, ISBN, year, ...), and other +data associated with each publication authors and their affiliations, funders and funder awards, up- +dates (e.g. retractions), subjects, licenses, and hyperlinks for text mining of the publication’s full +text.1 Details about the data available through Crossref are listed in extended Table 1. Note that +coverage is incomplete; for example, 39% of the publications have a reference list associated with +them, 70% of funders are uniquely identified with a DOI, while only 11% of the publications have +an abstract. For most types of records coverage is increasing over time. +Alexandria3k can link Crossref records to imported author metadata through ORCID (Open +Researcher and Contributor ID) a non-proprietary system developed to identify authors in schol- +arly communication.2 ORCID tables that Alexandria3k supports include those detailing an author’s +URLs, countries, keywords, external identifiers, distinctions, education, employment, invited po- +sitions, memberships, qualifications, services, fundings, peer reviews, used research resources, +and published works. Most of these tables contain details of the associated organization (name, +department, city, region, country), the role title, and the starting and end date. The currently avail- +able ORCID data set contains about 78 million records associated with 14 million authors. The +completeness of the ORCID records is low and uneven (see extended Table 2), which means that +research based on it must be carefully designed. +Alexandria3k can also import the Research Organization Registry data25 containing details +of 104 402 organizations, as well as related acronyms (43 862 records) and aliases (25 119 records). +Through the provided ROR identifier it can link unambiguously elements from a person’s employ- +ment and education ORCID records to the corresponding organization. Currently ORCID contains +such identifiers for 130 033 employment records and 133 066 education records. Given that only +4.6% of work author records have an ORCID and only 23% of ORCID records contain employ- +ment information, Alexandria3k also provides a performant facility to match the textual affiliation +information listed in works and link it to ROR identifiers. +3 + +Finally, Alexandria3k can import and link three reference tables: the names of journals as- +sociated with ISSNs (currently 109 971 records), the funder names associated with funder DOIs +(32 198 records), and the metadata of open access journals (18 717).27 Alexandria3k further disag- +gregates journal ISSNs according to their type (electronic, print, or alternative — 158 580 records). +The data used by Alexandria3k is openly distributed by diverse third parties (see the data +availability statement) in textual tree or flat format files: JSON for Crossref and ROR, XML for +ORCID, and CSV for the rest. Alexandria3k structures the data it offers in the relational schema of +45 tables linked through 47 relations. Stored in a relational database and combined with suitable +indexes, this allows performing sophisticated analyses via SQL (structured query language) queries +in an efficient manner. Records between diverse data sets are linked through standardized globally +unique identifiers: DOIs for published works and funders, ISSNs for journals, ORCIDs for authors, +and RORs for research organizations. +Alexandria3k is distributed as open source software in the form of a Python package, which +can be easily installed through the PyPI repository. It can be used either as a command-line tool, +where its operation (e.g. query to run) is specified through command-line arguments, or as a +Python library, which can be used interactively (e.g. by developing a Jupyter Notebook28) or +through scripts. +In its simplest form Alexandria3k can evaluate an SQL query directly on the Crossref dataset, +often to perform exploratory data analysis. Results can be saved as a CSV (comma-separated +values) file or iterated over through Python code for further processing. This mode has limitations +in terms of performance, aggregation of query results, and combination of data from multiple +sources. +In most cases Alexandria3k is used to populate an SQLite database.29 The SQLite database +can be easily used, because its engine is embedded into Alexandria3k, directly available in Python, +and easily installable as a command-line tool in all popular computing platforms. Transferring a +database between computers only involves copying the corresponding file. Consequently, there is +no need to setup, configure, and maintain a complex client-server relational or NoSQL database +management system. Despite its -Lite suffix, SQLite supports most of the SQL standard (including +window functions and recursive queries), and employs sophisticated query optimization methods; +these feature make it ideal for use in Alexandria3k. SQLite’s main downsides — lack of multi-user +and client-server support — are not relevant to common Alexandria3k use cases. +The latest version of the Crossref data is distributed as 26 thousand compressed container +files, each containing details about 5 000 works. A complete import of the Crossref data would +amount to a 520 GB database. Given the large amount of Crossref data, the population of a +database with it can be controlled in three ways. First, only a horizontal subset of records can +be imported, by specifying an SQL expression that will select a publication record only when it +4 + +evaluates to TRUE (e.g. published year BETWEEN 2017 AND 2021). To facilitate the +selection of records selected through other means, the expression can also refer to tables of exter- +nal databases. Second, only a subset of the Crossref 26 810 data containers can be processed for +import, by specifying a Python function that will import a container when it evaluates to True. +This is mostly useful for random sampling, e.g. using random.random() < 0.01 to sam- +ple approximately 1% of the containers. (A fixed seed value is used internally for initializing +the pseudo-random number generator to allow deterministic and therefore repeatable sampling.) +Third, the populated tables or columns of the Crossref data set can be vertically restricted by us- +ing the table-name.column-name or table-name.* SQL notation. The population of a +database with ORCID data can be also horizontally restricted to records associated with existing +Crossref authors or published works (probably selected in a previous population step) and verti- +cally restricted to include only specific tables or columns, as in the case of Crossref. Given their +small volume, no population controls are supported for the other data sets. +Application examples +The following paragraphs outline some simple proof-of-concept applications of Alexandria3k, +which demonstrate its use and motivate its adoption. All are exactly replicable through SQL +queries and relational online analytical processing Makefiles30–32 provided in the accompanying +materials. +For a start, all metrics provided in the preceding section and in extended Tables 1 and 2 were +obtained through simple SELECT Count(*) FROM table or SELECT Count(*) FROM +(SELECT DISTINCT ...) SQL queries. +Figure 1 showcases the use of Alexandria3k to chart a view of scientific publishing evolution +in the post-WW2 period. Despite the exponential increase in the number of published works (ac- +commodated by a corresponding swell in available journals), publications are becoming ever more +connected by citing each other. This can be seen in the rises of the references each work contains, +the citations works receive, the phenomenal proportion of all works ever published that are cited at +least once every year (20%), and corresponding rises to the 2-year and even 20-year global impact +factor. Authors appear to be collaborating more and on longer papers with only a slight decrease +in the mean number that they publish each year. The fall in the consolidation/destabilization (CD) +index is in line with recently published research reporting that papers are becoming less disruptive +over time.33 There is significant correlation (Spearman rank-order correlation coefficient 0.93; p- +value 3 × 10−29) between the CD5 index yearly averages obtained using Alexandria3k with those +available in the previously published dataset, which was obtained from data that are not openly +available. The sharp inflections in the Figure probably stem from artifacts of the underlying data +set, and indicate that obtaining scientifically robust results would require deeper analysis of the +data. +5 + +1945 +1950 +1955 +1960 +1965 +1970 +1975 +1980 +1985 +1990 +1995 +2000 +2005 +2010 +2015 +2020 +Year +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +Change (1945 = 1) +Authors per work 1.4 + 4.4 +Works per author 1.99 1.59 +References per work 13 46 +Pages per work 6.3 12.8 +CD index 0.88 + 0.22 +Works published 81k 7.0M (log) +Citations per work .009 + 1.14 (log) +Journals 958 + 68k (log) +Works cited at least once .3% 20% (log) +2-year IF .06 + 1.5 (log) +20-year IF .02 + 1.3 (log) +100 +101 +102 +Change (log scale; 1945 = 0) +Figure 1: Evolution of scientific publishing metrics in the post-WW2 period +Colloid and Surface Chemistry +Molecular Biology +Cell Biology +Genetics +Economics and Econometrics +General Biochemistry, Genetics and Molecular Biology +Cancer Research +Oncology +Statistics and Probability +General Chemistry +Biochemistry +Immunology +Mechanical Engineering +Reproductive Medicine +Physiology +General Physics and Astronomy +General Materials Science +General Neuroscience +Immunology and Allergy +Organic Chemistry +Inorganic Chemistry +Physical and Theoretical Chemistry +Pharmaceutical Science +General Economics, Econometrics and Finance +Spectroscopy +Drug Discovery +Analytical Chemistry +Ecology, Evolution, Behavior and Systematics +Economics, Econometrics and Finance (miscellaneous) +Chemical Engineering (miscellaneous) +Catalysis +Chemistry (miscellaneous) +Engineering (miscellaneous) +Materials Chemistry +Computer Science Applications +Animal Science and Zoology +Business, Management and Accounting (miscellaneous) +Education +Neuroscience (miscellaneous) +Atomic and Molecular Physics, and Optics +Figure 2: Strong relationships between citing and cited subjects +6 + +1950 +1960 +1970 +1980 +1990 +2000 +2010 +2020 +Year +0 +20 +40 +60 +80 +100 +Coverage percentage +Immunology and Microbiology +Pharmacology, Toxicology and Pharmaceutics +Energy +Earth and Planetary Sciences +Arts and Humanities +Agricultural and Biological Sciences +Computer Science +Materials Science +Social Sciences +Engineering +Others +0 +1 +2 +3 +4 +5 +6 +7 +Number of published works (millions) +Figure 3: Evolution of subject coverage and publications +Linking publications with their specific scientific field and their citations allows us to exam- +ine the structure of the corresponding graph. We can find fields that strongly depend on others by +defining the citation imbalance between two fields as the ratio of one field’s outgoing citations over +the total number of citations between them. Figure 2 shows the fifty strongest field relationships +in terms of imbalance. We note citing fields from the health and life sciences, numerous cited +fields associated with chemistry, and the large number of fields from which Oncology and Cancer +Research draw upon. +Associating publications with the general scientific field of the journal they were published +(according to the Scopus All Science Journal Classification Codes — ASJCs) provides us a view +of the evolution of the ASJC 27 fields over the years. Figure 3 shows the ten fields amounting to +more than 2% of publications in 2021 that had the largest change in their publication number in the +period 1950–2021. Clearly visible is the expected rise in Computer Science, Materials Science, +and Engineering, as well as the fall of Arts and Humanities and Social Sciences publications. +Owing to the exponential rise of published works (tallied on the Figure’s right-hand axis) the falls +are in relative terms: 2021 saw the publication of 291 366 Arts and Humanities articles against +31 716 in 1950. +To showcase how Alexandria3k could be used to study research related to COVID-1934 we +examined publications containing COVID in their title or abstract. We counted 491 945 publica- +tions from about 1.5 million authors. These covered 331 different topics demonstrating the many +disciplines associated with the research. Some noteworthy topics and work numbers among those +7 + +with more than one thousand publications include General Medicine (rank 1 — 70 609 works), Psy- +chiatry and Mental health (rank 4 — 10 404 works), Education (rank 5 — 9 590 works), Computer +Science Applications (rank 18 — 6 013 works), General Engineering (rank 20 — 5 942 works), +Strategy and Management (rank 42 — 3 208 works), Law (rank 57 — 2 557 works), History (rank +62 — 2 329 works), Cultural Studies (rank 76 — 1 893 works), Pollution (rank 97 — 1 549 works), +and Anthropology (rank 130 — 1 032 works). Looking at listed funders, we saw that the top three +in terms of associated publications were the National Natural Science Foundation of China (3 506 +works), followed by the (US) National Institutes of Health (2 316), the (US) National Science +Foundation (1 022), the Wellcome Trust (914), and the (UK) National Institute for Health Re- +search (661). We also examined the affiliations of COVID study authors, propagating them to the +highest parent organization (e.g. a university hospital to its university). Through this measure the +top five entities were the Government of the United States of America (1 465 works), the Uni- +versity of California System (925), University of Toronto (910), University of London (824), and +University of Oxford (660). +We also looked how long it took for COVID articles to start citing each other, building, as it +where, “on shoulders of giants”. As can be seen in extended Figure 9 a citation network ramped +up relatively quickly, surpassing ten thousand citations to COVID research by April 2020, and +reaching 118 thousand citations on March 2022. The large number of works published in January +2020 appears to be due to journals publishing later in the year volumes with that date. This was, +for example, the case with an overview article on COVID-19 and cerebrovascular diseases35 and +an eight month retrospective.36 (The same phenomenon may also explain the January 2021 rise.) +Notably, this backdating practice can distort the establishment of priority over scientific advances +based on a journal’s publication date. +We replicated diverse impact and productivity metrics typically calculated in a proprietary +fashion by commercial bibliometric data providers. We calculated the 2021 Journal Impact Factor +(JIF), and compared the 600 journals with the highest JIF available through Clarivate with 581 +we matched through their ISSNs, obtaining a Spearman rank-order correlation coefficient of 0.72 +with a p-value 3 × 10−93. We also calculated the journal h5-index, and compared the values of +the 100 “Top publications” venues listed in Google Scholar37 against those of Alexandria3k by +hand-matching the venue titles. The comparison of the 91 matched elements gave a Spearman +rank-order correlation coefficient of 0.75 with a p-value 6 × 10−18. Examining the same metric +in a field where considerable work is published in conferences, we compared the h5-index of the +13 common venues between the 20 Google Scholar reports in the “Software Systems” category38 +against a curated list of 32 software engineering venues,39 and obtained a Spearman rank-order +correlation coefficient of 0.83 with a p-value 0.0004. +Through the same data we also obtained the most cited articles published in the correspond- +ing period (predictably, the top-one, with 21 426 citations, was on COVID-19)40 and overall (the +top one was a 1996 article on the generalized gradient approximation method,41 which received +39 715 citations from a surprising diversity of fields). +8 + +We also obtained the h5-index on authors identified through ORCID. A noteworthy observa- +tion was the number of authors with a high metric: the top-ranked author has an h5-index of 76, +twelve authors have an h5-index larger than 60, and 100 larger than 38. We explored the phenom- +enal productivity and impact exhibited by the authors at the top of the distribution by examining +the clustering coefficient of the graph induced by incoming and outgoing citations of distance 2 +for a given work. The clustering coefficients of a random sample of 50 works from authors with +an h5-index larger than 50 (median coefficient 0.05) appear to be significantly different from a +random sample of other works of the same size with the same number of citations for each one +(median 0.03). Specifically, comparing the two populations we obtained a Mann-Whitney U statis- +tic measure of 781 for the coefficients of the top-ranked authors’ works with a p-value of 0.0006. +Conclusions +Alexandria3k builds on a rich and evolving ecosystem of publicly available data and sophisticated +open source software libraries coupled with exponential advances in computing power, enabling +scientists to perform reproducible bibliometric, scientometric, and research synthesis studies in a +transparent and repeatable manner on a personal computer. In our use of Alexandria3k we found +that over time its features and interface, as well as the methods we used to conduct the proof-of- +concept studies, crystallized into a form we employed for distributing the code of all the proof- +of-concept studies. Other researchers can readily build upon these examples to conduct their own +studies. +Alexandria3k is not a panacea for the issues we identified in the opening paragraphs. Cross- +ref’s coverage of citation links is lacking compared to commercial alternatives.42 The linkage of +publications to their authors through ORCID is thin, and ORCID author metadata is also sparse. +The string-based matching used to link textual author affiliations to RORs can result in mismatches. +The subjects associated with works are derived from the categorization of whole journals by Sco- +pus, and they are therefore incomplete and potentially inaccurate. The absolute values of derived +bibliometric measures do not match the proprietary ones, which employ different methods for data +collection (web crawling for Google Scholar) and for processing (hand-curation for Clarivate). In +general, many of the supported data elements can be used for making statistical observations, but +the validity of any findings needs to be verified, e.g. through sensitivity analysis based on the +manual examination of a sample. Finally, one should also take into account the epistemological +shortcoming of structured approaches that use publication databases.43 +Nevertheless, Alexandria3k opens many possibilities to conduct research that goes beyond +what can be easily done through the query and API-based approaches offered by existing propri- +etary7 and open44 databases. Examples include the study: of collaboration patterns between orga- +nizations and disciplines, of citation cliques and organizational inbreeding, of funder, publisher, +and organizational performance, of open access availability and its effects, of pre-print servers +as alternatives to peer-reviewed publications, of interdisciplinary connections, and of structural +9 + +publication differences between organizations or scientific fields. +Apart from making Alexandria3k available as open-source software we also structured it and +intend to run it as an open-source project, accepting and integrating contributions of additional +open-data sources (e.g. MEDLINE/PubMed, patent metadata), algorithms (e.g. publication topic +classification; matching of authors, affiliations, or citations), and example studies. We hope that +this will allow Alexandria3k to grow organically serving ever more needs of research synthesis and +analysis studies. +References +1. +Lammey, R. 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Postepy Mikrobiologii — Advancements of Microbiology 59. Published Online: 12 +October 2020, 207–225 (Jan. 2020). +37. +Google Scholar Top Publications — All Accessed 2022-06-23. 2022. https://scholar. +google.com/citations?view_op=top_venues. +38. +Google Scholar Top Publications — Software Systems Accessed 2022-06-23. 2022. https: +//scholar.google.com/citations?view_op=top_venues&vq=eng_ +softwaresystems. +39. +Mathew, G., Agrawal, A. & Menzies, T. Finding Trends in Software Research. IEEE Trans- +actions on Software Engineering (2019). +40. +Huang, C. et al. Clinical Features of Patients Infected With 2019 Novel Coronavirus in +Wuhan, China. The Lancet 395, 497–506 (Feb. 2020). +41. +Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. +Physical Review Letters 77, 3865–3868 (Oct. 1996). +42. +Visser, M., van Eck, N. J. & Waltman, L. Large-Scale Comparison of Bibliographic Data +Sources: Scopus, Web of Science, Dimensions, Crossref, and Microsoft Academic. Quanti- +tative Science Studies 2, 20–41 (2021). +43. +Boell, S. K. & Cecez-Kecmanovic, D. Literature Reviews and the Hermeneutic Circle. Aus- +tralian Academic & Research Libraries 41, 129–144 (June 2010). +44. +Priem, J., Piwowar, H. & Orr, R. OpenAlex: A Fully-Open Index of Scholarly Works, Authors, +Venues, Institutions, and Concepts 2022. +Acknowledgements +The author thanks Panos Louridas, Arie van Deursen, Theodoros Evgeniou, Alberto +Bacchelli, Dirk Beyer, and Dimitris Karlis for valuable advice and feedback. +12 + +Competing Interests +The author declares that he has no competing financial interests. +Correspondence +Correspondence and requests for materials should be addressed to D.S. (email: dds@aueb.gr). +13 + +Methods +The paragraphs below describe key elements behind the implementation of Alexandria3k and the +reported proof-of-concept studies. Complete details regarding these is the Alexandria3k source +code (also containing all studies as examples of its use), which is made available as open source +software.25 +Implementation. Alexandria3k is designed around modules that handle the Crossref, ORCID, +ROR, and flat file (journal names, funders, open access journals) data sources. Each module de- +fines the data source’s schema in terms of tables and their columns. The schema’s SQL DDL +implementation is used to define the corresponding tables in a populated database. The schema +is also used for analyzing and satisfying vertical slicing requests when populating databases with +Crossref and ORCID data, and for running Crossref queries without populating a database. Where +possible, data are on-the-fly decompressed, extracted from an archive (with a tar or zip structure), +and parsed in chunks, thus avoiding the storage cost of the entire decompressed data set (1 TB for +Crossref and 467 GB for ORCID). +Multiple alternatives were considered for the data back-end. The native Crossref tree format +would suggest a JSON-style NoSQL database. Redis1 would be an efficient back-end, but would +limit the amount of data to that that could be processed in memory. MondGB2 would address the +capacity limitation, at the cost of a more complex installation process due to licensing issues. A +client-server relational database system could also be used to offer improved integrity constraints +and more advanced query optimization facilities. However, these options require the installation, +configuration, connection, and maintenance of a separate database process, which is not a trivial +task, especially for researchers outside the computing field. Consequently, the adopted approach +uses the SQLite embedded database,3 which is installed as part of the Alexandria3k package, can +handle efficiently the provided data and queries on it, and has APIs for diverse programming lan- +guages and environments. +Direct queries on the Crossref data set (without populating a database) are implemented by +defining SQLite virtual tables that correspond to the offered schema through the Python apsw mod- +ule. Crossref data are distributed in the form of about 26 thousand compressed containers. Running +a query on them is trivial when the query accesses a single table: the virtual table implementation +moves from one container to the next as the table is scanned sequentially. +Direct Crossref queries involving multiple tables are handled differently. First, a dummy +query execution is traced to determine the tables and fields it requires. Then, in-memory tempo- +rary tables are populated with the required data from each container, and the query is executed +repeatedly on the instantiated tables. This approach works for all cases where relational joins +happen within a Crossref container (e.g. works with their authors or references). More complex +cases, such as relational joins between works or aggregations, require the population of an external +database. +14 + +Database population with vertical slicing is implemented by attaching the virtual database +to the one to be populated and running SELECT INTO populated-table SELECT FROM virtual- +table queries. A condition specifying a container-identifier is added to each query, so that all tables +for each decompressed container can be populated before moving to the next container. As before, +query tracing is used to identify the tables and fields that the user has asked to populate. +When the database population specifies horizontal slicing (through a row-selection SQL ex- +pression) the expression is evaluated sequentially on each container through the following steps. +1. Create a query based on the specified expression and trace it to determine the tables and +fields required for evaluating the expression. +2. Create the empty tables to be populated. +3. Calculate the topological ordering4 of the specified tables, which will be used to join them +in order to evaluate the expression. +4. Iterate through the Crossref containers doing the following. +(a) For each table to be populated or participating in the SQL expression create an in- +memory mirror temporary table with the table’s primary keys, foreign keys, and fields +participating in the query. +(b) Create another temporary table made by joining the mirror tables, evaluating the SQL +expression as a WHERE condition, and inserting the resulting rows. This table contains +the identifiers of matched works. +(c) Insert into each populated table records associated with the matched works (directly or +through JOIN relations) selected from the Crossref container being processed. +The database population design uses two techniques to improve its performance. +First, +records are bulk-inserted in batches by attaching directly the virtual tables to the database to be +populated, and by having the database engine perform the insert operations with a single internal +command for each batch. This avoids a round-trip cost of obtaining the data in Alexandria3k and +then storing them back in the database. Second, database indices over the containers in which +the data are split are implemented and used so as to access each file in turn for populating all re- +quired tables. The correspondingly improved locality of reference5 is then utilized by caching the +decompressed and parsed file contents. +The performant matching of author affiliations with RORs is based on multiple applications +of the Aho-Corasick string-matching algorithm.6 First, an Aho-Corasick automaton is created with +all unique organization names, aliases, and acronyms. Second, the automaton is used to find entries +in it that also match other entries (e.g. “ai”, the acronym of the “AI Corporation”, also matches the +organization name “Ministry of Foreign Affairs”), and mark them for removal to avoid ambiguous +matches. Finally, a new automaton constructed from the cleaned-up entries is used to find the +15 + +longest match associated with each author affiliation string. This is stored in the database as the +affiliation’s organization identifier. When the Alexandria3k user specifies that affiliations should +match the ultimate parent organization, a recursive SQL query adds a “generation” number to each +matched organization, an SQL window (analytic) function7 orders results by generation, and a final +selection query obtains the ROR identifier associated with the most senior generation. +Proof of concept studies framework. We structured most proof-of-concept studies we presented +as a series of queries that build on each other. This aids comprehensibility, testability, analysability, +and recoverability. We specified the corresponding workflow using Makefiles8 based on the simple- +rolap system, which manages relational online analytical processing tasks.9,10 The simple-rolap +system establishes the dependencies between queries and executes them in the required order. +Most studies start with a population phase, which fills a database with the required horizontal and +vertical data slices. In many cases, we used the rdbunit SQL unit-testing framework to test SQL +queries.10 We employed a shared Makefile with rules and configurations that satisfy dependencies +required by more than one study. For example, this Makefile contains rules to download required +data sets and to populate the database with the datasets that do not support slicing. The common +directory where the shared Makefile resides also hosts downloaded data sets to minimize useless +data duplication. +Research synthesis studies. We calculated the numbers associated with research synthesis studies +by processing the output of Alexandria3k run on the Crossref data with an SQL query that matches +specific words in publication titles. The query’s terms are structured to give precedence to the +characterization of titles indicating a systematic review as such, classifying the rest as (unspecified) +secondary studies. In the extended Figure 4 we combined the plotting of bibliometric (BM) and +scientometric (SM) studies, and did not plot figures for mapping reviews (MR), umbrella reviews +(UR), and tertiary studies (S3) — 6061 publications in total. We also did not plot the 400 identified +studies published before 1971 as well as studies published after 2021, as only part of the year 2022 +data were available. Note that works containing “bibliometric” or “scientometric” in their title may +either employ these methods11 or refer to them.12 We used another query with the same terms to +list the 30 studies published before 1950 and obtain the earliest one13 among them. +Crossref graph database. We created a vertical slice of the complete Crossref database, which we +used for a number of purposes. The slice contains mainly the primary and foreign keys (including +DOIs, and ORCIDs) of all entities, plus the publication year, author affiliation names, and work +subjects, which are not normalized. We also run Alexandria3k to populate the database with the +Scopus All Science Journal Classification Codes (ASJCs)14 and RORs, and linked work subjects +to ASJCs and author affiliations to RORs. +Scientific publishing evolution. In common with other studies15,16 we limited our examination to +works published after World War II, in order to avoid misleading comparisons with the markedly +different scientific and publication environment that preceded the war. We calculated most numbers +used for plotting Figure 1 from the populated Crossref graph database. We obtained the number of +published works and journals through SQL Count aggregations of the underlying data grouped +16 + +by year. We obtained the ratios of authors per work and references per work by counting the +corresponding elements of the associated detail tables and then obtaining SQL Avg aggregations +grouped by year. We joined papers with their citations using the document’s DOI as a key and used +this to calculate the two-year impact factor, the received citations per work, the twenty-year impact +factor, and the proportion of all published works cited at least once each year. For calculating the +last two we used SQL window (analytic) functions7 to obtain an accumulating sum and a twenty- +year rolling sum over the number of works published each year. +We calculated the number of pages per work by populating a database with the required +work and author details, and by extracting the starting and ending page number from Crossref +works that contain a dash in their pages field. This process excludes single-page works reported +with only a single page number (rather than as a range with the same starting and ending page). +We excluded from the data records with a zero or negative number of pages or those having more +than 1000 pages, because the latter (164387 records) were often derived from data-entry mis- +takes, such as repeated page digits (e.g. 234-2366), as well as unusable data formats, such as +1744-8069-5-32. +We calculated a measure that can be used to track author productivity (works per author) +despite clashes in author names, by taking advantage of the fact the we display productivity in +relative terms (adjusting it to be 1 in 1950), In absolute terms authors with same names increase +the productivity’s absolute value. (An author named Smith, Kim, or Zhu would appear extremely +prolific.) However, assuming that the ratio of clashing names in the population does not change, +the effect of duplicate names on the relative productivity measurements is cancelled out. +As an example consider that 50% of all authors are named Smith, all other authors have +distinct names, and in 1950 1000 authors write 2000 works. The actual productivity should be +2. By not distinguishing authors with clashing names the situation will appear as 500 authors +writing 2000 works, i.e. a productivity of 4. Consider now in 2020 10 000 authors writing 80 000 +works. The actual productivity should be 8. The situation will however appear as 5 000 authors +writing 80 000 works, i.e. a productivity of 16. While the derived absolute productivity numbers +are incorrect, the ratio of the correct productivity numbers is the same as the ones that do not take +name clashes into account: 2/8 = 4/16. +Note that an actual study of author productivity would need to test and control for the assump- +tion we made, because the population’s composition might change over the years to include authors +from ethnic backgrounds with more or fewer common name clashes, (for example, about 80% of +the China’s population shares the 100 most common Chinese surnames)17 making the phenomenon +more frequent or less frequent over time. Using Alexandria3k we obtained frequently-occurring +names at the two ends of the examined period and found that in 1950 the five most frequent names +were W. Beinhoff (161 works — 0.10% of the total names), E. Rosenberg (149 — 0.09%), F. De +Quervain (115 — 0.07%), G. Niemann (114 — 0.07%), and A. Eichler (105 — 0.06%); whereas in +2021 they were Y. Wang (63363 — 0.23%), Y. Zhang (57414 — 0.21%), Y. Li (51792 — 0.19%), +Y. Liu (46013 — 0.17%), and X. Wang (44805 — 0.16%). The different percentages at the two +17 + +period’s ends indicate that further controls for this change would be required, e.g. by measuring +name clashes through ORCIDs. +We calculated the CD5 index18 of Crossref publications by populating a database with their +publication date and DOI, as well as the DOIs of the corresponding references. Given the available +data, we were able to calculate the CD5 index for six additional years (until 2016) compared to +previous results,16 using the remaining five complete years we had at out disposal (2017–2021) to +obtain the required citations’ window. +The CD5 calculation proved to be computationally challenging. +The processing for the +already-published 1945–2010 range required more than five days of computing and about 40 GB +of RAM. Extending the range to 2016 increased the required time to 45 days. To address this +we enhanced the original CD5 algorithm implementation to use more efficient data structures and +algorithms converting it to C++.19 We employed a union of pointers and integers to efficiently +represent vertices internally and in Python code, and used C++ sorted vectors and sets to improve +memory allocation and searching for nodes. Furthermore, we rewrote the CD5 calculation process +in C++ in order to parallelize it while maintaining in memory a single copy of the 50 GB graph +data structure. (It turned out that this could not be done neither with Python’s threads nor with +forked processes.) The improvements in computational efficiency allowed us to perform the CD5 +index calculation in 9.5 hours of elapsed time using 67 hours of CPU time and 50 GB of main +memory. +To allow other researchers to build on this data without incurring the associated high com- +putational cost, we have made the resulting data set containing the DOI and the CD5 index for +11 568 934 publications in the range 1945–2016 openly available.20 This improves upon previ- +ously available data,21 which extends to 2010 and only provides the time-stamp of each publication, +without other uniquely distinguishing publication identifiers. +Field dependencies. We calculated strong dependencies between fields (Figure 2) by using work +references and subjects in the Crossref graph database to construct a table containing the number +of citations between fields. Based on it we calculated for each field pair its “strength” (sum of +incoming and outgoing citations) and its “fundamentalness” (ratio of a field’s outgoing citations +over the pair’s strength). In the plotted results we included the top 50 field associations in terms of +strength from the top 10% in terms of fundamentalness. We furthermore excluded pairs associated +with the “Multidisciplinary” subject and also links within fields. Both sides in the Figure represent +a small fraction of the total field citations, and are drawn on different scales: the outgoing citations +shown amount to 0.8% of the fields’ total and the incoming ones amount to 0.2% of it. +Field evolution. We calculated the evolution in the number of publications across scientific fields +(Figure 3) by propagating the specific fields associated with each work in the Crossref graph +database to the more general containing field. For that we used as general fields the Scopus ASJCs +that ended in “00”, and as their sub-fields those that started with the same numeric prefix. For +example, we allocated publications under the subject of “Catalysis” (1503) to “General Chemical +18 + +Engineering” (1500). We then calculated total publications in 1950 and 2021, changes in the per- +centages of a field’s publications in terms of the total at the two time points, and included in the +Figure the ten fields with the largest change whose publications amounted to more than 2% of the +2021 total. +COVID-19 metrics. To study COVID-19 publications we populated a database with a full hori- +zontal slice of the Crossref data by specifying as the row selection criterion works containing the +string “COVID” in their title or abstract (“covid” is not part of any English word). We also linked +works to their subjects and author affiliations to the corresponding RORs. We obtained organi- +zations publishing COVID-19 research by assigning author affiliations to works, and by counting +both ROR-matched affiliations and unmatched affiliations as simple text. +Number of COVID-19 study authors. We calculated the approximate number of researchers +who worked on all COVID-19 studies by starting with the number of unique (author given-name, +author surname) pairs in the set of all COVID study authors Nan. The number of true authors could +be higher if many authors share the same name or lower if the same author appears differently (e.g. +through the use of initials) in some publications. We address this by obtaining from the set of +authors with an ORCID the number of distinct ORCIDs No, which is the true number of authors +in that set, and the number of distinct names Non, which approximates any bias also found in Nan. +We then consider the true number of authors as +Nan +No +Non +Journal impact factor. We calculated the 2021 journal impact factor22 by populating a database +with the keys, ISSNs, publications years, and pages of works and their references published be- +tween 2019 and 2021. We then created a table associating works with journal ISSNs. From this +table we obtained citations published in 2021 to works published in 2019–2020 (the impact fac- +tor’s numerator). We further filtered works to identify “citable” items, which Clarivate defines as +those that make a substantial contribution to science and therefore do not include elements such as +editorials and letters. For this we used as a rough heuristic works longer than two pages. (We also +included works lacking a page range.) From the count of citable items per journal we obtained the +number of publications published in the 2019–2020 period (the denominator). Finally, to compare +our results with the numbers published by Clarivate we associated each impact factor metric with +all ISSNs known for a journal (electronic, print, alternative), excluding the “alternative” ISSNs of +one journal used as primary for another journal. +Productivity metrics. We obtained the h5-index23 productivity metrics by populating databases +with data sliced vertically to include the keys of works and references and horizontally to include +items published in the period 2017–2021. For the software engineering venue metrics we selected +the examined conferences based on the DOI prefix assigned to the conference publications each +year. (In retrospect, we could have used the container title.) For each entity we counted its citations +and then used an SQL window (analytic) function to partition the results over the entity’s key +19 + +(ISSN, ORCID, or conference acronym), number each set’s rows, and select only those with a +rank lower or or equal to the corresponding number of citations. +To study the citation graph of top-ranked authors we obtained a) a random sample of 50 +works written by top-ranked authors, and b) a random sample of 50 works from all other publi- +cations, paired with the ones selected from the top-ranked authors to have the same number of +citations as them. For each publication in the two samples, we created a separate graph containing +the work w, the set S of the works w cites and the works that cite w, and then again the set S′ of the +works w′ that cite or are cited by w ∈ S. The graph’s edges are citations from one work to another. +We created each citation-induced graph with a Python program querying the populated database +and employed the NetworkX24 Python package to calculate the graph’s average clustering. +SQLite lacks the ability to provide a seed to its Random() function, which is required for +obtaining random samples in a deterministic manner. We worked around this limitation by multi- +plying the identifier of each author (which is sequentially allocated and therefore not random) with +a seed value, and used the last decimal digits of the result to place each work in a pseudo-random +ordering, which we used to obtain the required works. +Statistical analysis. For reporting the correlation between the metrics obtained by Alexandria3k +and existing ones and for comparing the graph clustering coefficients between two populations +we used the functions spearmanr and mannwhitneyu from the Python package scipy.stats. All +calculations were performed with “two-sided” as the alternative hypothesis (the default). No other +options were provided to the function calls. For the analysis and charting we used Python 3.9.10 +with matplotlib 3.3.4, numpy 1.20.1, pandas 1.2.3, pySankey 0.0.1, and scipy 1.6.2. +Code and data availability. A versioned release of the source code of Alexandria3k and the +proof-of-concept examples presented in this study is available on Zenodo.25 A replication package +with the results data and scripts associated with the reported example studies is also available +on Zenodo.26 Current versions of Alexandria3k are made available for installation through PyPi +https://pypi.org/project/alexandria3k/ and for contributions, feature requests, +and issue reporting on GitHub https://github.com/dspinellis/alexandria3k. +The data used in the example studies are available as follows. +Crossref April 2022 Public Data File +DOI:10.13003/83b2gq +ORCID Public Data File 2022 version 4 +DOI:10.23640/07243.21220892.v4 +ROR Data v1.17.1 +DOI:10.5281/zenodo.7448410 +Open access journals +https://doaj.org/csv +Funders +https://doi.crossref.org/funderNames?mode=list +Journals +http://ftp.crossref.org/titlelist/titleFile.csv +20 + +The 2021 Journal Impact Factor data used for assessing the numbers obtained by Alexan- +dria3k are available from Clarivate, but restrictions apply to the availability of these data, which +were used under license for the current study, and so are not publicly available. These data are +however available from the authors upon reasonable request and with permission of Clarivate. +References +1. +Carlson, J. Redis in Action (Simon and Schuster, 2013). +2. +Banker, K., Garrett, D., Bakkum, P. & Verch, S. MongoDB in Action: Covers MongoDB +Version 3.0 (Simon and Schuster, 2016). +3. +Owens, M. The Definitive Guide to SQLite ISBN: 978-1-59059-673-9 (Apress, Berkeley, CA, +2006). +4. +Dial, R. B. Algorithm 360: Shortest-Path Forest With Topological Ordering [H]. Communi- +cations of the ACM 12, 632–633 (1969). +5. +Denning, P. J. The Locality Principle. Communications of the ACM 48, 19–24 (July 2005). +6. +Aho, A. V. & Corasick, M. J. Efficient String Matching: an Aid to Bibliographic Search. +Communications of the ACM 18, 333–340 (1975). +7. +Zemke, F. What’s New in SQL: 2011. ACM SIGMOD Record 41, 67–73 (2012). +8. +Feldman, S. I. Make—A Program for Maintaining Computer Programs. Software: Practice +and Experience 9, 255–265 (1979). +9. +Gousios, G. & Spinellis, D. Mining Software Engineering Data from GitHub in Proceedings +of the 39th International Conference on Software Engineering Companion Technical Briefing +(IEEE Press, Buenos Aires, Argentina, May 2017), 501–502. ISBN: 978-1-5386-1589-8. +10. +Gousios, G. & Spinellis, D. Modular Queries and Unit Testing Technical briefing notes; +available online https://doi.org/10.5281/zenodo.7513793. May 2017. +11. +Carpenter, K. et al. Social Perception and Academic Trends on Women in the Field of Neu- +rosurgery: A Bibliometric Analysis. World Neurosurgery 152, 154–161 (Aug. 2021). +12. +Braun, T. Bibliometrics in Research Evaluation. Journal of Studies in International Educa- +tion 14, 365–366 (Jan. 1988). +13. +Burmeister, H., Bell, T. & Forbes, E. The Organization of Trilobites, Deduced from their +Living Affinities; With a Systematic Review of the Species Hitherto Described (Printed for the +Ray Society, London, 1846). +14. +Schotten, M., Meester, W. J., Steiginga, S., Ross, C. A., et al. in Research Analytics 31–58 +(Auerbach Publications, 2017). +15. +Wuchty, S., Jones, B. F. & Uzzi, B. The Increasing Dominance of Teams in Production of +Knowledge. Science 316, 1036–1039 (May 2007). +16. +Park, M., Leahey, E. & Funk, R. J. Papers and Patents are Becoming Less Disruptive Over +Time. Nature 613, 138–144 (Jan. 2023). +21 + +17. +Wang, Z. Author Lists: Include Original Language Script and Unique Identifiers. Nature 567, +311–311 (Mar. 2019). +18. +Funk, R. J. & Owen-Smith, J. A Dynamic Network Measure of Technological Change. Man- +agement Science 63, 791–817 (Mar. 2017). +19. +Funk, R. J. & Spinellis, D. dspinellis/fast-cdindex: 1.2.0 version 1.2.0. Jan. 2023. https: +//doi.org/10.5281/zenodo.7585311. +20. +Spinellis, D. CD5 Index of Works Published in the Period 1945-2016 version 1.0.0 (Zenodo, +Jan. 2023). https://doi.org/10.5281/zenodo.7584373. +21. +Funk, R. J., Park, M. & Leahey, E. Papers and Patents are Becoming Less Disruptive Over +Time Oct. 2022. +22. +Garfield, E. The History and Meaning of the Journal Impact Factor. Journal of the American +Medical Association 295, 90–93 (2006). +23. +Hirsch, J. E. An Index to Quantify an Individual’s Scientific Research Output. Proceedings +of the National Academy of Sciences 102, 16569–16572 (Nov. 2005). +24. +Hagberg, A., Swart, P. & S Chult, D. Exploring Network Structure, Dynamics, and Function +Using NetworkX tech. rep. LA-UR-08-5495 (Los Alamos National Lab, Los Alamos, NM, +US, 2008). +25. +Spinellis, D. dspinellis/alexandria3k: 2.4.0 version 2.4.0. Jan. 2023. https://doi.org/ +10.5281/zenodo.7584743. +26. +Spinellis, D. Replication package for the Alexandria3k paper version 1.0.0. Jan. 2023. https: +//doi.org/10.5281/zenodo.7586574. +22 + +Extended data +1970 +1980 +1990 +2000 +2010 +2020 +Year +100 +101 +102 +103 +104 +Number of studies +Systematic +Scientometric / bibliometric +Meta-analysis +Other secondary +Figure 4: Number of research synthesis studies published each year in the period 1971–2021. +23 + +Table 1: Number of Crossref Records +Entity +Records +Total records +2 531 227 295 +Works (publications) +134 048 223 +Works with a text mining link +96 294 821 +Works with subject +81 210 089 +Works with references +52 907 361 +Works with affiliation +36 389 868 +Works with an abstract +15 367 820 +Works with funders +7 519 462 +Author records (linked to works) +359 556 891 +Author records with ORCID +16 745 506 +Distinct authors with ORCID +4 525 906 +Author affiliation records +76 759 875 +Distinct affiliation names +19 453 361 +Work subject records +182 858 177 +Distinct subject names +340 +Work funders +15 491 915 +Funder records with DOI +10 811 496 +Distinct funder DOIs +29 610 +Funder awards +14 090 597 +References +1 748 421 617 +References with DOI +1 255 033 889 +Distinct reference DOIs +61 609 121 +24 + +Table 2: Number of ORCID Records +Table +Persons with +Records +Such Records +Personal data +14 811 567 +14 811 567 +URLs +1 325 399 +892 528 +Countries +2 141 021 +2 021 218 +Keywords +2 230 569 +838 796 +External identifiers +2 287 063 +1 653 995 +Distinctions +345 967 +159 113 +Educations +5 610 051 +3 087 381 +Employments +5 932 529 +3 472 114 +Invited positions +263 315 +147 497 +Memberships +660 304 +357 799 +Qualifications +940 400 +585 298 +Services +221 860 +108 288 +Fundings +1 170 432 +328 958 +Peer reviews +6 270 131 +751 404 +Research resources +3 012 +1 779 +Person’s works +34 237 877 +3 224 068 +Total +78 451 497 +25 + +works +id +container_id +doi +title +published_year +published_month +published_day +short_container_title +container_title +publisher +abstract +type +subtype +page +volume +article_number +journal_issue +issn_print +issn_electronic +update_count +work_authors +id +container_id +work_id +orcid +suffix +given +family +name +authenticated_orcid +prefix +sequence +1…N +1 +work_references +work_id +container_id +issn +standards_body +issue +key +series_title +isbn_type +doi_asserted_by +first_page +isbn +doi +component +article_title +volume_title +volume +author +standard_designator +year +unstructured +edition +journal_title +issn_type +0…N +1 +work_updates +work_id +container_id +label +doi +timestamp +0…N +1 +work_subjects +work_id +container_id +name +0…N +1 +work_funders +id +container_id +work_id +doi +name +0…N +1 +work_licenses +work_id +container_id +url +start_timestamp +delay_in_days +0…N +1 +work_links +work_id +container_id +url +content_type +0…N +1 +author_affiliations +author_id +container_id +name +1…N +1 +funder_awards +funder_id +container_id +name +0…N +1 +Figure 5: Relational schema of Crossref tables in a fully-populated database. +26 + +persons +id +orcid +given_names +family_name +biography +person_researcher_urls +person_id +name +url +person_countries +person_id +country +person_keywords +person_id +keyword +person_external_identifiers +person_id +type +value +url +person_distinctions +person_id +organization_name +organization_city +organization_region +organization_country +organization_identifier +department_name +role_title +start_year +start_month +start_day +end_year +end_month +end_day +person_educations +person_id +organization_name +organization_city +organization_region +organization_country +organization_identifier +department_name +role_title +start_year +start_month +start_day +end_year +end_month +end_day +person_employments +person_id +organization_name +organization_city +organization_region +organization_country +organization_identifier +department_name +role_title +start_year +start_month +start_day +end_year +end_month +end_day +person_invited_positions +person_id +organization_name +organization_city +organization_region +organization_country +organization_identifier +department_name +role_title +start_year +start_month +start_day +end_year +end_month +end_day +person_memberships +person_id +organization_name +organization_city +organization_region +organization_country +organization_identifier +department_name +role_title +start_year +start_month +start_day +end_year +end_month +end_day +person_qualifications +person_id +organization_name +organization_city +organization_region +organization_country +organization_identifier +department_name +role_title +start_year +start_month +start_day +end_year +end_month +end_day +person_services +person_id +organization_name +organization_city +organization_region +organization_country +organization_identifier +department_name +role_title +start_year +start_month +start_day +end_year +end_month +end_day +person_fundings +person_id +title +type +short_description +amount +url +start_year +start_month +start_day +end_year +end_month +end_day +organization_name +organization_city +organization_region +organization_country +organization_identifier +person_peer_reviews +person_id +reviewer_role +review_type +subject_type +subject_name +subject_url +group_id +completion_year +completion_month +completion_day +organization_name +organization_city +organization_region +organization_country +person_research_resources +person_id +title +start_year +start_month +start_day +end_year +end_month +end_day +person_works +person_id +doi +Figure 6: Relational schema of ORCID tables in a fully-populated database. +27 + +research_organizations +id +ror_path +name +status +established +country_code +ror_types +ror_id +type +1…N +1 +ror_links +ror_id +link +0…N +1 +ror_aliases +ror_id +alias +0…N +1 +ror_acronyms +ror_id +acronym +0…N +1 +ror_relationships +ror_id +type +ror_path +0…N +1 +ror_addresses +ror_id +lat +lng +city +state +postcode +0…N +1 +ror_funder_ids +ror_id +funder_id +0…N +1 +ror_wikidata_ids +ror_id +wikidata_id +0…N +1 +ror_isnis +ror_id +isni +0…N +1 +Figure 7: Relational schema of ROR tables in a fully-populated database. +28 + +Alexandria3k tables +asjcs +id +field +subject_area_id +general_field_id +asjc_general_fields +id +name +1 +1…N +asjc_subject_areas +id +name +1 +1…N +works_asjcs +work_id +asjc_id +1 +0…N +work_authors_rors +ror_id +work_author_id +work_authors +1 +0…N +Main Crossref tables +Additional Crossref tables +works +0…N +1 +person_works +0…1 +0..1 +open_access_journals +name +url +doaj_ur +oaj_start +alternative_name +issn_print +issn_eprint +keywords +languages +publisher +pubisher_country +society +society_country +license +license_attributes +license_terms_url +license_embedded +example_license_embedded_url +author_copyright +copyright_info_url +review_process +review_process_url +plagiarism_screening +plagiarism_info_url +aims_scope_url +board_url +author_instructions_url +sub_pub_weeks +apc +apc_info_url +apc_amount +apc_waiver +apc_waiver_info_url +other_fees +other_fees_info_url +preservation_services +preservation_national_library +preservation_info_url +deposit_policy_directory +deposit_policy_directory_url +persistent_article_identifiers +orcid_in_metadata +i4oc_compliance +doaj_oa_compliance +oa_statement_url +continues +continued_by +lcc_codes +subjects +doaj_Seal +added_on +last_updated +article_records_number +most_recent_addition +0…1 +1 +journals_issns +journal_id +issn +issn_type +0…1 +1 +persons +0…1 +1 +work_subjects +1 +1 +work_funders +ror_funder_ids +0…1 +1 +funder_names +url +name +replaced +0…1 +1 +ORCID tables +ROR tables +research_organizations +0…N +1 +DOAJ table +journal_names +title +id +publisher +issn_print +issn_eprint +issns_additional +doi +volume_info +1…N +1..N +Figure 8: Relational schema of other tables and Alexandria3k-generated links in a fully-populated +database. +29 + +2020-1 +2020-2 +2020-3 +2020-4 +2020-5 +2020-6 +2020-7 +2020-8 +2020-9 +2020-10 +2020-11 +2020-12 +2021-1 +2021-2 +2021-3 +2021-4 +2021-5 +2021-6 +2021-7 +2021-8 +2021-9 +2021-10 +2021-11 +2021-12 +2022-1 +2022-2 +2022-3 +2022-4 +Citing publication month +0 +20 +40 +60 +80 +100 +120 +Number of citations to COVID publications (thousands) +Figure 9: Citations from COVID research to COVID research over time +30 + diff --git a/PdFQT4oBgHgl3EQfYjZ5/content/tmp_files/load_file.txt b/PdFQT4oBgHgl3EQfYjZ5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..612a807b13884a3d56dac69d0f1febabf41f2102 --- /dev/null +++ b/PdFQT4oBgHgl3EQfYjZ5/content/tmp_files/load_file.txt @@ -0,0 +1,1488 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf,len=1487 +page_content='Open reproducible systematic publication research Diomidis Spinellis1,2 1Department of Management Science and Technology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Athens University of Economics and Busi- ness 2Department of Software Technology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Delft University of Technology Considerable scientific work involves locating, analyzing, systematizing, and synthesizing other publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Its results end up in a paper’s “background” section or in standalone articles, which include meta-analyses and systematic literature reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The required re- search is aided through the use of online scientific publication databases and search engines, such as Web of Science, Scopus, and Google Scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' However, use of online databases suf- fers from a lack of repeatability and transparency, as well as from technical restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Thankfully, open data, powerful personal computers, and open source software now make it possible to run sophisticated publication studies on the desktop in a self-contained en- vironment that peers can readily reproduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Here we report a Python software package and an associated command-line tool that can populate embedded relational databases with slices from the complete set of Crossref publication metadata,1 ORCID author records,2 and other open data sets, for in-depth processing through performant queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We demonstrate the software’s utility by analyzing the underlying dataset’s contents, by visulizing the evolu- tion of publications in diverse scientific fields and relationships between them, by outlining scientometric facts associated with COVID-19 research, and by replicating commonly-used bibliometric measures of productivity and impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Research synthesis is becoming an increasingly important3 and popular scientific method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' By our own calculations about 437 thousand scientific studies published from 1846 — the year of the first one we found4 — onward are based on the analysis of previously published primary studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' (The Methods section provides details on how all figures appearing in this report where obtained in a repeatable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=') These studies are typically identified in their titles with terms such as “systematic review”, “systematic literature review”, or “systematic mapping study” (sec- ondary studies using methods that help make their findings unbiased and repeatable — 251 850 titles);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' “secondary study”, “literature survey”, or “literature review” (a not necessarily systematic study reviewing primary studies — 77 037 titles);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' “tertiary study” or “umbrella review”5 (a study reviewing secondary studies — 4 039 titles);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' “meta-analysis” (a systematic secondary study em- ploying statistical methods — 92 363 titles);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='6 as well as (systematic by definition) “scientometric” (employing quantitative methods to study scientific research — 2 769 titles) and “bibliometric” (using statistical methods to study written communications — 12 361 titles) studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='6 The number of such studies published each year has risen considerably over the past two decades, particularly so for systematic literature reviews (see extended Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' A major data source for research syn- thesis studies are online specialized and general purpose bibliographic and article databases,7 such as ERIC, Google Scholar, Inspec, Scopus, and Web of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='13312v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='DL] 30 Jan 2023 Performing systematic studies on published literature through the available online systems can be problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='8 First, their constantly updated contents, bubble effects,9 location- and license- dependent results,10 and periodic changes to their internal workings compromise reproducibil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='11,12 Even when the search strategy is well-documented to aid reproducibility, by following best practice reporting guidance such as PRISMA,13 which is often not the case,14,15 it is difficult to repeat a query to an online service, and obtain the same results as those that have been published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='8 An associated problem is the lack of transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='16 Most online services work with proprietary data collections and algorithms, making it difficult to understand and explain the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' As an example, Clarivate’s journal impact factor calculation depends on an opaque collection of journals17 and list of “citable items”18 tagged so by the vendor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' In addition the reproducibility of such studies is hampered in the short term by the fees required for accessing some online services and in the long term by the commercial survival of the corresponding companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='19 Also, service access costs on their own can restrict institutions with limited funding from conducting systematic literature studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Finally, there are technical limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Some services lack a way to access them programmatically (an application programming interface — API),8 forcing researchers to resort to tricky and unreliable contortions, such as screen scrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Both APIs and offered query languages are not standardized,20 and often restrict the allowed operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='8 In addition, the network-based APIs suffer from corresponding latency21 and also often from rate and ceiling limits to the number of allowed invocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='22 These restrictions make it difficult to run studies that require performing a large number or sophisticated queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The outlined problems can be addressed thanks to sustained exponential advances in com- puting power,23 drops in associated costs, and Open Science initiatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='24 The Alexandria3k system presented here is an open-source software library and command-line tool that builds on these ad- vances to allow the conduct of sophisticated systematic research of published literature, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' liter- ature reviews, meta-analyses, bibliometric and scientometric studies) in a transparent, repeatable, reproducible, and efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Alexandria3k allows researchers to process on a personal com- puter publications’ metadata (including citations) from most major international academic publish- ers as well as corresponding author, funder, organization, and journal details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Specifically, Alexan- dria3k works on data snapshots offered periodically by initiatives, such as Crossref (publication metadata, journal names, funder names),1 ORCID (author details),2 ROR (research organization registry),25 and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Using Alexandria3k researchers can query and process that data through SQL queries launched by means of command-line tool invocations or Python scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Researchers can ensure the transparency, reproducibility, and exact repeatability of their methods by document- ing or publishing the version of the data used and the employed commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='24 The primary data sets can be stored and processed locally on a modern laptop, because they amount to a few hundred gigabytes in their compressed format (157 GB for Crossref, 25 GB for ORCID data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' the download- ing of the Crossref data is facilitated by its availability through the BitTorrent protocol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='26 The data are decompressed in small chunks ensuring that both main and secondary memory requirements are kept within the limits of what a typical personal computer can accommodate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' (Keeping the data decompressed or populating a relational or graph database with all of it would require more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='5 TB of storage space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=') In addition, Alexandria3k offers facilities for running relational database 2 queries on data partitions, sampling records, and populating a relational database with a subset of records or fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' All these facilities help tasks to execute in reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' On a populated and suitably indexed database with millions of records many queries finish in minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Queries or database population tasks involving a full scan of the entire Crossref publication data set complete in less than five hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Contents, structure, and use In total, Alexandria3k offers relational query access to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='6 billion records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' These are organized in a relational schema illustrated in supplemental figures 5–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Most records are publication metadata obtained from the Crossref Public Data File.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' These contain publication details (DOI, title, abstract, date, venue, type, pages, volume, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='), a publica- tion’s references to other publications (DOI, title, author, page, ISSN, ISBN, year, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='), and other data associated with each publication authors and their affiliations, funders and funder awards, up- dates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' retractions), subjects, licenses, and hyperlinks for text mining of the publication’s full text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='1 Details about the data available through Crossref are listed in extended Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Note that coverage is incomplete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' for example, 39% of the publications have a reference list associated with them, 70% of funders are uniquely identified with a DOI, while only 11% of the publications have an abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' For most types of records coverage is increasing over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Alexandria3k can link Crossref records to imported author metadata through ORCID (Open Researcher and Contributor ID) a non-proprietary system developed to identify authors in schol- arly communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='2 ORCID tables that Alexandria3k supports include those detailing an author’s URLs, countries, keywords, external identifiers, distinctions, education, employment, invited po- sitions, memberships, qualifications, services, fundings, peer reviews, used research resources, and published works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Most of these tables contain details of the associated organization (name, department, city, region, country), the role title, and the starting and end date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The currently avail- able ORCID data set contains about 78 million records associated with 14 million authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The completeness of the ORCID records is low and uneven (see extended Table 2), which means that research based on it must be carefully designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Alexandria3k can also import the Research Organization Registry data25 containing details of 104 402 organizations, as well as related acronyms (43 862 records) and aliases (25 119 records).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Through the provided ROR identifier it can link unambiguously elements from a person’s employ- ment and education ORCID records to the corresponding organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Currently ORCID contains such identifiers for 130 033 employment records and 133 066 education records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Given that only 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='6% of work author records have an ORCID and only 23% of ORCID records contain employ- ment information, Alexandria3k also provides a performant facility to match the textual affiliation information listed in works and link it to ROR identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 3 Finally, Alexandria3k can import and link three reference tables: the names of journals as- sociated with ISSNs (currently 109 971 records), the funder names associated with funder DOIs (32 198 records), and the metadata of open access journals (18 717).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='27 Alexandria3k further disag- gregates journal ISSNs according to their type (electronic, print, or alternative — 158 580 records).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The data used by Alexandria3k is openly distributed by diverse third parties (see the data availability statement) in textual tree or flat format files: JSON for Crossref and ROR, XML for ORCID, and CSV for the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Alexandria3k structures the data it offers in the relational schema of 45 tables linked through 47 relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Stored in a relational database and combined with suitable indexes, this allows performing sophisticated analyses via SQL (structured query language) queries in an efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Records between diverse data sets are linked through standardized globally unique identifiers: DOIs for published works and funders, ISSNs for journals, ORCIDs for authors, and RORs for research organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Alexandria3k is distributed as open source software in the form of a Python package, which can be easily installed through the PyPI repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' It can be used either as a command-line tool, where its operation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' query to run) is specified through command-line arguments, or as a Python library, which can be used interactively (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' by developing a Jupyter Notebook28) or through scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' In its simplest form Alexandria3k can evaluate an SQL query directly on the Crossref dataset, often to perform exploratory data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Results can be saved as a CSV (comma-separated values) file or iterated over through Python code for further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' This mode has limitations in terms of performance, aggregation of query results, and combination of data from multiple sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' In most cases Alexandria3k is used to populate an SQLite database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='29 The SQLite database can be easily used, because its engine is embedded into Alexandria3k, directly available in Python, and easily installable as a command-line tool in all popular computing platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Transferring a database between computers only involves copying the corresponding file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Consequently, there is no need to setup, configure, and maintain a complex client-server relational or NoSQL database management system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Despite its -Lite suffix, SQLite supports most of the SQL standard (including window functions and recursive queries), and employs sophisticated query optimization methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' these feature make it ideal for use in Alexandria3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' SQLite’s main downsides — lack of multi-user and client-server support — are not relevant to common Alexandria3k use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The latest version of the Crossref data is distributed as 26 thousand compressed container files, each containing details about 5 000 works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' A complete import of the Crossref data would amount to a 520 GB database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Given the large amount of Crossref data, the population of a database with it can be controlled in three ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' First, only a horizontal subset of records can be imported, by specifying an SQL expression that will select a publication record only when it 4 evaluates to TRUE (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' published year BETWEEN 2017 AND 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' To facilitate the selection of records selected through other means, the expression can also refer to tables of exter- nal databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Second, only a subset of the Crossref 26 810 data containers can be processed for import, by specifying a Python function that will import a container when it evaluates to True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' This is mostly useful for random sampling, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' using random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='random() < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='01 to sam- ple approximately 1% of the containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' (A fixed seed value is used internally for initializing the pseudo-random number generator to allow deterministic and therefore repeatable sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=') Third, the populated tables or columns of the Crossref data set can be vertically restricted by us- ing the table-name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='column-name or table-name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' * SQL notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The population of a database with ORCID data can be also horizontally restricted to records associated with existing Crossref authors or published works (probably selected in a previous population step) and verti- cally restricted to include only specific tables or columns, as in the case of Crossref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Given their small volume, no population controls are supported for the other data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Application examples The following paragraphs outline some simple proof-of-concept applications of Alexandria3k, which demonstrate its use and motivate its adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' All are exactly replicable through SQL queries and relational online analytical processing Makefiles30–32 provided in the accompanying materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' For a start, all metrics provided in the preceding section and in extended Tables 1 and 2 were obtained through simple SELECT Count(*) FROM table or SELECT Count(*) FROM (SELECT DISTINCT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=') SQL queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Figure 1 showcases the use of Alexandria3k to chart a view of scientific publishing evolution in the post-WW2 period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Despite the exponential increase in the number of published works (ac- commodated by a corresponding swell in available journals), publications are becoming ever more connected by citing each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' This can be seen in the rises of the references each work contains, the citations works receive, the phenomenal proportion of all works ever published that are cited at least once every year (20%), and corresponding rises to the 2-year and even 20-year global impact factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Authors appear to be collaborating more and on longer papers with only a slight decrease in the mean number that they publish each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The fall in the consolidation/destabilization (CD) index is in line with recently published research reporting that papers are becoming less disruptive over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='33 There is significant correlation (Spearman rank-order correlation coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='93;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' p- value 3 × 10−29) between the CD5 index yearly averages obtained using Alexandria3k with those available in the previously published dataset, which was obtained from data that are not openly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The sharp inflections in the Figure probably stem from artifacts of the underlying data set, and indicate that obtaining scientifically robust results would require deeper analysis of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 5 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 Year 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='5 Change (1945 = 1) Authors per work 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='4 Works per author 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='59 References per work 13 46 Pages per work 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='8 CD index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='22 Works published 81k 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='0M (log) Citations per work .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='009 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='14 (log) Journals 958 68k (log) Works cited at least once .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='3% 20% (log) 2-year IF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='5 (log) 20-year IF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='3 (log) 100 101 102 Change (log scale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 1945 = 0) Figure 1: Evolution of scientific publishing metrics in the post-WW2 period Colloid and Surface Chemistry Molecular Biology Cell Biology Genetics Economics and Econometrics General Biochemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Genetics and Molecular Biology Cancer Research Oncology Statistics and Probability General Chemistry Biochemistry Immunology Mechanical Engineering Reproductive Medicine Physiology General Physics and Astronomy General Materials Science General Neuroscience Immunology and Allergy Organic Chemistry Inorganic Chemistry Physical and Theoretical Chemistry Pharmaceutical Science General Economics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Econometrics and Finance Spectroscopy Drug Discovery Analytical Chemistry Ecology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Evolution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Behavior and Systematics Economics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Econometrics and Finance (miscellaneous) Chemical Engineering (miscellaneous) Catalysis Chemistry (miscellaneous) Engineering (miscellaneous) Materials Chemistry Computer Science Applications Animal Science and Zoology Business,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Management and Accounting (miscellaneous) Education Neuroscience (miscellaneous) Atomic and Molecular Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' and Optics Figure 2: Strong relationships between citing and cited subjects 6 1950 1960 1970 1980 1990 2000 2010 2020 Year 0 20 40 60 80 100 Coverage percentage Immunology and Microbiology Pharmacology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Toxicology and Pharmaceutics Energy Earth and Planetary Sciences Arts and Humanities Agricultural and Biological Sciences Computer Science Materials Science Social Sciences Engineering Others 0 1 2 3 4 5 6 7 Number of published works (millions) Figure 3: Evolution of subject coverage and publications Linking publications with their specific scientific field and their citations allows us to exam- ine the structure of the corresponding graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We can find fields that strongly depend on others by defining the citation imbalance between two fields as the ratio of one field’s outgoing citations over the total number of citations between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Figure 2 shows the fifty strongest field relationships in terms of imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We note citing fields from the health and life sciences, numerous cited fields associated with chemistry, and the large number of fields from which Oncology and Cancer Research draw upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Associating publications with the general scientific field of the journal they were published (according to the Scopus All Science Journal Classification Codes — ASJCs) provides us a view of the evolution of the ASJC 27 fields over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Figure 3 shows the ten fields amounting to more than 2% of publications in 2021 that had the largest change in their publication number in the period 1950–2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Clearly visible is the expected rise in Computer Science, Materials Science, and Engineering, as well as the fall of Arts and Humanities and Social Sciences publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Owing to the exponential rise of published works (tallied on the Figure’s right-hand axis) the falls are in relative terms: 2021 saw the publication of 291 366 Arts and Humanities articles against 31 716 in 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' To showcase how Alexandria3k could be used to study research related to COVID-1934 we examined publications containing COVID in their title or abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We counted 491 945 publica- tions from about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='5 million authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' These covered 331 different topics demonstrating the many disciplines associated with the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Some noteworthy topics and work numbers among those 7 with more than one thousand publications include General Medicine (rank 1 — 70 609 works),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Psy- chiatry and Mental health (rank 4 — 10 404 works),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Education (rank 5 — 9 590 works),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Computer Science Applications (rank 18 — 6 013 works),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' General Engineering (rank 20 — 5 942 works),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Strategy and Management (rank 42 — 3 208 works),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Law (rank 57 — 2 557 works),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' History (rank 62 — 2 329 works),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Cultural Studies (rank 76 — 1 893 works),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Pollution (rank 97 — 1 549 works),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' and Anthropology (rank 130 — 1 032 works).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Looking at listed funders, we saw that the top three in terms of associated publications were the National Natural Science Foundation of China (3 506 works), followed by the (US) National Institutes of Health (2 316), the (US) National Science Foundation (1 022), the Wellcome Trust (914), and the (UK) National Institute for Health Re- search (661).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We also examined the affiliations of COVID study authors, propagating them to the highest parent organization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' a university hospital to its university).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Through this measure the top five entities were the Government of the United States of America (1 465 works), the Uni- versity of California System (925), University of Toronto (910), University of London (824), and University of Oxford (660).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We also looked how long it took for COVID articles to start citing each other, building, as it where, “on shoulders of giants”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' As can be seen in extended Figure 9 a citation network ramped up relatively quickly, surpassing ten thousand citations to COVID research by April 2020, and reaching 118 thousand citations on March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The large number of works published in January 2020 appears to be due to journals publishing later in the year volumes with that date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' This was, for example, the case with an overview article on COVID-19 and cerebrovascular diseases35 and an eight month retrospective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='36 (The same phenomenon may also explain the January 2021 rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=') Notably, this backdating practice can distort the establishment of priority over scientific advances based on a journal’s publication date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We replicated diverse impact and productivity metrics typically calculated in a proprietary fashion by commercial bibliometric data providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We calculated the 2021 Journal Impact Factor (JIF), and compared the 600 journals with the highest JIF available through Clarivate with 581 we matched through their ISSNs, obtaining a Spearman rank-order correlation coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='72 with a p-value 3 × 10−93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We also calculated the journal h5-index, and compared the values of the 100 “Top publications” venues listed in Google Scholar37 against those of Alexandria3k by hand-matching the venue titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The comparison of the 91 matched elements gave a Spearman rank-order correlation coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='75 with a p-value 6 × 10−18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Examining the same metric in a field where considerable work is published in conferences, we compared the h5-index of the 13 common venues between the 20 Google Scholar reports in the “Software Systems” category38 against a curated list of 32 software engineering venues,39 and obtained a Spearman rank-order correlation coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='83 with a p-value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='0004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Through the same data we also obtained the most cited articles published in the correspond- ing period (predictably, the top-one, with 21 426 citations, was on COVID-19)40 and overall (the top one was a 1996 article on the generalized gradient approximation method,41 which received 39 715 citations from a surprising diversity of fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 8 We also obtained the h5-index on authors identified through ORCID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' A noteworthy observa- tion was the number of authors with a high metric: the top-ranked author has an h5-index of 76, twelve authors have an h5-index larger than 60, and 100 larger than 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We explored the phenom- enal productivity and impact exhibited by the authors at the top of the distribution by examining the clustering coefficient of the graph induced by incoming and outgoing citations of distance 2 for a given work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The clustering coefficients of a random sample of 50 works from authors with an h5-index larger than 50 (median coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='05) appear to be significantly different from a random sample of other works of the same size with the same number of citations for each one (median 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Specifically, comparing the two populations we obtained a Mann-Whitney U statis- tic measure of 781 for the coefficients of the top-ranked authors’ works with a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='0006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Conclusions Alexandria3k builds on a rich and evolving ecosystem of publicly available data and sophisticated open source software libraries coupled with exponential advances in computing power, enabling scientists to perform reproducible bibliometric, scientometric, and research synthesis studies in a transparent and repeatable manner on a personal computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' In our use of Alexandria3k we found that over time its features and interface, as well as the methods we used to conduct the proof-of- concept studies, crystallized into a form we employed for distributing the code of all the proof- of-concept studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Other researchers can readily build upon these examples to conduct their own studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Alexandria3k is not a panacea for the issues we identified in the opening paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Cross- ref’s coverage of citation links is lacking compared to commercial alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='42 The linkage of publications to their authors through ORCID is thin, and ORCID author metadata is also sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The string-based matching used to link textual author affiliations to RORs can result in mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The subjects associated with works are derived from the categorization of whole journals by Sco- pus, and they are therefore incomplete and potentially inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The absolute values of derived bibliometric measures do not match the proprietary ones, which employ different methods for data collection (web crawling for Google Scholar) and for processing (hand-curation for Clarivate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' In general, many of the supported data elements can be used for making statistical observations, but the validity of any findings needs to be verified, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' through sensitivity analysis based on the manual examination of a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Finally, one should also take into account the epistemological shortcoming of structured approaches that use publication databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='43 Nevertheless, Alexandria3k opens many possibilities to conduct research that goes beyond what can be easily done through the query and API-based approaches offered by existing propri- etary7 and open44 databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Examples include the study: of collaboration patterns between orga- nizations and disciplines, of citation cliques and organizational inbreeding, of funder, publisher, and organizational performance, of open access availability and its effects, of pre-print servers as alternatives to peer-reviewed publications, of interdisciplinary connections, and of structural 9 publication differences between organizations or scientific fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Apart from making Alexandria3k available as open-source software we also structured it and intend to run it as an open-source project, accepting and integrating contributions of additional open-data sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' MEDLINE/PubMed, patent metadata), algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' publication topic classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' matching of authors, affiliations, or citations), and example studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We hope that this will allow Alexandria3k to grow organically serving ever more needs of research synthesis and analysis studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Lammey, R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Boell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' & Cecez-Kecmanovic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Literature Reviews and the Hermeneutic Circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Aus- tralian Academic & Research Libraries 41, 129–144 (June 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Priem, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=', Piwowar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' & Orr, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' OpenAlex: A Fully-Open Index of Scholarly Works, Authors, Venues, Institutions, and Concepts 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Acknowledgements The author thanks Panos Louridas, Arie van Deursen, Theodoros Evgeniou, Alberto Bacchelli, Dirk Beyer, and Dimitris Karlis for valuable advice and feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 12 Competing Interests The author declares that he has no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Correspondence Correspondence and requests for materials should be addressed to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' (email: dds@aueb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='gr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 13 Methods The paragraphs below describe key elements behind the implementation of Alexandria3k and the reported proof-of-concept studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Complete details regarding these is the Alexandria3k source code (also containing all studies as examples of its use), which is made available as open source software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='25 Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Alexandria3k is designed around modules that handle the Crossref, ORCID, ROR, and flat file (journal names, funders, open access journals) data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Each module de- fines the data source’s schema in terms of tables and their columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The schema’s SQL DDL implementation is used to define the corresponding tables in a populated database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The schema is also used for analyzing and satisfying vertical slicing requests when populating databases with Crossref and ORCID data, and for running Crossref queries without populating a database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Where possible, data are on-the-fly decompressed, extracted from an archive (with a tar or zip structure), and parsed in chunks, thus avoiding the storage cost of the entire decompressed data set (1 TB for Crossref and 467 GB for ORCID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Multiple alternatives were considered for the data back-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The native Crossref tree format would suggest a JSON-style NoSQL database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Redis1 would be an efficient back-end, but would limit the amount of data to that that could be processed in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' MondGB2 would address the capacity limitation, at the cost of a more complex installation process due to licensing issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' A client-server relational database system could also be used to offer improved integrity constraints and more advanced query optimization facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' However, these options require the installation, configuration, connection, and maintenance of a separate database process, which is not a trivial task, especially for researchers outside the computing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Consequently, the adopted approach uses the SQLite embedded database,3 which is installed as part of the Alexandria3k package, can handle efficiently the provided data and queries on it, and has APIs for diverse programming lan- guages and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Direct queries on the Crossref data set (without populating a database) are implemented by defining SQLite virtual tables that correspond to the offered schema through the Python apsw mod- ule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Crossref data are distributed in the form of about 26 thousand compressed containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Running a query on them is trivial when the query accesses a single table: the virtual table implementation moves from one container to the next as the table is scanned sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Direct Crossref queries involving multiple tables are handled differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' First, a dummy query execution is traced to determine the tables and fields it requires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Then, in-memory tempo- rary tables are populated with the required data from each container, and the query is executed repeatedly on the instantiated tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' This approach works for all cases where relational joins happen within a Crossref container (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' works with their authors or references).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' More complex cases, such as relational joins between works or aggregations, require the population of an external database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 14 Database population with vertical slicing is implemented by attaching the virtual database to the one to be populated and running SELECT INTO populated-table SELECT FROM virtual- table queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' A condition specifying a container-identifier is added to each query, so that all tables for each decompressed container can be populated before moving to the next container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' As before, query tracing is used to identify the tables and fields that the user has asked to populate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' When the database population specifies horizontal slicing (through a row-selection SQL ex- pression) the expression is evaluated sequentially on each container through the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Create a query based on the specified expression and trace it to determine the tables and fields required for evaluating the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Create the empty tables to be populated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Calculate the topological ordering4 of the specified tables, which will be used to join them in order to evaluate the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Iterate through the Crossref containers doing the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' (a) For each table to be populated or participating in the SQL expression create an in- memory mirror temporary table with the table’s primary keys, foreign keys, and fields participating in the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' (b) Create another temporary table made by joining the mirror tables, evaluating the SQL expression as a WHERE condition, and inserting the resulting rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' This table contains the identifiers of matched works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' (c) Insert into each populated table records associated with the matched works (directly or through JOIN relations) selected from the Crossref container being processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The database population design uses two techniques to improve its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' First, records are bulk-inserted in batches by attaching directly the virtual tables to the database to be populated, and by having the database engine perform the insert operations with a single internal command for each batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' This avoids a round-trip cost of obtaining the data in Alexandria3k and then storing them back in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Second, database indices over the containers in which the data are split are implemented and used so as to access each file in turn for populating all re- quired tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The correspondingly improved locality of reference5 is then utilized by caching the decompressed and parsed file contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The performant matching of author affiliations with RORs is based on multiple applications of the Aho-Corasick string-matching algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='6 First, an Aho-Corasick automaton is created with all unique organization names, aliases, and acronyms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Second, the automaton is used to find entries in it that also match other entries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' “ai”, the acronym of the “AI Corporation”, also matches the organization name “Ministry of Foreign Affairs”), and mark them for removal to avoid ambiguous matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Finally, a new automaton constructed from the cleaned-up entries is used to find the 15 longest match associated with each author affiliation string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' This is stored in the database as the affiliation’s organization identifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' When the Alexandria3k user specifies that affiliations should match the ultimate parent organization, a recursive SQL query adds a “generation” number to each matched organization, an SQL window (analytic) function7 orders results by generation, and a final selection query obtains the ROR identifier associated with the most senior generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Proof of concept studies framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We structured most proof-of-concept studies we presented as a series of queries that build on each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' This aids comprehensibility, testability, analysability, and recoverability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We specified the corresponding workflow using Makefiles8 based on the simple- rolap system, which manages relational online analytical processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='9,10 The simple-rolap system establishes the dependencies between queries and executes them in the required order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Most studies start with a population phase, which fills a database with the required horizontal and vertical data slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' In many cases, we used the rdbunit SQL unit-testing framework to test SQL queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='10 We employed a shared Makefile with rules and configurations that satisfy dependencies required by more than one study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' For example, this Makefile contains rules to download required data sets and to populate the database with the datasets that do not support slicing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The common directory where the shared Makefile resides also hosts downloaded data sets to minimize useless data duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Research synthesis studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We calculated the numbers associated with research synthesis studies by processing the output of Alexandria3k run on the Crossref data with an SQL query that matches specific words in publication titles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The query’s terms are structured to give precedence to the characterization of titles indicating a systematic review as such, classifying the rest as (unspecified) secondary studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' In the extended Figure 4 we combined the plotting of bibliometric (BM) and scientometric (SM) studies, and did not plot figures for mapping reviews (MR), umbrella reviews (UR), and tertiary studies (S3) — 6061 publications in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We also did not plot the 400 identified studies published before 1971 as well as studies published after 2021, as only part of the year 2022 data were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Note that works containing “bibliometric” or “scientometric” in their title may either employ these methods11 or refer to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='12 We used another query with the same terms to list the 30 studies published before 1950 and obtain the earliest one13 among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Crossref graph database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We created a vertical slice of the complete Crossref database, which we used for a number of purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The slice contains mainly the primary and foreign keys (including DOIs, and ORCIDs) of all entities, plus the publication year, author affiliation names, and work subjects, which are not normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We also run Alexandria3k to populate the database with the Scopus All Science Journal Classification Codes (ASJCs)14 and RORs, and linked work subjects to ASJCs and author affiliations to RORs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Scientific publishing evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' In common with other studies15,16 we limited our examination to works published after World War II, in order to avoid misleading comparisons with the markedly different scientific and publication environment that preceded the war.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We calculated most numbers used for plotting Figure 1 from the populated Crossref graph database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We obtained the number of published works and journals through SQL Count aggregations of the underlying data grouped 16 by year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We obtained the ratios of authors per work and references per work by counting the corresponding elements of the associated detail tables and then obtaining SQL Avg aggregations grouped by year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We joined papers with their citations using the document’s DOI as a key and used this to calculate the two-year impact factor, the received citations per work, the twenty-year impact factor, and the proportion of all published works cited at least once each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' For calculating the last two we used SQL window (analytic) functions7 to obtain an accumulating sum and a twenty- year rolling sum over the number of works published each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We calculated the number of pages per work by populating a database with the required work and author details, and by extracting the starting and ending page number from Crossref works that contain a dash in their pages field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' This process excludes single-page works reported with only a single page number (rather than as a range with the same starting and ending page).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We excluded from the data records with a zero or negative number of pages or those having more than 1000 pages, because the latter (164387 records) were often derived from data-entry mis- takes, such as repeated page digits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 234-2366), as well as unusable data formats, such as 1744-8069-5-32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We calculated a measure that can be used to track author productivity (works per author) despite clashes in author names, by taking advantage of the fact the we display productivity in relative terms (adjusting it to be 1 in 1950), In absolute terms authors with same names increase the productivity’s absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' (An author named Smith, Kim, or Zhu would appear extremely prolific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=') However, assuming that the ratio of clashing names in the population does not change, the effect of duplicate names on the relative productivity measurements is cancelled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' As an example consider that 50% of all authors are named Smith, all other authors have distinct names, and in 1950 1000 authors write 2000 works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The actual productivity should be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' By not distinguishing authors with clashing names the situation will appear as 500 authors writing 2000 works, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' a productivity of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Consider now in 2020 10 000 authors writing 80 000 works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The actual productivity should be 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The situation will however appear as 5 000 authors writing 80 000 works, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' a productivity of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' While the derived absolute productivity numbers are incorrect, the ratio of the correct productivity numbers is the same as the ones that do not take name clashes into account: 2/8 = 4/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Note that an actual study of author productivity would need to test and control for the assump- tion we made, because the population’s composition might change over the years to include authors from ethnic backgrounds with more or fewer common name clashes, (for example, about 80% of the China’s population shares the 100 most common Chinese surnames)17 making the phenomenon more frequent or less frequent over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Using Alexandria3k we obtained frequently-occurring names at the two ends of the examined period and found that in 1950 the five most frequent names were W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Beinhoff (161 works — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='10% of the total names), E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Rosenberg (149 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='09%), F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' De Quervain (115 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='07%), G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Niemann (114 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='07%), and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Eichler (105 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='06%);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' whereas in 2021 they were Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Wang (63363 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='23%), Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Zhang (57414 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='21%), Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Li (51792 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='19%), Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Liu (46013 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='17%), and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Wang (44805 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='16%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The different percentages at the two 17 period’s ends indicate that further controls for this change would be required, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' by measuring name clashes through ORCIDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We calculated the CD5 index18 of Crossref publications by populating a database with their publication date and DOI, as well as the DOIs of the corresponding references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Given the available data, we were able to calculate the CD5 index for six additional years (until 2016) compared to previous results,16 using the remaining five complete years we had at out disposal (2017–2021) to obtain the required citations’ window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The CD5 calculation proved to be computationally challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The processing for the already-published 1945–2010 range required more than five days of computing and about 40 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Extending the range to 2016 increased the required time to 45 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' To address this we enhanced the original CD5 algorithm implementation to use more efficient data structures and algorithms converting it to C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='19 We employed a union of pointers and integers to efficiently represent vertices internally and in Python code, and used C++ sorted vectors and sets to improve memory allocation and searching for nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Furthermore, we rewrote the CD5 calculation process in C++ in order to parallelize it while maintaining in memory a single copy of the 50 GB graph data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' (It turned out that this could not be done neither with Python’s threads nor with forked processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=') The improvements in computational efficiency allowed us to perform the CD5 index calculation in 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='5 hours of elapsed time using 67 hours of CPU time and 50 GB of main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' To allow other researchers to build on this data without incurring the associated high com- putational cost, we have made the resulting data set containing the DOI and the CD5 index for 11 568 934 publications in the range 1945–2016 openly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='20 This improves upon previ- ously available data,21 which extends to 2010 and only provides the time-stamp of each publication, without other uniquely distinguishing publication identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Field dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We calculated strong dependencies between fields (Figure 2) by using work references and subjects in the Crossref graph database to construct a table containing the number of citations between fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Based on it we calculated for each field pair its “strength” (sum of incoming and outgoing citations) and its “fundamentalness” (ratio of a field’s outgoing citations over the pair’s strength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' In the plotted results we included the top 50 field associations in terms of strength from the top 10% in terms of fundamentalness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We furthermore excluded pairs associated with the “Multidisciplinary” subject and also links within fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Both sides in the Figure represent a small fraction of the total field citations, and are drawn on different scales: the outgoing citations shown amount to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='8% of the fields’ total and the incoming ones amount to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='2% of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Field evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We calculated the evolution in the number of publications across scientific fields (Figure 3) by propagating the specific fields associated with each work in the Crossref graph database to the more general containing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' For that we used as general fields the Scopus ASJCs that ended in “00”, and as their sub-fields those that started with the same numeric prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' For example, we allocated publications under the subject of “Catalysis” (1503) to “General Chemical 18 Engineering” (1500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We then calculated total publications in 1950 and 2021, changes in the per- centages of a field’s publications in terms of the total at the two time points, and included in the Figure the ten fields with the largest change whose publications amounted to more than 2% of the 2021 total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' COVID-19 metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' To study COVID-19 publications we populated a database with a full hori- zontal slice of the Crossref data by specifying as the row selection criterion works containing the string “COVID” in their title or abstract (“covid” is not part of any English word).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We also linked works to their subjects and author affiliations to the corresponding RORs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We obtained organi- zations publishing COVID-19 research by assigning author affiliations to works, and by counting both ROR-matched affiliations and unmatched affiliations as simple text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Number of COVID-19 study authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We calculated the approximate number of researchers who worked on all COVID-19 studies by starting with the number of unique (author given-name, author surname) pairs in the set of all COVID study authors Nan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The number of true authors could be higher if many authors share the same name or lower if the same author appears differently (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' through the use of initials) in some publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We address this by obtaining from the set of authors with an ORCID the number of distinct ORCIDs No, which is the true number of authors in that set, and the number of distinct names Non, which approximates any bias also found in Nan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We then consider the true number of authors as Nan No Non Journal impact factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We calculated the 2021 journal impact factor22 by populating a database with the keys, ISSNs, publications years, and pages of works and their references published be- tween 2019 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We then created a table associating works with journal ISSNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' From this table we obtained citations published in 2021 to works published in 2019–2020 (the impact fac- tor’s numerator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We further filtered works to identify “citable” items, which Clarivate defines as those that make a substantial contribution to science and therefore do not include elements such as editorials and letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' For this we used as a rough heuristic works longer than two pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' (We also included works lacking a page range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=') From the count of citable items per journal we obtained the number of publications published in the 2019–2020 period (the denominator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Finally, to compare our results with the numbers published by Clarivate we associated each impact factor metric with all ISSNs known for a journal (electronic, print, alternative), excluding the “alternative” ISSNs of one journal used as primary for another journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Productivity metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We obtained the h5-index23 productivity metrics by populating databases with data sliced vertically to include the keys of works and references and horizontally to include items published in the period 2017–2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' For the software engineering venue metrics we selected the examined conferences based on the DOI prefix assigned to the conference publications each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' (In retrospect, we could have used the container title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=') For each entity we counted its citations and then used an SQL window (analytic) function to partition the results over the entity’s key 19 (ISSN, ORCID, or conference acronym), number each set’s rows, and select only those with a rank lower or or equal to the corresponding number of citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' To study the citation graph of top-ranked authors we obtained a) a random sample of 50 works written by top-ranked authors, and b) a random sample of 50 works from all other publi- cations, paired with the ones selected from the top-ranked authors to have the same number of citations as them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' For each publication in the two samples, we created a separate graph containing the work w, the set S of the works w cites and the works that cite w, and then again the set S′ of the works w′ that cite or are cited by w ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The graph’s edges are citations from one work to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We created each citation-induced graph with a Python program querying the populated database and employed the NetworkX24 Python package to calculate the graph’s average clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' SQLite lacks the ability to provide a seed to its Random() function, which is required for obtaining random samples in a deterministic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' We worked around this limitation by multi- plying the identifier of each author (which is sequentially allocated and therefore not random) with a seed value, and used the last decimal digits of the result to place each work in a pseudo-random ordering, which we used to obtain the required works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Statistical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' For reporting the correlation between the metrics obtained by Alexandria3k and existing ones and for comparing the graph clustering coefficients between two populations we used the functions spearmanr and mannwhitneyu from the Python package scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' All calculations were performed with “two-sided” as the alternative hypothesis (the default).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' No other options were provided to the function calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' For the analysis and charting we used Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='10 with matplotlib 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='4, numpy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='1, pandas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='3, pySankey 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='1, and scipy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Code and data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' A versioned release of the source code of Alexandria3k and the proof-of-concept examples presented in this study is available on Zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='25 A replication package with the results data and scripts associated with the reported example studies is also available on Zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='26 Current versions of Alexandria3k are made available for installation through PyPi https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='org/project/alexandria3k/ and for contributions, feature requests, and issue reporting on GitHub https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='com/dspinellis/alexandria3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' The data used in the example studies are available as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Crossref April 2022 Public Data File DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='13003/83b2gq ORCID Public Data File 2022 version 4 DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='23640/07243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='21220892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='v4 ROR Data v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='1 DOI:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='7448410 Open access journals https://doaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='org/csv Funders https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='crossref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='org/funderNames?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='mode=list Journals http://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='crossref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='org/titlelist/titleFile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='csv 20 The 2021 Journal Impact Factor data used for assessing the numbers obtained by Alexan- dria3k are available from Clarivate, but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' These data are however available from the authors upon reasonable request and with permission of Clarivate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' References 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Gousios, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' & Spinellis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Modular Queries and Unit Testing Technical briefing notes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' available online https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='org/10.' metadata={'source': 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Individual’s Scientific Research Output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Proceedings of the National Academy of Sciences 102, 16569–16572 (Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Hagberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=', Swart, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' & S Chult, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Exploring Network Structure, Dynamics, and Function Using NetworkX tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' LA-UR-08-5495 (Los Alamos National Lab, Los Alamos, NM, US, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Spinellis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' dspinellis/alexandria3k: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='0 version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='7584743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Spinellis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Replication package for the Alexandria3k paper version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='7586574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 22 Extended data 1970 1980 1990 2000 2010 2020 Year 100 101 102 103 104 Number of studies Systematic Scientometric / bibliometric Meta-analysis Other secondary Figure 4: Number of research synthesis studies published each year in the period 1971–2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Table 1: Number of Crossref Records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Entity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Total records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='2 531 227 295 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Works (publications) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='134 048 223 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Works with a text mining link ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='96 294 821 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Works with subject ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='81 210 089 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Works with references ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='52 907 361 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Works with affiliation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='36 389 868 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Works with an abstract ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='15 367 820 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Works with funders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='7 519 462 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Author records (linked to works) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='359 556 891 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Author records with ORCID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='16 745 506 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Distinct authors with ORCID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='4 525 906 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Author affiliation records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='76 759 875 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Distinct affiliation names ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='19 453 361 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Work subject records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='182 858 177 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Distinct subject names ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='340 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Work funders ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='15 491 915 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Funder records with DOI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='10 811 496 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Distinct funder DOIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='29 610 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Funder awards ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='14 090 597 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='References ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='1 748 421 617 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='References with DOI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='1 255 033 889 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Distinct reference DOIs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='61 609 121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Table 2: Number of ORCID Records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Table ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Persons with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Such Records ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Personal data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='14 811 567 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='14 811 567 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='URLs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='1 325 399 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='892 528 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Countries ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='2 141 021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='2 021 218 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Keywords ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='2 230 569 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='838 796 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='External identifiers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='2 287 063 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='1 653 995 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='Distinctions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='345 967 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content='159 113 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} +page_content=' 29 2020-1 2020-2 2020-3 2020-4 2020-5 2020-6 2020-7 2020-8 2020-9 2020-10 2020-11 2020-12 2021-1 2021-2 2021-3 2021-4 2021-5 2021-6 2021-7 2021-8 2021-9 2021-10 2021-11 2021-12 2022-1 2022-2 2022-3 2022-4 Citing publication month 0 20 40 60 80 100 120 Number of citations to COVID publications (thousands) Figure 9: Citations from COVID research to COVID research over time 30' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFQT4oBgHgl3EQfYjZ5/content/2301.13312v1.pdf'} diff --git a/PtA0T4oBgHgl3EQfDP9d/content/tmp_files/2301.02000v1.pdf.txt b/PtA0T4oBgHgl3EQfDP9d/content/tmp_files/2301.02000v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d38f63f93c835a93b7481ac29703f77f12daa8e0 --- /dev/null +++ b/PtA0T4oBgHgl3EQfDP9d/content/tmp_files/2301.02000v1.pdf.txt @@ -0,0 +1,2360 @@ +arXiv:2301.02000v1 [math.AP] 5 Jan 2023 +Fine asymptotic expansion of the ODE’s flow +Marc Briane & Lo¨ıc Herv´e +Univ Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France +mbriane@insa-rennes.fr & loic.herve@insa-rennes.fr +Friday 6th January, 2023 +Contents +1 +Introduction +2 +2 +Fine asymptotic expansion +6 +3 +The incommensurable case +11 +4 +The commensurable case +17 +5 +Examples +19 +5.1 +Cases with a non vanishing vector field . . . . . . . . . . . . . . . . . . . . . . . +19 +5.2 +Cases with a vanishing vector field +. . . . . . . . . . . . . . . . . . . . . . . . . +21 +Abstract +In this paper, we study the asymptotic expansion of the flow X(t, x) solution to the +nonlinear ODE: X′(t, x) = b +� +X(t, x) +� +with X(0, x) = x ∈ Rd, where b is a regular Zd- +periodic vector field in Rd. More precisely, we provide various conditions on b to obtain +a “fine” asymptotic expansion of X of the type: |X(t, x) − x − t ζ(x)| ≤ M < ∞, which +is uniform with respect to t ≥ 0 and x ∈ Rd (or at least in a subset of Rd), and where +ζ(x) for x ∈ Rd, are the rotation vectors induced by the flow X. +On the one hand, +we give a necessary and sufficient condition on the vector field b so that the expansion +X(t, x) − x − t ζ(x) reads as Φ +� +X(t, x) +� +− Φ(x), which yields immediately the desired ex- +pansion when the vector-valued function Φ is bounded. In return, we derive an admissible +class of vector fields b in terms of suitable diffeomorphisms on Yd and of vector-valued +functions Φ. On the other hand, assuming that the two-dimensional Kolmogorov theorem +and some extension in higher dimension hold, we establish different regimes depending +on the commensurability of the rotation vectors of the flow X for which the fine estimate +expansion of X is valid or not. It turns out that for any two-dimensional flow X associ- +ated with a non vanishing smooth vector field b and inducing a unique incommensurable +rotation vector ξ, the fine asymptotic expansion of X holds in R2 if, and only if, ξ1/ξ2 +is a Diophantine number. This result seems new in the setting of the ODE’s flow. The +case of commensurable rotation vectors ζ(x) is investigated in a similar way. Finally, sev- +eral examples and counter-examples illustrate the different results of the paper, including +the case of a vanishing vector field b which blows up the asymptotic expansion in some +direction. +1 + +Keywords: ODE’s flow, asymptotic expansion, rotation number, incommensurable vector, +Diophantine number, Liouville’s number +Mathematics Subject Classification: 34E05, 34E10, 37C10, 37C40 +1 +Introduction +Let b be a C1-regular vector field in Rd defined on the torus Yd := Rd/Zd. In this paper, we +study the ODE’s flow X(·, x) for x ∈ Yd, defined by + + + +∂X +∂t (t, x) = b(X(t, x)), +t ≥ 0 +X(0, x) = x. +(1.1) +Here, we are interested by the asymptotics of the flow X(t, x) as t → ∞ for a given x ∈ Rd. In +dimension two the nice result due to Peirone [19] (see also [21]) claims that if the vector field +b does not vanish in Y2, then one has +∀ x ∈ R2, +lim +t→∞ +X(t, x) +t += ζ(x) ∈ R2, +(1.2) +where the limit vector ζ(x) may depend on x. On the contrary, when either b does vanish +in Y2 (see [21, Theorem 6.1]), or when dimension d is greater than 2 (see [19, Theorem 4.10]), +limit (1.2) does not hold necessarily for any x ∈ Yd. More recently, using the two-dimensional +Peirone’s result among others, the authors have obtained various asymptotic results for the +flow (1.3) in any dimension with applications to the homogenization of linear transport equa- +tions [3, 4, 5]. Dimension two is very specific in ergodic theory, since Franks and Misiurewicz [9] +have proved that for any continuous flow X(t, x) the Herman rotation set [11] – derived from [18, +Corollary 2.6] as the convex combination of the limit points of all the sequences +� +X(n, x)/n +� +n∈N +for x ∈ Y2 – is actually a closed segment line of R2. In the case of a two-dimensional ODE’s +flow, the closed segment Cb is carried by a line passing through 0R2. For the ODE’s flow X +associated with the vector field b by (1.1), Herman’s rotation set may be equivalently defined +by +Cb := +�ˆ +Yd +b(x) µ(dx) : µ ∈ Mp(Yd) s.t. for any t ≥ 0, µ ◦ X(t, ·) = µ +� +, +(1.3) +i.e. µ in (1.3) is a probability measure on Yd which is invariant for the flow X. In dimension +three the situation is again completely different, since [5, Theorem 4.1] shows that the rotation +set (1.3) may be any convex polyhedron of R3 with rational vertices. +In this paper, we focus on a more precise asymptotics of the flow X (1.1). It is rather +natural to study beyond the limits of type (1.2) when they do exist, the asymptotic behavior +of the expansions +X(t, x) − x − t ζ(x) +as t → ∞ and for x ∈ Rd. +(1.4) +In the framework of ergodic theory, the problem of the dynamics of the iterates F n, n ∈ N, of +the lift F (1) obtained from some homeomorphism f homotopic to the identity on the torus Yd +(see, e.g., [18]), is extremely delicate. Indeed, only dimension two is investigated, the estimates +of the vector-valued expansion (1.4) for a general lift are only obtained in one direction, and +moreover the last developments are quite recent. More precisely (see, e.g., the introduction of +[13] and the references therein), the two following results hold: +1In the context of the ODE’s flow X defined by (1.1), we have F = X(1, ·), and due to the semi-group +property of X we get that F n = X(n, ·) for any n ∈ N. +2 + +• By virtue of [14] and [16, Theorem 1] there exists a homeomorphism f on Y2 homotopic +to the identity with a lift F on R2, such that the Herman rotation set Rf is reduced to +the unit set {ρf} and +∀ v ∈ S1, +sup +x∈R2, n∈N +�� +F n(x) − x − n ρf +� +· v +� += ∞. +(1.5) +In [16, Theorem 1] ρf is actually chosen to be 0R2. +• By virtue of [8, Theorem A], for any homeomorphism f on Y2 homotopic to the identity +with a lift F on R2 and the Herman rotation set Rf of which is a closed line segment +of R2 with an irrational slope containing several points of Q2, there exist a unit vector v +in (Rf)⊥ and a constant M > 0 such that +∀ ρ ∈ Rf, +sup +x∈R2, n∈Z +�� � +F n(x) − x − n ρ +� +· v +�� ≤ M. +(1.6) +In our setting, we have obtained an example of a two-dimensional flow X (1.1) associated +with a vanishing vector field b parallel to a fixed incommensurable vector ξ, which satisfies the +large deviation (1.5) except in the direction v := ξ⊥ (see Proposition 5.1), but whose Herman +rotation set Cb is a non degenerate closed line segment of R2 (see Remark 5.1). In contrast, +due to the differential structure we can hope better results than the two-dimensional bounded +deviation (1.6) in some direction. More precisely, assuming the existence of the limit (1.2) for +any point x in a subset A of Rd, we will prove in several situations a fine asymptotic expansion +of the type +sup +x∈A, t≥0 +�� X(t, x) − x − t ζ(x) +�� ≤ MA < ∞. +(1.7) +In Section 2 we prove a criterium (see Proposition 2.2) for which expression (1.4) reads as +∀ t ≥ 0, ∀ x ∈ Rd, +X(t, x) − x − t ζ(x) = Φ +� +X(t, x) +� +− Φ(x), +(1.8) +so that the boundedness of the vector-valued function Φ in Rd implies immediately the fine +asymptotic expansion (1.7) in the whole set Rd. The right-hand side of (1.8) can be regarded +as a continuous sum of coboundary terms (see Remark 2.1). In return, from expression (1.8) +we deduce (see Proposition 2.3) a general class of vectors fields b such that (1.7) holds in Rd. +Finally, assuming that there exists a Zd-periodic regular gradient ∇u satisfying the positivity +property b · ∇u > 0 in Yd, Theorem 2.1 provides sufficient conditions for which asymptotic the +expansion (1.7) is satisfied in Rd. +Section 3 deals with the case of a non vanishing vector field b in R2 such that Herman’s +rotation set (1.3) is a unit set of {ξ} of R2, where the rotation vector ξ = (ξ1, ξ2) is incommen- +surable in R2 (see (1.9)). This corresponds to the second case of the proof of [19, Theorem 3.1]. +Assuming in addition the existence of an invariant probability measure for the flow with a pos- +itive regular Lebesgue’s density, we prove (see Theorem 3.1) using the celebrated Kolmogorov +theorem [15] that if the irrational number ξ1/ξ2 is a Diophantine number (see (1.10)), then the +fine asymptotic fine expansion (1.7) is fulfilled in R2. In contrast, given a vector ξ in R2 such +that ξ1/ξ2 is a Liouville’s number (see (1.11)), we can construct a two-dimensional Stepanoff’s +flow [22], i.e. a flow associated with the unidirectional vector field b = a ξ, such that the fine +asymptotic expansion does not hold in R2. +At this point, note that the alternative between “commensurable and incommensurable” for +the rotation vector is well-known in ergodic theory to guarantee the uniqueness of the asymp- +totics (1.2) of the flow (see, e.g., [19]). Moreover, the alternative between “Diophantine and +3 + +Liouville” is essential in the conjugacy Denjoy theorem related to the dynamical properties of +the diffeomorphisms on the circle S1 with an irrational rotation number (see Remark 3.1 and +the references therein). In the present context of the fine asymptotic expansion (1.7) of a two- +dimensional ODE’s flow, the same alternative on the irrational number ξ1/ξ2 can be regarded, +up to our best knowledge, as a new example of the crucial role played by the Diophantine +property of the rotation number in a dynamical system. Finally, using the rather restrictive +extension [17, Theorems 1,2] (see also [2, Theorem 3.3] which was obtained and used in an +independent way) of Kolmogorov’s theorem to dimension d > 2) the previous two-dimensional +result can be also extended to higher dimension (see Remark 3.2). +In contrast with Section 3, Section 4 is devoted to the commensurable case in any dimension, +which is based on the existence of periodic solutions in the torus Yd to the ODE (1.1). Again +assuming that Kolmogorov’s theorem in dimension two and its extension [17, Theorems 1,2] +in higher dimension hold true, we get (see Theorem 4.1) the fine asymptotic expansion (1.7) +in R2, with an explicit non constant vector-valued function ζ in Rd. +The results stated above are based on the condition that the vector field b does not vanish +in Yd. +When b does vanish, the fine asymptotic expansion (1.7) may fail in Rd. +Indeed, +Proposition 5.1 shows that the two-dimensional Stepanoff flow associated with the vector field +b = a ξ, where a vanishes at one point in Y2 and ξ is any incommensurable vector in R2, does +not satisfy the fine asymptotic expansion (1.7) in the set A = R ξ+Z2. In contrast, Example 5.4 +provides a two-dimensional Stepanoff’s flow which satisfies the fine asymptotic expansion in R2 +for any vector ξ in R2, but the function a then has an infinite number of roots in Y2. +Other examples illustrate the results of the paper in Section 5. +To conclude, we have not succeeded for the moment to derive a fine asymptotic expan- +sion (1.7) of the flow either without using the bounded coboundary sum of (1.8), or without +the conditions supporting Kolmogorov’s theorem in dimension two and its extension in higher +dimension. For instance, when b is only a non vanishing regular two-dimensional vector field, +namely the framework of [19], we do not know if the fine asymptotic expansion (1.7) holds in +the whole set R2, while however the asymptotics (1.2) is satisfied at each point of R2. +Definitions and notations +• d ∈ N denotes the space dimension. +• S1 denotes the unit sphere of R2. +• A vector ξ in Rd is said to be incommensurable in Rd if +∀ k ∈ Zd \ {0Rd}, +ξ · k ̸= 0. +(1.9) +Otherwise, the vector ξ is said to be commensurable in Rd. +• A Diophantine number is an irrational real number λ with the property that there exists +m ∈ N satisfying +# +�� +(p, q) ∈ Z × N : +���� λ − p +q +���� ≤ 1 +qm +�� +< ∞, +(1.10) +i.e. λ is badly approximated by rational numbers. +• On the contrary, a Liouville number is an irrational number λ with the property that for +any n ∈ N, there exists a pair of integers (pn, qn) with qn > 1, such that +0 < +���� λ − pn +qn +���� < +1 +(qn)n , +(1.11) +4 + +i.e. λ is closely approximated by a sequence of rational numbers. +• (e1, . . . , ed) denotes the canonical basis of Rd, and 0Rd denotes the null vector of Rd. +• Id denotes the unit matrix of Rd×d. +• “ · ” denotes the scalar product and | · | the euclidean norm in Rd. +• × denotes the cross product in R3. +• |A| denotes the Lebesgue measure of any measurable set in Rd or Yd. +• Yd denotes the d-dimensional torus Rd/Zd (which may be identified to the unit cube [0, 1)d +in Rd), and 0Yd denotes the null vector of Yd. +• Π denotes the canonical surjection from Rd on Yd. +• Ck +c (Rd), k ∈ N ∪ {∞}, denotes the space of the real-valued functions in Ck(Rd) with +compact support in Rd. +• Ck +♯ (Yd), k ∈ N ∪ {∞}, denotes the space of the real-valued functions f ∈ Ck(Rd) which +are Zd-periodic, i.e. +∀ k ∈ Zd, ∀ x ∈ Rd, +f(x + k) = f(x). +(1.12) +• The jacobian matrix of a C1-mapping F : Rd → Rd is denoted by the matrix-valued +function ∇F with entries ∂Fi +∂xj +for i, j ∈ {1, . . . , d}. +• The abbreviation “a.e.” for almost everywhere, will be used throughout the paper. The +simple mention “a.e.” refers to the Lebesgue measure on Rd. +• dx or dy denotes the Lebesgue measure on Rd. +• For a Borel measure µ on Yd, extended by Zd-periodicity to a Borel measure ˜µ on Rd, a +˜µ-measurable function f : Rd → R is said to be Zd-periodic ˜µ-a.e. in Rd, if +∀ k ∈ Zd, +f(· + k) = f(·) ˜µ-a.e. in Rd. +(1.13) +• For a Borel measure µ on Yd, Lp +♯(Yd, µ), p ≥ 1, denotes the space of the µ-measurable +functions f : Yd → C such that +ˆ +Yd +|f(x)|p µ(dx) < ∞. +• Lp +♯(Yd), p ≥ 1, simply denotes the space of the Lebesgue measurable functions f in Lp +loc(Rd), +which are Zd-periodic dx-a.e. in Rd. +• Mloc(Rd) denotes the space of the non negative Borel measures on Rd, which are finite +on any compact set of Rd. +• M♯(Yd) denotes the space of the non negative Radon measures on Yd, and Mp(Yd) denotes +the space of the probability measures on Yd. +• D′(Rd) denotes the space of the distributions on Rd. +5 + +• For a Borel measure µ on Yd and for f ∈ L1 +♯(Yd, µ), we denote +µ(f) := +ˆ +Yd +f(x) µ(dx), +(1.14) +which is simply denoted by f when µ is Lebesgue’s measure. The same notation is used +for a vector-valued function in L1 +♯(Yd, µ)d. If f is non negative, its harmonic mean f is +defined by +f := +�ˆ +Yd +dy +f(y) +�−1 +. +• For a given measure λ ∈ M♯(Yd), the Fourier coefficients of λ are defined by +ˆλ(n) := +ˆ +Yd +e−2iπ n·x λ(dx) +for n ∈ Zd. +The same notation is used for a vector-valued measure in M♯(Yd)d. +• c denotes a positive constant which may vary from line to line. +2 +Fine asymptotic expansion +Definition 2.1 A flow X associated with a vector field b ∈ C1 +♯ (Yd)d by (1.1) is said to admit +a fine asymptotic expansion if there exists a Zd-periodic vector-valued function ζ such that +∀ t ≥ 0, ∀ x ∈ Rd, +X(t, x) = x + t ζ(x) + O(1), +(2.1) +where O(1) denotes a vector-valued function which is bounded uniformly with respect to t and x. +More precisely, the flow X is said to admit a fine asymptotic expansion in the subset A of Rd +if there exists a constant CA > 0 only depending on A, such that +∀ t ≥ 0, ∀ x ∈ A, +��X(t, x) − x − t ζ(x) +�� ≤ CA. +(2.2) +The following result gives a way for a flow to admit a fine asymptotic expansion (2.1). +Proposition 2.2 Let b, ζ be two vector fields in C1 +♯ (Yd)d, and let Φ be a vector-valued function +in C1(Rd)d. Then, the following assertions are equivalent : +∀ t ≥ 0, ∀ x ∈ Rd, +X(t, x) = x + t ζ(x) + Φ +� +X(t, x) +� +− Φ(x), +(2.3) +(Id − ∇Φ) b = ζ in Rd +and +∀ t ≥ 0, ζ +� +X(t, ·) +� += ζ in Yd, +(2.4) +The last property in (2.4) means that ζ is invariant for the flow X. If one of these two assertions +is satisfied and Φ is bounded in Rd, then ζ is Zd-periodic, the Herman rotation set is given by +Cb = +� +conv +� +ζ(Yd) +� +if d ≥ 3 +ζ(Y2) +if d = 2, +(2.5) +and the flow X admits a fine asymptotic expansion in the sense of (2.1). +6 + +Remark 2.1 If the flow X satisfies the expression (2.3), then the function Φ is not necessarily +periodic. +However, for any t ≥ 0, the function Φ +� +X(t, ·) +� +− Φ(·) is Zd-periodic, since the +functions +� +x �→ X(t, x) − x +� +and ζ are Zd-periodic. The function Φ +� +X(t, ·) +� +−Φ(·) can be +regarded as a “continuous coboundary sum”, since we have +Φ +� +X(n, ·) +� +− Φ(·) = +n−1 +� +i=0 +� +Φ +� +X(i + 1, ·) +� +− Φ +� +X(i, ·) +�� +for n ∈ N, +where each term of the sum is a coboundary term. +In the sequel we will construct such continuous coboundary sums possibly uniformly bounded in +various situations, so that the fine asymptotic expansion (2.1) will follow immediately. +Based on Proposition 2.2 the following result allows us to construct a general family of flows +which satisfy the fine asymptotic expansion (2.1). +Proposition 2.3 Let Ψ be a C2-diffeomorphism on Yd satisfying the conditions +Φ : +� +x ∈ Rd �→ x − Ψ(x) +� +∈ C2 +♯ (Yd)d +and +det (∇Ψ) ̸= 0 in Yd. +(2.6) +Let ζ be a vector field in C1 +♯ (Yd)d satisfying the equality +∇ζ (∇Ψ)−1 ζ = 0 in Yd. +(2.7) +Then, the flow X associated with the vector field b ∈ C1 +♯ (Yd)d defined by +b := (∇Ψ)−1 ζ = (Id − ∇Φ)−1 ζ in Yd, +(2.8) +fulfills both the expression (2.3) and the fine asymptotic expansion (2.1). +Proof of Proposition 2.2. First, assume that assertion (2.3) holds. Then, by the boundedness +of the vector field Φ and by the semi-group property of the flow X, we deduce from (2.3) that +for any t ≥ 0 and any x ∈ Rd, +lim +s→∞ +X(s, x) +s += ζ(x) = lim +s→∞ +X(s + t, x) +s += lim +s→∞ +X(s, X(t, x)) +s += ζ +� +X(t, x) +� +, +(2.9) +which shows that the vector-valued function ζ is invariant for the flow X. Moreover, we have +∀ x ∈ Rd, ∀ k ∈ Rd, +ζ(x + k) = lim +t→∞ +X(t, x + k) +t += lim +t→∞ +X(t, x) + k +t += ζ(x), +which shows that ζ is Zd-periodic. +Now, let us determine the Herman rotation set Cb. By [18, Corollary 2.6] combined with (2.9) +we have +Cb = conv +� � +x∈Rd +� � +n∈N +�X(k, x) − x +k +: k ≥ n +� �� += conv +� +ζ(Yd) +� +. +(2.10) +In dimension two the first equality of (2.5) can be refined. Indeed, by virtue of [9, Theorem 1.2] +for two-dimensional continuous flows, Herman’s rotation set Cb is a closed line segment of R2, +and by the continuity of ζ the subset ζ(Y2) of R2 is a connected compact set. +Therefore, +it is enough to prove that the extremal points of Cb belong to ζ(Y2). To this end, by [18, +Remark 2.5] (see [5, Section 6.1] for a proof) each extremal point of Cb is a vector ν(b) for some +7 + +ergodic invariant probability measure ν. Then, by Birkhoff’s ergodic theorem there exists a +point x ∈ Y2 such that +ζ(x) = lim +t→∞ +X(t, x) +t += ν(b) ∈ ζ(Y2), +which thus implies the second equality of (2.5). +Next, we have for any t ≥ 0 and any x ∈ Rd, +∂ +∂t +� +X(t, x) − x − t ζ(x) − Φ +� +X(t, x) +� ++ Φ(x) +� += +� +b − ∇Φ b +�� +X(t, x) +� +− ζ(x). +(2.11) +Since the assertion (2.3) holds and ζ is invariant for X, the equality (2.11) is reduced to +∀ t ≥ 0, ∀ x ∈ Rd, +� +b − ∇Φ b +�� +X(t, x) +� += ζ +� +X(t, x) +� +. +Therefore, taking t = 0 in the previous equality we get the relation (2.4). +Conversely, if the assertion (2.4) is satisfied, then the right hand side of (2.11) is zero, which +implies that or any t ≥ 0 and any x ∈ Rd, +X(t, x) − x − t ζ(x) − Φ +� +X(t, x) +� ++ Φ(x) = X(0, x) − x − Φ +� +X(0, x) +� ++ Φ(x) = 0, +which yields assertion (2.3). +Finally, note that the expression (2.3) of the flow X combined with the boundedness of the +vector field Φ provides immediately the fine asymptotic expansion (2.1) of X, which concludes +the proof of Proposition 2.2. +□ +Proof of Proposition 2.3. Define the mapping X by +X(t, x) := Ψ−1� +t ζ(x) + Ψ(x) +� +for (t, x) ∈ [0, ∞) × Rd. +(2.12) +First of all, let us prove that the vector-valued function ζ is invariant for X. Using the equalities +(2.12) and +Id = ∇(Ψ−1 ◦ Ψ) = +� +∇(Ψ−1) ◦ Ψ +� +∇Ψ +in Rd, +(2.13) +we have for any (t, x) ∈ [0, ∞) × Rd, +∂ +∂t +� +ζ +� +X(t, x) +�� += (∇ζ) +� +X(t, x) +� ∂ +∂t +� +X(t, x) +� += (∇ζ) +� +X(t, x) +� +∇(Ψ−1) +� +t ζ(x) + Ψ(x) +� +ζ(x) += (∇ζ) +� +X(t, x) +� +(∇Ψ)−1� +X(t, x) +� +ζ(x). +This combined with equality (2.7) yields that for a fixed x ∈ Rd and any t ≥ 0, +f ′ +x(t) = − +� +∇ζ (∇Ψ)−1�� +X(t, x) +� +fx(t) +where +fx(t) := ζ +� +X(t, x) +� +− ζ(x). +(2.14) +Hence, by the continuity of the Zd-periodic matrix-valued function ∇ζ (∇Ψ)−1 in Rd, for any +T ∈ (0, ∞) there exists a constant cT ≥ 0 such that +∀ t ∈ [0, T], +|fx(t)| ≤ cT +ˆ t +0 +|fx(s)| ds, +which by Gr¨onwall’s inequality applied in [0, T] implies that fx = 0 in [0, T]. Therefore, the +vector field ζ is invariant for the mapping X. +8 + +Now, consider the vector field b ∈ C1 +♯ (Yd)d defined by (2.8). Hence, due to (2.13) and the +invariance of ζ combined with equality (2.8), we have for any (t, x) ∈ [0, ∞) × Rd, +∂ +∂t +� +X(t, x) +� += ∇(Ψ−1) +� +t ζ(x) + Ψ(x) +� +ζ(x) += +� +∇(Ψ−1) ◦ Ψ +�� +X(t, x) +� +ζ +� +X(t, x) +� += (∇Ψ)−1� +X(t, x) +� +ζ +� +X(t, x) +� += b +� +X(t, x) +� +. +Therefore, the mapping X defined by (2.12) is actually the flow associated with the vector +field b defined by (2.8) through the ODE (1.1). +Finally, since Ψ(x) = x−Φ(x) for x ∈ Rd, the desired expression (2.3) of the flow X directly +follows from the composition of equality (2.12) by Ψ, and the fine asymptotic expansion (2.1) +is an immediate consequence of the Zd-periodicity of the vector-valued Φ. +This concludes the proof of Proposition 2.3. +□ +Finally, the following result provides sufficient conditions to obtain two vector-valued func- +tions ζ and Φ satisfying the expression (2.3) of the flow X, and to also derive fine asymptotic +expansion (2.2) in some sets of Rd. +Theorem 2.1 Let b ∈ C1 +♯ (Yd)d be a vector field in Rd, d ≥ 2. +i) Assume that the vector field b satisfies the positivity condition +∃ ∇u ∈ C0 +♯ (Yd)d, +b · ∇u > 0 in Yd. +(2.15) +Also assume that there exists a vector-valued function ζ such that X satisfies the asymp- +totics +∀ x ∈ Yd, +lim +t→∞ +X(t, x) +t += ζ(x). +(2.16) +Then, the vector field ζ is invariant for the flow X, and there exists Φ ∈ C1(Rd)d such +that the expression (2.3) of the flow X holds. +ii) Replace in part i) condition (2.15) by the stronger gradient invertibility condition +∃ ∇u1 ∈ C0 +♯ (Yd)d, +b · ∇u1 = 1 in Yd. +(2.17) +Then, the fine asymptotic expansion (2.2) holds in any strip of Rd orthogonal to the +direction ξ := ∇u1 of type +� +x ∈ Rd : x · ξ ∈ [a, b] +� +for − ∞ < a < b < +∞. +(2.18) +iii) Replace in part ii) condition (2.17) by the existence of a vector field U = (u1, . . . , ud) +satisfying +∇U ∈ C0 +♯ (Yd)d×d +with + + + + + + + +b · ∇u1 = 1, +b · ∇u2 = · · · = b · ∇ud = 0, +det (∇U) ̸= 0, +in Yd. +(2.19) +Then, the fine asymptotic expansion (2.1) is satisfied through the expression (2.3) obtained +with the vector field +Φ(x) := x − +� +∇U +�−1U(x) for x ∈ Rd +and +ζ := +� +∇U +�−1e1. +(2.20) +9 + +Remark 2.2 In dimension two Peirone [19, Theorem 3.1] proved remarkably that the asymp- +totics (2.16) of the flow X is always satisfied when the vector field b does not vanish in Y2, +while this asymptotics is generally false in higher dimension [19, Section 4] and in dimension +two with a vanishing vector field b [20]. +Proof of Theorem 2.1. +Proof of part i). First of all, due to the asymptotics (2.16) the invariance of the vector-valued +function ζ for the flow X follows from the equalities (2.9). +Next, following [4, Remark 3.6] we can consider for each x ∈ Rd the unique times τ(x) for +the orbit X(·, x) to meet the equipotential {u = 0}, i.e. +u +� +X(τ(x), x) +� += 0. +(2.21) +Using the positivity (2.15) and the C1-regularity of the flow X, the implicit function theorem +implies that the function τ belongs to C1(Rd). By the uniqueness of τ combined with the +semi-group property of X we also have +∀ t ≥ 0, +τ +� +X(t, x) +� += τ(x) − t. +(2.22) +Now, consider the vector-valued function Φ (not necessarily bounded in Rd nor Zd-periodic) +defined by +Φ(x) = +ˆ τ(x) +0 +� +ζ(x) − b +� +X(s, x) +�� +ds +for x ∈ Rd. +(2.23) +Then, we have for any t ≥ 0 and any x ∈ Rd, +Φ +� +X(t, x) +� += +ˆ τ(x)−t +0 +� +ζ(x) − b +� +X(s + t, x) +�� +ds = +ˆ τ(x) +t +� +ζ(x) − b +� +X(s, x) +�� +ds. +Hence, taking the t-derivative of Φ +� +X(t, x) +� +at point t = 0, we get that +∀ x ∈ Rd, +∇Φ(x) b(x) = b(x) − ζ(x), +which is exactly the first equality of (2.4). +This combined with the invariance of ζ for X +yields (2.4). Therefore, by virtue of Proposition 2.2 we deduce the equivalent expression (2.3) +of the flow X. +Proof of part ii). From equation (2.17) we deduce that +∀ (t, x) ∈ [0, ∞) × Rd, +u1 +� +X(t, x) +� += t + u1(x). +Then, the solution τ(x) to the equation (2.21) with the function u1 is given by τ(x) = − u1(x), +and the vector-valued function Φ defined by (2.23) reads as for any x ∈ Rd, +Φ(x) = +ˆ − u1(x) +0 +� +ζ(x) − ∂X +∂s (s, x) +� +ds = − u1(x) ζ(x) − X(−u1(x), x) + x. +Since ∇u1 is in C0 +♯ (Yd)d, the function u1 can be written u1(x) = ξ · x − v1(x) where ξ = ∇u1 +and v1 ∈ C1 +♯ (Yd). Then, we have for any point x in the affine hyperplane x · ξ = c, +Φ(x) = +� +v1(x)−c +� +ζ(x)+x−X +� +v1(x)−c, x +� += +� +v1(x)−c +� +ζ(x)− +ˆ v1(x)−c +0 +b +� +X(s, x) +� +ds, (2.24) +10 + +and for any t ≥ 0, +Φ +� +X(t, x) +� += +� +v1 +� +X(t, x) +� +− c +� +ζ(x) − +ˆ v1(X(t,x))−c +0 +b +� +X(s + t, x) +� +ds. +Hence, since the functions v1 and ζ are Zd-periodic and continuous in Yd, we get that for any +t ≥ 0 and any x in the affine hyperplane x · ξ = c, +��Φ +� +X(t, x) +� +− Φ(x) +�� ≤ 2 +� +|c| + ∥v1∥L∞ +♯ (Yd) +�� +∥ζ∥L∞ +♯ (Yd)d + ∥b∥L∞ +♯ (Yd)d +� +. +Therefore, taking into account the expression (2.3) of the flow given by the part i), we obtain +the fine asymptotic expansion (2.2) in any strip defined by (2.18). +Proof of part iii). This result has been obtained in [2, Theorem 3.3] for obtaining a class of +ODE’s flows whose Herman’s rotation sets are reduced to a unit set. In the present context, +by (2.19) and (2.20) we get immediately the equality +(Id − ∇Φ) b = +� +∇U +�−1DU b = +� +∇U +�−1e1 = ζ +in Yd, +which by virtue of Proposition 2.2 implies the fine asymptotic expansion (2.1). +The proof of Theorem 2.1 is done. +□ +3 +The incommensurable case +We have the following result. +Theorem 3.1 +I) Let b be a non vanishing vector field at least in C2 +♯ (Y2)2 admitting an invariant probability +measure σ(x) dx where σ is a positive function at least in C5 +♯ (Y2), such that +σb is incommensurable in R2 +and +the ratio σb1 +σb2 +is a Diophantine number. +(3.1) +Then, provided that b and σ are regular enough, the flow X defined by (1.1) satisfies the +fine asymptotic expansion +∀ t ≥ 0, ∀ x ∈ Rd, +X(t, x) = x + t σb + O(1), +(3.2) +where O(1) is a vector-valued function which is bounded uniformly with respect to t and x. +II) Let ξ be a unit vector of R2 such that ξ1/ξ2 is a Liouville’s number. Then, there exists +a positive function a ∈ C∞ +♯ (Y2) such that the Stepanoff flow X associated with the vector +field b = a ξ does not satisfies the fine asymptotic expansion (2.1). +Remark 3.1 In view of the two cases of Theorem 3.1, restricting ourselves to the class of +smooth two-dimensional vector fields b and assuming for each b the existence of an invariant +probability measure σ(x) dx for the flow with a smooth Lebesgue’s density σ > 0 and an incom- +mensurable rotation vector ξ (= σb in (3.1)), we obtain that a necessary and sufficient condition +to derive systematically the fine asymptotic assumption (2.1) in R2 with ζ(x) = ξ, is that the +ratio ξ1/ξ2 is a Diophantine number. +11 + +On the one hand, by virtue of the Kolmogorov theorem [15] (see also [23, Lecture 11]) the +Diophantine property of some rotation number permits to prove that the two-dimensional ODE +(1.1) can be mapped to a linear ODE through a suitable diffeomorphism on Y2, provided that the +vector field b is smooth and non vanishing in Y2 and that the associated flow X has an invariant +probability measure with a smooth Lebesgue’s density. On the other hand, the conjugacy Denjoy +theorem (see [12, Section 12.1]) claims that any smooth diffeomorphism on the circle S1 with an +irrational rotation number ρ is topologically equivalent to the rotation of angle ρ. It turns out +that the Arnold theorem [1] (see [12, Sections 12.3 and 12.5] and [7, Chapter 3, §5]) shows that +the Diophantine property of the rotation number is essential to show that the conjugating map +involved in Denjoy’s theorem is smooth (at last differentiable). The construction of the Peirone +two-dimensional counterexample [20] (recall Remark 2.2) is also based on some Diophantine +rotation number for the ODE’s flow. Alternatively, Theorem 3.1 seems to be, up to our best +knowledge, a new example of the essential role played by the Diophantine property of the rotation +number. +Proof of Theorem 3.1. +Proof of part I). +First step: Reduction to a Stepanoff flow. +By the Kolmogorov theorem [15] combined with enough regularity for the vector field b (at +least C2) and the invariant probability measure σ(x) dx (at least C5) (2), there exists a diffeo- +morphism Ψ on the torus Y2 (see, e.g., [4, Remark 2.1]) of class C2 (at least) satisfying +∀ x ∈ Rd, +Ψ(x) = Mx + Ψ♯(x), +(3.3) +where M ∈ SL± +2 (Z) (i.e. M is a unimodular matrix) and Ψ♯ ∈ C2 +♯ (Y2)2, such that the flow � +X +obtained from the flow X through the diffeomorphism Ψ by +� +X(t, y) := Ψ +� +X(t, Ψ−1(y)) +� +for (t, y) ∈ R × Y2, +(3.4) +is actually the flow associated with the vector field ˆb ∈ C1 +♯ (Y2)2 defined by +ˆb(y) = +� +(∇Ψ b) ◦ Ψ−1� +(y) = a(y) ξ +for y ∈ Y2, +(3.5) +where a is a non vanishing function in C1 +♯ (Y2) (at least) and ξ a non null vector of R2. Moreover, +we easily check that +∀ y ∈ Y2, +lim +t→∞ +� +X(t, y) +t += M +� +lim +t→∞ +X(t, Ψ−1(y)) +t +� +, +(3.6) +if one of the two limits does exist. However, by virtue of Liouville’s theorem (see, e.g., [4, +Proposition 2.2]) the vector field σ b is divergence free in Y2, so that there exists u ∈ C2 +♯ (Y2) +satisfying +σ b = R⊥∇u +or equivalently +b = σ−1R⊥∇u +in Y2. +By hypothesis the mean value of σ b is incommensurable, so is the mean value of ∇u. Then, +by virtue of [4, Corollary 3.4] the Herman rotation set associated with the vector field b is the +unit set +Cb = +� +σb +� +. +2See the remark of [10, p. 8-9] in connection with the Denjoy counterexample (see, e.g., [11]). +12 + +By [4, Proposition 2.1] this combined with (3.6) implies that +∀ y ∈ Yd, +lim +t→∞ +� +X(t, y) +t += M +� +lim +t→∞ +X(t, Ψ−1(y)) +t +� += M σb +which is also an incommensurable vector due to M ∈ SL± +2 (Z). Hence, again applying [4, Propo- +sition 2.1] but with the Stepanoff flow � +X, using the results [5, Section 2.4] on the asymptotics +of Stepanoff’s flows, and recalling (3.5) we get that +Cˆb = {a ξ} = +� +M σb +� +. +(3.7) +Hence, due to M ∈ SL± +2 (Z) it follows that ξ is an incommensurable vector of R2 as σb, and +ξ1/ξ2 is a Diophantine number as the equivalent number σb1/σb2. Therefore, we are led to a +Stepanoff’s flow satisfying the same assumption (3.1) as the original flow X. +Now, it remains to derive the asymptotic (3.2) for any Stepanoff’s flow satisfying condi- +tion (3.1) with σ = a/a and a regular enough. This is the aim of the following step. +Second step: The Stepanoff flow in the incommensurable case. +Assume that ˆb = a ξ where a is a positive function in C1 +♯ (R2) and ξ is an incommensurable +vector of R2 such that ξ1/ξ2 is a Diophantine number. +First of all, following [5, Section 2.4] recall some general results about the Stepanoff flow [22] +in the incommensurable case, namely associated with the vector field ˆb = a ξ where a is a +positive function in C1 +♯ (Yd) and ξ is an incommensurable unit vector of Rd for d ≥ 2. Let θ be +the function defined by +θ(y) +:= +ˆ y·ξ +0 +� +a +a +� +t ξ + (y · ξi) ξi� − 1 +� +dt +(s = t − y · ξ) += +ˆ 0 +−y·ξ +� +a +a +� +s ξ + y +� − 1 +� +ds +for y ∈ Rd, +(3.8) +where (ξ2, . . . , ξd) is an orthonormal basis of (R ξ)⊥ so that for any y ∈ Rd, +y = (y · ξ) ξ + (y · ξi) ξi +with +(y · ξi) ξi = (ξ2 · y) ξ2 + · · · + (ξd · y) ξd, +according to Einstein’s convention. The function θ is in C1(Rd) and satisfies for any y ∈ Rd, +∇θ(y) · ξ += +� a +a(y) − 1 +� +ξ · ξ + +ˆ y·ξ +0 +� +(ξi ⊗ ξi) ∇ +�a +a +�� +t ξ + (y · ξi) ξi�� +· ξ dt += +a +a(y) − 1 + +ˆ y·ξ +0 +� +ξi · ∇ +�a +a +� � +t ξ + (y · ξi) ξi�� +(ξi · ξ) +� �� � +=0 +dt += +a +a(y) − 1. +(3.9) +On the other hand, the two-dimensional flow � +X associated with the vector field ˆb explicitly +reads as +� +X(t, y) = F −1 +y (t) ξ + y +where +Fy(t) := +ˆ t +0 +ds +a(s ξ + y), +(3.10) +and F −1 +y +denotes the reciprocal function of Fy. By (3.9) we have +a Fy(t) = t + +ˆ t +0 +∂ +∂s +� +θ(s ξ + y) +� +ds = t + θ(t ξ + y) − θ(y). +13 + +Therefore, replacing t by F −1 +y (t) in the previous equality and using the expression (3.10) of the +flow, we get that +∀ y ∈ Rd, + + + + + +∀ t ≥ 0, +�X(t, y) = a t ξ + y + θ(y) ξ − θ +� � +X(t, y) +� +ξ +lim +t→∞ +� +X(t, y) +t += a ξ. +(3.11) +Now, assume that d = 2 and that ξ1/ξ2 is a Diophantine number. Consider the function +α ∈ C1 +♯ (Y2) and its Fourier expansion defined by +α(y) := +a +a(y) − 1 = +� +n∈Z2\{0R2} +ˆα(n) e2iπ (y·n) +for y ∈ Y2, +(3.12) +where ˆα(n) denote the Fourier coefficients of α. Then, putting the Fourier expansion (3.12) +in the second integral of (3.8), we may permute the integral and the series due to ˆα ∈ ℓ1(Z2), +which implies that for any x ∈ Y2, +θ(y) = +� +n∈Z2\{0R2} +ˆα(n) +2iπ (ξ · n) +� +e2iπ (y·n) − e2iπ (y−(y·ξ) ξ)·n� +. +(3.13) +Next, since ξ1/ξ2 is a Diophantine number, by (1.10) there exists a non negative integer mξ +such that +# +�� +(p, q) ∈ Z × N : +���� +ξ1 +ξ2 +− p +q +���� ≤ +1 +qmξ+1 +�� +< ∞. +(3.14) +Also assume that a ∈ C +mξ+2 +♯ +(Y2). Then, by the Cauchy-Schwarz inequality we get that +� +n ∈ Z2\{0R2} �−→ |n|mξ |ˆα(n)| = |ˆα(n)| |n|mξ+2 +|n|2 +� +∈ ℓ1(Z2\{0R2}), +(3.15) +since by the Parseval identity applied with the tensor-valued function ∇(mξ+2)α ∈ C0 +♯ (Y2)2(mξ+2) +we have +� +n∈Z2\{0R2} +1 +|n|4 < ∞ +and +� +n∈Z2\{0R2} +|n|2(mξ+2) |ˆα(n)|2 ≤ c ∥∇(mξ+2)α∥2 +ℓ2(Z)2(mξ +2). +Moreover, by (3.14) we have for any n = (n1, n2) ∈ Z2\{0R2} with |n| ≥ N large enough, +|ξ · n| = +� +|ξ2 n2| ≥ |ξ2| +if n1 = 0 +|ξ2| |n1| |ξ1/ξ2 + n2/n1| ≥ |ξ2|/|n1|mξ +if n1 ̸= 0, +which implies that +∃ c > 0, ∀ n ∈ Z2\{0R2}, +|ξ · n| ≥ +c +|n|mξ . +(3.16) +This combined with (3.15) thus yields +∀ n ∈ Z2\{0R2} with |n| ≥ N, +|ˆα(n)| +|ξ · n| ≤ C |ˆα(n)| |n|mξ = |ˆα(n)| |n|mξ+2 +|n|2 +∈ ℓ1(Z2\{0R2}). +Therefore, we deduce that the asymptotic expansion of (3.11) satisfies the uniform estimate +∀ t ≥ 0, ∀ y ∈ R2, +�� � +X(t, y) − t a ξ − y +�� ≤ c +� +n∈Z2\{0R2} +|ˆα(n)| +|ξ · n| < ∞, +(3.17) +14 + +which establishes the asymptotic expansion (3.2) for the Stepanoff flow in the Diophantine case. +Let us conclude the proof of part I). Starting from formula (3.4), multiplying formula (3.3) +by the matrix M−1, and using the estimate (3.17) of � +X combined with the equality (3.7) and +the boundedness of Ψ♯, we get that for any t ≥ 0 and any x ∈ Y2, +X(t, x) = Ψ−1� � +X(t, Ψ(x)) +� += M−1� � +X(t, Ψ(x)) +� +− M−1� +Ψ♯ ◦ Ψ−1�� � +X(t, Ψ(x)) +� += M−1� +t a ξ + Mx + Ψ♯(x) + O(1) +� +− O(1) += t σb + x + O(1), +which finally yields the desired fine asymptotic expansion (3.2). +Proof of part II). +Since ξ1/ξ2 is a Liouville’s number, by (1.11) there exist two sequences of integers (pn)n∈N in ZN +and (qn)n∈N in NN satisfying +∀ n ∈ N, +���� +ξ1 +ξ2 +− pn +qn +���� < +1 +(qn)n , +(3.18) +or equivalently, +∀ n ∈ N, +|ξ · kn| < +|ξ2| +(qn)n−1 +where +kn := qn e1 − pn e2 ∈ Z2. +(3.19) +Up to extract a subsequence of the sequence (qn)n∈N (which converges to ∞) still denoted by +(qn)n∈N, we can assume in addition that +∀ n ≥ 3, +qn ≥ |ξ · kn−1| +1 +3−n + n + +n−1 +� +i=1 +qi +and +∞ +� +n=3 +2π |ξ2| +(qn)n−2 < 1, +(3.20) +which implies in particular that (qn)n∈N is increasing. Then, define the positive function a +in C∞ +♯ (Y2) by its inverse +1 +a(x) := 1 + +∞ +� +n=3 +αn cos (2π kn · x) +for x ∈ Y2, +where +αn := 2π qn ξ · kn. +(3.21) +The function a is well defined and positive due to the second inequality of (3.20) combined +with inequality (3.19). Moreover, since by (3.19) and (3.21) we have for any m ∈ N, +∞ +� +n=m+2 +αn |kn|m ≤ +∞ +� +n=m+2 +2π |ξ2| qn (|pn| + qn)m +(qn)n−1 +≤ c +∞ +� +n=m+2 +1 +(qn)n−m−2 < ∞, +the function a belongs to C∞ +♯ (Y2). +On the other hand, define the sequence (τn)n∈N by +τn := +1 +4 ξ · kn +for n ∈ N. +(3.22) +15 + +Then, the function θ defined by the first integral of (3.8) with 1/a defined by the series expansion +(3.21), satisfies for any integer m ≥ 4 (note that a = 1) +θ(τm ξ) += +ˆ τm +0 +� ∞ +� +n=3 +αn cos +� +2π (ξ · kn) t +� +� +dt += +∞ +� +n=3 +αn +sin +� +2π (ξ · kn) τm +� +2π (ξ · kn) += qm + +m−1 +� +n=3 +αn +sin +� +2π (ξ · kn) τm +� +2π (ξ · kn) ++ +∞ +� +n=m+1 +αn +sin +� +2π (ξ · kn) τm +� +2π (ξ · kn) +, +which by the first inequalities of (3.20) and (3.19) implies that +θ(τm ξ) +≥ qm − +m−1 +� +n=3 +|αn| +2π |ξ · kn| − +∞ +� +n=m+1 +|τm| |αn| +≥ qm − +m−1 +� +n=3 +qn − π +2 +∞ +� +n=m+1 +qn +|ξ · kn| +|ξ · km| +≥ m − π |ξ2| +2 +∞ +� +n=m+1 +1 +(qn)n−2 +1 +|ξ · km|. +(3.23) +Moreover, applying the first inequality of (3.20) with n = m+1, we get that for any n ≥ m+1, +qn ≥ qm+1 ≥ |ξ · km| +1 +2−m +so that +1 +(qn)n−m ≥ +1 +(qn)n−2 +1 +|ξ · km|. +This combined with (3.23) and the increase of (qn)n∈N thus yields +θ(τm ξ) ≥ m − π |ξ2| +2 +∞ +� +n=m+1 +1 +(qn)n−m = m − π |ξ2| +2 +∞ +� +n=1 +1 +(qn+m)n ≥ m − π |ξ2| +2 +∞ +� +n=1 +1 +(qn)n +� +�� +� +<∞ +. +Hence, we deduce that +lim +m→∞ θ(τm ξ) = ∞. +(3.24) +Finally, by the expression (3.10) of the Stepanoff flow for y = 0R2, we have for any m ∈ N, +�X(tm, 0R2) = τm ξ +where +tm := F0R2(τm). +Therefore, using the expression (3.11) of the flow � +X for y = 0R2 and limit (3.24), we obtain +that +�� � +X(tm, 0R2) − tm ξ +�� = +��θ(0R2) − θ +� +τm ξ) +�� −→ +m→∞ ∞, +which shows that the fine asymptotic expansion (2.1) does not hold for the Stepanoff flow �X. +The proof of part II) is done, which also concludes the proof of Theorem 3.1. +□ +16 + +Remark 3.2 In higher dimension and in spirit of the case iii) of Theorem 2.1, assume that +there exists a vector-valued function U := (u1, . . . , ud) satisfying besides condition (2.19) the +following one +∇U ∈ C1 +♯ (Yd)d×d +with + + + + + + + +b · ∇u1 > 0, +b · ∇u2 = · · · = b · ∇ud = 0, +det (∇U) ̸= 0, +in Yd. +(3.25) +Then, following [2, Theorem 3.3] the matrix ∇U is invertible and the diffeomorphism on the +torus Ψ := MU with M := +� +∇U +�−1 (3), satisfies +∇Ψ ∈ C1(Yd)d×d, +∇Ψ = Id, +∇Ψ b = (b · ∇u1) ξ in Yd, +with ξ := Me1. +(3.26) +Hence, Ψ is a C2-diffeomorphism on the torus Yd (recall (3.3)) which maps the flow X associated +with b to the Stepanoff flow � +X (3.4) associated with the vector field +ˆb := a ξ +where +a(y) := +� +(b · ∇u1) ◦ Ψ−1� +(y) > 0 +for y ∈ Yd. +(3.27) +When the vector ξ satisfies the extension of (3.16) +∃ c > 0, ∃ mξ ∈ N, ∀ n ∈ Zd \ {0Rd}, +|ξ · n| ≥ +c +|n|mξ , +(3.28) +and a ∈ C +mξ+p +♯ +(Yd) for some integer p > d/2, we get similarly to the proof of the second part of +Theorem 3.1, that the flow X satisfies the fine asymptotic expansion (2.1). +In the part iii) of Theorem 4.1 below we will again use the previous diffeomorphism Ψ on Yd +with d > 2, in the case where the vector ξ is commensurable in Rd. +4 +The commensurable case +We have the following result. +Theorem 4.1 Let b ∈ C1 +♯ (Yd)d be a vector field in Rd. +i) Let A be a non-empty subset of Rd. Assume that there exist TA, kA ∈ (0, ∞) such that the +flow X satisfies the periodicity property +∀ x ∈ A, ∃ T(x) ∈ (0, TA], ∃ k(x) ∈ Zd with |k(x)| ≤ kA, ∀ t ≥ 0, +X +� +t + T(x), x +� += X(t, x) + k(x). +(4.1) +Then, the flow X associated with b satisfies the fine asymptotic expansion (2.2) in A with +ζ(x) := k(x)/T(x) for x ∈ A. +3Actually, the authors have recently discovered that the mapping Ψ used in [2] was previously introduced +by Kozlov in [17, Theorems 1,2] to extend in some way the two-dimensional Kolmogorov theorem [15] to higher +dimension. +17 + +ii) Assume that b is a non vanishing vector field in C2 +♯ (Y2)2 admitting an invariant probability +measure σ(x) dx, where σ is a positive function in C5 +♯ (Y2) with mean value 1, such that +σb is commensurable in R2. +(4.2) +Then, the flow X satisfies the fine asymptotic expansion (2.1) with +ζ +� +Ψ(x) +� +:= +� +1 +T +ˆ T +0 +dt +a +� +t ξ + Ψ(x) +� +�−1 +ξ +for x ∈ Y2, +(4.3) +where the C2-diffeomorphism Ψ on Y2 maps the flow X on the Stepanoff flow � +X associated +with the vector field ˆb through equalities (3.3), (3.4), (3.5). +iii) Assume that for d > 2, the vector field b satisfies (3.25) with DU ∈ C1 +♯ (Yd)d×d, and +that the vector ξ := +� +∇U +�−1 e1 in (3.26) is commensurable, i.e. there exists T > 0 such +that T ξ ∈ Zd. +Then, the flow X still satisfies the fine asymptotic expansion (2.1) with the vector-valued +function ζ defined by (4.3) in Yd, where the C2-diffeomorphism Ψ = MU on Yd maps the +flow X on the Stepanoff flow associated with the vector field ˆb through equalities (3.25), +(3.26), (3.27). +Remark 4.1 By virtue of [9, Theorem 1.2] it is known that the rotation set Cb of the ODE’s +flow (1.1) associated with a vector field b ∈ C1 +♯ (Y2) is always a closed line segment of R2 carried +by a line passing through 0R2. This combined with [8, Theorem B] implies that if Cb contains a +non null commensurable vector ζ, then the flow X satisfies a fine asymptotic expansion in the +direction ζ⊥, i.e. there exists a constant C ≥ 0 such that +∀ t ≥ 0, ∀ x ∈ R2, +�� � +X(t, x) − x +� +· ζ⊥ �� ≤ C, +(4.4) +where the first-order term t ζ(x) does not appear due to ζ(x) ∈ Cb ⊂ R ζ. Estimate (4.4) +extends the one obtained in the first case of the proof of [19, Theorem 3.1] where the constant +does depend on x a priori. +Proof of Theorem 4.1. +Proof of part i). First of all, for t ≥ 0 and x ∈ A, let nt,x be the integer satisfying +nt,x T(x) ≤ t < (nt,x + 1) T(x). +(4.5) +Reiterating equality (4.1) we get that +X(t, x) += X +� +t − nt,x T(x), x +� ++ nt,x k(x) += x + t k(x) +T(x) + +� +nt,x − +t +T(x) +� +k(x) + X +� +t − nt,x T(x), x +� +− x, +and by (4.5) we have +���� +� +nt,x − +t +T(x) +� +k(x) + X +� +t − nt,x T(x), x +� +− x +���� +≤ |k(x)| + +����� +ˆ t−nt,x T(x) +0 +b +� +X(s, x) +� +ds +����� +≤ kA + TA ∥b∥L∞(Yd)d. +18 + +Therefore, we obtain the fine asymptotic expansion (2.2) for the flow X in the subset A with +ζ(x) := k(x)/T(x) for x ∈ A. +Proof of part ii). Proceeding as the first step of Theorem 3.1 thanks to Kolmogorov’s theorem +we are led to Stepanoff flow associated with the vector field ˆb = a ξ, where a is a positive +function in C1 +♯ (Y2) and ξ is a vector of R2 such that T ξ = k ∈ Z2 for some T ∈ (0, ∞). Indeed, +due to (3.7) with M ∈ SL± +2 (Z) and to condition (4.2), the vector +ξ := 1 +a M σb is commensurable in R2. +(4.6) +Moreover, by the expression (3.10) of the Stepanoff flow � +X combined with the Zd-periodicity +of a, we have for any t ≥ 0 and any y ∈ Rd, +Fy(t + T) = Fy(t) + +ˆ T +0 +ds +a(s ξ + y) = Fy(t) + +T +m(y) +where +m(y) := +� 1 +T +ˆ T +0 +ds +a(s ξ + y) +�−1 +Hence, replacing t by F −1 +y (t) in the previous equality we obtain that +�X +� +t + +T +m(y), y +� += F −1 +y +� +t + +T +m(y), y +� +ξ + y = F −1 +y (t) ξ + T ξ + y = � +X(t, y) + k, +which implies condition (4.1) with A := Rd, T(y) := T/m(y) bounded by TA := T ∥a−1∥L∞(Y2), +and k(x) := k. Therefore, the fine asymptotic expansion (2.1) holds with the vector-valued +function ζ defined by (4.3), i.e. +ζ(y) = m(y) ξ +and +� +X(t, y) = y + t ζ(y) + O(1). +Hence, since the vector-valued functions +� +y �→ Ψ−1(y) − y +� +and +� +x �→ Ψ(x) − x +� +are Z2- +periodic and continuous thus bounded in R2, mapping the previous equality by Ψ−1 and using +the relation (3.4) between the two flows X and � +X, we deduce that for any t ≥ 0 and any +x := Ψ−1(y) ∈ R2, +X(t, x) = Ψ−1� +y + t ζ(y) + O(1) +� += Ψ(x) + t ζ +� +Ψ(x) +� ++ O(1) = x + t ζ +� +Ψ(x) +� ++ O(1), +which is the desired fine asymptotic expansion (2.1) satisfied by X. +Proof of part iii). The proof is quite similar to the one of case ii), which concludes the proof +of Theorem 4.1. +□ +5 +Examples +5.1 +Cases with a non vanishing vector field +Let us start by a very simple example illustrating explicitly Theorem 3.1. +Example 5.1 Let ξ be an incommensurable vector of R2, and let b be the vector field +b(x) := +ξ +2 + cos(2πx1) +for x ∈ Y2. +19 + +Then, an explicit computation of formulas (3.8), (3.10) and (3.11) leads us to + + + + + + + + + + + +X(t, x) = x + +� +1 +2 t + sin(2πx1) +4πξ1 +− sin +� +2π(x1 + F −1 +x (t) ξ1) +� +4πξ1 +� +ξ +Fx(t) := 2 t + sin +� +2π(x1 + t ξ1) +� +− sin(2πx1) +4πξ1 +, +for t ≥ 0, x ∈ Y2. +Therefore, the flow X associated with the vector field b satisfies Theorem 3.1, and consequently +the fine asymptotic expansion (2.1) with the vector-valued function ζ(x) ≡ 1 +2 ξ. +The following example revisits the two-dimensional flow of [6, Example 2.7] in the light of +the fine asymptotic expansion (2.1). +Example 5.2 Consider the non vanishing two-dimensional vector field b defined by +b(x) := e1 + 2π sin(2πx2) e2 = ∇u(x) +where +u(x) := x1 − cos(2πx2) +for x ∈ R2. +(5.1) +By [6, Example 2.12] a tedious but easy computation shows that the flow X associated with +the vector field (5.1) is given explicitly by +X(t, x) = + + + +(t + x1) e1 + +� +n + 1 +π arctan +� +e4π2t tan(πx2) +�� +e2, +|x2 − n| < 1 +2 +(t + x1) e1 + +� +n + 1 +2 +� +e2, +x2 = n + 1 +2, +for n ∈ Z. +(5.2) +Condition (2.15) is clearly satisfied with u(x) = x1. +Moreover, we have +∀ x ∈ Y2, +lim +t→∞ +X(t, x) +t += e1, +(5.3) +so that by [4, Proposition 2.1] Herman’s rotation set is the unit set Cb = {e1}. By the analysis +of [6, Example 2.12] it is surprising to observe that the flow X (5.2) has no invariant measure of +type σ(x) dx where σ is a positive function in C0 +♯ (Y2). However, the Radon measure dx1 ⊗δx2=0 +on Y2 is invariant for the flow X. Indeed, we have +∀ ϕ ∈ C1 +♯ (Y2), +ˆ +Y2 +b(x) · ∇ϕ(x) (dx1⊗δx2=0) = +ˆ 1 +0 +∂ϕ +∂x1 +(x1, 0) dx1 = 0, +which owing to Liouville’s theorem (see, e.g., [4, Proposition 2.2]) yields the invariance. +Finally, the expression (5.2) of the flow shows directly that for any t ≥ 0 and any x ∈ R2 +such that x2 ∈ +� +n − 1 +2, n − 1 +2 +� +with n ∈ Z, +��X(t, x) − x − t e1 +�� ≤ |n − x2| + 1 +2 ≤ 1. +(5.4) +Therefore, the flow X satisfies the fine asymptotic expansion (2.1) with ζ = e1 and a uniformly +bounded term. +However, following Proposition 2.2 it is interesting to recover the fine asymptotic expan- +sion (2.3) from a suitable bounded vector-valued function Φ. To this end, the general defini- +tion (2.23) with asymptotics (5.3) leads us to the vector field Φ defined for x ∈ R2, by +Φ(x) := +ˆ τ(x) +0 +� +e1 − b(X(t, x)) +� +dt +where τ(x) is solution to +u +� +X(τ(x), x) +� += 0, +(5.5) +20 + +which similarly to (2.3) yields the expression of the flow +∀ t ≥ 0, ∀ x ∈ Rd, +X(t, x) = x + t e1 + Φ +� +X(t, x) +� +− Φ(x). +(5.6) +Then, due to (5.2) we have +0 = u +� +X(τ(x), x) +� += X1(τ(x), x) − cos +� +2πX2(τ(x), x) +� += τ(x) + x1 − cos +� +2πX2(τ(x), x) +� +, +which implies that +Φ(x) = − 2π e2 +ˆ −x1+cos(2πX2(τ(x),x)) +0 +sin +� +2πX2(t, x) +� +dt +(5.7) +Noting that by (5.2) we have for any t ≥ 0 and any x ∈ R2 such that x2 ∈ +� +n − 1 +2, n + 1 +2 +� +with +n ∈ Z, +sin +� +2πX2(t, x) +� += sin +� +2 arctan +� +e4π2t tan(πx2) +�� += +2 e4π2t tan(πx2) +1 + e8π2t tan2(πx2). +(5.8) +Therefore, we deduce the inequality +∀ x ∈ R2, +|Φ(x)| ≤ +ˆ ∞ +−∞ +4π e4π2t tan(πx2) +1 + e8π2t tan2(πx2) dt = 1 +π +� +arctan +� +e4π2t tan(πx2) +��∞ +−∞ = 1. +which yields the uniform boundedness of Φ +� +X(t, x) +� +−Φ(x) with respect to t and x in (5.6). +5.2 +Cases with a vanishing vector field +In the first example a vector field with separate variables is investigated. +Example 5.3 Let vector field b(x) = +� +b1(x1), . . . , bd(xd) +� +∈ C1 +♯ (Yd)d having 0Yd as unique root +in Yd, so that 0 is the unique common root of the functions b1, . . . , bd in Y1. +First of all, it is clear that property (2.15) does not hold, since the vector field b does vanish. +Then, the flow X = (X1, . . . , Xd) associated with b is given for i = 1, . . . , d and x ∈ Yd, by (see, +e.g., [5, Section 2.4]) + + + + + +Xi(t, x) = F −1 +i,x (t) + xi +for t ≥ 0 +Fi,x(t) := +ˆ t +0 +ds +bi(s + xi) +for t ∈ +� +[xi] − xi, 1 + [xi] − xi +� +, +(5.9) +where F −1 +i,x is the reciprocal function of Fi,x, and [xi] is the integer satisfying [xi] ≤ xi < [xi]+1. +Since the zero set of b is Zd, each function bi has a constant sign in the interval +� +[xi], 1 + [xi] +� +, +and for any +� +[xi], 1 + [xi] +� +, +ˆ [xi]−xi +0 +ds +bi(s + xi) = − +ˆ 1+[xi]−xi +0 +ds +bi(s + xi) ∈ {−∞, ∞}. +Hence, the function F −1 +i,x is a bijection from R on the interval +� +[xi] − xi, 1 + [xi] − xi +� +⊂ [−1, 1]. +Therefore, the range of the flow X is contained in [−1, 1]d, so that X satisfies the fine asymptotic +expansion (2.1) with the vector-valued function ζ(x) ≡ 0. +The following example deals with a two-dimensional Stepanoff flow associated with a vector +field which has isolated roots in Y2. +21 + +Example 5.4 Let b ∈ C∞ +♯ (Y2)2 be the vector field defined by +b(x) := cos2(πx1) (e1 + γ e2) +for x ∈ Y2, +with γ ∈ R. +The flow X associated with b is given by the explicit formula +X(t, x) = +� +x + +� 1 +π arctan +� +π t + tan(π(x1 − n)) +� ++ n − x1 +� +(e1 + γ e2) +if |x1 − n| < 1 +2 +x +if x1 = n + 1 +2, +n ∈ Z. +Therefore, the flow X satisfies the inequality +∀ t ≥ 0, ∀ x ∈ R2, +|X(t, x) − x| ≤ +� +1 + γ2, +which provides the fine asymptotic expansion (2.1) with the vector-valued function ζ(x) ≡ 0. +The following general result shows that any two-dimensional Stepanoff flow associated with +a vector field having one root in Y2 and an incommensurable direction ξ in R2, does not satisfy +the fine asymptotic expansion (2.2) in the set A := R ξ + Zd. +Proposition 5.1 Let b = a ξ be a two-dimensional vector field such that a ∈ C1 +♯ (Y2) has 0Y2 +as unique root in Y2, and ξ is any incommensurable unit vector of R2. +Then, the flow X satisfies the asymptotics +∀ x ∈ R2, +ζ(x) := lim +|t|→∞ +X(t, x) +t += + + + + + + + +a ξ +if x ∈ R2\(R ξ+Z2) +a ξ +if x ∈ R ξ+Z2, τx < 0 +0R2 +if x ∈ R ξ+Z2, τx ≥ 0, +(5.10) +where τx is the unique real number satisfying +x + τx ξ = kx ∈ Z2. +(5.11) +Moreover, the fine asymptotic expansion (2.2) is not fulfilled in the set A := R ξ+Z2, and the +following large deviation holds +∀ v ∈ S1 s.t. ξ · v ̸= 0, +sup +t∈R, x∈A +� +X(t, x) − x − t ζ(x) +� +· v = ∞. +(5.12) +Remark 5.1 Taking into account the asymptotics of the flow (5.10), by virtue of [18, Theo- +rem 2.4, Remark 2.5, Corollary 2.6] the Herman rotation set is given by the non degenerate +closed line segment +Cb = conv +� +ζ(R2) +� += [0, a] ξ. +Therefore, in the present case of a Stepanoff flow associated with a vanishing vector field and +an incommensurable vector, we recover directly from the asymptotics of the flow the result of +[5, Section 2.4] obtained by a perturbation result. +Contrary to the hypothesis of Proposition 5.1, the function a of the Stepanoff vector field +b = a ξ, has non isolated roots in Example 5.4. It turns out that the fine asymptotic expansion +(2.1) holds in Example 5.4 for any vector ξ in R2, while it fails in Proposition 5.1 for any +incommensurable vector ξ in R2. +22 + +Proof of Proposition 5.1. First of all, make some considerations on the set R ξ+Z2. By the +incommensurability of ξ, for any x ∈ R ξ + Z2 there exists a unique τx ∈ R satisfying (5.11). +Let y be a point in R2 \ (R ξ + Z2). Since ξ is incommensurable, the set R ξ + Z2 is dense +into R2. Then, there exists a sequence (xn)n∈N in (R ξ+Z2)N which converges to y. We have +xn = − τxn ξ + kxn +with +τxn ∈ R and kxn ∈ Z2, +(5.13) +where +lim +n→∞ |kxn| = ∞ +and consequently +lim +n→∞ |τxn| = ∞. +(5.14) +Indeed, assume that the first limit of (5.14) does not hold. Then, there exists a subsequence of +the integer vectors sequence (kxn)n∈N which is stationary, so that by (5.13) the corresponding +subsequence of (τxn)n∈N converges, which implies that y ∈ R ξ +Z2, a contradiction. Up to +consider − y with τ−y = − τy, and to extract a subsequence we can assume that τxn > 0 for any +n ∈ N. We have just established the existence of a sequence (xn)n∈N in (R ξ+Z2)N satisfying +∀ n ∈ N, xn + τxn ξ ∈ Z2, +lim +n→∞ xn = y +and +lim +n→∞ τxn = ∞. +(5.15) +On the other hand, due the uniqueness of the representation (5.11) τx is the unique root of +the function +� +t �→ a(t ξ + x) +� +in R. Moreover, since the continuous function a does not vanish +in the connected set R2 \ Z2, it has a constant sign in R2 \ Z2. Without loss of generality we +can assume that a is positive in R2 \ Z2. Then, defining for each x ∈ R2 the function Fx by +Fx(t) := + + + + + + + + + + + + + + + + + + + + + + + + + +ˆ t +0 +ds +a(s ξ + x) +for t ∈ R, +if x ∈ R2\(R ξ+Z2) +ˆ t +0 +ds +a(s ξ + x) +for t ∈ (−∞, τx), +if x ∈ R ξ+Z2, τx > 0 +ˆ t +0 +ds +a(s ξ + x) +for t ∈ (τx, ∞), +if x ∈ R ξ+Z2, τx < 0 +0 +for t ∈ R, +if x ∈ Z2 (i.e. τx = 0), +(5.16) +the function Fx is increasing in the first cases of (5.16) due to the positivity of a. Then, the +reciprocal application F −1 +x +is an increasing bijection from R onto (−∞, τx) if τx > 0, and from +R onto (τx, ∞) if τx < 0. Hence, by formula (3.10) the flow X associated with the vector field +b = a ξ satisfies +∀ t ∈ R, +X(t, x) = +� +F −1 +x (t) ξ + x +if x ∈ R2\ Z2 +x +if x ∈ Z2 (i.e. τx = 0), +(5.17) +which combined with (5.16) and τxn > 0, implies in particular that +∀ n ∈ N, +lim +t→∞ X(t, xn) = τxn ξ + xn. +(5.18) +Therefore, the formula (5.17) of the flow X together with the formula (5.16) of the function Fx +(see also the positive case of [5, Section 2.4]) yield the desired asymptotics (5.10), which in +return implies that +∀ n ∈ N, +lim +t→−∞ +�X(t, xn) +t +� += a ξ. +(5.19) +23 + +Finally, applying (5.18) and (5.19) with the sequence (xn)n∈N satisfying (5.15), we get that +for any vector v ∈ S1 such that ξ · v ̸= 0, +∀ n ∈ N, +ζ(xn) = 0R2 +and + + + +lim +t→∞ +� +X(t, xn) − xn +� +· v = τxn ξ · v − xn · v +if ξ · v > 0 +lim +t→−∞ +� +X(t, xn) − xn +� +· v = ∞ +if ξ · v < 0. +Hence, it follows that the fine asymptotic expansion (2.2) is not fulfilled in the set A := R ξ+Z2, +and that the following large deviation in any direction v ∈ S1 such that ξ · v ̸= 0, holds + + + + + +sup +t∈R, x∈A +� +X(t, x) − x − t ζ(x) +� +· v ≥ lim +n→∞ +� +τxn ξ · v − xn · v +� += ∞ +if ξ · v > 0 +sup +t∈R, x∈A +� +X(t, x) − x − t ζ(x) +� +· v ≥ lim +t→−∞ +� +X(t, xn) − xn +� +· v = ∞ +if ξ · v < 0, +(5.20) +which yields equality (5.12). +This concludes the proof of Proposition 5.1. +□ +References +[1] V.I. Arnol’d: “Small denominators I. Mapping the circle onto itself” (Russian), Izv. +Akad. Nauk SSSR Ser. Mat., 25 (1961), 21-86. English translation: Trans. Amer. Math. +Soc. (Series 2), 46 (1965), 213-284. +[2] M. Briane: “Isotropic realizability of fields and reconstruction of invariant measures +under positivity properties. Asymptotics of the flow by a nonergodic approach”, SIAM J. +App. Dyn. Sys., 18 (4) (2019), 1846-1866. +[3] M. Briane & L. Herv´e: “A picture of the ODE’s flow in the torus: from everywhere or +almost-everywhere asymptotics to homogenization of transport equations”, J. Differential +Equations, 304 (2021), 165-190. +[4] M. Briane & L. Herv´e: “Asymptotics of ODE’s flow on the torus through a singleton +condition and a perturbation result. Applications”, Dis. Con. Dyn. Sys., 42 (7) (2022), +3431-3463. +[5] M. Briane & L. Herv´e: “Specific properties of the ODE’s flow in dimension two versus +dimension three”, accepted and to appear in J. Dyn. Diff. Equa., arXiv:2111.02090 (2021), +pp. 42. +[6] M. Briane, G.W. Milton & A. Treibergs : “Which electric fields are realizable in +conducting materials?”, ESAIM: Math. Model. Numer. Anal., 48 (2) (2014), 307-323. +[7] I.P. Cornfeld, S.V. Fomin & Ya.G. Sina˘ı: Ergodic Theory, translated from the +Russian by A.B. Sosinskii, Grundlehren der Mathematischen Wissenschaften [Fundamental +Principles of Mathematical Sciences] 245, Springer-Verlag, New York, 1982, 486 pp. +[8] P. D`avalos: “On annular maps of the torus and sublinear diffusion”, J. Inst. Math. +Jussieu, 17 (4) (2018), 913-978. +[9] J. Franks & M. Misiurewicz: “Rotation sets of toral flows”, Proc. Amer. Math. Soc., +109 (1) (1990), 243-249. +24 + +[10] F. Golse: “Moyennisation des champs de vecteurs et EDP” (French), [The averaging of +vector fields and PDEs], Journ´ees ´Equations aux D´eriv´ees Partielles, Saint Jean de Monts +1990, Exp. no. XVI, ´Ecole Polytech. Palaiseau, 1990, 17 pp. +[11] M.R. Herman: “Existence et non existence de tores invariants par des diff´eomorphismes +symplectiques” (French), [Existence and nonexistence of tori invariant under symplectic +diffeomorphisms], S´eminaire sur les ´Equations aux D´eriv´ees Partielles 1987-1988, XIV, +´Ecole Polytech. Palaiseau, 1988, 24 pp. +[12] Katok & B. Hasselblatt: Introduction to the modern theory of dynamical systems. +With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathe- +matics and its Applications, 54, Cambridge University Press, Cambridge, 1995, pp. 802. +[13] A. Kocsard: “On the dynamics of minimal homeomorphisms of T2 which are not pseudo- +rotations”, Ann. Scient. ´Ec. Norm. Sup., 4e s´erie 54 (4) (2021), 991-1034. +[14] A. Kocsard & A. Koropecki: “A mixing-like property and inexistence of invariant +foliations for minimal diffeomorphisms of the 2-torus”, Proc. Amer. Math. Soc., 137 (10) +(2009), 3379-3386. +[15] A.N. Kolmogorov: “On dynamical systems with an integral invariant on the torus” +(Russian), Doklady Akad. Nauk SSSR (N.S.), 93 (1953), 763-766. +[16] A. Koropecki & F. A. Tal: “Area-preserving irrotational diffeomorphisms of the torus +with sublinear diffusion”, Proc. Amer. Math. Soc., 142 (10) (2014), 3483-3490. +[17] V. V. Kozlov: “Dynamical systems on a torus with multivalued integrals”, (Russian) Tr. +Mat. Inst. Steklova 256 (2007), Din. Sist. i Optim., 201-218, translation in Proc. Steklov +Inst. Math., 256 (1) (2007), 188-205. +[18] M. Misiurewicz & K. Ziemian: “Rotation sets for maps of tori”, J. London Math. +Soc.(2), 40 (3) (1989), 490-506. +[19] R. Peirone: “Convergence of solutions of linear transport equations”, Ergodic Theory +Dynam. Systems, 23 (3) (2003), 919-933. +[20] R. Peirone: “A nonhomogenizable linear transport equation in R2”, Ann. Sc. Norm. +Super. Pisa Cl. Sci. (5) 8 (1) (2009), 175-206. +[21] R. Peirone: “Homogenization of ODE’s in RN”, Ann. Mat. Pura Appl. (4) 198 (3) +(2019), 869-879. +[22] W. Stepanoff: “Sur une extension du th´eor`eme ergodique”, Compositio Math., 3 (1936), +pp. 239-253. +[23] Ya.G. Sinai: Introduction to Ergodic Theory, Translated by V. Scheffer, Mathematical +Notes 18, Princeton University Press, Princeton, N.J., 1976, 144 pp. +25 + diff --git a/PtA0T4oBgHgl3EQfDP9d/content/tmp_files/load_file.txt b/PtA0T4oBgHgl3EQfDP9d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e14c03bd5ff3e381b685e5b5531e72c2fd8d055e --- /dev/null +++ b/PtA0T4oBgHgl3EQfDP9d/content/tmp_files/load_file.txt @@ -0,0 +1,987 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf,len=986 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='02000v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='AP] 5 Jan 2023 Fine asymptotic expansion of the ODE’s flow Marc Briane & Lo¨ıc Herv´e Univ Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France mbriane@insa-rennes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='fr & loic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='herve@insa-rennes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='fr Friday 6th January, 2023 Contents 1 Introduction 2 2 Fine asymptotic expansion 6 3 The incommensurable case 11 4 The commensurable case 17 5 Examples 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 Cases with a non vanishing vector field .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2 Cases with a vanishing vector field .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 21 Abstract In this paper, we study the asymptotic expansion of the flow X(t, x) solution to the nonlinear ODE: X′(t, x) = b � X(t, x) � with X(0, x) = x ∈ Rd, where b is a regular Zd- periodic vector field in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' More precisely, we provide various conditions on b to obtain a “fine” asymptotic expansion of X of the type: |X(t, x) − x − t ζ(x)| ≤ M < ∞, which is uniform with respect to t ≥ 0 and x ∈ Rd (or at least in a subset of Rd), and where ζ(x) for x ∈ Rd, are the rotation vectors induced by the flow X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' On the one hand, we give a necessary and sufficient condition on the vector field b so that the expansion X(t, x) − x − t ζ(x) reads as Φ � X(t, x) � − Φ(x), which yields immediately the desired ex- pansion when the vector-valued function Φ is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In return, we derive an admissible class of vector fields b in terms of suitable diffeomorphisms on Yd and of vector-valued functions Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' On the other hand, assuming that the two-dimensional Kolmogorov theorem and some extension in higher dimension hold, we establish different regimes depending on the commensurability of the rotation vectors of the flow X for which the fine estimate expansion of X is valid or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' It turns out that for any two-dimensional flow X associ- ated with a non vanishing smooth vector field b and inducing a unique incommensurable rotation vector ξ, the fine asymptotic expansion of X holds in R2 if, and only if, ξ1/ξ2 is a Diophantine number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' This result seems new in the setting of the ODE’s flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The case of commensurable rotation vectors ζ(x) is investigated in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Finally, sev- eral examples and counter-examples illustrate the different results of the paper, including the case of a vanishing vector field b which blows up the asymptotic expansion in some direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 1 Keywords: ODE’s flow, asymptotic expansion, rotation number, incommensurable vector, Diophantine number, Liouville’s number Mathematics Subject Classification: 34E05, 34E10, 37C10, 37C40 1 Introduction Let b be a C1-regular vector field in Rd defined on the torus Yd := Rd/Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In this paper, we study the ODE’s flow X(·, x) for x ∈ Yd, defined by \uf8f1 \uf8f2 \uf8f3 ∂X ∂t (t, x) = b(X(t, x)), t ≥ 0 X(0, x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) Here, we are interested by the asymptotics of the flow X(t, x) as t → ∞ for a given x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In dimension two the nice result due to Peirone [19] (see also [21]) claims that if the vector field b does not vanish in Y2, then one has ∀ x ∈ R2, lim t→∞ X(t, x) t = ζ(x) ∈ R2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) where the limit vector ζ(x) may depend on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' On the contrary, when either b does vanish in Y2 (see [21, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1]), or when dimension d is greater than 2 (see [19, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10]), limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) does not hold necessarily for any x ∈ Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' More recently, using the two-dimensional Peirone’s result among others, the authors have obtained various asymptotic results for the flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) in any dimension with applications to the homogenization of linear transport equa- tions [3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Dimension two is very specific in ergodic theory, since Franks and Misiurewicz [9] have proved that for any continuous flow X(t, x) the Herman rotation set [11] – derived from [18, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='6] as the convex combination of the limit points of all the sequences � X(n, x)/n � n∈N for x ∈ Y2 – is actually a closed segment line of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In the case of a two-dimensional ODE’s flow, the closed segment Cb is carried by a line passing through 0R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' For the ODE’s flow X associated with the vector field b by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1), Herman’s rotation set may be equivalently defined by Cb := �ˆ Yd b(x) µ(dx) : µ ∈ Mp(Yd) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' for any t ≥ 0, µ ◦ X(t, ·) = µ � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' µ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) is a probability measure on Yd which is invariant for the flow X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In dimension three the situation is again completely different, since [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1] shows that the rotation set (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) may be any convex polyhedron of R3 with rational vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In this paper, we focus on a more precise asymptotics of the flow X (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' It is rather natural to study beyond the limits of type (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) when they do exist, the asymptotic behavior of the expansions X(t, x) − x − t ζ(x) as t → ∞ and for x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4) In the framework of ergodic theory, the problem of the dynamics of the iterates F n, n ∈ N, of the lift F (1) obtained from some homeomorphism f homotopic to the identity on the torus Yd (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=', [18]), is extremely delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Indeed, only dimension two is investigated, the estimates of the vector-valued expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4) for a general lift are only obtained in one direction, and moreover the last developments are quite recent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' More precisely (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=', the introduction of [13] and the references therein), the two following results hold: 1In the context of the ODE’s flow X defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1), we have F = X(1, ·), and due to the semi-group property of X we get that F n = X(n, ·) for any n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 2 By virtue of [14] and [16, Theorem 1] there exists a homeomorphism f on Y2 homotopic to the identity with a lift F on R2, such that the Herman rotation set Rf is reduced to the unit set {ρf} and ∀ v ∈ S1, sup x∈R2, n∈N �� F n(x) − x − n ρf � v � = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5) In [16, Theorem 1] ρf is actually chosen to be 0R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' By virtue of [8, Theorem A], for any homeomorphism f on Y2 homotopic to the identity with a lift F on R2 and the Herman rotation set Rf of which is a closed line segment of R2 with an irrational slope containing several points of Q2, there exist a unit vector v in (Rf)⊥ and a constant M > 0 such that ∀ ρ ∈ Rf, sup x∈R2, n∈Z �� � F n(x) − x − n ρ � v �� ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='6) In our setting, we have obtained an example of a two-dimensional flow X (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) associated with a vanishing vector field b parallel to a fixed incommensurable vector ξ, which satisfies the large deviation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5) except in the direction v := ξ⊥ (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1), but whose Herman rotation set Cb is a non degenerate closed line segment of R2 (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In contrast, due to the differential structure we can hope better results than the two-dimensional bounded deviation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='6) in some direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' More precisely, assuming the existence of the limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) for any point x in a subset A of Rd, we will prove in several situations a fine asymptotic expansion of the type sup x∈A, t≥0 �� X(t, x) − x − t ζ(x) �� ≤ MA < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) In Section 2 we prove a criterium (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) for which expression (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4) reads as ∀ t ≥ 0, ∀ x ∈ Rd, X(t, x) − x − t ζ(x) = Φ � X(t, x) � − Φ(x), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8) so that the boundedness of the vector-valued function Φ in Rd implies immediately the fine asymptotic expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) in the whole set Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8) can be regarded as a continuous sum of coboundary terms (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In return, from expression (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8) we deduce (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) a general class of vectors fields b such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) holds in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Finally, assuming that there exists a Zd-periodic regular gradient ∇u satisfying the positivity property b · ∇u > 0 in Yd, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 provides sufficient conditions for which asymptotic the expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) is satisfied in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Section 3 deals with the case of a non vanishing vector field b in R2 such that Herman’s rotation set (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) is a unit set of {ξ} of R2, where the rotation vector ξ = (ξ1, ξ2) is incommen- surable in R2 (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' This corresponds to the second case of the proof of [19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Assuming in addition the existence of an invariant probability measure for the flow with a pos- itive regular Lebesgue’s density, we prove (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) using the celebrated Kolmogorov theorem [15] that if the irrational number ξ1/ξ2 is a Diophantine number (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10)), then the fine asymptotic fine expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) is fulfilled in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In contrast, given a vector ξ in R2 such that ξ1/ξ2 is a Liouville’s number (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11)), we can construct a two-dimensional Stepanoff’s flow [22], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' a flow associated with the unidirectional vector field b = a ξ, such that the fine asymptotic expansion does not hold in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' At this point, note that the alternative between “commensurable and incommensurable” for the rotation vector is well-known in ergodic theory to guarantee the uniqueness of the asymp- totics (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) of the flow (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=', [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Moreover, the alternative between “Diophantine and 3 Liouville” is essential in the conjugacy Denjoy theorem related to the dynamical properties of the diffeomorphisms on the circle S1 with an irrational rotation number (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In the present context of the fine asymptotic expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) of a two- dimensional ODE’s flow, the same alternative on the irrational number ξ1/ξ2 can be regarded, up to our best knowledge, as a new example of the crucial role played by the Diophantine property of the rotation number in a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Finally, using the rather restrictive extension [17, Theorems 1,2] (see also [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3] which was obtained and used in an independent way) of Kolmogorov’s theorem to dimension d > 2) the previous two-dimensional result can be also extended to higher dimension (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In contrast with Section 3, Section 4 is devoted to the commensurable case in any dimension, which is based on the existence of periodic solutions in the torus Yd to the ODE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Again assuming that Kolmogorov’s theorem in dimension two and its extension [17, Theorems 1,2] in higher dimension hold true, we get (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) the fine asymptotic expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) in R2, with an explicit non constant vector-valued function ζ in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The results stated above are based on the condition that the vector field b does not vanish in Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' When b does vanish, the fine asymptotic expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) may fail in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Indeed, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 shows that the two-dimensional Stepanoff flow associated with the vector field b = a ξ, where a vanishes at one point in Y2 and ξ is any incommensurable vector in R2, does not satisfy the fine asymptotic expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) in the set A = R ξ+Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In contrast, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4 provides a two-dimensional Stepanoff’s flow which satisfies the fine asymptotic expansion in R2 for any vector ξ in R2, but the function a then has an infinite number of roots in Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Other examples illustrate the results of the paper in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' To conclude, we have not succeeded for the moment to derive a fine asymptotic expan- sion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) of the flow either without using the bounded coboundary sum of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8), or without the conditions supporting Kolmogorov’s theorem in dimension two and its extension in higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' For instance, when b is only a non vanishing regular two-dimensional vector field, namely the framework of [19], we do not know if the fine asymptotic expansion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) holds in the whole set R2, while however the asymptotics (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) is satisfied at each point of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Definitions and notations d ∈ N denotes the space dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' S1 denotes the unit sphere of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' A vector ξ in Rd is said to be incommensurable in Rd if ∀ k ∈ Zd \\ {0Rd}, ξ · k ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='9) Otherwise, the vector ξ is said to be commensurable in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' A Diophantine number is an irrational real number λ with the property that there exists m ∈ N satisfying # �� (p, q) ∈ Z × N : ���� λ − p q ���� ≤ 1 qm �� < ∞, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' λ is badly approximated by rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' On the contrary, a Liouville number is an irrational number λ with the property that for any n ∈ N, there exists a pair of integers (pn, qn) with qn > 1, such that 0 < ���� λ − pn qn ���� < 1 (qn)n , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11) 4 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' λ is closely approximated by a sequence of rational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' , ed) denotes the canonical basis of Rd, and 0Rd denotes the null vector of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Id denotes the unit matrix of Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' “ · ” denotes the scalar product and | · | the euclidean norm in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' × denotes the cross product in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' |A| denotes the Lebesgue measure of any measurable set in Rd or Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Yd denotes the d-dimensional torus Rd/Zd (which may be identified to the unit cube [0, 1)d in Rd), and 0Yd denotes the null vector of Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Π denotes the canonical surjection from Rd on Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Ck c (Rd), k ∈ N ∪ {∞}, denotes the space of the real-valued functions in Ck(Rd) with compact support in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Ck ♯ (Yd), k ∈ N ∪ {∞}, denotes the space of the real-valued functions f ∈ Ck(Rd) which are Zd-periodic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' ∀ k ∈ Zd, ∀ x ∈ Rd, f(x + k) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='12) The jacobian matrix of a C1-mapping F : Rd → Rd is denoted by the matrix-valued function ∇F with entries ∂Fi ∂xj for i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The abbreviation “a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.” for almost everywhere, will be used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The simple mention “a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.” refers to the Lebesgue measure on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' dx or dy denotes the Lebesgue measure on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' For a Borel measure µ on Yd, extended by Zd-periodicity to a Borel measure ˜µ on Rd, a ˜µ-measurable function f : Rd → R is said to be Zd-periodic ˜µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' in Rd, if ∀ k ∈ Zd, f(· + k) = f(·) ˜µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='13) For a Borel measure µ on Yd, Lp ♯(Yd, µ), p ≥ 1, denotes the space of the µ-measurable functions f : Yd → C such that ˆ Yd |f(x)|p µ(dx) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Lp ♯(Yd), p ≥ 1, simply denotes the space of the Lebesgue measurable functions f in Lp loc(Rd), which are Zd-periodic dx-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Mloc(Rd) denotes the space of the non negative Borel measures on Rd, which are finite on any compact set of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' M♯(Yd) denotes the space of the non negative Radon measures on Yd, and Mp(Yd) denotes the space of the probability measures on Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' D′(Rd) denotes the space of the distributions on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 5 For a Borel measure µ on Yd and for f ∈ L1 ♯(Yd, µ), we denote µ(f) := ˆ Yd f(x) µ(dx), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='14) which is simply denoted by f when µ is Lebesgue’s measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The same notation is used for a vector-valued function in L1 ♯(Yd, µ)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' If f is non negative, its harmonic mean f is defined by f := �ˆ Yd dy f(y) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' For a given measure λ ∈ M♯(Yd), the Fourier coefficients of λ are defined by ˆλ(n) := ˆ Yd e−2iπ n·x λ(dx) for n ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The same notation is used for a vector-valued measure in M♯(Yd)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' c denotes a positive constant which may vary from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 2 Fine asymptotic expansion Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 A flow X associated with a vector field b ∈ C1 ♯ (Yd)d by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) is said to admit a fine asymptotic expansion if there exists a Zd-periodic vector-valued function ζ such that ∀ t ≥ 0, ∀ x ∈ Rd, X(t, x) = x + t ζ(x) + O(1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) where O(1) denotes a vector-valued function which is bounded uniformly with respect to t and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' More precisely, the flow X is said to admit a fine asymptotic expansion in the subset A of Rd if there exists a constant CA > 0 only depending on A, such that ∀ t ≥ 0, ∀ x ∈ A, ��X(t, x) − x − t ζ(x) �� ≤ CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) The following result gives a way for a flow to admit a fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2 Let b, ζ be two vector fields in C1 ♯ (Yd)d, and let Φ be a vector-valued function in C1(Rd)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, the following assertions are equivalent : ∀ t ≥ 0, ∀ x ∈ Rd, X(t, x) = x + t ζ(x) + Φ � X(t, x) � − Φ(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) (Id − ∇Φ) b = ζ in Rd and ∀ t ≥ 0, ζ � X(t, ·) � = ζ in Yd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4) The last property in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4) means that ζ is invariant for the flow X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' If one of these two assertions is satisfied and Φ is bounded in Rd, then ζ is Zd-periodic, the Herman rotation set is given by Cb = � conv � ζ(Yd) � if d ≥ 3 ζ(Y2) if d = 2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5) and the flow X admits a fine asymptotic expansion in the sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 6 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 If the flow X satisfies the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3), then the function Φ is not necessarily periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' However, for any t ≥ 0, the function Φ � X(t, ·) � − Φ(·) is Zd-periodic, since the functions � x �→ X(t, x) − x � and ζ are Zd-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The function Φ � X(t, ·) � −Φ(·) can be regarded as a “continuous coboundary sum”, since we have Φ � X(n, ·) � − Φ(·) = n−1 � i=0 � Φ � X(i + 1, ·) � − Φ � X(i, ·) �� for n ∈ N, where each term of the sum is a coboundary term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In the sequel we will construct such continuous coboundary sums possibly uniformly bounded in various situations, so that the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) will follow immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Based on Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2 the following result allows us to construct a general family of flows which satisfy the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3 Let Ψ be a C2-diffeomorphism on Yd satisfying the conditions Φ : � x ∈ Rd �→ x − Ψ(x) � ∈ C2 ♯ (Yd)d and det (∇Ψ) ̸= 0 in Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='6) Let ζ be a vector field in C1 ♯ (Yd)d satisfying the equality ∇ζ (∇Ψ)−1 ζ = 0 in Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) Then, the flow X associated with the vector field b ∈ C1 ♯ (Yd)d defined by b := (∇Ψ)−1 ζ = (Id − ∇Φ)−1 ζ in Yd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8) fulfills both the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) and the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' First, assume that assertion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, by the boundedness of the vector field Φ and by the semi-group property of the flow X, we deduce from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) that for any t ≥ 0 and any x ∈ Rd, lim s→∞ X(s, x) s = ζ(x) = lim s→∞ X(s + t, x) s = lim s→∞ X(s, X(t, x)) s = ζ � X(t, x) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='9) which shows that the vector-valued function ζ is invariant for the flow X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Moreover, we have ∀ x ∈ Rd, ∀ k ∈ Rd, ζ(x + k) = lim t→∞ X(t, x + k) t = lim t→∞ X(t, x) + k t = ζ(x), which shows that ζ is Zd-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Now, let us determine the Herman rotation set Cb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' By [18, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='6] combined with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='9) we have Cb = conv � � x∈Rd � � n∈N �X(k, x) − x k : k ≥ n � �� = conv � ζ(Yd) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10) In dimension two the first equality of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5) can be refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Indeed, by virtue of [9, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2] for two-dimensional continuous flows, Herman’s rotation set Cb is a closed line segment of R2, and by the continuity of ζ the subset ζ(Y2) of R2 is a connected compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, it is enough to prove that the extremal points of Cb belong to ζ(Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' To this end, by [18, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5] (see [5, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1] for a proof) each extremal point of Cb is a vector ν(b) for some 7 ergodic invariant probability measure ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, by Birkhoff’s ergodic theorem there exists a point x ∈ Y2 such that ζ(x) = lim t→∞ X(t, x) t = ν(b) ∈ ζ(Y2), which thus implies the second equality of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Next, we have for any t ≥ 0 and any x ∈ Rd, ∂ ∂t � X(t, x) − x − t ζ(x) − Φ � X(t, x) � + Φ(x) � = � b − ∇Φ b �� X(t, x) � − ζ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11) Since the assertion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) holds and ζ is invariant for X, the equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11) is reduced to ∀ t ≥ 0, ∀ x ∈ Rd, � b − ∇Φ b �� X(t, x) � = ζ � X(t, x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, taking t = 0 in the previous equality we get the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Conversely, if the assertion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4) is satisfied, then the right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11) is zero, which implies that or any t ≥ 0 and any x ∈ Rd, X(t, x) − x − t ζ(x) − Φ � X(t, x) � + Φ(x) = X(0, x) − x − Φ � X(0, x) � + Φ(x) = 0, which yields assertion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Finally, note that the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) of the flow X combined with the boundedness of the vector field Φ provides immediately the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) of X, which concludes the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' □ Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Define the mapping X by X(t, x) := Ψ−1� t ζ(x) + Ψ(x) � for (t, x) ∈ [0, ∞) × Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='12) First of all, let us prove that the vector-valued function ζ is invariant for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Using the equalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='12) and Id = ∇(Ψ−1 ◦ Ψ) = � ∇(Ψ−1) ◦ Ψ � ∇Ψ in Rd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='13) we have for any (t, x) ∈ [0, ∞) × Rd, ∂ ∂t � ζ � X(t, x) �� = (∇ζ) � X(t, x) � ∂ ∂t � X(t, x) � = (∇ζ) � X(t, x) � ∇(Ψ−1) � t ζ(x) + Ψ(x) � ζ(x) = (∇ζ) � X(t, x) � (∇Ψ)−1� X(t, x) � ζ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' This combined with equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) yields that for a fixed x ∈ Rd and any t ≥ 0, f ′ x(t) = − � ∇ζ (∇Ψ)−1�� X(t, x) � fx(t) where fx(t) := ζ � X(t, x) � − ζ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='14) Hence, by the continuity of the Zd-periodic matrix-valued function ∇ζ (∇Ψ)−1 in Rd, for any T ∈ (0, ∞) there exists a constant cT ≥ 0 such that ∀ t ∈ [0, T], |fx(t)| ≤ cT ˆ t 0 |fx(s)| ds, which by Gr¨onwall’s inequality applied in [0, T] implies that fx = 0 in [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, the vector field ζ is invariant for the mapping X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 8 Now, consider the vector field b ∈ C1 ♯ (Yd)d defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Hence, due to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='13) and the invariance of ζ combined with equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8), we have for any (t, x) ∈ [0, ∞) × Rd, ∂ ∂t � X(t, x) � = ∇(Ψ−1) � t ζ(x) + Ψ(x) � ζ(x) = � ∇(Ψ−1) ◦ Ψ �� X(t, x) � ζ � X(t, x) � = (∇Ψ)−1� X(t, x) � ζ � X(t, x) � = b � X(t, x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, the mapping X defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='12) is actually the flow associated with the vector field b defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8) through the ODE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Finally, since Ψ(x) = x−Φ(x) for x ∈ Rd, the desired expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) of the flow X directly follows from the composition of equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='12) by Ψ, and the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) is an immediate consequence of the Zd-periodicity of the vector-valued Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' This concludes the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' □ Finally, the following result provides sufficient conditions to obtain two vector-valued func- tions ζ and Φ satisfying the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) of the flow X, and to also derive fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) in some sets of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 Let b ∈ C1 ♯ (Yd)d be a vector field in Rd, d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' i) Assume that the vector field b satisfies the positivity condition ∃ ∇u ∈ C0 ♯ (Yd)d, b · ∇u > 0 in Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='15) Also assume that there exists a vector-valued function ζ such that X satisfies the asymp- totics ∀ x ∈ Yd, lim t→∞ X(t, x) t = ζ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='16) Then, the vector field ζ is invariant for the flow X, and there exists Φ ∈ C1(Rd)d such that the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) of the flow X holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' ii) Replace in part i) condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='15) by the stronger gradient invertibility condition ∃ ∇u1 ∈ C0 ♯ (Yd)d, b · ∇u1 = 1 in Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='17) Then, the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) holds in any strip of Rd orthogonal to the direction ξ := ∇u1 of type � x ∈ Rd : x · ξ ∈ [a, b] � for − ∞ < a < b < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='18) iii) Replace in part ii) condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='17) by the existence of a vector field U = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' , ud) satisfying ∇U ∈ C0 ♯ (Yd)d×d with \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 b · ∇u1 = 1, b · ∇u2 = · · · = b · ∇ud = 0, det (∇U) ̸= 0, in Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='19) Then, the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) is satisfied through the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) obtained with the vector field Φ(x) := x − � ∇U �−1U(x) for x ∈ Rd and ζ := � ∇U �−1e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='20) 9 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2 In dimension two Peirone [19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1] proved remarkably that the asymp- totics (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='16) of the flow X is always satisfied when the vector field b does not vanish in Y2, while this asymptotics is generally false in higher dimension [19, Section 4] and in dimension two with a vanishing vector field b [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proof of part i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' First of all, due to the asymptotics (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='16) the invariance of the vector-valued function ζ for the flow X follows from the equalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Next, following [4, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='6] we can consider for each x ∈ Rd the unique times τ(x) for the orbit X(·, x) to meet the equipotential {u = 0}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' u � X(τ(x), x) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='21) Using the positivity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='15) and the C1-regularity of the flow X, the implicit function theorem implies that the function τ belongs to C1(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' By the uniqueness of τ combined with the semi-group property of X we also have ∀ t ≥ 0, τ � X(t, x) � = τ(x) − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='22) Now, consider the vector-valued function Φ (not necessarily bounded in Rd nor Zd-periodic) defined by Φ(x) = ˆ τ(x) 0 � ζ(x) − b � X(s, x) �� ds for x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='23) Then, we have for any t ≥ 0 and any x ∈ Rd, Φ � X(t, x) � = ˆ τ(x)−t 0 � ζ(x) − b � X(s + t, x) �� ds = ˆ τ(x) t � ζ(x) − b � X(s, x) �� ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Hence, taking the t-derivative of Φ � X(t, x) � at point t = 0, we get that ∀ x ∈ Rd, ∇Φ(x) b(x) = b(x) − ζ(x), which is exactly the first equality of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' This combined with the invariance of ζ for X yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, by virtue of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2 we deduce the equivalent expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) of the flow X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proof of part ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' From equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='17) we deduce that ∀ (t, x) ∈ [0, ∞) × Rd, u1 � X(t, x) � = t + u1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, the solution τ(x) to the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='21) with the function u1 is given by τ(x) = − u1(x), and the vector-valued function Φ defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='23) reads as for any x ∈ Rd, Φ(x) = ˆ − u1(x) 0 � ζ(x) − ∂X ∂s (s, x) � ds = − u1(x) ζ(x) − X(−u1(x), x) + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Since ∇u1 is in C0 ♯ (Yd)d, the function u1 can be written u1(x) = ξ · x − v1(x) where ξ = ∇u1 and v1 ∈ C1 ♯ (Yd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, we have for any point x in the affine hyperplane x · ξ = c, Φ(x) = � v1(x)−c � ζ(x)+x−X � v1(x)−c, x � = � v1(x)−c � ζ(x)− ˆ v1(x)−c 0 b � X(s, x) � ds, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='24) 10 and for any t ≥ 0, Φ � X(t, x) � = � v1 � X(t, x) � − c � ζ(x) − ˆ v1(X(t,x))−c 0 b � X(s + t, x) � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Hence, since the functions v1 and ζ are Zd-periodic and continuous in Yd, we get that for any t ≥ 0 and any x in the affine hyperplane x · ξ = c, ��Φ � X(t, x) � − Φ(x) �� ≤ 2 � |c| + ∥v1∥L∞ ♯ (Yd) �� ∥ζ∥L∞ ♯ (Yd)d + ∥b∥L∞ ♯ (Yd)d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, taking into account the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) of the flow given by the part i), we obtain the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) in any strip defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proof of part iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' This result has been obtained in [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3] for obtaining a class of ODE’s flows whose Herman’s rotation sets are reduced to a unit set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In the present context, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='19) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='20) we get immediately the equality (Id − ∇Φ) b = � ∇U �−1DU b = � ∇U �−1e1 = ζ in Yd, which by virtue of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2 implies the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' □ 3 The incommensurable case We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 I) Let b be a non vanishing vector field at least in C2 ♯ (Y2)2 admitting an invariant probability measure σ(x) dx where σ is a positive function at least in C5 ♯ (Y2), such that σb is incommensurable in R2 and the ratio σb1 σb2 is a Diophantine number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) Then, provided that b and σ are regular enough, the flow X defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) satisfies the fine asymptotic expansion ∀ t ≥ 0, ∀ x ∈ Rd, X(t, x) = x + t σb + O(1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) where O(1) is a vector-valued function which is bounded uniformly with respect to t and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' II) Let ξ be a unit vector of R2 such that ξ1/ξ2 is a Liouville’s number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, there exists a positive function a ∈ C∞ ♯ (Y2) such that the Stepanoff flow X associated with the vector field b = a ξ does not satisfies the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 In view of the two cases of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1, restricting ourselves to the class of smooth two-dimensional vector fields b and assuming for each b the existence of an invariant probability measure σ(x) dx for the flow with a smooth Lebesgue’s density σ > 0 and an incom- mensurable rotation vector ξ (= σb in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1)), we obtain that a necessary and sufficient condition to derive systematically the fine asymptotic assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) in R2 with ζ(x) = ξ, is that the ratio ξ1/ξ2 is a Diophantine number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 11 On the one hand, by virtue of the Kolmogorov theorem [15] (see also [23, Lecture 11]) the Diophantine property of some rotation number permits to prove that the two-dimensional ODE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) can be mapped to a linear ODE through a suitable diffeomorphism on Y2, provided that the vector field b is smooth and non vanishing in Y2 and that the associated flow X has an invariant probability measure with a smooth Lebesgue’s density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' On the other hand, the conjugacy Denjoy theorem (see [12, Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1]) claims that any smooth diffeomorphism on the circle S1 with an irrational rotation number ρ is topologically equivalent to the rotation of angle ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' It turns out that the Arnold theorem [1] (see [12, Sections 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5] and [7, Chapter 3, §5]) shows that the Diophantine property of the rotation number is essential to show that the conjugating map involved in Denjoy’s theorem is smooth (at last differentiable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The construction of the Peirone two-dimensional counterexample [20] (recall Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) is also based on some Diophantine rotation number for the ODE’s flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Alternatively, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 seems to be, up to our best knowledge, a new example of the essential role played by the Diophantine property of the rotation number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proof of part I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' First step: Reduction to a Stepanoff flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' By the Kolmogorov theorem [15] combined with enough regularity for the vector field b (at least C2) and the invariant probability measure σ(x) dx (at least C5) (2), there exists a diffeo- morphism Ψ on the torus Y2 (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=', [4, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1]) of class C2 (at least) satisfying ∀ x ∈ Rd, Ψ(x) = Mx + Ψ♯(x), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) where M ∈ SL± 2 (Z) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' M is a unimodular matrix) and Ψ♯ ∈ C2 ♯ (Y2)2, such that the flow � X obtained from the flow X through the diffeomorphism Ψ by � X(t, y) := Ψ � X(t, Ψ−1(y)) � for (t, y) ∈ R × Y2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4) is actually the flow associated with the vector field ˆb ∈ C1 ♯ (Y2)2 defined by ˆb(y) = � (∇Ψ b) ◦ Ψ−1� (y) = a(y) ξ for y ∈ Y2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5) where a is a non vanishing function in C1 ♯ (Y2) (at least) and ξ a non null vector of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Moreover, we easily check that ∀ y ∈ Y2, lim t→∞ � X(t, y) t = M � lim t→∞ X(t, Ψ−1(y)) t � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='6) if one of the two limits does exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' However, by virtue of Liouville’s theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=', [4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2]) the vector field σ b is divergence free in Y2, so that there exists u ∈ C2 ♯ (Y2) satisfying σ b = R⊥∇u or equivalently b = σ−1R⊥∇u in Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' By hypothesis the mean value of σ b is incommensurable, so is the mean value of ∇u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, by virtue of [4, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4] the Herman rotation set associated with the vector field b is the unit set Cb = � σb � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 2See the remark of [10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 8-9] in connection with the Denjoy counterexample (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=', [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 12 By [4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1] this combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='6) implies that ∀ y ∈ Yd, lim t→∞ � X(t, y) t = M � lim t→∞ X(t, Ψ−1(y)) t � = M σb which is also an incommensurable vector due to M ∈ SL± 2 (Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Hence, again applying [4, Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1] but with the Stepanoff flow � X, using the results [5, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4] on the asymptotics of Stepanoff’s flows, and recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5) we get that Cˆb = {a ξ} = � M σb � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) Hence, due to M ∈ SL± 2 (Z) it follows that ξ is an incommensurable vector of R2 as σb, and ξ1/ξ2 is a Diophantine number as the equivalent number σb1/σb2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, we are led to a Stepanoff’s flow satisfying the same assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) as the original flow X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Now, it remains to derive the asymptotic (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) for any Stepanoff’s flow satisfying condi- tion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) with σ = a/a and a regular enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' This is the aim of the following step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Second step: The Stepanoff flow in the incommensurable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Assume that ˆb = a ξ where a is a positive function in C1 ♯ (R2) and ξ is an incommensurable vector of R2 such that ξ1/ξ2 is a Diophantine number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' First of all, following [5, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4] recall some general results about the Stepanoff flow [22] in the incommensurable case, namely associated with the vector field ˆb = a ξ where a is a positive function in C1 ♯ (Yd) and ξ is an incommensurable unit vector of Rd for d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Let θ be the function defined by θ(y) := ˆ y·ξ 0 � a a � t ξ + (y · ξi) ξi� − 1 � dt (s = t − y · ξ) = ˆ 0 −y·ξ � a a � s ξ + y � − 1 � ds for y ∈ Rd, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8) where (ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' , ξd) is an orthonormal basis of (R ξ)⊥ so that for any y ∈ Rd, y = (y · ξ) ξ + (y · ξi) ξi with (y · ξi) ξi = (ξ2 · y) ξ2 + · · · + (ξd · y) ξd, according to Einstein’s convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The function θ is in C1(Rd) and satisfies for any y ∈ Rd, ∇θ(y) · ξ = � a a(y) − 1 � ξ · ξ + ˆ y·ξ 0 � (ξi ⊗ ξi) ∇ �a a �� t ξ + (y · ξi) ξi�� ξ dt = a a(y) − 1 + ˆ y·ξ 0 � ξi · ∇ �a a � � t ξ + (y · ξi) ξi�� (ξi · ξ) � �� � =0 dt = a a(y) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='9) On the other hand, the two-dimensional flow � X associated with the vector field ˆb explicitly reads as � X(t, y) = F −1 y (t) ξ + y where Fy(t) := ˆ t 0 ds a(s ξ + y), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10) and F −1 y denotes the reciprocal function of Fy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='9) we have a Fy(t) = t + ˆ t 0 ∂ ∂s � θ(s ξ + y) � ds = t + θ(t ξ + y) − θ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 13 Therefore, replacing t by F −1 y (t) in the previous equality and using the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10) of the flow, we get that ∀ y ∈ Rd, \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∀ t ≥ 0, �X(t, y) = a t ξ + y + θ(y) ξ − θ � � X(t, y) � ξ lim t→∞ � X(t, y) t = a ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11) Now, assume that d = 2 and that ξ1/ξ2 is a Diophantine number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Consider the function α ∈ C1 ♯ (Y2) and its Fourier expansion defined by α(y) := a a(y) − 1 = � n∈Z2\\{0R2} ˆα(n) e2iπ (y·n) for y ∈ Y2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='12) where ˆα(n) denote the Fourier coefficients of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, putting the Fourier expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='12) in the second integral of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8), we may permute the integral and the series due to ˆα ∈ ℓ1(Z2), which implies that for any x ∈ Y2, θ(y) = � n∈Z2\\{0R2} ˆα(n) 2iπ (ξ · n) � e2iπ (y·n) − e2iπ (y−(y·ξ) ξ)·n� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='13) Next, since ξ1/ξ2 is a Diophantine number, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10) there exists a non negative integer mξ such that # �� (p, q) ∈ Z × N : ���� ξ1 ξ2 − p q ���� ≤ 1 qmξ+1 �� < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='14) Also assume that a ∈ C mξ+2 ♯ (Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, by the Cauchy-Schwarz inequality we get that � n ∈ Z2\\{0R2} �−→ |n|mξ |ˆα(n)| = |ˆα(n)| |n|mξ+2 |n|2 � ∈ ℓ1(Z2\\{0R2}), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='15) since by the Parseval identity applied with the tensor-valued function ∇(mξ+2)α ∈ C0 ♯ (Y2)2(mξ+2) we have � n∈Z2\\{0R2} 1 |n|4 < ∞ and � n∈Z2\\{0R2} |n|2(mξ+2) |ˆα(n)|2 ≤ c ∥∇(mξ+2)α∥2 ℓ2(Z)2(mξ +2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Moreover, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='14) we have for any n = (n1, n2) ∈ Z2\\{0R2} with |n| ≥ N large enough, |ξ · n| = � |ξ2 n2| ≥ |ξ2| if n1 = 0 |ξ2| |n1| |ξ1/ξ2 + n2/n1| ≥ |ξ2|/|n1|mξ if n1 ̸= 0, which implies that ∃ c > 0, ∀ n ∈ Z2\\{0R2}, |ξ · n| ≥ c |n|mξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='16) This combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='15) thus yields ∀ n ∈ Z2\\{0R2} with |n| ≥ N, |ˆα(n)| |ξ · n| ≤ C |ˆα(n)| |n|mξ = |ˆα(n)| |n|mξ+2 |n|2 ∈ ℓ1(Z2\\{0R2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, we deduce that the asymptotic expansion of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11) satisfies the uniform estimate ∀ t ≥ 0, ∀ y ∈ R2, �� � X(t, y) − t a ξ − y �� ≤ c � n∈Z2\\{0R2} |ˆα(n)| |ξ · n| < ∞, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='17) 14 which establishes the asymptotic expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) for the Stepanoff flow in the Diophantine case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Let us conclude the proof of part I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Starting from formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4), multiplying formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) by the matrix M−1, and using the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='17) of � X combined with the equality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) and the boundedness of Ψ♯, we get that for any t ≥ 0 and any x ∈ Y2, X(t, x) = Ψ−1� � X(t, Ψ(x)) � = M−1� � X(t, Ψ(x)) � − M−1� Ψ♯ ◦ Ψ−1�� � X(t, Ψ(x)) � = M−1� t a ξ + Mx + Ψ♯(x) + O(1) � − O(1) = t σb + x + O(1), which finally yields the desired fine asymptotic expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proof of part II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Since ξ1/ξ2 is a Liouville’s number, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11) there exist two sequences of integers (pn)n∈N in ZN and (qn)n∈N in NN satisfying ∀ n ∈ N, ���� ξ1 ξ2 − pn qn ���� < 1 (qn)n , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='18) or equivalently, ∀ n ∈ N, |ξ · kn| < |ξ2| (qn)n−1 where kn := qn e1 − pn e2 ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='19) Up to extract a subsequence of the sequence (qn)n∈N (which converges to ∞) still denoted by (qn)n∈N, we can assume in addition that ∀ n ≥ 3, qn ≥ |ξ · kn−1| 1 3−n + n + n−1 � i=1 qi and ∞ � n=3 2π |ξ2| (qn)n−2 < 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='20) which implies in particular that (qn)n∈N is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, define the positive function a in C∞ ♯ (Y2) by its inverse 1 a(x) := 1 + ∞ � n=3 αn cos (2π kn · x) for x ∈ Y2, where αn := 2π qn ξ · kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='21) The function a is well defined and positive due to the second inequality of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='20) combined with inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Moreover, since by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='21) we have for any m ∈ N, ∞ � n=m+2 αn |kn|m ≤ ∞ � n=m+2 2π |ξ2| qn (|pn| + qn)m (qn)n−1 ≤ c ∞ � n=m+2 1 (qn)n−m−2 < ∞, the function a belongs to C∞ ♯ (Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' On the other hand, define the sequence (τn)n∈N by τn := 1 4 ξ · kn for n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='22) 15 Then, the function θ defined by the first integral of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8) with 1/a defined by the series expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='21), satisfies for any integer m ≥ 4 (note that a = 1) θ(τm ξ) = ˆ τm 0 � ∞ � n=3 αn cos � 2π (ξ · kn) t � � dt = ∞ � n=3 αn sin � 2π (ξ · kn) τm � 2π (ξ · kn) = qm + m−1 � n=3 αn sin � 2π (ξ · kn) τm � 2π (ξ · kn) + ∞ � n=m+1 αn sin � 2π (ξ · kn) τm � 2π (ξ · kn) , which by the first inequalities of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='20) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='19) implies that θ(τm ξ) ≥ qm − m−1 � n=3 |αn| 2π |ξ · kn| − ∞ � n=m+1 |τm| |αn| ≥ qm − m−1 � n=3 qn − π 2 ∞ � n=m+1 qn |ξ · kn| |ξ · km| ≥ m − π |ξ2| 2 ∞ � n=m+1 1 (qn)n−2 1 |ξ · km|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='23) Moreover, applying the first inequality of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='20) with n = m+1, we get that for any n ≥ m+1, qn ≥ qm+1 ≥ |ξ · km| 1 2−m so that 1 (qn)n−m ≥ 1 (qn)n−2 1 |ξ · km|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' This combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='23) and the increase of (qn)n∈N thus yields θ(τm ξ) ≥ m − π |ξ2| 2 ∞ � n=m+1 1 (qn)n−m = m − π |ξ2| 2 ∞ � n=1 1 (qn+m)n ≥ m − π |ξ2| 2 ∞ � n=1 1 (qn)n � �� � <∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Hence, we deduce that lim m→∞ θ(τm ξ) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='24) Finally, by the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10) of the Stepanoff flow for y = 0R2, we have for any m ∈ N, �X(tm, 0R2) = τm ξ where tm := F0R2(τm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, using the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11) of the flow � X for y = 0R2 and limit (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='24), we obtain that �� � X(tm, 0R2) − tm ξ �� = ��θ(0R2) − θ � τm ξ) �� −→ m→∞ ∞, which shows that the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) does not hold for the Stepanoff flow �X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The proof of part II) is done, which also concludes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' □ 16 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2 In higher dimension and in spirit of the case iii) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1, assume that there exists a vector-valued function U := (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' , ud) satisfying besides condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='19) the following one ∇U ∈ C1 ♯ (Yd)d×d with \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 b · ∇u1 > 0, b · ∇u2 = · · · = b · ∇ud = 0, det (∇U) ̸= 0, in Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='25) Then, following [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3] the matrix ∇U is invertible and the diffeomorphism on the torus Ψ := MU with M := � ∇U �−1 (3), satisfies ∇Ψ ∈ C1(Yd)d×d, ∇Ψ = Id, ∇Ψ b = (b · ∇u1) ξ in Yd, with ξ := Me1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='26) Hence, Ψ is a C2-diffeomorphism on the torus Yd (recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3)) which maps the flow X associated with b to the Stepanoff flow � X (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4) associated with the vector field ˆb := a ξ where a(y) := � (b · ∇u1) ◦ Ψ−1� (y) > 0 for y ∈ Yd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='27) When the vector ξ satisfies the extension of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='16) ∃ c > 0, ∃ mξ ∈ N, ∀ n ∈ Zd \\ {0Rd}, |ξ · n| ≥ c |n|mξ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='28) and a ∈ C mξ+p ♯ (Yd) for some integer p > d/2, we get similarly to the proof of the second part of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1, that the flow X satisfies the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' In the part iii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 below we will again use the previous diffeomorphism Ψ on Yd with d > 2, in the case where the vector ξ is commensurable in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 4 The commensurable case We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 Let b ∈ C1 ♯ (Yd)d be a vector field in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' i) Let A be a non-empty subset of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Assume that there exist TA, kA ∈ (0, ∞) such that the flow X satisfies the periodicity property ∀ x ∈ A, ∃ T(x) ∈ (0, TA], ∃ k(x) ∈ Zd with |k(x)| ≤ kA, ∀ t ≥ 0, X � t + T(x), x � = X(t, x) + k(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) Then, the flow X associated with b satisfies the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) in A with ζ(x) := k(x)/T(x) for x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 3Actually, the authors have recently discovered that the mapping Ψ used in [2] was previously introduced by Kozlov in [17, Theorems 1,2] to extend in some way the two-dimensional Kolmogorov theorem [15] to higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 17 ii) Assume that b is a non vanishing vector field in C2 ♯ (Y2)2 admitting an invariant probability measure σ(x) dx, where σ is a positive function in C5 ♯ (Y2) with mean value 1, such that σb is commensurable in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) Then, the flow X satisfies the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) with ζ � Ψ(x) � := � 1 T ˆ T 0 dt a � t ξ + Ψ(x) � �−1 ξ for x ∈ Y2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) where the C2-diffeomorphism Ψ on Y2 maps the flow X on the Stepanoff flow � X associated with the vector field ˆb through equalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' iii) Assume that for d > 2, the vector field b satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='25) with DU ∈ C1 ♯ (Yd)d×d, and that the vector ξ := � ∇U �−1 e1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='26) is commensurable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' there exists T > 0 such that T ξ ∈ Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, the flow X still satisfies the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) with the vector-valued function ζ defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) in Yd, where the C2-diffeomorphism Ψ = MU on Yd maps the flow X on the Stepanoff flow associated with the vector field ˆb through equalities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='25), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='26), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 By virtue of [9, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2] it is known that the rotation set Cb of the ODE’s flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) associated with a vector field b ∈ C1 ♯ (Y2) is always a closed line segment of R2 carried by a line passing through 0R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' This combined with [8, Theorem B] implies that if Cb contains a non null commensurable vector ζ, then the flow X satisfies a fine asymptotic expansion in the direction ζ⊥, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' there exists a constant C ≥ 0 such that ∀ t ≥ 0, ∀ x ∈ R2, �� � X(t, x) − x � ζ⊥ �� ≤ C, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4) where the first-order term t ζ(x) does not appear due to ζ(x) ∈ Cb ⊂ R ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4) extends the one obtained in the first case of the proof of [19, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1] where the constant does depend on x a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proof of part i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' First of all, for t ≥ 0 and x ∈ A, let nt,x be the integer satisfying nt,x T(x) ≤ t < (nt,x + 1) T(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5) Reiterating equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) we get that X(t, x) = X � t − nt,x T(x), x � + nt,x k(x) = x + t k(x) T(x) + � nt,x − t T(x) � k(x) + X � t − nt,x T(x), x � − x, and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5) we have ���� � nt,x − t T(x) � k(x) + X � t − nt,x T(x), x � − x ���� ≤ |k(x)| + ����� ˆ t−nt,x T(x) 0 b � X(s, x) � ds ����� ≤ kA + TA ∥b∥L∞(Yd)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 18 Therefore, we obtain the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) for the flow X in the subset A with ζ(x) := k(x)/T(x) for x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proof of part ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proceeding as the first step of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 thanks to Kolmogorov’s theorem we are led to Stepanoff flow associated with the vector field ˆb = a ξ, where a is a positive function in C1 ♯ (Y2) and ξ is a vector of R2 such that T ξ = k ∈ Z2 for some T ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Indeed, due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) with M ∈ SL± 2 (Z) and to condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2), the vector ξ := 1 a M σb is commensurable in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='6) Moreover, by the expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10) of the Stepanoff flow � X combined with the Zd-periodicity of a, we have for any t ≥ 0 and any y ∈ Rd, Fy(t + T) = Fy(t) + ˆ T 0 ds a(s ξ + y) = Fy(t) + T m(y) where m(y) := � 1 T ˆ T 0 ds a(s ξ + y) �−1 Hence, replacing t by F −1 y (t) in the previous equality we obtain that �X � t + T m(y), y � = F −1 y � t + T m(y), y � ξ + y = F −1 y (t) ξ + T ξ + y = � X(t, y) + k, which implies condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) with A := Rd, T(y) := T/m(y) bounded by TA := T ∥a−1∥L∞(Y2), and k(x) := k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) holds with the vector-valued function ζ defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' ζ(y) = m(y) ξ and � X(t, y) = y + t ζ(y) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Hence, since the vector-valued functions � y �→ Ψ−1(y) − y � and � x �→ Ψ(x) − x � are Z2- periodic and continuous thus bounded in R2, mapping the previous equality by Ψ−1 and using the relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4) between the two flows X and � X, we deduce that for any t ≥ 0 and any x := Ψ−1(y) ∈ R2, X(t, x) = Ψ−1� y + t ζ(y) + O(1) � = Ψ(x) + t ζ � Ψ(x) � + O(1) = x + t ζ � Ψ(x) � + O(1), which is the desired fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) satisfied by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proof of part iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The proof is quite similar to the one of case ii), which concludes the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' □ 5 Examples 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 Cases with a non vanishing vector field Let us start by a very simple example illustrating explicitly Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 Let ξ be an incommensurable vector of R2, and let b be the vector field b(x) := ξ 2 + cos(2πx1) for x ∈ Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 19 Then, an explicit computation of formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11) leads us to \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 X(t, x) = x + � 1 2 t + sin(2πx1) 4πξ1 − sin � 2π(x1 + F −1 x (t) ξ1) � 4πξ1 � ξ Fx(t) := 2 t + sin � 2π(x1 + t ξ1) � − sin(2πx1) 4πξ1 , for t ≥ 0, x ∈ Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, the flow X associated with the vector field b satisfies Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1, and consequently the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) with the vector-valued function ζ(x) ≡ 1 2 ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The following example revisits the two-dimensional flow of [6, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7] in the light of the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2 Consider the non vanishing two-dimensional vector field b defined by b(x) := e1 + 2π sin(2πx2) e2 = ∇u(x) where u(x) := x1 − cos(2πx2) for x ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) By [6, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='12] a tedious but easy computation shows that the flow X associated with the vector field (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) is given explicitly by X(t, x) = \uf8f1 \uf8f2 \uf8f3 (t + x1) e1 + � n + 1 π arctan � e4π2t tan(πx2) �� e2, |x2 − n| < 1 2 (t + x1) e1 + � n + 1 2 � e2, x2 = n + 1 2, for n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) Condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='15) is clearly satisfied with u(x) = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Moreover, we have ∀ x ∈ Y2, lim t→∞ X(t, x) t = e1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) so that by [4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1] Herman’s rotation set is the unit set Cb = {e1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' By the analysis of [6, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='12] it is surprising to observe that the flow X (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) has no invariant measure of type σ(x) dx where σ is a positive function in C0 ♯ (Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' However, the Radon measure dx1 ⊗δx2=0 on Y2 is invariant for the flow X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Indeed, we have ∀ ϕ ∈ C1 ♯ (Y2), ˆ Y2 b(x) · ∇ϕ(x) (dx1⊗δx2=0) = ˆ 1 0 ∂ϕ ∂x1 (x1, 0) dx1 = 0, which owing to Liouville’s theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=', [4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2]) yields the invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Finally, the expression (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) of the flow shows directly that for any t ≥ 0 and any x ∈ R2 such that x2 ∈ � n − 1 2, n − 1 2 � with n ∈ Z, ��X(t, x) − x − t e1 �� ≤ |n − x2| + 1 2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4) Therefore, the flow X satisfies the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) with ζ = e1 and a uniformly bounded term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' However, following Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2 it is interesting to recover the fine asymptotic expan- sion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) from a suitable bounded vector-valued function Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' To this end, the general defini- tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='23) with asymptotics (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) leads us to the vector field Φ defined for x ∈ R2, by Φ(x) := ˆ τ(x) 0 � e1 − b(X(t, x)) � dt where τ(x) is solution to u � X(τ(x), x) � = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5) 20 which similarly to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3) yields the expression of the flow ∀ t ≥ 0, ∀ x ∈ Rd, X(t, x) = x + t e1 + Φ � X(t, x) � − Φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='6) Then, due to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) we have 0 = u � X(τ(x), x) � = X1(τ(x), x) − cos � 2πX2(τ(x), x) � = τ(x) + x1 − cos � 2πX2(τ(x), x) � , which implies that Φ(x) = − 2π e2 ˆ −x1+cos(2πX2(τ(x),x)) 0 sin � 2πX2(t, x) � dt (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='7) Noting that by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) we have for any t ≥ 0 and any x ∈ R2 such that x2 ∈ � n − 1 2, n + 1 2 � with n ∈ Z, sin � 2πX2(t, x) � = sin � 2 arctan � e4π2t tan(πx2) �� = 2 e4π2t tan(πx2) 1 + e8π2t tan2(πx2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='8) Therefore, we deduce the inequality ∀ x ∈ R2, |Φ(x)| ≤ ˆ ∞ −∞ 4π e4π2t tan(πx2) 1 + e8π2t tan2(πx2) dt = 1 π � arctan � e4π2t tan(πx2) ��∞ −∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' which yields the uniform boundedness of Φ � X(t, x) � −Φ(x) with respect to t and x in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2 Cases with a vanishing vector field In the first example a vector field with separate variables is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='3 Let vector field b(x) = � b1(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' , bd(xd) � ∈ C1 ♯ (Yd)d having 0Yd as unique root in Yd, so that 0 is the unique common root of the functions b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' , bd in Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' First of all, it is clear that property (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='15) does not hold, since the vector field b does vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, the flow X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' , Xd) associated with b is given for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' , d and x ∈ Yd, by (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=', [5, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4]) \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Xi(t, x) = F −1 i,x (t) + xi for t ≥ 0 Fi,x(t) := ˆ t 0 ds bi(s + xi) for t ∈ � [xi] − xi, 1 + [xi] − xi � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='9) where F −1 i,x is the reciprocal function of Fi,x, and [xi] is the integer satisfying [xi] ≤ xi < [xi]+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Since the zero set of b is Zd, each function bi has a constant sign in the interval � [xi], 1 + [xi] � , and for any � [xi], 1 + [xi] � , ˆ [xi]−xi 0 ds bi(s + xi) = − ˆ 1+[xi]−xi 0 ds bi(s + xi) ∈ {−∞, ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Hence, the function F −1 i,x is a bijection from R on the interval � [xi] − xi, 1 + [xi] − xi � ⊂ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, the range of the flow X is contained in [−1, 1]d, so that X satisfies the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) with the vector-valued function ζ(x) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The following example deals with a two-dimensional Stepanoff flow associated with a vector field which has isolated roots in Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 21 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4 Let b ∈ C∞ ♯ (Y2)2 be the vector field defined by b(x) := cos2(πx1) (e1 + γ e2) for x ∈ Y2, with γ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The flow X associated with b is given by the explicit formula X(t, x) = � x + � 1 π arctan � π t + tan(π(x1 − n)) � + n − x1 � (e1 + γ e2) if |x1 − n| < 1 2 x if x1 = n + 1 2, n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, the flow X satisfies the inequality ∀ t ≥ 0, ∀ x ∈ R2, |X(t, x) − x| ≤ � 1 + γ2, which provides the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) with the vector-valued function ζ(x) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' The following general result shows that any two-dimensional Stepanoff flow associated with a vector field having one root in Y2 and an incommensurable direction ξ in R2, does not satisfy the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) in the set A := R ξ + Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 Let b = a ξ be a two-dimensional vector field such that a ∈ C1 ♯ (Y2) has 0Y2 as unique root in Y2, and ξ is any incommensurable unit vector of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, the flow X satisfies the asymptotics ∀ x ∈ R2, ζ(x) := lim |t|→∞ X(t, x) t = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 a ξ if x ∈ R2\\(R ξ+Z2) a ξ if x ∈ R ξ+Z2, τx < 0 0R2 if x ∈ R ξ+Z2, τx ≥ 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10) where τx is the unique real number satisfying x + τx ξ = kx ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11) Moreover, the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) is not fulfilled in the set A := R ξ+Z2, and the following large deviation holds ∀ v ∈ S1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' ξ · v ̸= 0, sup t∈R, x∈A � X(t, x) − x − t ζ(x) � v = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='12) Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 Taking into account the asymptotics of the flow (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10), by virtue of [18, Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='5, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='6] the Herman rotation set is given by the non degenerate closed line segment Cb = conv � ζ(R2) � = [0, a] ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Therefore, in the present case of a Stepanoff flow associated with a vanishing vector field and an incommensurable vector, we recover directly from the asymptotics of the flow the result of [5, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4] obtained by a perturbation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Contrary to the hypothesis of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1, the function a of the Stepanoff vector field b = a ξ, has non isolated roots in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' It turns out that the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1) holds in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4 for any vector ξ in R2, while it fails in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1 for any incommensurable vector ξ in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' 22 Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' First of all, make some considerations on the set R ξ+Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' By the incommensurability of ξ, for any x ∈ R ξ + Z2 there exists a unique τx ∈ R satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Let y be a point in R2 \\ (R ξ + Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Since ξ is incommensurable, the set R ξ + Z2 is dense into R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, there exists a sequence (xn)n∈N in (R ξ+Z2)N which converges to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' We have xn = − τxn ξ + kxn with τxn ∈ R and kxn ∈ Z2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='13) where lim n→∞ |kxn| = ∞ and consequently lim n→∞ |τxn| = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='14) Indeed, assume that the first limit of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='14) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, there exists a subsequence of the integer vectors sequence (kxn)n∈N which is stationary, so that by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='13) the corresponding subsequence of (τxn)n∈N converges, which implies that y ∈ R ξ +Z2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Up to consider − y with τ−y = − τy, and to extract a subsequence we can assume that τxn > 0 for any n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' We have just established the existence of a sequence (xn)n∈N in (R ξ+Z2)N satisfying ∀ n ∈ N, xn + τxn ξ ∈ Z2, lim n→∞ xn = y and lim n→∞ τxn = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='15) On the other hand, due the uniqueness of the representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='11) τx is the unique root of the function � t �→ a(t ξ + x) � in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Moreover, since the continuous function a does not vanish in the connected set R2 \\ Z2, it has a constant sign in R2 \\ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Without loss of generality we can assume that a is positive in R2 \\ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, defining for each x ∈ R2 the function Fx by Fx(t) := \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ˆ t 0 ds a(s ξ + x) for t ∈ R, if x ∈ R2\\(R ξ+Z2) ˆ t 0 ds a(s ξ + x) for t ∈ (−∞, τx), if x ∈ R ξ+Z2, τx > 0 ˆ t 0 ds a(s ξ + x) for t ∈ (τx, ∞), if x ∈ R ξ+Z2, τx < 0 0 for t ∈ R, if x ∈ Z2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' τx = 0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='16) the function Fx is increasing in the first cases of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='16) due to the positivity of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Then, the reciprocal application F −1 x is an increasing bijection from R onto (−∞, τx) if τx > 0, and from R onto (τx, ∞) if τx < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Hence, by formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10) the flow X associated with the vector field b = a ξ satisfies ∀ t ∈ R, X(t, x) = � F −1 x (t) ξ + x if x ∈ R2\\ Z2 x if x ∈ Z2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' τx = 0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='17) which combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='16) and τxn > 0, implies in particular that ∀ n ∈ N, lim t→∞ X(t, xn) = τxn ξ + xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='18) Therefore, the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='17) of the flow X together with the formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='16) of the function Fx (see also the positive case of [5, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='4]) yield the desired asymptotics (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='10), which in return implies that ∀ n ∈ N, lim t→−∞ �X(t, xn) t � = a ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='19) 23 Finally, applying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='18) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='19) with the sequence (xn)n∈N satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='15), we get that for any vector v ∈ S1 such that ξ · v ̸= 0, ∀ n ∈ N, ζ(xn) = 0R2 and \uf8f1 \uf8f2 \uf8f3 lim t→∞ � X(t, xn) − xn � v = τxn ξ · v − xn · v if ξ · v > 0 lim t→−∞ � X(t, xn) − xn � v = ∞ if ξ · v < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' Hence, it follows that the fine asymptotic expansion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='2) is not fulfilled in the set A := R ξ+Z2, and that the following large deviation in any direction v ∈ S1 such that ξ · v ̸= 0, holds \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 sup t∈R, x∈A � X(t, x) − x − t ζ(x) � v ≥ lim n→∞ � τxn ξ · v − xn · v � = ∞ if ξ · v > 0 sup t∈R, x∈A � X(t, x) − x − t ζ(x) � v ≥ lim t→−∞ � X(t, xn) − xn � v = ∞ if ξ · v < 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='20) which yields equality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' This concludes the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content=' □ References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} +page_content='I.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtA0T4oBgHgl3EQfDP9d/content/2301.02000v1.pdf'} diff --git a/PtAyT4oBgHgl3EQfUvdy/content/tmp_files/2301.00131v1.pdf.txt b/PtAyT4oBgHgl3EQfUvdy/content/tmp_files/2301.00131v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1dbc16276e4721cb2c0158778b5eb62162eaec5 --- /dev/null +++ b/PtAyT4oBgHgl3EQfUvdy/content/tmp_files/2301.00131v1.pdf.txt @@ -0,0 +1,2023 @@ +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. X, NO. X, X 2022 +1 +Guided Hybrid Quantization for Object Detection in +Multimodal Remote Sensing Imagery via +One-to-one Self-teaching +Jiaqing Zhang, Jie Lei, Member, IEEE, Weiying Xie, Member, IEEE, Yunsong Li, Member, IEEE, and Xiuping +Jia, Fellow, IEEE +Abstract—Recently, deep convolution neural networks (CNNs) +have promoted accuracy in the computer vision field. However, +the high computation and memory cost prevents its development +in edge devices with limited resources, such as intelligent satellites +and unmanned aerial vehicles. Considering the computation +complexity, we propose a Guided Hybrid Quantization with +One-to-one Self-Teaching (GHOST) framework. More concretely, +we first design a structure called guided quantization self- +distillation (GQSD), which is an innovative idea for realizing +lightweight through the synergy of quantization and distillation. +The training process of the quantization model is guided by its +full-precision model, which is time-saving and cost-saving without +preparing a huge pre-trained model in advance. Second, we +put forward a hybrid quantization (HQ) module to obtain the +optimal bit width automatically under a constrained condition +where a threshold for distribution distance between the center +and samples is applied in the weight value search space. Third, +in order to improve information transformation, we propose +a one-to-one self-teaching (OST) module to give the student +network a ability of self-judgment. A switch control machine +(SCM) builds a bridge between the student network and teacher +network in the same location to help the teacher to reduce +wrong guidance and impart vital knowledge to the student. +This distillation method allows a model to learn from itself and +gain substantial improvement without any additional supervision. +Extensive experiments on a multimodal dataset (VEDAI) and +single-modality datasets (DOTA, NWPU, and DIOR) show that +object detection based on GHOST outperforms the existing detec- +tors. The tiny parameters (<9.7 MB) and Bit-Operations (BOPs) +(<2158 G) compared with any remote sensing-based, lightweight +or distillation-based algorithms demonstrate the superiority in +the lightweight design domain. Our code and model will be +released at https://github.com/icey-zhang/GHOST. +Index Terms—Object detection, remote sensing image, Quan- +tization, Distillation. +I. INTRODUCTION +O +BJECT detection in aerial images plays an important role +in military security aiming to locate interested objects +(e.g., vehicles, airplanes) on the ground and identifying their +categories [1]. From universal detectors for natural images +This work was supported in part by the National Natural Science Foundation +of China under Grant 62071360. +J. Zhang, J. Lei, W. Xie, Y. Li are with the State Key Laboratory +of +Integrated +Services +Networks, +Xidian +University, +Xi’an +710071, +China +(e-mail: +jqzhang 2@stu.xidian.edu.cn; +jielei@mail.xidian.edu.cn; +wyxie@xidian.edu.cn; ysli@mail.xidian.edu.cn). +X. Jia is with the School of Engineering and Information Technology, +The University of New South Wales, Canberra, ACT 2600, Australia (e-mail: +xp.jia@ieee.org). +such as YOLOv3 [2], Faster R-CNN [3], FCOS [4] are +widely introduced in the field of remote sensing (RS); more +and more dedicated detectors for RS scene are designed and +improved with the requirements of objects tasks. However, +the large complexity of the object detection network is under- +investigated, which limits the practical deployment under +resource-limited scenarios and bring a heavy burden to process +massive multimodal images collected from satellites, drone, +and airplanes. Hence, a series of compression schemes have +been proposed to settle this problem, such as pruning [5], +quantization [6], [7], [8] and distillation [9], [10]. +Step1: Train a full- +precision model as the +pretrained model +Step2: Get a small N- +bit model via +completing n-bit +quantization under the +pretrained model +Step1:Train a large +full-precision model as +the pretrained teacher +model +Step2: Get a small +full-precision model +under the guidance of +pretrained teacher via +distillation +Step1:Train a small full- +precision model as the +pretrained teacher model +Step2: Get a small +mixed-bit model by +completing n-bit +quantization +under the guidance of +pretrained teacher via +distillation +Target +Conclusion +SuperYOLO +(Standard Training) +BOPs:17024G +Params:19.3MB +mAP50:80.93% +6-bit SuperYOLO +(via Quantization) +BOPs:727G +Params:3.6MB +mAP50:75.92% +GHOST +(via Hybrid Quantization +and Self Distillation) +BOPs:692G +Params:2.5MB +mAP50:80.31 +SuperYOLO +(via Distillation) +BOPs:17024G +Params:19.3MB +mAP50:79.78% +SuperYOLO +(Standard Training) +BOPs:17024G +Params:19.3MB +mAP50:80.93% +SuperYOLOl +(Standard Training) +BOPs:125031G +Params:125.2MB +mAP50:81.68% + Mixed-bit self distillation outperformances the results of +both the traditional distillation and quantization with less +computation cost + Train a small quantization model as possible +Traditional Model +Quantization +Traditional Model Distillation +Proposed Mixed-bit self +Distillation +Fig. 1. Comparison of training complexity, and accuracy between traditional +distillation, traditional quantization and proposed mixed-bit self distillation +(reported on VEDAI). +Quantization algorithms [11], [12] directly compress the +cumbersome network, effectively reducing the computation +cost and model size with a great compression potential. How- +ever, trivially applying quantization to CNNs usually leads to +inferior performance if the compression bit decreases to a low +level. Some knowledge distillation methods [13], [14], [15] are +arXiv:2301.00131v1 [cs.CV] 31 Dec 2022 + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. X, NO. X, X 2022 +2 +proven to be valid to elevate the performance of the lightweight +model but have to pre-train a huge teacher model as a +guidance of the student model which is time-consuming and +resource-consuming [16]. Self-distillation methods [17], [16], +[18] overcome this problem via the transfer of information +inside the model itself without introducing extra huge storage +and time consuming from the teacher model. +The above view naturally leads to a question: What re- +search results will we get if we combine quantization and +distillation by using a small full-precise network to guide +the learning process of a quantization for this full-precision +network? In this way, the tremendous compression capacity +of quantified networks and the performance of full precision +networks can be collaborative and cooperative. +In this paper, we design an adaptive one-to-one educational +policy pertaining the full-precision network and the quantiza- +tion network. We propose a simple yet novel approach that +allows quantization network to reinforce presentation learning +of itself relative full-precision network without the need of +additional labels and external supervision. Our approach is +named as Guided Hybrid Quantization with One-to-one Self- +Teaching (GHOST) based on the guided quantization self- +distillation (GQSD) framework. As the name implies, GHOST +allows a network to exploit useful and vital knowledge derived +from its own full-precision layers as the distillation targets for +its quantization layers. GHOST opens a new possibility of +training accurate tiny object detection networks. +As shown in Fig. 1, in order to train a small compact model +to achieve as high accuracy as possible with less computation +cost, we propose mixed-bit self distillation framework. Instead +of implementing two steps in traditional distillation, which +means that to train a large teacher model comes first, following +by distilling the knowledge from it to the student model, we +propose a two-step mixed-bit self distillation framework, in +which the training process of the second quantization step +is based on the pretrained small full-precision model. The +proposed framework not only requires less computation cost +(from 20797 G BOPs to 692 G BOPs on VEDAI dataset, a +30X faster training cost), but also can accomplish much higher +accuracy (from 75.84% in traditional quantization to 80.31% +on SuperYOLO) The main contributions of our work are as +follows: +• We propose a unified guided quantization thought based +on self-distillation called GQSD, which can tackle the +lightweight object detectors’ quantization optimization +problem in remote sensing. We are the first to formulate +an adaptive one-to-one education policy between the full- +precision network and the quantization network at the +same structure in object detection. +• For the finding of weight value distribution features of +remote sensing images, we design a hybrid quantization +module, whose adaptive selection of the core information +of the weights for quantization with a constrained preset +condition can keep the balance of accuracy and efficiency. +• Aiming to offset the loss of the quantization information, +the switch control machine is adopted to enable the +student to distinguish and close the teacher’s wrong +guidance and mine the correct and vital knowledge from +self-distillation. +The rest of this paper is organized as follows: Section II give +a rough overview of the spacific related work to this paper. +Section III presents our proposed method in detail. Section +IV introduces experimental results and analysis. Section V +concludes this paper and discusses the future work. +II. RELATED WORK +In this section, we reviewed related work from object +detection and network compression and acceleration in detail. +A. Object Detection with Deep Learning +Various CNN-based object detection architectures have +shown promising performance, bringing the field to a new +level. The architectures can be roughly divided into two main +domains: two-stage, and one-stage according to the change +process of proposals. +Two-stage Detectors: A typical method is selective search +work [19], where the first stage is to generate a large set of +proposed region candidates that are required to cover the whole +objects and then filter out most negative positions, and then +the second stage is to complete classification for each region. +R-CNN [20] creates a new era as one of the most successive +two-stage algorithms owing to the upgrading of the second- +stage classifier to a convolution network. Fast-RCNN [21] +extractes features over the images before proposing regions +and integrates the extractor and classifier by employing a soft +layer rather than SVM classifier. Faster R-CNN [3] introduces +a CNN-based region proposal network to further integrate +proposal generation with the second-stage classifier into a +single convolution network. +One-stage Detectors: One-stage detectors aim to jointly +predict the classification and location of objects by integrating +the detection and classification process. Recently, a series +of SSD [22], [23], [24] and YOLO [25], [26], [2], [27] +have renewed interest in the one-stage object detection. SSD +implements independent detection on multiscale feature maps, +while the YOLO utilizes combined detection. These methods +have paid more attention to speed, but their accuracy trails +behind that of tow-stage methods. YOLOv2 [26] modifies +the location regression pattern depending on bounding boxes. +YOLOv3 [2] considers multiscale objects and detects them in +the three scales, which can realize the detection of multiple +sizes of objects. YOLOv4 [27] introduces more data augmen- +tation tricks, activation functions, backbone structures, and +IoU loss metrics to enhance the robustness of the network. +YOLOv5 [28] releases four different size models, where the +basic structures are identical, which allows YOLOv5 to have +higher flexibility and versatility in practical applications. To +solve the dilemma of the category imbalance, RetinaNet [29] +reduces the weight of massive amounts of simple negative +samples in training by designing a focal term for cross-entropy +loss. As FCOS [4] which belongs to anchor-free methods is +proposed, adjusting hyperparameters and calculations related +to anchor boxes has been avoided. ATSS [30] selects positive +samples adaptively to enhance the detection performance. + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. X, NO. X, X 2022 +3 +Teacher +Student +8s +9s +…… +…… +…… +…… +Input Samples +Full-precision Network Features +Quantization Network Features +Distillation +Switch +1 +0 +0 +0 +Self Distillation with +Same Structure +Mixed Bit +Width +…… +…… +8t +9t +1t +2t +3t +4t +1t +2t +3t +4t +5t +6t +7t +8t +1s +2s +3s +9t +4s +5s +6s +7s +8s +9s +10 +s +10 +t +Attention Map +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +Switch Control Machine +1 +1 1 +1 1 +0 +0 0 +0 0 +1 +1 +1 +…… +…… +…… + : Cluster Number +…… +Hybrid Quantization +lt +ls +ls +lt +1 +ls − +1 +lt − +1s +2s +3s +4s +0 +min +2b +min 1 +2b ++ +1B +2B +3B +4B +8B +9B +1 +lB − +lB +l +W + : Distance +l +W +T : Threshold +min +max +82 +lB +One-to-one self-teaching +2n +2 +2n− +1 +2n− +2n +( ) +ld n +T + +( ) +ld n +( ) +ld n +Fig. 2. Overview of our proposed framework. An attention-based model determines similarities between the teacher and student features. Knowledge from +each teacher feature is transferred to the student with similarities identified by Switch control machine (SCM) by self-distillation with the same structure. The +mixed bit widths of the student network for quantization are based on the search results of the full-precision weights research space of teacher network in +the same layer. +B. Deep Network Compression and Acceleration +Although the speed of the one-stage detection network is +superior, its large model and high computation complexity +still deserves to explore. Some researches focus on the design +of a lightweight backbone. MobileNetV2 [31] utilizes the +depthwise separable convolutions to build a lightweight model. +ShuffleNet [32], and SqueezeNet [33] also effectively reduces +the memory footprint during inference and speed up the +detection. In the literature, a potential direction of model com- +pression is knowledge distillation (KD) which concentrates on +transferring knowledge from a heavy model (teacher) to a light +one (teacher) to improve the light model’s performance with- +out introducing extra costs [34], [35]. Whereas the knowledge +distillation enables utilizing the larger network in a condensed +manner, the pretraining of the large network requires extra +substantial computation resources to prepare the teacher net- +work [17]. The preparation of the pretrained teacher network +is time-consuming and cost-consuming. The self-knowledge +distillation [17], [16], [18] can overcome this problem by +distilling its own knowledge without prior preparation of the +teacher network. Quantization is another way to compact the +model directly and compress the ponderous network by using +low-bit representation. Mixed-precision quantization method +uses different numbers of bits for a given data type to represent +values in weight tensor. Many works [11], [36], [37], [37] +have shown that the mixed-precision method is efficient for +quantizing network layers that have different importance and +sensitiveness for the bit width. However, trivially applying + +biomgi2 (s) +(p) Qnwp6J-2!8wo!g +0 +5 +e +8 +0 +5 +e +8 +00.0 +0.0 +8 +S0.0 +8 +0'S +0'04 +e- +e +a0.0 +0'4 +4 - +80.0 +4 +a.0 +o1.0 +5- +O'JS +5 +8.0 - +0- +0'J4 +0 +0.1IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. X, NO. X, X 2022 +4 +quantization to CNNs usually leads to inferior performance +if the compression bit decreases to a low level. +III. NETWORK ARCHITECTURE +In this section, we first revisit conventional KD and de- +scribe the proposed GHOST framework in Sec. III-A. Then, +we present the details of the inspired hybrid quantization +algorithm (Sec. III-B) and this quantization training process +is guided by a one-to-one self-teaching method illustrated in +Sec. III-C. +A. Overview +KD is a widely-applied method that can be expressed as +a knowledge transformer from teacher to student. Given a +teacher model T and a student model S, the x is the data +examples of models, here they can be the same for the teacher +and the student model. In general, the KD machine can be +uniformly expressed as: +min LKD = min +� +xi∈x +L (T (xi) , S (xi)) , +(1) +where L is the loss function that penalizes the differences +between the teacher and the student. +The student model size is commonly designed in a small +size to achieve the purpose of model compression in which the +performance of the student can chase the teacher but consumes +a computing-friendly resource. Nonetheless, the computation +cost of the student model is larger than the pruning method +directly completed on the teacher model. This demonstrates +that the existing KD-based quantization algorithms still have +great potential room for improvement. +We aim at developing a novel and generic baseline network +with a focus on the learnable knowledge characteristics, mak- +ing it well-applicable to the highly accurate and fine object +detection of RS images with less computational costs. The key +to model quantization with knowledge learning is to reduce the +discrepancy which can be punished by distance or angle loss +function between full-precision model P (teacher) and low- +precision model Q (student) through optimizing Q, which can +be expressed as: +Q∗ = min +Q +� +xi∈x +L(P(xi), Q(xi)). +(2) +The weights of the teacher are frozen without gradient propa- +gation when the teacher network guides the training of the stu- +dent network. Based on the above presentation, we design an +effective teacher-student distillation framework called GQSD +which can be represented as: +min LKD = min +� +xi∈x +L(P(xi), Q(xi), R(xi)), +s.t. +WQ = Winit, BQ = B. +(3) +Specifically, the full precision network is the import funda- +mental teacher which not only provides the initial weights +Winit and bit width B of the quantization model but also +guides the quantization process to mine the vital knowledge +from the teacher in specific features selected by a control R. +As shown in Fig 2, we propose a GHOST framework that +concludes a hybrid quantization (HQ) module and a one-to- +one self-teaching (OST) module. The mixed bit widths of +the student network for quantization are based on the search +results of the full-precision weights research space of the +teacher network in the same layer. Inspired by the idea of +harnessing intermediate features to improve performance in +knowledge distillation [38], [39], [40], we design a Switch +Control Machine (SCM) as R to generate an attention map +that gains intermediate feature similarities between the teacher +and student. The SCM controls the distillation switch and +determines which knowledge should be delivered dynamically. +Knowledge from each teacher feature is transferred to the +student with similarities identified by SCM by self-distillation +with the same structure. With a pretrained full-precision model +as a initial weight, the quantization and distillation processes +are conducted simultaneously to ultimately obtain a small +lightweight model with little loss of accuracy. The details of +the modules will be described separately as follows. +B. Hybrid Quantization +Powerful deep networks normally benefit from large model +capacities but induce high computational and storage costs. +Modal quantization is a promising approach to compress deep +neural networks, making it possible to be deployed on edge +devices. The quantization operator divides the weight into +different fixed values by a quantization function which can +be regarded as a cluster of convolution kernels in substance. +The different scale weights are clustered to a certain value. +To illustrate this intuition explicitly, a SuperYOLO [41] +network model which consists of 60 convolutional layers +is trained based on the VEDAI dataset. After training, test +samples are fed into the model. The convolution weight is +firstly clustered into different categories by k-means and then +transformed into 2 dimensions by t-SNE [42] to realize the +visualization. As shown in Fig. 3, the convolution kernel +weight in the (a) 0th, (b) 26th and (c) 52nd convolutional layer +are clustered in different numbers. In Fig. 3 (a), the distance +between different categories is relatively far which indicates +that the weight distribution is dispersed and complicated in +the initial layer. This is due to the fact that the color and +texture features, which are detailed and multifarious, are +captured in the shallow layer. As the layer propagates forward +(Fig. 3 (b) and Fig. 3 (c)), the convolution weight becomes +converging gradually. In other words, the semantic features in +the deep layer are more robust and condensed so that with the +deepening of the network layers, the clustering categories of +weights can be relatively reduced. +Based on this finding, the hybrid quantization idea is +introduced to search for the optimal bit width definition in +the weight value space. We initially design a hyperparameter +T as a threshold to constrain the research space to control +the compression rate of the quantization model. The search +strategy can be described as: +B =argmax(d(n)) +s.t. +d(n) < T. +(4) + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. X, NO. X, X 2022 +5 +(a) +(b) +(c) +(a) +(b) +(c) +Fig. 3. In the bottom layer of a trained network, the feature distributions of different categories intervene with each other severely as (a) shows. Many delicate +neurons are needed to distinguish the overlapped distributions. And as the network propagates forward, the feature distribution of the same category gathers +gradually in (b) and (c). At the end of the hidden layers, there exist clear margins between the semantic feature distributions of different classes in (d). With +the improvement of separability among the feature manifolds, a neuron with lower-precision parameters is able to extract robust features. +where the function d(x) denotes the measurement of clus- +tering extent at the n bit width for each convolution layer. +This definition aims to find the limited minimum clustering +categories (maximum clustering extent) for each layer, hence +the smallest quantization model with a minimum bit width is +obtained at the preset ratio constraint. The hybrid quantization +of the whole network definitely can be collected as: +B = [B1, B2, ..., Bl] +(5) +where the l is the total convolution layer and the Bl is the lth +bit width of each layer weight parameter, and the bit width +decreases progressively as the network propagates forward. We +utilize the distribution distance defined as follows to determine +the final bit width for quantization of the lth convolution layer +weight: +dl(n) = 1 +M +M +� +j=0 +2n +� +i=0 +(wl +ij − cl +i) +2, +(6) +where the M is the total number of kernel weights which +correspond to M = Cin × Cout × K × K. The Cin, Cout, +and K are the input channels, output channels and kernel size +of the convolution layer. The whole weight values of each +convolution layer complete the kmeans++ algorithm on the +different cluster numbers. While the 2n represents the cluster +number. cl +i and wl +ij are the cluster centers and samples, as +shown respectively in Fig. 4. We set the initial bit width as 8 +and then select the superior and adaptive bit width by +Bl = min(n|dl(n) < T) +n = bmin, bmin + 1, ..., 8, +(7) +where the bmin is a limit of the minimum bit width in +the quantization process. When the dl(n) is smaller than a +manual threshold T set in advance, the bit width of the current +convolution layer is updated as Bl. The activation following +this convolution layer keeps the same bit-width. +Take the distance threshold T = 50 as an example, the +Fig. 5 demonstrates the judgment results of bit width for +each convolution layer. It can be indicated that the values of +bit width progressively decrease with the deepening of the +network layer. In addition, the bit width of the convolution +layer before the detection process may be relevantly large to +maintain more location discrimination information. +We use a simple-yet-effective quantization method which +refers to [8] for both weights and activations. The uniform +quantization function q(�) is defined as: +q (v, k) = +1 +2k − 1round((2k − 1)v), +(8) +where v is a real number indicating the full-precision (float32) +value, v ∈ [0, 1]. the output q(v, k) of quantization function +is a k bits real number, q(v, k) ∈ [0, 1]. The quantization +calculations of lth convolution layer weight and activation are + +-O'JO +-0'02 +00.0 +0'02 +O'TO +-O'JO +-0'02 +00.0 +0'02 +O'JO01.0- +-0'02 +00.0 +0'02 +O'TO +-O'JO +-0'02 +00.0 +0'02 +O'JO-012 +-0'20-0'52 +00.0 +0'52 +0'20 +0'12 +J'O0 +J'S2 +J'52 +-J'00 +-0'12 +-0'20 +-0'52 +00.0 +0'52 +02.0-012 +-0'20-0'52 +00.0 +0'52 +0'20 +0'12 +1'00 +J'S2 +J'52 +-J'00 +-0'12 +-0'20 +-0'52 +00.0 +0'52 +02.0-012 +-0'20-0'52 +00.0 +0'52 +0'20 +0'12 +J'O0 +J'S2 +-J'S2 +00.1- +-0'12 +-0'20 +-0'52 +00.0 +0'52 +0'200'50 +-0'J2-0'J0-0'02 +00.0 +0'02 +O'TO +0'J2 +0'50 +-0'5 +-O'T +0.0 +O'J +o'S-0'50 +-012 +-010 +-0'02 +00.0 +0'02 +01.0 +0'J2 +0'50 +-0'5 +-0'T +0.0 +O'T +o'S-0'50-0'J2-0'J0-0'02 +00.0 +0'02 +O'TO +1.0 +0'50 +-0'5 +-O'J +0.0 +O'J +o'S-O'TO +-0'02 +00.0 +0'02 +O'TO +-O'JO +-0'02 +00.0 +0'02 +O'JOIEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. X, NO. X, X 2022 +6 +Center +Distance +Distribution Distance +iC +ij +w +Fig. 4. The distribution distance of kmeans++ cluster method. +10 +20 +30 +40 +50 +Convolution Layer Index +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +6 +Bit Width +10 +20 +30 +40 +50 +Convolution Layer Index +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +6 +Bit Width +Fig. 5. The bit width results of each convolution layer at the threshold T = +50. The values of bit width progressively decrease with the deepening of +the network layer. In addition, the bit width of the convolution layer before +the detection module may be relevant large to maintaining more location +discrimination information. +defined respectively as follows: +wl +o = 2q( +tanh(wl +i) +2 max( +��tanh(wl +i) +��) + 1 +2, Bl) − 1, +(9) +al +o = q(al +i, Bl). +(10) +The activation al +i is the range in [0, 1] determined by a bounded +activation function while the weight wl +i is not restricted in a +limit boundary. Here, the quantization result of weight wl +o is +the range in [−1, 1], and the quantization result of activation +al +o is the range in [0, 1]. The Algorithm 1 clarifies the process +of the hybrid quantization method. As described in [8], the +first and last layers in the network are sensitive to performance +during the process of quantization. Based on this intuition, the +last detection layer keeps intact to avoid potential degradation +of detection performance. +Algorithm 1 The Hybrid Quantization Method +Input: The weights of lth certain convolution layer W ∈ +RH×W ×K×K, The manual distance threshold T and the +minimum bit width bmin +Output: The bit width of the current convolution layer and +activation Bl +1: Initialize the Bl as 8 +2: for n in range (bmin,8) do +3: +Cluster weights into 2n clusters via the kmeans++ +algorithm and then get the centers ci and samples wij +of the ith cluster. +4: +Calculate the distribution distance according to Eq. 6. +5: +Update the bit width Bl by Eq. 7 +6: Complete the quantization for the convolution layer weight +and activation by Eq. 9 and Eq. 10, respectively. +C. One-to-one Self-teaching +Previous mixed quantization approaches pay more attention +to the bit-width selection [7] which costs a lot of resources to +obtain the optimal decision. Our hybrid quantization method +can make a quick decision with less computation cost. The loss +of performance is fixed by the guide of distillation. In general, +previous distillation algorithms are a full precision network, +so the network weights are in the same order of magnitude, +and the feature maps generated by the teacher network or +the student network are similar. However, for the quantitative +network, the feature map generated by the quantitative network +as a student network will have obvious weight information +loss due to the increase of the zero content, resulting in some +differences between the feature map of the teacher network and +the feature map of the student network, which makes it difficult +for the teacher network to directly restrict the quantitative +student network from the feature layer. Therefore we proposed +an OST to conquer this question. SCM first calculates the +inner connected relationship between the full-precision and +quantization network. The distillation switch (DS) chooses the +core information between matched student and teacher features +by this relationship matrix. We sketch the architecture of self- +feature distillation in Fig. 2. +Let s = s1, s2, ..., sl represent a set of multiscale feature +maps for the student network and t = t1, t2, ..., tl for the +teacher. To calculate the attention map similar to [39] between +the student feature and teacher feature, we define that each +teacher feature generates a query qi, and each student feature +produces a key kj: +qi = Wi · GAP(si), +(11) +kj = Wj · GAP(tj). +(12) +GAP(·) represents a global average pooling. Wt and Ws are +the liner transition parameters for the ith query and the jth +key. Then the attention map that reveals the inner relationship +between teacher and student features is defined as: +a = (q · kT)/ +√ +d. +(13) + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. X, NO. X, X 2022 +7 +Here, we introduce the Gumble-Softmax trick [43] to convert +the values greater than the threshold to 1 and the rest to +0. Formally, the decision at ith entry of a is derived in the +following way: +Ai,k = +exp(log(ai,k + Gi,k)/τ) +K +� +j=1 +exp(log(ai,j + Gi,j)/τ) +k = 1, 2, . . . , K, (14) +where K is set as 2 for binary decision [44] in our case. +Gi is the Gumbel distribution. Temperature τ is used to +control the smoothness of Ai. With a better attention map +to mining the internal correlation of features, we generate a +DS mask that can automatically determine whether to transfer +the information from the teacher to the student at the same +site in the network. +α = Diag(A). +(15) +where SCM digs out the diagonal elements from the attention +map matrix A. We devise the self-feature distillation loss as +follows: +LF = +m +� +i=0 +αi∥CAP(ti) − CAP(si)∥2, +(16) +where CAP represents a channel-wise average pooling. m is +the total features utilized for distillation. +Ltotal = βLF + Ldec, +(17) +Finally, the distillation loss terms are combined with detection +loss and minimized in an end-to-end manner as Eq. 17. +Ldec includes the objectness, location, and classification. The +hyper-parameter β indicates the impact balance between the +detection and distillation. +IV. EXPERIMENTAL RESULTS +In this section, we evaluate the proposed method on the +multimodal dataset for remote sensing object detection and +four widely adopted CNNs with different scales. We first +demonstrate the experience set up, including the introductions +of datasets and networks, implementation details, and evalua- +tion metrics. Then, we report the performance of our method +on a dataset in detail, mean average precision and compression +ratio are calculated to measure the comprehensive performance +in the accuracy and computation cost. +A. Dataset Description +The publicly available dataset VEDAI [45] designed for +multimodal remote sensing image object detection is adopted +in our experiments. In addition to validation on the multimodal +object detection dataset, three single modal datasets (DOTA +[46], NWPU [47] and DIOR [48]) are utilized in experiments +to verify the generation of our proposed algorithm. +1) VEDAI: The VEDAI dataset consists of 1246 smaller +images cropped from the much larger Utah Automated Geo- +graphic Reference Center (AGRC) dataset. Each image col- +lected from the same altitude in AGRC has approximately +16, 000 × 16, 000 pixels, with a resolution of about 12.5cm × +TABLE I. Training Strategy +Dataset +Image Size +Batch Size +Lr +Epoch +VEDAI +512 +2 +0.01 +300 +DOTA +512 +16 +0.01 +100 +NWPU +512 +8 +0.01 +150 +DIOR +512 +16 +0.01 +150 +12.5cm per pixel. The main scenes of VEDAI include grass, +highway, mountains, and urban areas. The size of the image +is fixed to 512 × 512. +2) DOTA: The DOTA dataset was proposed by Xia et al. in +2018 for object detection of remote sensing. It contains 2806 +large images and 188 282 instances, which are divided into +15 categories. The size of each original image is 4000×4000, +and the images are cropped into 1024 × 1024 pixels with an +overlap of 200 pixels in the experiment. We select half of the +original images as the training set, 1/6 as the validation set, +and 1/3 as the testing set. The size of the image is fixed to +512 × 512. +3) NWPU VHR-10: The dataset of NWPU VHR-10 was +proposed by Cheng et al. in 2016. It contains 800 images, of +which 650 pictures contain objects, so we use 520 images as +the training set and 130 images as the testing set. The dataset +contains 10 categories, and the size of the image is fixed to +512 × 512. +4) DIOR: The DIOR dataset was proposed by Li et al. in +2020 for the task of object detection, which involves 23 463 +images and 192 472 instances. The size of each image is 800× +800. We choose 11 725 images as the training set and 11 738 +images as the testing set. +B. Implementation Details +1) Networks: To demonstrate superior performance, Su- +perYOLO [41] is tested as a teacher model with our method. +For the multimodal VEDAI dataset, the number of convolution +layers is 47 including one detection layer on the small scale. +For a single modal dataset (DOTA, NWPU, and DIOR), the +number of convolution layers is 61 including three detection +layers on the small, medium, and large scale. To verify the +superiority of the GHOST proposed in this paper, we selected +12 generic methods for comparison: +one-stage +algorithms +(YOLOv3 +[2], +YOLOv4 +[27], +YOLOv5 [28], SuperYOLO, FCOS [4], ATSS [30], Retain- +Net [29], GFL [49]); +two-stage method (Faster R-CNN [3]); +lightweight models (MobileNetV2 [31] and ShuffleNet +[32]); +distillation-based methods (ARSD [15]); +remote sensing designed approaches (FMSSD [50] and +O2DNet [51]). +2) Training Strategy: Our proposed framework is imple- +mented in PyTorch and runs on a workstation with an NVIDIA +A100-SXM4-80GB GPU. We also use different training strate- +gies for different datasets, and the detail is illustrated in + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. X, NO. X, X 2022 +8 +(a) +(b) +Fig. 6. Comparisons of the teacher model and lightweight model by parameters and BOPs on the four datasets (VEDAI, DOTA, NWPU, and DIOR). The +BOPs and parameters of the lightweight model are smaller, and the inference speed is faster. (a) Params (MB). (b) BOPs +(a) +(b) +(c) +Fig. 7. Comparison of the efficiency between the current SOTA methods and our method on the three datasets. The bigger size of cycles represents costing +more parameters. (a) DOTA. (b) NWPU, and (c) DIOR. +TABLE II. The comparison result of the tranditonal quantization method and our mixed-bit quantization and we use the same abbreviation in the following +sections. +Bit Width +T +Max Min +Car +Pickup Camping Truck Other Tractor +Boat +Van +mAP50 mAP Params(MB) BOPs(G) +32W32A +- +32 +32 +89.20 +87.10 +79.50 +86.80 58.20 +88.00 +70.30 88.30 +80.93 +50.80 +19.30 +17023.76 +8W8A +- +8 +8 +88.25 +85.86 +75.78 +69.94 48.42 +80.29 +66.10 92.32 +75.87 +47.08 +4.83 +1201.10 +HQ +0.1 +7 +8 +86.58 +83.06 +69.90 +76.45 69.77 +76.75 +69.77 99.51 +79.57 +48.55 +4.34 +1123.12 +6W6A +- +6 +6 +86.81 +87.16 +69.77 +75.72 61.58 +81.99 +60.46 83.87 +75.92 +45.63 +3.63 +727.04 +HQ +4 +3 +8 +90.94 +86.57 +74.45 +73.35 55.92 +79.70 +68.63 97.84 +78.42 +46.32 +2.49 +691.88 +4W4A +- +4 +4 +88.74 +82.65 +71.71 +58.14 61.32 +86.58 +59.03 84.99 +74.14 +44.85 +2.43 +382.91 +HQ +70 +3 +8 +84.72 +83.41 +75.13 +65.39 61.41 +87.77 +63.52 84.73 +75.76 +46.00 +1.87 +371.23 + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. X, NO. X, X 2022 +9 +(a) +(b) +(b) +(c) +Fig. 8. Three sets of visualization results. (a) DOTA. (b) NWPU, and (c) DIOR. +TABLE III. The validation result of the self-destillation method in the different model size. +HQ +T +Max +Min +OST +Car +Pickup +Camping +Truck +Other +Tractor +Boat +Van +mAP50 +mAP +✓ +0.1 +2 +8 +89.00 +87.28 +77.30 +69.80 +59.79 +84.12 +64.91 +91.77 +78.00 +46.59 +✓ +0.1 +2 +8 +✓ +91.14 +87.72 +74.85 +82.22 +64.57 +84.99 +60.21 +82.98 +78.59 +47.14 +✓ +4 +2 +8 +90.94 +86.57 +74.45 +73.35 +55.92 +79.7 +68.63 +97.84 +78.42 +46.32 +✓ +4 +2 +8 +✓ +88.58 +86.16 +71.84 +75.18 +68.57 +88.14 +70.44 +86.80 +79.46 +48.57 +✓ +52 +2 +8 +89.46 +80.71 +68.41 +72.16 +66.65 +88.02 +53.56 +78.18 +74.64 +44.29 +✓ +52 +2 +8 +✓ +88.47 +83.47 +71.4 +72.45 +60.2 +83.66 +66.14 +89.82 +76.99 +46.91 +TABLE IV. The comparison with sota distillation method for detectors on the VEDAI . +Distillation +HQ +Car +Pickup +Camping +Truck +Other +Tractor +Boat +Van +mAP50 +mAP +- +✓ +89.00 +87.28 +77.30 +69.80 +59.79 +84.12 +64.91 +91.77 +78.00 +46.59 +ZAQ [38] +✓ +88.04 +85.86 +70.51 +79.45 +45.16 +88.14 +67.76 +84.03 +76.12 +46.48 +AFD [39] +✓ +88.52 +85.56 +71.35 +73.35 +58.71 +89.14 +59.76 +80.49 +75.86 +45.44 +ReviewKD [40] +✓ +85.11 +84.52 +72.89 +73.69 +58.46 +84.36 +68.88 +94.06 +77.75 +47.35 +OST +✓ +91.14 +87.72 +74.85 +82.22 +64.57 +84.99 +60.21 +82.98 +78.59 +47.14 + +C中 +口IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. X, NO. X, X 2022 +10 +TABLE V. mAP comparisons of different β value. +β +0 +100 +200 +300 +400 +500 +T +0.1 +46.59 +47.17 +48.26 +46.72 +49.17 +46.29 +4 +46.32 +48.57 +47.05 +47.96 +47.29 +49.05 +52 +44.29 +46.91 +44.20 +45.14 +46.48 +47.03 +TABLE I. In addition, data is augmented with Hue Saturation +Value (HSV), multi-scale, translation, left-right flip, and mo- +saic. The augmentation strategy is canceled in the test stage. +The standard Stochastic Gradient Descent (SGD) is used to +train the network with a momentum of 0.937, weight decay +of 0.0005 for the Nesterov accelerated gradients utilized, and +a batch size of 2. The learning rate is set to 0.01 initially. +All the baseline training process is completed from scratch +without any pre-trained model while the GHOST is carried on +the baseline model. In the test stage, the IoU threshold of non- +maximum suppression is 0.6 on NWPU VHR-10 and VEDAI, +and it is 0.4 on DOTA and DIOR. +3) Evaluation Metric: For the detection result, the IoU is +defined as the ratio of the intersection and union of two boxes. +During the evaluation, according to the IoU of predicted boxes +and ground truths, each sample will be assigned attributes: +true positive (TP) for correctly matching, false positive (FP) +for wrongly predicting the background as an object, and false +negative (FN) for the undetected object. During the evaluation, +all the detection boxes are sorted in order of confidence score +from high two low and then traversed. In the traversed process, +the calculations of the precision and recall metrics can be +defined as: +Precision = +TP +TP + FP , +(18) +Recall = +TP +TP + FN . +(19) +The precision and recall are correlated with the commission +and omission errors, respectively. The AP values use an +integral method to calculate the area enclosed by the Precision- +Recall curve and coordinate axis of all categories. Hence, the +AP can be calculated by +AP = +� 1 +0 +p(r)dr, +(20) +where p denotes Precision, r denotes Recall. The mAP is a +comprehensive indicator obtained by averaging APs for all +classes. Moreover, we choose Bit-Operations (BOPs) count +[52] and parameters to measure the compression performance. +The Bops of convolution are calculated as: +BOPsl = cl−1 ×cl ×wl ×hl ×kw ×kh ×bw,l ×ba,l−1. (21) +The hl, wl, and cl are the with, height, and several channels +of the l − th layer output feature map, respectively. bw,l and +ba,l denote l − th layer weight and activation bit-weight. The +parameters (params) are defined as: +params = cl−1 × cl × kh × kw × bw,l +8bit +(B). +(22) +C. Ablation Study +In this section, we conduct the ablation experiments of our +GHOST framework. We explore how each module (HQ and +OST) compresses the model and promotes the performance of +the small student model. Besides, the experiments of different +distillation algorithms and distillation hyperparameters’ opti- +mization are carried out. We conduct ablation experiments on +the dataset of VEDAI for object detection. +Validation of HQ: Distribution distance hybrid quantization +can integrate device n-bit settings for the network, so we +experiment with such variations of integrated hybrid n-bit +quantization. To analyze the performance of differences, we +compare fixed DoReFa-Net [8] and various hybrid DoReFa- +Net quantization methods on the SuperYOLO detection net- +work. As illustrated in TABLE II, the HQ is the proposed +hybrid quantization method, and the ·W · A presents the +traditional unified bit width quantization algorithm except +32W32A represents the full precision network. Max and Min +denote the maximum and minimum bit width in the quan- +tization model. Hybrid quantization enables the quantization +model to preserve the significant information to achieve the +minimal accuracy loss possible. As shown in TABLE II, the +hybrid quantization module accomplishes the optimal conse- +quence and detection accuracy has reached 79.57%, 78.42%, +and 75.76% respectively, which are more 3.7%, 2.5%, and +1.62% than fixed quantization at the different computation +orders. The hybrid quantization achieves better accuracy in +the VEDAI dataset than the accuracy of fixed quantization +costing fewer computation resources (parameters and BOPs). +Effect of OST Module: After the HQ module has been +added to the network, we also adopt one-to-one self-teaching +within the three-quantization scale. Table III is based on +SuperYOLO which is used as the teacher network and student +network simultaneously. The experiments are carried out on +the VEDAI dataset. The OST module enables the typical de- +tection network to recover the performance of the quantization +detection network improved by 0.59%, 1.04%, and 2.35%, +respectively no matter what kind of bit width. +Comparison with the SOTA Distillation Method: In +addition, Table IV shows the comparisons between the pro- +posed OST module and existing distillation frameworks. OST +achieves superior performance in the field of remote sensing +under the premise of the same computation cost while the +ZAQ, AFD, and ReviewKD lead to an accuracy degradation +of the quantization network. And also it can be proved that +OST distillation is a benefit for the guidance between the full- +precision model and the quantization model. +Hyperparameters Optimization: As presented in Section +III-C, β is the distillation weights of the OST module, so we +compare the performance of the distillation in the different +weights which are shown in TABLE V. We conduct the +hyperparameter experiment on the VEDAI dataset on GHOST +to find the best β. As shown in the TABLE V shows, the +model reaches the best when β = 400 at the T = 0.1 and +when β = 400 at the T = 4 or T = 52. +Lightweight Analysis: Owning to the GQSD design idea, +our student model is very lightweight. We compare the + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. X, NO. X, X 2022 +11 +TABLE VI. Performance of different algorithms on VEDAI testing set. +Method +Car +Pickup +Camping +Truck +Other +Tractor +Boat +Van +mAP50 +mAP +Params(MB) +BOPs(G) +YOLOv3 +83.5 +71.7 +64.2 +67.5 +45.5 +62.8 +42.0 +63.4 +62.6 +37.2 +246 +50,749 +YOLOv4 +86.2 +70.9 +71.9 +75.3 +54.9 +69.3 +30.7 +66.6 +65.7 +40.9 +210 +39,076 +YOLOv5s +81.1 +71.3 +70.8 +66.4 +58.1 +67.3 +27.0 +55.7 +62.2 +34.6 +28 +5,427 +YOLOv5m +81.6 +73.9 +59.0 +70.0 +57.2 +77.7 +30.5 +65.5 +64.4 +38.2 +84 +16,599 +YOLOv5l +84.3 +76.8 +74.0 +75.0 +51.5 +61.3 +30.3 +57.4 +63.9 +37.9 +186 +37,530 +YOLOv5x +84.7 +66.4 +66.8 +72.4 +65.8 +67.2 +29.2 +58.8 +63.9 +37.8 +349 +71,301 +Teacher +89.2 +87.1 +79.5 +86.8 +58.2 +88.0 +70.3 +88.3 +80.93 +50.80 +19.3 +17,024 +GHOST +88.82 +86.06 +74.33 +86.96 +61.28 +86.00 +71.76 +87.26 +80.31 +49.05 +2.5 +692 +TABLE VII. Performance of different algorithms on DOTA, NWPU and DIOR testing set. +DOTA-v1.0 +NWPU +DIOR +Method +mAP50 +Params(MB) BOPs(G) +mAP50 +Params(MB) BOPs(G) +mAP50 +Params(MB) BOPs(G) +Faster R-CNN +60.64 +240 +296,192 +77.80 +164 +130,764 +54.10 +240 +186,572 +RetainNet +50.39 +221 +300,400 +89.40 +145 +126,228 +65.70 +221 +184,954 +YOLOv3 +60.00 +246 +203,694 +88.30 +246 +124,180 +57.10 +247 +125,153 +GFL +66.53 +76 +163,000 +88.80 +76 +93,931 +68.00 +76 +99,768 +FCOS +67.72 +126 +207,001 +89.65 +127 +119,429 +67.60 +127 +126,474 +ATSS +66.84 +75 +159,754 +90.50 +75 +92,057 +67.70 +75 +97,792 +MobileNetV2 +56.91 +41 +127,221 +76.90 +41 +73,205 +58.20 +41 +77,926 +ShuffleNet +57.73 +48 +146,022 +83.00 +48 +84,142 +61.30 +48 +89,405 +O2-DNet +71.10 +836 +- +- +- +- +68.3 +836 +- +FMSSD +72.43 +544 +- +- +- +- +69.5 +544 +- +Teacher +71.65 +203 +291,491 +93.21 +127 +122,234 +71.7 +203 +178,176 +ARSD +68.28 -3.37 +52 +69,662 +90.92 -2.29 +46 +27,289 +70.10 -1.6 +52 +42,598 +Teacher +69.99 +30.8 +21,390 +93.30 +30.7 +21,357 +71.95 +30.8 +21,428 +GHOST +69.02 -0.97 +9.7 +2,146 +91.97 -1.33 +8.5 +1,927 +71.53 -0.4 +9.3 +2158 +GHOST and the teacher model in terms of parameters and +BOPs on the four datasets. As Fig. 6 shows, GHOST has +smaller model parameters, and fewer BOPs compared with the +teacher model no matter in which dataset. Hence, our com- +pression strategy for the lightweight model is more practical +to be deployed on intelligent terminals. +D. Comparison with the SOTA Detectors +In this part, we compare our lightweight model with other +classic heavy object detection methods. As shown in TABLE +VI and TABLE VII. Experiments on the four datasets prove +the efficiency and efficacy of the GHOST framework. Not +only does our lightweight have higher accuracy but also it has +strong information retention capability under extreme model +compression. +1) VEDAI: Our GHOST achieves 80.31% mAP50 compared +with other detectors, surpassing the one-stage series mentioned +in TABLE VI. Our model achieves the lowest model parame- +ters (2.5 MB) and BOPs (692 G). +2) DOTA: As presented in TABLE VII, our GHOST +achieves the optimal detection result (69.02% mAP50) and +the model parameters (9.7 MB) and BOPs (2,146 G) are much +smaller than other SOTA detectors regardless of the two-stage, +one-stage, anchor-free or distillation-based method. We also +compare two detectors designed for remote sensing imagery +such as FMSSD [50] and O2DNet [51]. Although these models +have a close performance with our lightweight model, the +huger parameters and BOPs seem to be a massive cost in +computation resources. Hence, our model has a better balance +in consideration of detection efficiency and efficacy. Compare +to the distillation-based method ARSD (-3.37%), the GHOST +obtains an even smaller accuracy gap (-0.97%) between the +student network and the teacher network. It demonstrates that +our GHOST method can transfer sufficient knowledge to guide +the learning of the student model and the misunderstanding +between both models can be reduced by the SCM training +strategy. +3) NWPU: We compare the results of our method with other +approaches on the NWPU dataset. As shown in TABLE VII, + +IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. X, NO. X, X 2022 +12 +our GHOST obtains the best result (91.97% mAP50) with the +smallest amount of model parameters (8.5 MB) and the fewer +BOPs (1,927 G). +4) DIOR: As illustrated in TABLE VII, our GHOST +achieves the optimal detection result (69.02% mAP50) and +the model parameters (9.3 MB) and BOPs (2,158 G) are much +smaller than other SOTA detectors regardless of the two-stage, +one-stage, anchor-free lightweight, distillation-based methods. +It reveals the strong ability to compress models and the power +capacity of object detection in remote sensing imagery. The +accuracy of the student GHOST is only a bit less 0.4% than +teacher network, compared with ARSD (1.6%). +In order to show the performance of our algorithm more +intuitively, We compare the detection accuracy, parameters, +and BOPs of various algorithms in Fig. 7. It can be obvious +that GHOST has a better trade-off between performance and +lightweight. The visualization results on the three datasets are +illustrated in Fig. 8 in which we can see that the GHOST can +have an outstanding detection of objects at different scales. +V. CONCLUSION +In this paper, we propose a GHOST framework for a +lightweight object detection method in remote sensing im- +agery. We first design a guided quantization self-distillation +structure which is not only a training technique to preserve +model performance but also a method to compress and accel- +erate models. Although most of the previous research focuses +on knowledge transfer among different models, we believe +that inside distillation is also very promising. Secondly, we +propose a hybrid quantization that captures the optimal bit +width selection based on an adaptive way in the weight value +research space to break the limit of the fixed quantization +model accuracy. Thirdly, the proposed one-to-one self-teaching +module gives the student network of self-judgment through a +switch control machine that accurately handles the knowledge +transformation. It can dynamically discriminate the wrong +guidance and mine the effective knowledge from the teacher. +The experiments based on the VEDAI, DOTA, NWPU, and +DIOR datasets certify that our GHOST achieves SOTA perfor- +mance compared with other detectors. It can well balance the +tradeoff between accuracy and specific resource constraints. +REFERENCES +[1] J. Ding, N. Xue, Y. Long, G.-S. Xia, and Q. 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Vis. +Springer, 2020, pp. 259–277. + diff --git a/PtAyT4oBgHgl3EQfUvdy/content/tmp_files/load_file.txt b/PtAyT4oBgHgl3EQfUvdy/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..017faa8fc81b7c34eeebede27ffcf8f6bd6fbc5f --- /dev/null +++ b/PtAyT4oBgHgl3EQfUvdy/content/tmp_files/load_file.txt @@ -0,0 +1,1496 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf,len=1495 +page_content='IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, X 2022 1 Guided Hybrid Quantization for Object Detection in Multimodal Remote Sensing Imagery via One-to-one Self-teaching Jiaqing Zhang, Jie Lei, Member, IEEE, Weiying Xie, Member, IEEE, Yunsong Li, Member, IEEE, and Xiuping Jia, Fellow, IEEE Abstract—Recently, deep convolution neural networks (CNNs) have promoted accuracy in the computer vision field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' However, the high computation and memory cost prevents its development in edge devices with limited resources, such as intelligent satellites and unmanned aerial vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Considering the computation complexity, we propose a Guided Hybrid Quantization with One-to-one Self-Teaching (GHOST) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' More concretely, we first design a structure called guided quantization self- distillation (GQSD), which is an innovative idea for realizing lightweight through the synergy of quantization and distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The training process of the quantization model is guided by its full-precision model, which is time-saving and cost-saving without preparing a huge pre-trained model in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Second, we put forward a hybrid quantization (HQ) module to obtain the optimal bit width automatically under a constrained condition where a threshold for distribution distance between the center and samples is applied in the weight value search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Third, in order to improve information transformation, we propose a one-to-one self-teaching (OST) module to give the student network a ability of self-judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' A switch control machine (SCM) builds a bridge between the student network and teacher network in the same location to help the teacher to reduce wrong guidance and impart vital knowledge to the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' This distillation method allows a model to learn from itself and gain substantial improvement without any additional supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Extensive experiments on a multimodal dataset (VEDAI) and single-modality datasets (DOTA, NWPU, and DIOR) show that object detection based on GHOST outperforms the existing detec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The tiny parameters (<9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='7 MB) and Bit-Operations (BOPs) (<2158 G) compared with any remote sensing-based, lightweight or distillation-based algorithms demonstrate the superiority in the lightweight design domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Our code and model will be released at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='com/icey-zhang/GHOST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Index Terms—Object detection, remote sensing image, Quan- tization, Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' INTRODUCTION O BJECT detection in aerial images plays an important role in military security aiming to locate interested objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=', vehicles, airplanes) on the ground and identifying their categories [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' From universal detectors for natural images This work was supported in part by the National Natural Science Foundation of China under Grant 62071360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Lei, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Li are with the State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an 710071, China (e-mail: jqzhang 2@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' jielei@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' wyxie@xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' ysli@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='xidian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Jia is with the School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT 2600, Australia (e-mail: xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='jia@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' such as YOLOv3 [2], Faster R-CNN [3], FCOS [4] are widely introduced in the field of remote sensing (RS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' more and more dedicated detectors for RS scene are designed and improved with the requirements of objects tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' However, the large complexity of the object detection network is under- investigated, which limits the practical deployment under resource-limited scenarios and bring a heavy burden to process massive multimodal images collected from satellites, drone, and airplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Hence, a series of compression schemes have been proposed to settle this problem, such as pruning [5], quantization [6], [7], [8] and distillation [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='Step1: Train a full- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='precision model as the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='pretrained model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='Step2: Get a small N- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='bit model via ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='completing n-bit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='quantization under the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='pretrained model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='Step1:Train a large ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='full-precision model as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='the pretrained teacher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='Step2: Get a small ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='full-precision model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='under the guidance of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='pretrained teacher via ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='distillation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='Step1:Train a small full- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='precision model as the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='pretrained teacher model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='Step2: Get a small ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='mixed-bit model by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='completing n-bit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='quantization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='under the guidance of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='pretrained teacher via ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='distillation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='Conclusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='SuperYOLO ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='(Standard Training) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='BOPs:17024G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='Params:19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='3MB mAP50:80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='93% 6-bit SuperYOLO (via Quantization) BOPs:727G Params:3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='6MB mAP50:75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='92% GHOST (via Hybrid Quantization and Self Distillation) BOPs:692G Params:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5MB mAP50:80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='31 SuperYOLO (via Distillation) BOPs:17024G Params:19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='3MB mAP50:79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='78% SuperYOLO (Standard Training) BOPs:17024G Params:19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='3MB mAP50:80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='93% SuperYOLOl (Standard Training) BOPs:125031G Params:125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='2MB mAP50:81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='68% Mixed-bit self distillation outperformances the results of both the traditional distillation and quantization with less computation cost Train a small quantization model as possible Traditional Model Quantization Traditional Model Distillation Proposed Mixed-bit self Distillation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Comparison of training complexity, and accuracy between traditional distillation, traditional quantization and proposed mixed-bit self distillation (reported on VEDAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Quantization algorithms [11], [12] directly compress the cumbersome network, effectively reducing the computation cost and model size with a great compression potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' How- ever, trivially applying quantization to CNNs usually leads to inferior performance if the compression bit decreases to a low level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Some knowledge distillation methods [13], [14], [15] are arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='00131v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='CV] 31 Dec 2022 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, X 2022 2 proven to be valid to elevate the performance of the lightweight model but have to pre-train a huge teacher model as a guidance of the student model which is time-consuming and resource-consuming [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Self-distillation methods [17], [16], [18] overcome this problem via the transfer of information inside the model itself without introducing extra huge storage and time consuming from the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The above view naturally leads to a question: What re- search results will we get if we combine quantization and distillation by using a small full-precise network to guide the learning process of a quantization for this full-precision network?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In this way, the tremendous compression capacity of quantified networks and the performance of full precision networks can be collaborative and cooperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In this paper, we design an adaptive one-to-one educational policy pertaining the full-precision network and the quantiza- tion network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We propose a simple yet novel approach that allows quantization network to reinforce presentation learning of itself relative full-precision network without the need of additional labels and external supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Our approach is named as Guided Hybrid Quantization with One-to-one Self- Teaching (GHOST) based on the guided quantization self- distillation (GQSD) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As the name implies, GHOST allows a network to exploit useful and vital knowledge derived from its own full-precision layers as the distillation targets for its quantization layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' GHOST opens a new possibility of training accurate tiny object detection networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 1, in order to train a small compact model to achieve as high accuracy as possible with less computation cost, we propose mixed-bit self distillation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Instead of implementing two steps in traditional distillation, which means that to train a large teacher model comes first, following by distilling the knowledge from it to the student model, we propose a two-step mixed-bit self distillation framework, in which the training process of the second quantization step is based on the pretrained small full-precision model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The proposed framework not only requires less computation cost (from 20797 G BOPs to 692 G BOPs on VEDAI dataset, a 30X faster training cost), but also can accomplish much higher accuracy (from 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='84% in traditional quantization to 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='31% on SuperYOLO) The main contributions of our work are as follows: We propose a unified guided quantization thought based on self-distillation called GQSD, which can tackle the lightweight object detectors’ quantization optimization problem in remote sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We are the first to formulate an adaptive one-to-one education policy between the full- precision network and the quantization network at the same structure in object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' For the finding of weight value distribution features of remote sensing images, we design a hybrid quantization module, whose adaptive selection of the core information of the weights for quantization with a constrained preset condition can keep the balance of accuracy and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Aiming to offset the loss of the quantization information, the switch control machine is adopted to enable the student to distinguish and close the teacher’s wrong guidance and mine the correct and vital knowledge from self-distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The rest of this paper is organized as follows: Section II give a rough overview of the spacific related work to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Section III presents our proposed method in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Section IV introduces experimental results and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Section V concludes this paper and discusses the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' RELATED WORK In this section, we reviewed related work from object detection and network compression and acceleration in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Object Detection with Deep Learning Various CNN-based object detection architectures have shown promising performance, bringing the field to a new level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The architectures can be roughly divided into two main domains: two-stage, and one-stage according to the change process of proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Two-stage Detectors: A typical method is selective search work [19], where the first stage is to generate a large set of proposed region candidates that are required to cover the whole objects and then filter out most negative positions, and then the second stage is to complete classification for each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' R-CNN [20] creates a new era as one of the most successive two-stage algorithms owing to the upgrading of the second- stage classifier to a convolution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Fast-RCNN [21] extractes features over the images before proposing regions and integrates the extractor and classifier by employing a soft layer rather than SVM classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Faster R-CNN [3] introduces a CNN-based region proposal network to further integrate proposal generation with the second-stage classifier into a single convolution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' One-stage Detectors: One-stage detectors aim to jointly predict the classification and location of objects by integrating the detection and classification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Recently, a series of SSD [22], [23], [24] and YOLO [25], [26], [2], [27] have renewed interest in the one-stage object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' SSD implements independent detection on multiscale feature maps, while the YOLO utilizes combined detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' These methods have paid more attention to speed, but their accuracy trails behind that of tow-stage methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' YOLOv2 [26] modifies the location regression pattern depending on bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' YOLOv3 [2] considers multiscale objects and detects them in the three scales, which can realize the detection of multiple sizes of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' YOLOv4 [27] introduces more data augmen- tation tricks, activation functions, backbone structures, and IoU loss metrics to enhance the robustness of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' YOLOv5 [28] releases four different size models, where the basic structures are identical, which allows YOLOv5 to have higher flexibility and versatility in practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' To solve the dilemma of the category imbalance, RetinaNet [29] reduces the weight of massive amounts of simple negative samples in training by designing a focal term for cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As FCOS [4] which belongs to anchor-free methods is proposed, adjusting hyperparameters and calculations related to anchor boxes has been avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' ATSS [30] selects positive samples adaptively to enhance the detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, X 2022 3 Teacher Student 8s 9s ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Input Samples Full-precision Network Features Quantization Network Features Distillation Switch 1 0 0 0 Self Distillation with Same Structure Mixed Bit Width ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 8t 9t 1t 2t 3t 4t 1t 2t 3t 4t 5t 6t 7t 8t 1s 2s 3s 9t 4s 5s 6s 7s 8s 9s 10 s 10 t Attention Map 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='0 Switch Control Machine 1 1 1 1 1 0 0 0 0 0 1 1 1 ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' : Cluster Number ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Hybrid Quantization lt ls ls lt 1 ls − 1 lt − 1s 2s 3s 4s 0 min 2b min 1 2b + 1B 2B 3B 4B 8B 9B 1 lB − lB l W : Distance l W T : Threshold min max 82 lB One-to-one self-teaching 2n 2 2n− 1 2n− 2n ( ) ld n T \uf03c ( ) ld n ( ) ld n Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Overview of our proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' An attention-based model determines similarities between the teacher and student features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Knowledge from each teacher feature is transferred to the student with similarities identified by Switch control machine (SCM) by self-distillation with the same structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The mixed bit widths of the student network for quantization are based on the search results of the full-precision weights research space of teacher network in the same layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Deep Network Compression and Acceleration Although the speed of the one-stage detection network is superior, its large model and high computation complexity still deserves to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Some researches focus on the design of a lightweight backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' MobileNetV2 [31] utilizes the depthwise separable convolutions to build a lightweight model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' ShuffleNet [32], and SqueezeNet [33] also effectively reduces the memory footprint during inference and speed up the detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In the literature, a potential direction of model com- pression is knowledge distillation (KD) which concentrates on transferring knowledge from a heavy model (teacher) to a light one (teacher) to improve the light model’s performance with- out introducing extra costs [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Whereas the knowledge distillation enables utilizing the larger network in a condensed manner, the pretraining of the large network requires extra substantial computation resources to prepare the teacher net- work [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The preparation of the pretrained teacher network is time-consuming and cost-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The self-knowledge distillation [17], [16], [18] can overcome this problem by distilling its own knowledge without prior preparation of the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Quantization is another way to compact the model directly and compress the ponderous network by using low-bit representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Mixed-precision quantization method uses different numbers of bits for a given data type to represent values in weight tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Many works [11], [36], [37], [37] have shown that the mixed-precision method is efficient for quantizing network layers that have different importance and sensitiveness for the bit width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' However, trivially applying biomgi2 (s) (p) Qnwp6J-2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='8wo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='g 0 5 e 8 0 5 e 8 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='0 8 S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content="0 8 0'S 0'04 e- e a0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content="0 0'4 4 - 80." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='0 4 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='0 o1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content="0 5- O'JS 5 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content="0 - 0- 0'J4 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='1IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, X 2022 4 quantization to CNNs usually leads to inferior performance if the compression bit decreases to a low level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' NETWORK ARCHITECTURE In this section, we first revisit conventional KD and de- scribe the proposed GHOST framework in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Then, we present the details of the inspired hybrid quantization algorithm (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' III-B) and this quantization training process is guided by a one-to-one self-teaching method illustrated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Overview KD is a widely-applied method that can be expressed as a knowledge transformer from teacher to student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Given a teacher model T and a student model S, the x is the data examples of models, here they can be the same for the teacher and the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In general, the KD machine can be uniformly expressed as: min LKD = min � xi∈x L (T (xi) , S (xi)) , (1) where L is the loss function that penalizes the differences between the teacher and the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The student model size is commonly designed in a small size to achieve the purpose of model compression in which the performance of the student can chase the teacher but consumes a computing-friendly resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Nonetheless, the computation cost of the student model is larger than the pruning method directly completed on the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' This demonstrates that the existing KD-based quantization algorithms still have great potential room for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We aim at developing a novel and generic baseline network with a focus on the learnable knowledge characteristics, mak- ing it well-applicable to the highly accurate and fine object detection of RS images with less computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The key to model quantization with knowledge learning is to reduce the discrepancy which can be punished by distance or angle loss function between full-precision model P (teacher) and low- precision model Q (student) through optimizing Q, which can be expressed as: Q∗ = min Q � xi∈x L(P(xi), Q(xi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (2) The weights of the teacher are frozen without gradient propa- gation when the teacher network guides the training of the stu- dent network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Based on the above presentation, we design an effective teacher-student distillation framework called GQSD which can be represented as: min LKD = min � xi∈x L(P(xi), Q(xi), R(xi)), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' WQ = Winit, BQ = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (3) Specifically, the full precision network is the import funda- mental teacher which not only provides the initial weights Winit and bit width B of the quantization model but also guides the quantization process to mine the vital knowledge from the teacher in specific features selected by a control R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As shown in Fig 2, we propose a GHOST framework that concludes a hybrid quantization (HQ) module and a one-to- one self-teaching (OST) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The mixed bit widths of the student network for quantization are based on the search results of the full-precision weights research space of the teacher network in the same layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Inspired by the idea of harnessing intermediate features to improve performance in knowledge distillation [38], [39], [40], we design a Switch Control Machine (SCM) as R to generate an attention map that gains intermediate feature similarities between the teacher and student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The SCM controls the distillation switch and determines which knowledge should be delivered dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Knowledge from each teacher feature is transferred to the student with similarities identified by SCM by self-distillation with the same structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' With a pretrained full-precision model as a initial weight, the quantization and distillation processes are conducted simultaneously to ultimately obtain a small lightweight model with little loss of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The details of the modules will be described separately as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Hybrid Quantization Powerful deep networks normally benefit from large model capacities but induce high computational and storage costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Modal quantization is a promising approach to compress deep neural networks, making it possible to be deployed on edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The quantization operator divides the weight into different fixed values by a quantization function which can be regarded as a cluster of convolution kernels in substance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The different scale weights are clustered to a certain value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' To illustrate this intuition explicitly, a SuperYOLO [41] network model which consists of 60 convolutional layers is trained based on the VEDAI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' After training, test samples are fed into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The convolution weight is firstly clustered into different categories by k-means and then transformed into 2 dimensions by t-SNE [42] to realize the visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 3, the convolution kernel weight in the (a) 0th, (b) 26th and (c) 52nd convolutional layer are clustered in different numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 3 (a), the distance between different categories is relatively far which indicates that the weight distribution is dispersed and complicated in the initial layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' This is due to the fact that the color and texture features, which are detailed and multifarious, are captured in the shallow layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As the layer propagates forward (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 3 (b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 3 (c)), the convolution weight becomes converging gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In other words, the semantic features in the deep layer are more robust and condensed so that with the deepening of the network layers, the clustering categories of weights can be relatively reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Based on this finding, the hybrid quantization idea is introduced to search for the optimal bit width definition in the weight value space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We initially design a hyperparameter T as a threshold to constrain the research space to control the compression rate of the quantization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The search strategy can be described as: B =argmax(d(n)) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' d(n) < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (4) IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, X 2022 5 (a) (b) (c) (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In the bottom layer of a trained network, the feature distributions of different categories intervene with each other severely as (a) shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Many delicate neurons are needed to distinguish the overlapped distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' And as the network propagates forward, the feature distribution of the same category gathers gradually in (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' At the end of the hidden layers, there exist clear margins between the semantic feature distributions of different classes in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' With the improvement of separability among the feature manifolds, a neuron with lower-precision parameters is able to extract robust features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' where the function d(x) denotes the measurement of clus- tering extent at the n bit width for each convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' This definition aims to find the limited minimum clustering categories (maximum clustering extent) for each layer, hence the smallest quantization model with a minimum bit width is obtained at the preset ratio constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The hybrid quantization of the whole network definitely can be collected as: B = [B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=', Bl] (5) where the l is the total convolution layer and the Bl is the lth bit width of each layer weight parameter, and the bit width decreases progressively as the network propagates forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We utilize the distribution distance defined as follows to determine the final bit width for quantization of the lth convolution layer weight: dl(n) = 1 M M � j=0 2n � i=0 (wl ij − cl i) 2, (6) where the M is the total number of kernel weights which correspond to M = Cin × Cout × K × K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The Cin, Cout, and K are the input channels, output channels and kernel size of the convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The whole weight values of each convolution layer complete the kmeans++ algorithm on the different cluster numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' While the 2n represents the cluster number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' cl i and wl ij are the cluster centers and samples, as shown respectively in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We set the initial bit width as 8 and then select the superior and adaptive bit width by Bl = min(n|dl(n) < T) n = bmin, bmin + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=', 8, (7) where the bmin is a limit of the minimum bit width in the quantization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' When the dl(n) is smaller than a manual threshold T set in advance, the bit width of the current convolution layer is updated as Bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The activation following this convolution layer keeps the same bit-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Take the distance threshold T = 50 as an example, the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 5 demonstrates the judgment results of bit width for each convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' It can be indicated that the values of bit width progressively decrease with the deepening of the network layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In addition, the bit width of the convolution layer before the detection process may be relevantly large to maintain more location discrimination information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We use a simple-yet-effective quantization method which refers to [8] for both weights and activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The uniform quantization function q(�) is defined as: q (v, k) = 1 2k − 1round((2k − 1)v), (8) where v is a real number indicating the full-precision (float32) value, v ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' the output q(v, k) of quantization function is a k bits real number, q(v, k) ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=" The quantization calculations of lth convolution layer weight and activation are O'JO 0'02 00." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content="0 0'02 O'TO O'JO 0'02 00." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content="0 0'02 O'JO01." metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content="0 0'50 0'5 O'J 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content="0 O'J o'S-O'TO 0'02 00." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content="0 0'02 O'TO O'JO 0'02 00." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content="0 0'02 O'JOIEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, X 2022 6 Center Distance Distribution Distance iC ij w Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The distribution distance of kmeans++ cluster method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 10 20 30 40 50 Convolution Layer Index 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 6 Bit Width 10 20 30 40 50 Convolution Layer Index 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 6 Bit Width Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The bit width results of each convolution layer at the threshold T = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The values of bit width progressively decrease with the deepening of the network layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In addition, the bit width of the convolution layer before the detection module may be relevant large to maintaining more location discrimination information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' defined respectively as follows: wl o = 2q( tanh(wl i) 2 max( ��tanh(wl i) ��) + 1 2, Bl) − 1, (9) al o = q(al i, Bl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (10) The activation al i is the range in [0, 1] determined by a bounded activation function while the weight wl i is not restricted in a limit boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Here, the quantization result of weight wl o is the range in [−1, 1], and the quantization result of activation al o is the range in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The Algorithm 1 clarifies the process of the hybrid quantization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As described in [8], the first and last layers in the network are sensitive to performance during the process of quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Based on this intuition, the last detection layer keeps intact to avoid potential degradation of detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Algorithm 1 The Hybrid Quantization Method Input: The weights of lth certain convolution layer W ∈ RH×W ×K×K, The manual distance threshold T and the minimum bit width bmin Output: The bit width of the current convolution layer and activation Bl 1: Initialize the Bl as 8 2: for n in range (bmin,8) do 3: Cluster weights into 2n clusters via the kmeans++ algorithm and then get the centers ci and samples wij of the ith cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 4: Calculate the distribution distance according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 5: Update the bit width Bl by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 7 6: Complete the quantization for the convolution layer weight and activation by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 9 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' One-to-one Self-teaching Previous mixed quantization approaches pay more attention to the bit-width selection [7] which costs a lot of resources to obtain the optimal decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Our hybrid quantization method can make a quick decision with less computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The loss of performance is fixed by the guide of distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In general, previous distillation algorithms are a full precision network, so the network weights are in the same order of magnitude, and the feature maps generated by the teacher network or the student network are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' However, for the quantitative network, the feature map generated by the quantitative network as a student network will have obvious weight information loss due to the increase of the zero content, resulting in some differences between the feature map of the teacher network and the feature map of the student network, which makes it difficult for the teacher network to directly restrict the quantitative student network from the feature layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Therefore we proposed an OST to conquer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' SCM first calculates the inner connected relationship between the full-precision and quantization network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The distillation switch (DS) chooses the core information between matched student and teacher features by this relationship matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We sketch the architecture of self- feature distillation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Let s = s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=', sl represent a set of multiscale feature maps for the student network and t = t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=', tl for the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' To calculate the attention map similar to [39] between the student feature and teacher feature, we define that each teacher feature generates a query qi, and each student feature produces a key kj: qi = Wi · GAP(si), (11) kj = Wj · GAP(tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (12) GAP(·) represents a global average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Wt and Ws are the liner transition parameters for the ith query and the jth key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Then the attention map that reveals the inner relationship between teacher and student features is defined as: a = (q · kT)/ √ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (13) IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, X 2022 7 Here, we introduce the Gumble-Softmax trick [43] to convert the values greater than the threshold to 1 and the rest to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Formally, the decision at ith entry of a is derived in the following way: Ai,k = exp(log(ai,k + Gi,k)/τ) K � j=1 exp(log(ai,j + Gi,j)/τ) k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' , K, (14) where K is set as 2 for binary decision [44] in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Gi is the Gumbel distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Temperature τ is used to control the smoothness of Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' With a better attention map to mining the internal correlation of features, we generate a DS mask that can automatically determine whether to transfer the information from the teacher to the student at the same site in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' α = Diag(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (15) where SCM digs out the diagonal elements from the attention map matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We devise the self-feature distillation loss as follows: LF = m � i=0 αi∥CAP(ti) − CAP(si)∥2, (16) where CAP represents a channel-wise average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' m is the total features utilized for distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Ltotal = βLF + Ldec, (17) Finally, the distillation loss terms are combined with detection loss and minimized in an end-to-end manner as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Ldec includes the objectness, location, and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The hyper-parameter β indicates the impact balance between the detection and distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' EXPERIMENTAL RESULTS In this section, we evaluate the proposed method on the multimodal dataset for remote sensing object detection and four widely adopted CNNs with different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We first demonstrate the experience set up, including the introductions of datasets and networks, implementation details, and evalua- tion metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Then, we report the performance of our method on a dataset in detail, mean average precision and compression ratio are calculated to measure the comprehensive performance in the accuracy and computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Dataset Description The publicly available dataset VEDAI [45] designed for multimodal remote sensing image object detection is adopted in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In addition to validation on the multimodal object detection dataset, three single modal datasets (DOTA [46], NWPU [47] and DIOR [48]) are utilized in experiments to verify the generation of our proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 1) VEDAI: The VEDAI dataset consists of 1246 smaller images cropped from the much larger Utah Automated Geo- graphic Reference Center (AGRC) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Each image col- lected from the same altitude in AGRC has approximately 16, 000 × 16, 000 pixels, with a resolution of about 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5cm × TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Training Strategy Dataset Image Size Batch Size Lr Epoch VEDAI 512 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='01 300 DOTA 512 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='01 100 NWPU 512 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='01 150 DIOR 512 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='01 150 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5cm per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The main scenes of VEDAI include grass, highway, mountains, and urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The size of the image is fixed to 512 × 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 2) DOTA: The DOTA dataset was proposed by Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' in 2018 for object detection of remote sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' It contains 2806 large images and 188 282 instances, which are divided into 15 categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The size of each original image is 4000×4000, and the images are cropped into 1024 × 1024 pixels with an overlap of 200 pixels in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We select half of the original images as the training set, 1/6 as the validation set, and 1/3 as the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The size of the image is fixed to 512 × 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 3) NWPU VHR-10: The dataset of NWPU VHR-10 was proposed by Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' It contains 800 images, of which 650 pictures contain objects, so we use 520 images as the training set and 130 images as the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The dataset contains 10 categories, and the size of the image is fixed to 512 × 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 4) DIOR: The DIOR dataset was proposed by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' in 2020 for the task of object detection, which involves 23 463 images and 192 472 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The size of each image is 800× 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We choose 11 725 images as the training set and 11 738 images as the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Implementation Details 1) Networks: To demonstrate superior performance, Su- perYOLO [41] is tested as a teacher model with our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' For the multimodal VEDAI dataset, the number of convolution layers is 47 including one detection layer on the small scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' For a single modal dataset (DOTA, NWPU, and DIOR), the number of convolution layers is 61 including three detection layers on the small, medium, and large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' To verify the superiority of the GHOST proposed in this paper, we selected 12 generic methods for comparison: one-stage algorithms (YOLOv3 [2], YOLOv4 [27], YOLOv5 [28], SuperYOLO, FCOS [4], ATSS [30], Retain- Net [29], GFL [49]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' two-stage method (Faster R-CNN [3]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' lightweight models (MobileNetV2 [31] and ShuffleNet [32]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' distillation-based methods (ARSD [15]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' remote sensing designed approaches (FMSSD [50] and O2DNet [51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 2) Training Strategy: Our proposed framework is imple- mented in PyTorch and runs on a workstation with an NVIDIA A100-SXM4-80GB GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We also use different training strate- gies for different datasets, and the detail is illustrated in IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, X 2022 8 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Comparisons of the teacher model and lightweight model by parameters and BOPs on the four datasets (VEDAI, DOTA, NWPU, and DIOR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The BOPs and parameters of the lightweight model are smaller, and the inference speed is faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (a) Params (MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (b) BOPs (a) (b) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Comparison of the efficiency between the current SOTA methods and our method on the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The bigger size of cycles represents costing more parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (a) DOTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (b) NWPU, and (c) DIOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The comparison result of the tranditonal quantization method and our mixed-bit quantization and we use the same abbreviation in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Bit Width T Max Min Car Pickup Camping Truck Other Tractor Boat Van mAP50 mAP Params(MB) BOPs(G) 32W32A 32 32 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='20 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='10 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='50 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='80 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='20 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='00 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='30 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='30 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='93 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='80 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='30 17023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='76 8W8A 8 8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='25 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='86 75.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='14 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='82 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='99 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='91 TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The comparison with sota distillation method for detectors on the VEDAI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Distillation HQ Car Pickup Camping Truck Other Tractor Boat Van mAP50 mAP ✓ 89.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='14 C中 口IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, X 2022 10 TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' mAP comparisons of different β value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' β 0 100 200 300 400 500 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='1 46.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='20 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='14 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='48 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='03 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In addition, data is augmented with Hue Saturation Value (HSV), multi-scale, translation, left-right flip, and mo- saic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The augmentation strategy is canceled in the test stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The standard Stochastic Gradient Descent (SGD) is used to train the network with a momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='937, weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='0005 for the Nesterov accelerated gradients utilized, and a batch size of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='01 initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' All the baseline training process is completed from scratch without any pre-trained model while the GHOST is carried on the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In the test stage, the IoU threshold of non- maximum suppression is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='6 on NWPU VHR-10 and VEDAI, and it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='4 on DOTA and DIOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 3) Evaluation Metric: For the detection result, the IoU is defined as the ratio of the intersection and union of two boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' During the evaluation, according to the IoU of predicted boxes and ground truths, each sample will be assigned attributes: true positive (TP) for correctly matching, false positive (FP) for wrongly predicting the background as an object, and false negative (FN) for the undetected object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' During the evaluation, all the detection boxes are sorted in order of confidence score from high two low and then traversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In the traversed process, the calculations of the precision and recall metrics can be defined as: Precision = TP TP + FP , (18) Recall = TP TP + FN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (19) The precision and recall are correlated with the commission and omission errors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The AP values use an integral method to calculate the area enclosed by the Precision- Recall curve and coordinate axis of all categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Hence, the AP can be calculated by AP = � 1 0 p(r)dr, (20) where p denotes Precision, r denotes Recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The mAP is a comprehensive indicator obtained by averaging APs for all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Moreover, we choose Bit-Operations (BOPs) count [52] and parameters to measure the compression performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The Bops of convolution are calculated as: BOPsl = cl−1 ×cl ×wl ×hl ×kw ×kh ×bw,l ×ba,l−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (21) The hl, wl, and cl are the with, height, and several channels of the l − th layer output feature map, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' bw,l and ba,l denote l − th layer weight and activation bit-weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The parameters (params) are defined as: params = cl−1 × cl × kh × kw × bw,l 8bit (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' (22) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Ablation Study In this section, we conduct the ablation experiments of our GHOST framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We explore how each module (HQ and OST) compresses the model and promotes the performance of the small student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Besides, the experiments of different distillation algorithms and distillation hyperparameters’ opti- mization are carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We conduct ablation experiments on the dataset of VEDAI for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Validation of HQ: Distribution distance hybrid quantization can integrate device n-bit settings for the network, so we experiment with such variations of integrated hybrid n-bit quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' To analyze the performance of differences, we compare fixed DoReFa-Net [8] and various hybrid DoReFa- Net quantization methods on the SuperYOLO detection net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As illustrated in TABLE II, the HQ is the proposed hybrid quantization method, and the ·W · A presents the traditional unified bit width quantization algorithm except 32W32A represents the full precision network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Max and Min denote the maximum and minimum bit width in the quan- tization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Hybrid quantization enables the quantization model to preserve the significant information to achieve the minimal accuracy loss possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As shown in TABLE II, the hybrid quantization module accomplishes the optimal conse- quence and detection accuracy has reached 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='57%, 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='42%, and 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='76% respectively, which are more 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='7%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5%, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='62% than fixed quantization at the different computation orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The hybrid quantization achieves better accuracy in the VEDAI dataset than the accuracy of fixed quantization costing fewer computation resources (parameters and BOPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Effect of OST Module: After the HQ module has been added to the network, we also adopt one-to-one self-teaching within the three-quantization scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Table III is based on SuperYOLO which is used as the teacher network and student network simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The experiments are carried out on the VEDAI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The OST module enables the typical de- tection network to recover the performance of the quantization detection network improved by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='59%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='04%, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='35%, respectively no matter what kind of bit width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Comparison with the SOTA Distillation Method: In addition, Table IV shows the comparisons between the pro- posed OST module and existing distillation frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' OST achieves superior performance in the field of remote sensing under the premise of the same computation cost while the ZAQ, AFD, and ReviewKD lead to an accuracy degradation of the quantization network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' And also it can be proved that OST distillation is a benefit for the guidance between the full- precision model and the quantization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Hyperparameters Optimization: As presented in Section III-C, β is the distillation weights of the OST module, so we compare the performance of the distillation in the different weights which are shown in TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We conduct the hyperparameter experiment on the VEDAI dataset on GHOST to find the best β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As shown in the TABLE V shows, the model reaches the best when β = 400 at the T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='1 and when β = 400 at the T = 4 or T = 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Lightweight Analysis: Owning to the GQSD design idea, our student model is very lightweight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We compare the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, X 2022 11 TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Performance of different algorithms on VEDAI testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Method Car Pickup Camping Truck Other Tractor Boat Van mAP50 mAP Params(MB) BOPs(G) YOLOv3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 71.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='3 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='3 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='93 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='80 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='3 17,024 GHOST 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='82 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='06 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='33 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='96 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='28 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='00 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='76 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='26 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='31 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 692 TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Performance of different algorithms on DOTA, NWPU and DIOR testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' DOTA-v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='0 NWPU DIOR Method mAP50 Params(MB) BOPs(G) mAP50 Params(MB) BOPs(G) mAP50 Params(MB) BOPs(G) Faster R-CNN 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='64 240 296,192 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='80 164 130,764 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='10 240 186,572 RetainNet 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='39 221 300,400 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='40 145 126,228 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='70 221 184,954 YOLOv3 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='00 246 203,694 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='30 246 124,180 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='10 247 125,153 GFL 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='53 76 163,000 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='80 76 93,931 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='00 76 99,768 FCOS 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='72 126 207,001 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='65 127 119,429 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='60 127 126,474 ATSS 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='84 75 159,754 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='50 75 92,057 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='70 75 97,792 MobileNetV2 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='91 41 127,221 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='90 41 73,205 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='20 41 77,926 ShuffleNet 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='73 48 146,022 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='00 48 84,142 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='30 48 89,405 O2-DNet 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='10 836 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='3 836 FMSSD 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='43 544 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 544 Teacher 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='65 203 291,491 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='21 127 122,234 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='7 203 178,176 ARSD 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='28 -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='37 52 69,662 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='92 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='29 46 27,289 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='10 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='6 52 42,598 Teacher 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='99 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='8 21,390 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='30 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='7 21,357 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='95 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='8 21,428 GHOST 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='97 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='7 2,146 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='97 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='33 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 1,927 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='53 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='3 2158 GHOST and the teacher model in terms of parameters and BOPs on the four datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 6 shows, GHOST has smaller model parameters, and fewer BOPs compared with the teacher model no matter in which dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Hence, our com- pression strategy for the lightweight model is more practical to be deployed on intelligent terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Comparison with the SOTA Detectors In this part, we compare our lightweight model with other classic heavy object detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As shown in TABLE VI and TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Experiments on the four datasets prove the efficiency and efficacy of the GHOST framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Not only does our lightweight have higher accuracy but also it has strong information retention capability under extreme model compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 1) VEDAI: Our GHOST achieves 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='31% mAP50 compared with other detectors, surpassing the one-stage series mentioned in TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Our model achieves the lowest model parame- ters (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 MB) and BOPs (692 G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 2) DOTA: As presented in TABLE VII, our GHOST achieves the optimal detection result (69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='02% mAP50) and the model parameters (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='7 MB) and BOPs (2,146 G) are much smaller than other SOTA detectors regardless of the two-stage, one-stage, anchor-free or distillation-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We also compare two detectors designed for remote sensing imagery such as FMSSD [50] and O2DNet [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Although these models have a close performance with our lightweight model, the huger parameters and BOPs seem to be a massive cost in computation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Hence, our model has a better balance in consideration of detection efficiency and efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Compare to the distillation-based method ARSD (-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='37%), the GHOST obtains an even smaller accuracy gap (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='97%) between the student network and the teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' It demonstrates that our GHOST method can transfer sufficient knowledge to guide the learning of the student model and the misunderstanding between both models can be reduced by the SCM training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 3) NWPU: We compare the results of our method with other approaches on the NWPU dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' As shown in TABLE VII, IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' X, X 2022 12 our GHOST obtains the best result (91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='97% mAP50) with the smallest amount of model parameters (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='5 MB) and the fewer BOPs (1,927 G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 4) DIOR: As illustrated in TABLE VII, our GHOST achieves the optimal detection result (69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='02% mAP50) and the model parameters (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='3 MB) and BOPs (2,158 G) are much smaller than other SOTA detectors regardless of the two-stage, one-stage, anchor-free lightweight, distillation-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' It reveals the strong ability to compress models and the power capacity of object detection in remote sensing imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The accuracy of the student GHOST is only a bit less 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='4% than teacher network, compared with ARSD (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content='6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' In order to show the performance of our algorithm more intuitively, We compare the detection accuracy, parameters, and BOPs of various algorithms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' It can be obvious that GHOST has a better trade-off between performance and lightweight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The visualization results on the three datasets are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' 8 in which we can see that the GHOST can have an outstanding detection of objects at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' CONCLUSION In this paper, we propose a GHOST framework for a lightweight object detection method in remote sensing im- agery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' We first design a guided quantization self-distillation structure which is not only a training technique to preserve model performance but also a method to compress and accel- erate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Although most of the previous research focuses on knowledge transfer among different models, we believe that inside distillation is also very promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Secondly, we propose a hybrid quantization that captures the optimal bit width selection based on an adaptive way in the weight value research space to break the limit of the fixed quantization model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Thirdly, the proposed one-to-one self-teaching module gives the student network of self-judgment through a switch control machine that accurately handles the knowledge transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' It can dynamically discriminate the wrong guidance and mine the effective knowledge from the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' The experiments based on the VEDAI, DOTA, NWPU, and DIOR datasets certify that our GHOST achieves SOTA perfor- mance compared with other detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' It can well balance the tradeoff between accuracy and specific resource constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Ding, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtAyT4oBgHgl3EQfUvdy/content/2301.00131v1.pdf'} +page_content=' Xue, Y.' metadata={'source': 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a/PtFPT4oBgHgl3EQfnzX3/content/tmp_files/2301.13132v1.pdf.txt b/PtFPT4oBgHgl3EQfnzX3/content/tmp_files/2301.13132v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6aac850523b549ab7c57b8a34238ea2b354a00b7 --- /dev/null +++ b/PtFPT4oBgHgl3EQfnzX3/content/tmp_files/2301.13132v1.pdf.txt @@ -0,0 +1,1481 @@ +Synthetic-reflection self-injection-locked microcombs +Alexander E. Ulanov,1 Thibault Wildi,1 Nikolay G. Pavlov,2 +John D. Jost,2 Maxim Karpov,2 Tobias Herr1,3,* +1Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany +2Enlightra Sarl, Rue de Lausanne 64, 1020 Renens, Switzerland +3Physics Department, Universit¨at Hamburg UHH, Luruper Chaussee 149, 22607 Hamburg, Germany +∗tobias.herr@desy.de +Laser-driven +microresonators +have +enabled +chip-integrated light sources with unique prop- +erties, including the self-organized formation of +ultrashort soliton pulses and frequency combs +(microcombs). +While +poised +to +impact +ma- +jor photonic applications, such as spectroscopy, +sensing and optical data processing, microcombs +still necessitate complex scientific equipment to +achieve and maintain suitable single-pulse opera- +tion. Here, to address this challenge, we demon- +strate microresonators with programmable syn- +thetic reflection providing an injection-feedback to +the driving laser. +When designed appropriately, +the synthetic reflection enables robust access to +self-injection-locked microcombs operating exclu- +sively in the single-soliton regime and with low- +threshold power. These results provide a route to +easily-operable microcombs for portable sensors, +autonomous +navigation, +or +extreme-bandwidth +data processing and represent a novel paradigm +that can be generalized to other integrated pho- +tonic systems. +Laser-driven microresonators provide access to non- +linear +optical +phenomena, +already +with +low-power +continuous-wave excitation [1]. Leveraging efficient non- +linear frequency conversion, they have enabled novel +sources of coherent laser radiation across broad spectral +span [2, 3]. +Soliton microcombs[4–6] are an important +representative of such sources, providing frequency comb +spectra of mutually coherent laser lines, based on self- +organized dissipative Kerr solitons (DKSs) in resonators +with anomalous dispersion [7]. Such DKS microcombs can +be integrated on photonic chips [8, 9] and have demon- +strated their disruptive potential in many emerging and +ground-breaking applications, e.g. high-throughput opti- +cal data transmission [10] reaching Pbit-per-second data +rates [11], ultrafast laser ranging [12, 13], precision as- +tronomy in support of exo-planet searches [14, 15], high- +acquisition rate dual-comb spectroscopy [16], ultra-low +noise microwave photonics [17, 18], photonic computing +and all-optical neural networks [19–21]. To leverage mi- +crocomb technology in out-of-lab applications, it is critical +to reliably access the DKS regime and ideally single-DKS +operation [22], ensuring well-defined temporal and spec- +tral characteristics. While routine in research laborato- +ries, achieving such a state outside such environments is +challenging. +A critical challenge lays in the initiation and sustained +operation of DKS, requiring the detuning ∆ω0 = ω0 − ωp +of the pump laser ωp (with respect to the pumped res- +onance ω0) to be controlled and stabilized. While this is +common to all resonant approaches, it is particularly chal- +lenging during DKS initiation, when thermo-optic effects +can cause a rapid (∼ µs) change in resonance frequency +[4]. +To overcome this challenge, a number of methods +have been developed, involving rapid laser actuation [4, +8], auxiliary lasers [23] and/or auxiliary resonances [24, +25], laser modulation [26], additional nonlinearities [27–29] +or, pulsed driving [30]. Many of these methods are now +routinely used in research. However, they cannot easily be +transferred to out-of-the-lab scenarios, as they require sig- +nificant experimental skills and scientific instrumentation. +In contrast, self-injection locking (SIL) [31–33], has been +demonstrated as an approach that can intrinsically follow +the rapid changes in resonance frequency and elegantly +stabilize the laser detuning for stable DKS operation[17, +34–38]. +Usually, SIL is based on Rayleigh backscatter- +ing from random fabrication imperfections or material de- +fects in the microresonator [39]. The backscattered wave +provides feedback (injection) to the driving diode laser +and effectively locks the laser frequency to the microres- +onator. However, backscattering random defects are nei- +ther wanted nor can they yield predictable sample charac- +teristics. Relying on random defects is also fundamentally +incompatible with the intense efforts towards improved +materials and fabrication techniques (targeting material +absorption limited performance with negligible scattering +similar to optical fiber technology). Already now, fabrica- +tion techniques have advanced to a level, where identifying +samples with accidental scattering suitable for SIL-based +DKS often requires careful and tedious screening. +In this work, we demonstrate SIL and robust access to +self-injection locked DKS microcombs without relying on +random resonator defects. Instead of random backscatter- +1 +arXiv:2301.13132v1 [physics.optics] 30 Jan 2023 + +ing, we achieve SIL via programmable synthetic reflection, +dramatically increasing the access to laser detunings that +support DKS. The synthetic reflection is generated via +photonic crystal ring resonators (PhCR) [40], which have +recently received growing attention in integrated nonlin- +ear photonics [41–45]. In addition, we show that robust +access to SIL-based DKS can be combined with recent +results of spontaneous single-DKS generation in PhCRs +(avoiding non-solitonic states) [41]. Based on analytic cri- +teria, we design the synthetic reflection to ensure exclu- +sive operation in the single-DKS regime as well as low- +threshold power. Resulting from the synthetic reflection +we also observe DKS breathing [41, 46] in a limited range +of operating parameters, which can readily be avoided, if +necessary. These results provide a route to easily-operable +microcombs for out-of-lab applications. +Results +To gain independence from backscattering random de- +fects and imperfections, we use PhCRs that enable syn- +thetic reflection. The reflection is controlled by periodic +nano-patterned corrugations of the ring-resonators’ inner +walls. The angular corrugation period is θ0 = 2π/(2m0), +where m0 is the angular (azimuthal) mode number, for +which a deliberate coupling between forward and back- +ward propagating waves with a coupling rate γ is induced +(see Fig. 1a). Besides inducing the desired synthetic reflec- +tion, the coupling leads to mode hybridization resulting in +a split resonance lineshape (frequency splitting 2γ) in both +transmission and reflection (see Fig. 1b). Here, we only +consider the lower frequency hybrid mode for pumping, as +it corresponds to strong (spectrally local) anomalous dis- +persion, which prevents high-noise comb states [47]. For +choosing γ we balance multiple criteria. +First, a strong reflection can significantly extend the +range of normalized detunings ζ0 = 2∆ω0/κ (κ is the mi- +croresonator linewidth) accessible via SIL (SIL range) in a +nonlinear microresonator [37]. This is crucial, as it permits +robust access to detunings where DKS can exist (DKS ex- +istence range). In conventional resonators the normalized +forward-backward coupling is usually small 2γ/κ < 1 and +the intersection between SIL and DKS ranges is limited +to small detunings, complicating access to DKS states. In +contrast, strong forward-backward coupling could enable +robust access to DKS states over a wide range of detun- +ings. This is exemplified in Fig. 1c, where the SIL range +[37] is shown along with the conventional analytic DKS ex- +istence range (valid for small γ) and the numerically com- +puted DKS existence range for large γ, obtained through +numeric integration of the coupled mode equations (cf. +Methods). Note, that in a resonator with a shifted pump +mode [49], the existence range of DKS deviates strongly +from that known from resonators without a shifted pump +mode [48] and can currently only be obtained numerically +(Fig. 1c). +Second, while advantageous for an extended SIL range, +ω0 +PhCR +γ +θ0 +laser diode +T +R +synthetic +reflection +a +b +microcomb +CW +T +R +m0 + 1 +m0 +m0 - 1 +FSR 2π +2γ +2 mm +d +c +1 +3 +4 +0 +5 +10 +Backscattering 2γ/κ +Laser detuning ζ0 +DKS range (num.) +DKS range (no splitting, analyt.) +SIL range +2 +Angular frequency ω +Figure 1 | Self-injection locking with synthetic reflection. a, +An integrated photonic crystal ring-microresonator (PhCR) with a +periodic corrugation (angular period θ0), which induces coupling +at a rate γ between forward and backward-propagating waves for +a mode m0 = π/θ0. In addition to a transmission signal (T), this +leads to a well-defined synthetic resonant reflection (R), which +can be programmed for self-injection locking with a laser diode +driving the system. b, Indicative transmission and reflection spec- +trum for the resonance with mode number m0 and two adjacent +resonances m0±1, separated by ±1 free-spectral range (FSR). For +γ ̸= 0, the lineshape at mode number m0 exhibits a split lineshape +(frequency splitting 2γ), and shows non-zero resonant reflection. +c, Comparison of nonlinear SIL [37] and DKS existence ranges +computed numerically for a critically coupled microresonator with +total linewidth κ/2π = 120 MHz, dispersion D2/2π = 8 MHz, +driven with a normalized pump power of f 2 = 9. For large γ, +the DKS existence range deviates significantly from the analytical +estimation for zero-γ [47, 48]. d, Photograph of the experimen- +tal system showing the semiconductor laser diode butt-coupled +to the photonic chip carrying the PhCRs. +Transmitted light is +out-coupled using a lensed fiber. +stronger forward-backward coupling will also result in an +increased parametric threshold (modulation instability, +2 + +MI) pump power, as detailed in the Supplemental Infor- +mation (SI). Below this threshold DKS cannot form inside +the resonator, without external stimuli (such as triggering +pulses [7]). The threshold power is different from that in +a conventional ring resonator and its derivation critically +requires consideration of the backward wave. For strong +forward-backward coupling (2γ/κ > 1), the following ap- +proximation is derived (cf. SI): +f 2 +th = 4γ +κ + κ +γ +(1) +where f += +� +8ηω0cn2P/(κ2n2Veff) is the normalized +pump power, with the coupling coefficient η = 1/2 (criti- +cal coupling), ω0 the resonance frequency of the pumped +mode, c the speed of light, P the input pump power, n +the refractive index, n2 the nonlinear refractive index and +Veff the effective mode volume. The value of f 2 +th must not +exceed the available pump power f 2. If the MI threshold +is reached at a detuning within the DKS existence range, +then the MI state may be only transient and DKS can +form spontaneously [41]. In both conventional and pump +mode-shifted resonators the DKS regime overlaps with the +MI regime and extends further towards larger detunings +ζ0. +Third, with regard to practical applications single-DKS +states, as opposed to states with multiple solitons, are +highly desirable owing to their smooth squared hyperbolic +secant spectral envelope and well-defined temporal out- +put. In their formation process, DKS are seeded by MI, +where the separation of the first pair of sidebands from the +pump laser in units of the resonator’s FSR determines the +number of generated DKS [4, 41, 47]. A conservative cri- +terion that guarantees single-DKS formation (i.e. modu- +lation instability sidebands separated from the pump laser +by 1 FSR) is derived in the SI: +γ +κ > f 2 +8 +(2) +The presented considerations can inform the design of +a suitable PhCR for SIL-based DKS. +In preparation of the experiments, a range of critically +coupled resonators with varying corrugation amplitude +and a free-spectral range (FSR) of 300 GHz (radius 75 µm) +are fabricated in a commercial foundry process (Ligentec). +We characterize the fabricated resonators via frequency +comb-calibrated laser scans [50], permitting to retrieve the +coupling rates γ, the resonance widths κ, and the disper- +sion D2, over a broad spectral bandwidth. An example +is shown in Fig. 2a, where indeed the forward-backward +coupling is random and 2γ/κ ≪ 1 for most resonances. In +marked contrast, a single pre-defined resonance to which +the PhCR’s corrugation is matched, exhibits significant +forward-backward coupling. Fig. 2b shows the dependence +of γ and the Q-factor (Q = ω0/κ) on the corrugation am- +plitude. No noticeable degradation of the Q factor is ob- +served up to γ ≲ 5 GHz, and critically coupled linewidth +are +κ +2π ≈ 120 MHz; even for large coupling γ ≈ 45 GHz , +the Q-factor is only halved. For the experiments, a semi- +conductor distributed feedback laser diode (DFB) is butt- +coupled to a waveguide on the photonic chip, permitting +an estimated on-chip pump power of P = 30 mW, cor- +responding to f 2 ≈ 9. +From Eqs. 1 and 2, we obtain +an ideal 2γ/κ ∈ (2.26, 4.26), ensuring MI-based sponta- +neous comb initiation and deterministic generation of sin- +gle DKS. Based on these considerations we choose a PhCR +with a synthetic coupling for the pump mode at 1557 nm of +2γ/κ ≈ 4.2 ( γ +2π ≈ 250 MHz), within the ideal range. This +PhCR is critically coupled and exhibits anomalous group +velocity dispersion (D2 ≈ 8 MHz). As shown for those val- +ues in Fig. 1c, numeric simulation confirms that the DKS +existence and SIL ranges have significant overlap. We note +that another band of DKS existence may exist [49], how- +ever, it is inaccessible for spontaneous MI-assisted comb +initiation and not considered here. The DFB pump laser +diode is mounted on a piezo translation stage to adjust +the injection phase [33], an actuator which can readily be +achieved through on-chip heaters [38]; to reduce the de- +vice footprint and allow for more resonators on the chip, +we have omitted this feature. +The transmitted light is +collected by a lensed-fiber for further analysis as shown in +Fig. 2f. +In a first experiment, we validate the basic SIL dynamics +below parametric threshold at a coupled pump power of +25 mW (f 2 = 7.3). As long as the laser diode does not re- +ceive a resonant injection from the microresonator it is free +running. When the laser’s emission wavelength is tuned +(via its drive current) close to the lower-frequency pump +resonance, a strong resonant backward wave is generated, +providing frequency-selective optical feedback resulting in +SIL. The SIL regime manifests itself as a rectangular- +shaped dip in the transmission signal and, after optimiz- +ing the injection phase, extends over a wide range of elec- +trical drive current values. The optical spectrum of the +DFB laser in the SIL regime is shown in Fig. 2e, show- +ing a single-mode suppression ratio (SMSR) >60 dB. The +beatnote of the SIL laser with a table-top low-noise CW +laser is shown in Fig. 2d. In addition, we record the SIL- +laser phase noise (Fig. 2c), which is drastically lower than +that of the free-running DFB laser diode outside the SIL +regime. +In a second experiment, utilizing the same setup as +in Fig. 2f, we explore DKS-based microcomb generation +with the full available pump power (30 mW, f 2 ≈ 9). +Similar to the previous lower power SIL experiment, we +slowly (within ca. 10 s) tune the DFB’s electrical drive +current to scan the emission wavelength across the lower +frequency pump resonance, with increasing and then de- +creasing wavelength. During this scan we monitor the op- +tical spectrum in transmission. We note that the exact +tuning curve in the nonlinear SIL regime when increasing +(decreasing) the DFB pump current follows a nontrivial +behavior that may include non-monotonic sections [37]; +the scan outside the SIL range is however monotonic in +frequency. Upon entering the SIL regime (again marked +3 + +a +b +Heterodyne detection +CW +PD +ESA +LD +CC +PhCR +SIL Setup +PD +OSC +OSA +Wavelength (nm) +1552 +1556 +1560 +Power (20 dB/div) +~60 dB +SIL laser +f +Frequency +10 MHz / div +Power (20 dB/div) +RBW +5 KHz +Free +SIL +SSB Phase Noise (dBc/Hz) +10 KHz +100 KHz +1 MHz +Frequency +c +d +e +Corrugation amplitude +1 +2 +0 +0 +15 +30 +Coupling rate γ/2π (GHz) +Q-factor (million) +45 +Frequency +500 MHz / div +Wavelength (nm) +PhCR +induced +Backscattering 2γ/κ +random +scattering +1540 +1560 +1580 +1600 +0 +5 +3 +1 +transmission +reflection +1 +0 +1 +free-running +SIL laser +ref. laser +−120 +−80 +−40 +0 +Figure 2 | Resonator characterization and low-power SIL. a, The single predefined resonance (red dot) to which the corrugation +pattern is matched, exhibits significant forward-backward coupling unlike the other modes (blue dots) where it is weak and random. +b, Measured forward-backward coupling rates (red, left axis) and Q-factors (blue, right axis) of PhCRs with increasing corrugation +amplitude. c, Phase noise of the DFB laser in free-running and SIL regimes measured through heterodyne detection with a reference +laser. The phase noise of the reference laser is provided as a baseline. d, Heterodyne beatnote signal between the reference oscillator +and DFB laser in the free-running and SIL states. RBW, resolution bandwidth. e, Optical spectrum of the DFB laser in the SIL state. +f, Experimental setup. CC, current controller; LD, laser diode; OSA and ESA, optical and electrical spectrum analyzers respectively; +OSC, oscilloscope; CW, continuous-wave laser; PD, photodiode. +by pronounced dip of the transmitted power after opti- +mization of the injection phase), we observe at first only +the single optical frequency of the SIL pump laser, as in +the lower power experiment before (Fig. 3a 1 ). +Con- +tinuing the scan we next observe an abrupt transition +into a single-DKS microcomb state (Fig. 3a 2 ). +Such +single-DKS states are characterized by a smooth squared +hyperbolic-secant amplitude and a pulse repetition rate +that corresponds to the resonator’s FSR; these properties +are highly-desirable for applications. Further continuing +the scan induces a surprising second abrupt transition into +a different single-DKS state (Fig. 3a 3 ). Scanning even +further causes the DKS to disappear, with the system re- +turning to CW SIL (spectrum similar to Fig. 3a 1 ), before +eventually exiting the SIL regime entirely. When repeated, +each scan shows the same SIL dynamics, including deter- +ministic single-DKS generation. Reversing the scan direc- +tion qualitatively yields the same phenomena in reversed +order. Turing patterns, noisy comb-states and multi-DKS +regimes are absent in stark contrast to previous SIL-based +microcomb generation, but consistent with spontaneous +single-DKS formation in conventionally-driven (non-SIL) +PhCR [41]. Although not pursued here, we note that the +pump to DKS conversion efficiency in the states 2 and 3 +is 13.8 % and 15.2 %, resp., significantly higher than what +would be expected in conventional resonators. This is a +consequence of the mode splitting, shifting the pumped +resonance effectively closer to the pump laser as explored +previously in coupled ring-resonators [49]. +Different +from +DKS +generation +experiments +in +conventionally-driven PhCR, laser tuning speed and +even tuning direction are irrelevant and do not notice- +ably impact the observed dynamics. +Owing to the SIL +mechanism, switching between the system’s states is +readily possible and without the risk of ‘dropping’ out of +resonance. Each state of the system can be maintained +without requiring external stabilizing feedback. +All +observations are reproduced in all 4 tested copies of the +PhCR (fabricated on 4 different chips). As such, through +synthetic reflection, our system not only achieves robust +and predictable SIL operation, but also leverages the +advantages of spontaneous and deterministic single-DKS +generation, observed in conventionally driven PhCRs [41]. +To further investigate the SIL dynamics and DKS gen- +4 + +3 +2 +3 +2 +0.4 +0.2 +0 +-0.2 +-0.3 +-0.6 +Rep. rate − 300.3 (GHz) +Power (a.u.) +0 +1 +Current + 20 mA/div +1 +Time (2 s/div) +1 +0 +Power (a.u.) +transmission +filtered transm. +DFB current +transmission +filtered transm. +DFB current +Time (2 s/div) +multi- +DKS +single-DKS +Current + 20 mA/div +c +d +e +1700 +Wavelength (nm) +1450 +1500 +1550 +1600 +EOM +ESA +PD +SIL Setup +FBG +repetition rate detection +1 +2 +3 +Power (20 dB/div) +SIL DFB +Single DKS +Breather DKS +a +b +sech2 Fit +increasing current +decreasing current +Eff. ~ 13.8 % +sech2 Fit +Eff. ~ 15.2 % +Figure 3 | SIL-based DKS generation. a, Optical spectra measured during a laser scan towards longer wavelength within the SIL +range: CW SIL 1 ; SIL-based single DKS states 2 and 3 . b, Experimental setup for repetition rate detection. EOM - electro- +optical modulator, FBG - fiber Bragg grating, PD - photodetector, ESA - electrical spectrum analyzer. c, Measured SIL-microcomb +repetition rate signal representing single DKS 2 and breather 3 states. d Total transmission (blue) and bandpass-filtered power +(red; filter offset from the pump, indicates comb formation) measured during a high-power laser scan with a PhCR. The orange line +corresponds to the driving current. e, same as d, but with a conventional resonator that has been selected for a relatively strong +random backscattering (cf. main text for details). +eration, we record the 300 GHz DKS repetition rate beat- +note via the setup shown in Fig. 3b. +As this signal +would not be directly detectable, modulation sidebands +around a pair of adjacent DKS comb lines are generated +electro-optically. Their beating creates a signal at lower +frequency, from which the repetition rate can be recon- +structed [51]. Fig. 3c shows the reconstructed repetition +rate signal obtained during the DFB laser scan in both di- +rections. The two distinct spectral regimes are also man- +ifest in this signal: in regime 2 a single low-noise repe- +tition rate beatnote is present, whereas in regime 3 ad- +ditional sidebands (ca. ±200 MHz) indicate a breathing +soliton, similar to so far unexplained breathing phenom- +ena in conventionally-driven PhCRs at large detuning [41]. +In the wavelength-decreasing scan, the reversed dynamics +is observed (the additional breathing towards the end of +the scan is well-known from conventionally driven DKS +[46]). +During the laser scan, we also record the transmitted +power as well as the power of a filtered spectral portion +of the long-wavelength wing of the generated microcombs +(as an indicator for comb formation). +Both are shown +along with the DFB diode’s drive current in Fig. 3d. Here, +the SIL regime is clearly evidenced by sharp drops of the +(full) transmission from the base level in both scan direc- +tions. The DKS regime is marked by the non-zero filtered +power within the SIL regime. Note that in Fig. 3d the +breathing oscillation is only visible in the recorded power +for the lowest breathing frequencies, due to the limited +100 MHz bandwidth of the utilized photo-detectors. For +comparison, Fig. 3e shows a similar transmission and fil- +tered power trace obtained with a non-PhCR microres- +onator of the same FSR. To achieve SIL-based DKS in +a non-PhCR, we screened a large number of resonators +to find one with large random reflection (2γ/κ ≈ 0.6) and +validated numerically that it can meet the criteria for SIL- +based DKS (similar to Fig. 1c). In contrast to the PhCR, +it does however not respect the criterion for deterministic +single-DKS. Indeed, different from the SIL dynamics in +the PhCR, noisy comb states before (after) DKS genera- +tion are observable in the laser scan and the DKS-regime +is dominated by less desirable multi-DKS states, showing +the characteristic multi-step features [52]; the single-DKS +regime is marginal and only visible as a narrow step when +the scan time is reduced to the millisecond level. +To explore the emergence of the breathing soliton at +5 + +a +b +c +Comb repetition rate (a.u.) +Kerr shiftts\detuning +(units of κ/2) +−6 +12 +Time (a.u.) +Intracavity +power (a.u.) +comb fwd +pump fwd +comb bwd +pump bwd +laser detuning +Kerr shift (bwd) +Kerr shift (fwd) +Kerr shift (hybrid modes) +0 +Figure 4 | SIL-microcomb dynamics simulation. a, The time- +frequency spectrogram reconstructed from the numerical simula- +tion during a wavelength-increasing scan of the pump laser. b, +Corresponding intracavity powers for forward (beige) and back- +ward (red) propagating microcomb modes (µ ̸= 0) as well as +forward (orange) and backward (blue) propagating pump modes +(µ = 0). c, Kerr-shifts of forward (orange) and backward (blue) +propagating pump modes with respect to the ‘cold’ resonance +frequency ω0. Detuning of pump laser (beige). The dark lines +indicate the effective hybrid modes frequencies (cf. main text and +SI for details). +higher DFB currents (regime 3 in Fig. 3c,d), we per- +form numeric simulation [53, 54], where we include forward +and backward propagating waves (cf. Methods). Starting +from a single DKS state at small detuning (within the +well-known breather regime [46]), we integrate the cou- +pled mode equations while imposing a slow laser scan to- +wards longer wavelength. The spectrogram of the repe- +tition rate signal obtained from the numerical simulation +is shown in Fig. 4a; it reproduces an abrupt appearance +of sidebands in agreement with the experimental results +of Fig. 3c. +The average power levels of the pump and +comb (full spectrum without pump) in both forward- and +backward-direction show rapid oscillations Fig. 4b, coin- +ciding with the breathing state. The phase-lag between +comb and pump mode suggesting an oscillatory power flow +between both, reminiscent of the detuning dependent in- +termode breathing dynamics [55], observed in DKS in the +presence of coupling between a DKS mode to a frequency +degenerate higher-order transverse mode. +The latter is +absent in our case, however, by design a strong coupling +exists between forward and backward pump modes. Al- +though, up to a certain power level, the strongly coupled +forward and backward modes are frequency degenerate nu- +meric simulations suggest that this degeneracy is not gen- +erally preserved (similar to CW symmetry breaking [56]) +in the DKS regime and dependent on the state of opera- +tion. Based on the simulation, we derive the nonlinear fre- +quency shifts of forward and backward propagating pump +waves as well as those of the hybridized modes (cf. SI) +(Fig. 4c). Strikingly, find that the regime of the breathing +oscillation coincides with a situation where the differen- +tial nonlinear shifts between forward and backward modes +are small compared to the resonance width, suggesting +forward-backward coupling as its origin. Importantly, the +phenomenon is restricted to a specific parameter regime, +and can therefore easily be circumvented by choosing the +operating parameters (notably pump power or detuning) +accordingly. +Conclusion +In conclusion, we have demonstrated microresonators with +synthetic reflection for robust SIL and SIL-based micro- +comb generation. By designing the synthetic reflection we +enable deterministic low-threshold, single-DKS operation, +highly-favorable for applications and address a major chal- +lenge of this technology. For certain combinations of pump +power and detuning we observe DKS breathing, which can +readily be avoided by small changes in the operating pa- +rameters, if necessary. The presented results in conjunc- +tion with the scalable, widely accessible fabrication pro- +cess, the low-cost components, and, notably, its ease of +operation meet important requirements of out-of-lab ap- +plications and extension to other integrated photonic sys- +tems, including normal dispersion combs [52, 57] and novel +quantum light sources [58, 59]. Further research may ex- +plore extending the presented results to combs in the back- +ward direction (cf. SI) [60], effectively blue-detuned DKS +combs with potentially even higher conversion efficiency +[49] or multiple pump wavelengths [61]. +Materials +Numerical model. +To simulate the nonlinear dynamics +we consider a system of coupled mode equations [53, 54] for +forward aµ and backward bµ mode amplitudes, where µ denotes +the relative (longitudinal) mode number with respect to the +pump mode (m0 ↔ µ = 0): +∂taµ = − (1 + iζµ)aµ + i +� +µ′=ν+η−µ +aνaηa∗ +µ′ + 2iaµ +� +η +|bη|2+ ++ iδµ0 2γ +κ bµ + fδµ0 +∂tbµ = − (1 + iζµ)bµ + i +� +µ′=ν+η−µ +bνbηb∗ +µ′ + 2ibµ +� +η +|aη|2+ ++ iδµ0 2γ +κ aµ +where ζµ = +2 +κ(ωµ − ωp − µD1) is a dimensionless detuning +between the pump laser frequency ωp and cold resonance fre- +quencies ωµ = ω0 + D1µ + 1 +2D2µ2 (with D1/2π and D2/2π +6 + +being microresonator FSR and second-order dispersion respec- +tively). The third term in each equation corresponds to the +cross-phase modulation by the respective counter-propagating +waves, while the fourth term represents the coupling between +forward and backward propagating waves. Instead of modeling +the SIL dynamics by including laser rate equations, we numer- +ically define the detuning. This approach cannot describe the +abrupt transition from the free-running laser to the SIL state, +it remains however valid for the specified detuning and can +qualitatively capture the features observed in the experiment. +Simulation parameters similar to those of the experimental sys- +tem are used. +Funding +This project has received funding from the European Research +Council (ERC) under the EU’s Horizon 2020 research and inno- +vation program (grant agreement No 853564), from the EU’s +Horizon 2020 research and innovation program (grant agree- +ment No 965124) and through the Helmholtz Young Investiga- +tors Group VH-NG-1404; the work was supported through the +Maxwell computational resources operated at DESY. +Data availability +The datasets generated and analysed during the current study +are available from the corresponding author on reasonable re- +quest. +Code availability +Numeric simulation codes used in the current study are avail- +able from the corresponding author on reasonable request. +Disclosures +The authors declare no competing interests, however disclose +that J.D.J. and M.K. are cofounders of Enlightra. +References +1. +Vahala, K. 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Optics Ex- +press 30, 17094 (2022). +9 + +Synthetic-self-injection locked microcombs - +Supplemental Information +Alexander E. Ulanov,1 Thibault Wildi,1 Nikolay G. Pavlov,2 +John D. Jost,2 Maxim Karpov,2 Tobias Herr1,3,* +1Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany +2Enlightra Sarl, Rue de Lausanne 64, 1020 Renens, Switzerland +3Physics Department, Universit¨at Hamburg UHH, Luruper Chaussee 149, 22607 Hamburg, Germany +∗tobias.herr@desy.de +1 +Coupled mode equations and pump mode hybridization +The normalized coupled mode equations (CMEs) for the pump in forward and backward directions read +∂ta0 = −(1 + iζ0)a0 + i|a0|2a0 + 2i|b0|2a0 + iβb0 + f +(1) +∂tb0 = −(1 + iζ0)b0 + i|b0|2b0 + 2i|a0|2b0 + iβa0 +(2) +where for convenience the normalized coupling rate β = 2γ/κ ≥ 0 has been introduced. Without loss of generality, f ≥ 0. +The coefficient matrix of the system of equations (without the pump) +M = +�−(1 + iζ0) + i|a0|2 + 2i|b0|2 +iβ +iβ +−(1 + iζ0) + i|b0|2 + 2i|a0|2 +� +(3) +has the following Eigenvalues +λ± = − +� +�1 + iζ0 − 3 +2i(|a0|2 + |b0|2) ± i +� +β2 + +�1 +2(|a0|2 − |b0|2) +�2 +� +� +(4) += − +� +1 + i(ζ0 − δζNL ± +� +β2 + δβ2 +NL) +� +(5) +and is diagonalized in the following Eigenbasis of hybridized forward-backward modes +� +� +� +1 +2(|a0|2 − |b0|2) ± +� +β2 + +�1 +2(|a0|2 − |b0|2) +�2 +; −β +� +� +� +(6) += +� +δβNL ± +� +β2 + δβ2 +NL; −β +� +(7) +where δβNL = 1 +2(|a0|2 − |b0|2) and δζNL = 3 +2(|a0|2 + |b0|2). The transformation matrices are: +T = +� +δβNL + +� +β2 + δβ2 +NL +δβNL − +� +β2 + δβ2 +NL +−β +−β +� +(8) +and +T −1 = +� +� +� +1 +2√ +β2+δβ2 +NL +δβNL−√ +β2+δβ2 +NL +2β√ +β2+δβ2 +NL +− +1 +2√ +β2+δβ2 +NL +−δβNL−√ +β2+δβ2 +NL +2β√ +β2+δβ2 +NL +� +� +� +(9) +so that +�a0 +b0 +� += T +�a+ +a− +� +(10) +10 + +and +�a+ +a− +� += T −1 +�a0 +b0 +� +(11) +where a± denote the (not specifically normalized) field amplitudes of the hybrid modes. The steady state equations for the +hybrid modes are +0 = − +� +1 + i(ζ0 − δζNL + +� +β2 + δβ2 +NL) +� +a+ + +1 +2 +� +β2 + δβ2 +NL +f +(12) +0 = − +� +1 + i(ζ0 − δζNL − +� +β2 + δβ2 +NL) +� +a− − +1 +2 +� +β2 + δβ2 +NL +f +(13) +In non-normalized units, the effective resonance frequencies of the hybridized modes are +ω±,eff = ω0 − κ +2 +� +δNL ± +� +β2 + δβ2 +NL +� +(14) +2 +Approximations for the forward pump mode under strong coupling +In what follows, it is assumed that +• the coupling is strong β > 1 +• due to the strong coupling, the power levels in forward and backward directions are approximately equal |a0|2 = |b0|2. Note +that due to symmetry breaking [56] this is only valid up to a certain power level. We validated, by numeric integration of +the CMEs 1 and 2, that this approximation is valid. +• the detuning ζ0 is such that approximately only the lower frequency hybrid mode a− is driven, i.e. |a−| ≫ |a+|. +Under these assumptions, +a0 = +� +δβNL − +� +β2 + δβ2 +NL +� +a− ≈ −βa− +(15) +b0 = −βa− +(16) +and in consequence +(1 + i(ζ0 − δζNL − β))a0 = f +2 +(17) +Multiplying each side of the equation with its complex conjugate results in +(1 + (ζ0 − δζNL − β)2)|a0|2 = f 2 +4 +(18) +An immediate insight is that the strong coupling between forward and backward waves limits the power in the forward (or +backward) wave to values of +|a0|2 ≤ f 2/4 +(19) +Expressing δζNL via the field amplitudes gives +(1 + (ζ0 − 3|a0|2 − β)2)|a0|2 = f 2 +4 +(20) +and for the detuning +ζ0,± = β + 3|a0|2 ± +� +f 2 +4|a0|2 − 1 +(21) +where ζ0,+ corresponds to an effective red-detuning and ζ0,− to an effective blue-detuning with regard to the lower-frequency +hybrid mode a−. +11 + +3 +Threshold condition and first oscillating sideband +We consider two initially zero-power (except for vaccuum fluctuations) sidebands with mode number ±µ relative to the pumped +mode. Their CMEs are +∂ta+µ = −(1 + i(ζµ − 4|a0|2))a+µ + ia2 +0a∗ +−µ +(22) +∂ta∗ +−µ = −(1 − i(ζµ − 4|a0|2))a∗ +−µ − ia∗2 +0 a+µ +(23) +where again |a0|2 ≈ |b0|2 was assumed and ζµ = 2 +κ(ω0 − ωp + 1 +2D2µ2) = ζ0 + D2 +κ µ2. The Eigenvalues of this set of equations are +λ± = −1 ± +� +|a0|4 − (ζµ − 4|a0|2)2 +(24) +The parametric gain experienced by the two sidebands therefore is +G = κ +� +|a0|4 − +� +ζ0 + D2 +κ µ2 − 4|a0|2 +�2 +(25) +At least a intracavity power of |a0|2 = 1 is required to reach threshold. With Eq. 19 it follows that for strong coupling the +threshold pump power f 2 ≥ 4|a0| is at least four times the threshold power of a resonator without forward-backward coupling. +3.1 +First oscillating sideband +The phase mismatch between the pump wave and the resonator modes can be quantified via their effective (including nonlinear +frequency shifts) detuning ζµ,eff from an equidistant D1-space frequency grid. A smaller ζµ,eff implies better phase matching. +ζµ,eff = ζ0 + D2 +κ µ2 − 4|a0|2 +(26) += β − |a0|2 ± +� +f 2 +4|a0|2 − 1 + D2 +κ µ2 +(27) +For DKS the resonator is characterized by anomalous dispersion D2 > 0 (β ≫ D2/κ). It can therefore be guaranteed, that +the first generated sideband pair (best phase matching) will be µ ± 1, if +β − |a0|2 − +� +f 2 +4|a0|2 − 1 > 0 +(28) +Assuming |a0|2 ≤ f 2/4 we find +β > f 2/4 +⇔ +γ/κ > f 2/8 +(29) +as a condition that guarantees that the first sideband pair will be generated at µ = ±1. +3.2 +Threshold power +The threshold power fth is the power level where the parametric threshold G > κ can be reached. Inserting Eq. 21 for the +detuning into Eq. 25, we obtain for the threshold condition +|a0|4 − +� +β + 3|a0|2 ± +� +f 2 +th +4|a0|2 − 1 + D2 +κ µ2 − 4|a0|2 +�2 += 1 +(30) +⇔ |a0|4 − 1 = +� +β ± +� +f 2 +th +4|a0|2 − 1 + D2 +κ µ2 − |a0|2 +�2 +(31) +Under the assumption that β − |a0|2 − +� +f2 +th +4|a0|2 − 1 > 0 (condition for first oscillating sidebands µ = ±1), this results in +f 2 +th = 4|a0|2 + 4|a0|2 +� +β − |a0|2 + D2 +κ µ2 − +� +|a0|4 − 1 +�2 +, +(|a0|4 > 1) +(32) +This equation can be solved numerically. For example, β = 4 results in f 2 +th ≈ 8.4 +12 + +3.3 +Threshold power assuming zero-effective detuning +A simplified threshold condition may be derived assuming that the threshold will be reached at zero effective detuning so that +ζ0 = β + 3|a0|2 +and +(33) +f 2 +th = 4|a0|2 +(34) +In this case the threshold condition is +|a0|4 − +� +β + 3|a0|2 + D2 +κ µ2 − 4|a0|2 +�2 += 1 +(35) +⇔ f 4 +th +16 − +� +β − 1 +4f 2 +th + D2 +κ µ2 +�2 += 1 +(36) +⇔ f 2 +th = 2 + 2 +� +β + D2 +κ µ2�2 +β + D2 +κ µ2 +(37) +Assuming that β ≫ D2 +κ µ2 this simplifies to +f 2 +th ≈ 2 + 2β2 +β += 2β + 2 +β +⇔ +f 2 +th ≈ 4γ +κ + κ +γ +(38) +For β = 4 we find f 2 +th = 8.5, almost equal to what is obtained through Eq. 32. +4 +Pump mode hybridization above threshold +The derivations of Section 1 are only valid below threshold, where only the forward and backward pump mode are excited. +Above threshold, and in particular in presence of DKS, the effective frequencies of the hybrid mode resulting from the avoided +mode crossing of the coupled forward and backward modes are +ω±,eff = 1 +2(ωa,eff + ωb,eff) + 1 +2 +� +(ωa,eff − ωb,eff)2 + 4γ2 +(39) +where ωa,eff and ωb,eff are the effective (i.e. taking nonlinear frequency shifts into account) resonance frequencies of the forward +and backward modes, respectively. +ωa,eff = ω0 − κ +2 +� +Re( ˆF[|ψa(θ)|2ψa(θ)]µ=0/a0) + 2 +� +η +|bη|2 +� +(40) +ωb,eff = ω0 − κ +2 +� +Re( ˆF[|ψb(θ)|2ψb(θ)]µ=0/b0) + 2 +� +η +|aη|2 +� +(41) +(42) +where ψa(θ) = ˆF −1[aµ] and ψb(θ) = ˆF −1[bµ] are the spatio-temporal field profiles and ˆF[ . ]0 stands for the component +corresponding to the pump mode ( ˆF denotes the Fourier transform). +5 +Comb generation in the backward direction +Due to the initially similar power-levels in forward and backward pump modes, combs may in principle not only be generated +in the forward, but also in the backward direction. Indeed, when the pump laser detuning is between the (effective) resonance +frequencies of the hybridized modes, the backward-wave is usually stronger (despite the forward-pumping). For the parameters +considered in this work, we found that this range of detuning does not overlap with the soliton existence range and backward +combs were not observed. However, backward combs represent an interesting opportunity for additional research. Aside from +backward comb generation we note, that backward modulation instability, can trigger forward modulation instability (and comb +generation) and vice versa, through a non-zero forward-backward coupling of the modulation instability sidebands. This can +readily be included in the numeric model by introducing non-zero γ also for modes with µ ̸= 0. +13 + diff --git a/PtFPT4oBgHgl3EQfnzX3/content/tmp_files/load_file.txt b/PtFPT4oBgHgl3EQfnzX3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6795df5695519ab14aa6237f068b0d818bea9d3f --- /dev/null +++ b/PtFPT4oBgHgl3EQfnzX3/content/tmp_files/load_file.txt @@ -0,0 +1,908 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf,len=907 +page_content='Synthetic-reflection self-injection-locked microcombs Alexander E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Ulanov,1 Thibault Wildi,1 Nikolay G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Pavlov,2 John D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Jost,2 Maxim Karpov,2 Tobias Herr1,3,* 1Deutsches Elektronen-Synchrotron DESY, Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 85, 22607 Hamburg, Germany 2Enlightra Sarl, Rue de Lausanne 64, 1020 Renens, Switzerland 3Physics Department, Universit¨at Hamburg UHH, Luruper Chaussee 149, 22607 Hamburg, Germany ∗tobias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='herr@desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='de Laser-driven microresonators have enabled chip-integrated light sources with unique prop- erties, including the self-organized formation of ultrashort soliton pulses and frequency combs (microcombs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' While poised to impact ma- jor photonic applications, such as spectroscopy, sensing and optical data processing, microcombs still necessitate complex scientific equipment to achieve and maintain suitable single-pulse opera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Here, to address this challenge, we demon- strate microresonators with programmable syn- thetic reflection providing an injection-feedback to the driving laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' When designed appropriately, the synthetic reflection enables robust access to self-injection-locked microcombs operating exclu- sively in the single-soliton regime and with low- threshold power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' These results provide a route to easily-operable microcombs for portable sensors, autonomous navigation, or extreme-bandwidth data processing and represent a novel paradigm that can be generalized to other integrated pho- tonic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Laser-driven microresonators provide access to non- linear optical phenomena, already with low-power continuous-wave excitation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Leveraging efficient non- linear frequency conversion, they have enabled novel sources of coherent laser radiation across broad spectral span [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Soliton microcombs[4–6] are an important representative of such sources, providing frequency comb spectra of mutually coherent laser lines, based on self- organized dissipative Kerr solitons (DKSs) in resonators with anomalous dispersion [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Such DKS microcombs can be integrated on photonic chips [8, 9] and have demon- strated their disruptive potential in many emerging and ground-breaking applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' high-throughput opti- cal data transmission [10] reaching Pbit-per-second data rates [11], ultrafast laser ranging [12, 13], precision as- tronomy in support of exo-planet searches [14, 15], high- acquisition rate dual-comb spectroscopy [16], ultra-low noise microwave photonics [17, 18], photonic computing and all-optical neural networks [19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' To leverage mi- crocomb technology in out-of-lab applications, it is critical to reliably access the DKS regime and ideally single-DKS operation [22], ensuring well-defined temporal and spec- tral characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' While routine in research laborato- ries, achieving such a state outside such environments is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' A critical challenge lays in the initiation and sustained operation of DKS, requiring the detuning ∆ω0 = ω0 − ωp of the pump laser ωp (with respect to the pumped res- onance ω0) to be controlled and stabilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' While this is common to all resonant approaches, it is particularly chal- lenging during DKS initiation, when thermo-optic effects can cause a rapid (∼ µs) change in resonance frequency [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' To overcome this challenge, a number of methods have been developed, involving rapid laser actuation [4, 8], auxiliary lasers [23] and/or auxiliary resonances [24, 25], laser modulation [26], additional nonlinearities [27–29] or, pulsed driving [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Many of these methods are now routinely used in research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' However, they cannot easily be transferred to out-of-the-lab scenarios, as they require sig- nificant experimental skills and scientific instrumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In contrast, self-injection locking (SIL) [31–33], has been demonstrated as an approach that can intrinsically follow the rapid changes in resonance frequency and elegantly stabilize the laser detuning for stable DKS operation[17, 34–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Usually, SIL is based on Rayleigh backscatter- ing from random fabrication imperfections or material de- fects in the microresonator [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The backscattered wave provides feedback (injection) to the driving diode laser and effectively locks the laser frequency to the microres- onator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' However, backscattering random defects are nei- ther wanted nor can they yield predictable sample charac- teristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Relying on random defects is also fundamentally incompatible with the intense efforts towards improved materials and fabrication techniques (targeting material absorption limited performance with negligible scattering similar to optical fiber technology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Already now, fabrica- tion techniques have advanced to a level, where identifying samples with accidental scattering suitable for SIL-based DKS often requires careful and tedious screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In this work, we demonstrate SIL and robust access to self-injection locked DKS microcombs without relying on random resonator defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Instead of random backscatter- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='13132v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='optics] 30 Jan 2023 ing, we achieve SIL via programmable synthetic reflection, dramatically increasing the access to laser detunings that support DKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The synthetic reflection is generated via photonic crystal ring resonators (PhCR) [40], which have recently received growing attention in integrated nonlin- ear photonics [41–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In addition, we show that robust access to SIL-based DKS can be combined with recent results of spontaneous single-DKS generation in PhCRs (avoiding non-solitonic states) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Based on analytic cri- teria, we design the synthetic reflection to ensure exclu- sive operation in the single-DKS regime as well as low- threshold power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Resulting from the synthetic reflection we also observe DKS breathing [41, 46] in a limited range of operating parameters, which can readily be avoided, if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' These results provide a route to easily-operable microcombs for out-of-lab applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Results To gain independence from backscattering random de- fects and imperfections, we use PhCRs that enable syn- thetic reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The reflection is controlled by periodic nano-patterned corrugations of the ring-resonators’ inner walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The angular corrugation period is θ0 = 2π/(2m0), where m0 is the angular (azimuthal) mode number, for which a deliberate coupling between forward and back- ward propagating waves with a coupling rate γ is induced (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Besides inducing the desired synthetic reflec- tion, the coupling leads to mode hybridization resulting in a split resonance lineshape (frequency splitting 2γ) in both transmission and reflection (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Here, we only consider the lower frequency hybrid mode for pumping, as it corresponds to strong (spectrally local) anomalous dis- persion, which prevents high-noise comb states [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' For choosing γ we balance multiple criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' First, a strong reflection can significantly extend the range of normalized detunings ζ0 = 2∆ω0/κ (κ is the mi- croresonator linewidth) accessible via SIL (SIL range) in a nonlinear microresonator [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' This is crucial, as it permits robust access to detunings where DKS can exist (DKS ex- istence range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In conventional resonators the normalized forward-backward coupling is usually small 2γ/κ < 1 and the intersection between SIL and DKS ranges is limited to small detunings, complicating access to DKS states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In contrast, strong forward-backward coupling could enable robust access to DKS states over a wide range of detun- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' This is exemplified in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 1c, where the SIL range [37] is shown along with the conventional analytic DKS ex- istence range (valid for small γ) and the numerically com- puted DKS existence range for large γ, obtained through numeric integration of the coupled mode equations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Note, that in a resonator with a shifted pump mode [49], the existence range of DKS deviates strongly from that known from resonators without a shifted pump mode [48] and can currently only be obtained numerically (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Second, while advantageous for an extended SIL range, ω0 PhCR γ θ0 laser diode T R synthetic reflection a b microcomb CW T R m0 + 1 m0 m0 - 1 FSR 2π 2γ 2 mm d c 1 3 4 0 5 10 Backscattering 2γ/κ Laser detuning ζ0 DKS range (num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=') DKS range (no splitting, analyt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=') SIL range 2 Angular frequency ω Figure 1 | Self-injection locking with synthetic reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' a, An integrated photonic crystal ring-microresonator (PhCR) with a periodic corrugation (angular period θ0), which induces coupling at a rate γ between forward and backward-propagating waves for a mode m0 = π/θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In addition to a transmission signal (T), this leads to a well-defined synthetic resonant reflection (R), which can be programmed for self-injection locking with a laser diode driving the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' b, Indicative transmission and reflection spec- trum for the resonance with mode number m0 and two adjacent resonances m0±1, separated by ±1 free-spectral range (FSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' For γ ̸= 0, the lineshape at mode number m0 exhibits a split lineshape (frequency splitting 2γ), and shows non-zero resonant reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' c, Comparison of nonlinear SIL [37] and DKS existence ranges computed numerically for a critically coupled microresonator with total linewidth κ/2π = 120 MHz, dispersion D2/2π = 8 MHz, driven with a normalized pump power of f 2 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' For large γ, the DKS existence range deviates significantly from the analytical estimation for zero-γ [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' d, Photograph of the experimen- tal system showing the semiconductor laser diode butt-coupled to the photonic chip carrying the PhCRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Transmitted light is out-coupled using a lensed fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' stronger forward-backward coupling will also result in an increased parametric threshold (modulation instability, 2 MI) pump power, as detailed in the Supplemental Infor- mation (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Below this threshold DKS cannot form inside the resonator, without external stimuli (such as triggering pulses [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The threshold power is different from that in a conventional ring resonator and its derivation critically requires consideration of the backward wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' For strong forward-backward coupling (2γ/κ > 1), the following ap- proximation is derived (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' SI): f 2 th = 4γ κ + κ γ (1) where f = � 8ηω0cn2P/(κ2n2Veff) is the normalized pump power, with the coupling coefficient η = 1/2 (criti- cal coupling), ω0 the resonance frequency of the pumped mode, c the speed of light, P the input pump power, n the refractive index, n2 the nonlinear refractive index and Veff the effective mode volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The value of f 2 th must not exceed the available pump power f 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' If the MI threshold is reached at a detuning within the DKS existence range, then the MI state may be only transient and DKS can form spontaneously [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In both conventional and pump mode-shifted resonators the DKS regime overlaps with the MI regime and extends further towards larger detunings ζ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Third, with regard to practical applications single-DKS states, as opposed to states with multiple solitons, are highly desirable owing to their smooth squared hyperbolic secant spectral envelope and well-defined temporal out- put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In their formation process, DKS are seeded by MI, where the separation of the first pair of sidebands from the pump laser in units of the resonator’s FSR determines the number of generated DKS [4, 41, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' A conservative cri- terion that guarantees single-DKS formation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' modu- lation instability sidebands separated from the pump laser by 1 FSR) is derived in the SI: γ κ > f 2 8 (2) The presented considerations can inform the design of a suitable PhCR for SIL-based DKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In preparation of the experiments, a range of critically coupled resonators with varying corrugation amplitude and a free-spectral range (FSR) of 300 GHz (radius 75 µm) are fabricated in a commercial foundry process (Ligentec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' We characterize the fabricated resonators via frequency comb-calibrated laser scans [50], permitting to retrieve the coupling rates γ, the resonance widths κ, and the disper- sion D2, over a broad spectral bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' An example is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 2a, where indeed the forward-backward coupling is random and 2γ/κ ≪ 1 for most resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In marked contrast, a single pre-defined resonance to which the PhCR’s corrugation is matched, exhibits significant forward-backward coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 2b shows the dependence of γ and the Q-factor (Q = ω0/κ) on the corrugation am- plitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' No noticeable degradation of the Q factor is ob- served up to γ ≲ 5 GHz, and critically coupled linewidth are κ 2π ≈ 120 MHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' even for large coupling γ ≈ 45 GHz , the Q-factor is only halved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' For the experiments, a semi- conductor distributed feedback laser diode (DFB) is butt- coupled to a waveguide on the photonic chip, permitting an estimated on-chip pump power of P = 30 mW, cor- responding to f 2 ≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 1 and 2, we obtain an ideal 2γ/κ ∈ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='26, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='26), ensuring MI-based sponta- neous comb initiation and deterministic generation of sin- gle DKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Based on these considerations we choose a PhCR with a synthetic coupling for the pump mode at 1557 nm of 2γ/κ ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='2 ( γ 2π ≈ 250 MHz), within the ideal range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' This PhCR is critically coupled and exhibits anomalous group velocity dispersion (D2 ≈ 8 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' As shown for those val- ues in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 1c, numeric simulation confirms that the DKS existence and SIL ranges have significant overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' We note that another band of DKS existence may exist [49], how- ever, it is inaccessible for spontaneous MI-assisted comb initiation and not considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The DFB pump laser diode is mounted on a piezo translation stage to adjust the injection phase [33], an actuator which can readily be achieved through on-chip heaters [38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' to reduce the de- vice footprint and allow for more resonators on the chip, we have omitted this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The transmitted light is collected by a lensed-fiber for further analysis as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In a first experiment, we validate the basic SIL dynamics below parametric threshold at a coupled pump power of 25 mW (f 2 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' As long as the laser diode does not re- ceive a resonant injection from the microresonator it is free running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' When the laser’s emission wavelength is tuned (via its drive current) close to the lower-frequency pump resonance, a strong resonant backward wave is generated, providing frequency-selective optical feedback resulting in SIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The SIL regime manifests itself as a rectangular- shaped dip in the transmission signal and, after optimiz- ing the injection phase, extends over a wide range of elec- trical drive current values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The optical spectrum of the DFB laser in the SIL regime is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 2e, show- ing a single-mode suppression ratio (SMSR) >60 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The beatnote of the SIL laser with a table-top low-noise CW laser is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In addition, we record the SIL- laser phase noise (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 2c), which is drastically lower than that of the free-running DFB laser diode outside the SIL regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In a second experiment, utilizing the same setup as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 2f, we explore DKS-based microcomb generation with the full available pump power (30 mW, f 2 ≈ 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Similar to the previous lower power SIL experiment, we slowly (within ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 10 s) tune the DFB’s electrical drive current to scan the emission wavelength across the lower frequency pump resonance, with increasing and then de- creasing wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' During this scan we monitor the op- tical spectrum in transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' We note that the exact tuning curve in the nonlinear SIL regime when increasing (decreasing) the DFB pump current follows a nontrivial behavior that may include non-monotonic sections [37];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' the scan outside the SIL range is however monotonic in frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Upon entering the SIL regime (again marked ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Heterodyne detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='CW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='PD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='ESA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='LD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='CC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='PhCR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='SIL Setup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='PD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='OSC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='OSA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Wavelength (nm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1552 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1556 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1560 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Power (20 dB/div) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='~60 dB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='SIL laser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='10 MHz / div ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Power (20 dB/div) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='RBW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='5 KHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Free ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='SIL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='SSB Phase Noise (dBc/Hz) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='10 KHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='100 KHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1 MHz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Corrugation amplitude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Coupling rate γ/2π (GHz) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Q-factor (million) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='500 MHz / div ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Wavelength (nm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='PhCR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='induced ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Backscattering 2γ/κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='random ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='scattering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1540 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1560 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1580 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='transmission ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='reflection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='free-running ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='SIL laser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' laser −120 −80 −40 0 Figure 2 | Resonator characterization and low-power SIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' a, The single predefined resonance (red dot) to which the corrugation pattern is matched, exhibits significant forward-backward coupling unlike the other modes (blue dots) where it is weak and random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' b, Measured forward-backward coupling rates (red, left axis) and Q-factors (blue, right axis) of PhCRs with increasing corrugation amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' c, Phase noise of the DFB laser in free-running and SIL regimes measured through heterodyne detection with a reference laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The phase noise of the reference laser is provided as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' d, Heterodyne beatnote signal between the reference oscillator and DFB laser in the free-running and SIL states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' RBW, resolution bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' e, Optical spectrum of the DFB laser in the SIL state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' f, Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' CC, current controller;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' LD, laser diode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' OSA and ESA, optical and electrical spectrum analyzers respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' OSC, oscilloscope;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' CW, continuous-wave laser;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' PD, photodiode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' by pronounced dip of the transmitted power after opti- mization of the injection phase), we observe at first only the single optical frequency of the SIL pump laser, as in the lower power experiment before (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3a 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Con- tinuing the scan we next observe an abrupt transition into a single-DKS microcomb state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3a 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Such single-DKS states are characterized by a smooth squared hyperbolic-secant amplitude and a pulse repetition rate that corresponds to the resonator’s FSR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' these properties are highly-desirable for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Further continuing the scan induces a surprising second abrupt transition into a different single-DKS state (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3a 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Scanning even further causes the DKS to disappear, with the system re- turning to CW SIL (spectrum similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3a 1 ), before eventually exiting the SIL regime entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' When repeated, each scan shows the same SIL dynamics, including deter- ministic single-DKS generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Reversing the scan direc- tion qualitatively yields the same phenomena in reversed order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Turing patterns, noisy comb-states and multi-DKS regimes are absent in stark contrast to previous SIL-based microcomb generation, but consistent with spontaneous single-DKS formation in conventionally-driven (non-SIL) PhCR [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Although not pursued here, we note that the pump to DKS conversion efficiency in the states 2 and 3 is 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='8 % and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='2 %, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=', significantly higher than what would be expected in conventional resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' This is a consequence of the mode splitting, shifting the pumped resonance effectively closer to the pump laser as explored previously in coupled ring-resonators [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Different from DKS generation experiments in conventionally-driven PhCR, laser tuning speed and even tuning direction are irrelevant and do not notice- ably impact the observed dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Owing to the SIL mechanism, switching between the system’s states is readily possible and without the risk of ‘dropping’ out of resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Each state of the system can be maintained without requiring external stabilizing feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' All observations are reproduced in all 4 tested copies of the PhCR (fabricated on 4 different chips).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' As such, through synthetic reflection, our system not only achieves robust and predictable SIL operation, but also leverages the advantages of spontaneous and deterministic single-DKS generation, observed in conventionally driven PhCRs [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' To further investigate the SIL dynamics and DKS gen- 4 3 2 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='6 Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' rate − 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='3 (GHz) Power (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=') 0 1 Current 20 mA/div 1 Time (2 s/div) 1 0 Power (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=') transmission filtered transm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' DFB current transmission filtered transm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' DFB current Time (2 s/div) multi- DKS single-DKS Current 20 mA/div c d e 1700 Wavelength (nm) 1450 1500 1550 1600 EOM ESA PD SIL Setup FBG repetition rate detection 1 2 3 Power (20 dB/div) SIL DFB Single DKS Breather DKS a b sech2 Fit increasing current decreasing current Eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' ~ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='8 % sech2 Fit Eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' ~ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='2 % Figure 3 | SIL-based DKS generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' a, Optical spectra measured during a laser scan towards longer wavelength within the SIL range: CW SIL 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' SIL-based single DKS states 2 and 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' b, Experimental setup for repetition rate detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' EOM - electro- optical modulator, FBG - fiber Bragg grating, PD - photodetector, ESA - electrical spectrum analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' c, Measured SIL-microcomb repetition rate signal representing single DKS 2 and breather 3 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' d Total transmission (blue) and bandpass-filtered power (red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' filter offset from the pump, indicates comb formation) measured during a high-power laser scan with a PhCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The orange line corresponds to the driving current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' e, same as d, but with a conventional resonator that has been selected for a relatively strong random backscattering (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' main text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' eration, we record the 300 GHz DKS repetition rate beat- note via the setup shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' As this signal would not be directly detectable, modulation sidebands around a pair of adjacent DKS comb lines are generated electro-optically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Their beating creates a signal at lower frequency, from which the repetition rate can be recon- structed [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3c shows the reconstructed repetition rate signal obtained during the DFB laser scan in both di- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The two distinct spectral regimes are also man- ifest in this signal: in regime 2 a single low-noise repe- tition rate beatnote is present, whereas in regime 3 ad- ditional sidebands (ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' ±200 MHz) indicate a breathing soliton, similar to so far unexplained breathing phenom- ena in conventionally-driven PhCRs at large detuning [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In the wavelength-decreasing scan, the reversed dynamics is observed (the additional breathing towards the end of the scan is well-known from conventionally driven DKS [46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' During the laser scan, we also record the transmitted power as well as the power of a filtered spectral portion of the long-wavelength wing of the generated microcombs (as an indicator for comb formation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Both are shown along with the DFB diode’s drive current in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Here, the SIL regime is clearly evidenced by sharp drops of the (full) transmission from the base level in both scan direc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The DKS regime is marked by the non-zero filtered power within the SIL regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Note that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3d the breathing oscillation is only visible in the recorded power for the lowest breathing frequencies, due to the limited 100 MHz bandwidth of the utilized photo-detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' For comparison, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3e shows a similar transmission and fil- tered power trace obtained with a non-PhCR microres- onator of the same FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' To achieve SIL-based DKS in a non-PhCR, we screened a large number of resonators to find one with large random reflection (2γ/κ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='6) and validated numerically that it can meet the criteria for SIL- based DKS (similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' In contrast to the PhCR, it does however not respect the criterion for deterministic single-DKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Indeed, different from the SIL dynamics in the PhCR, noisy comb states before (after) DKS genera- tion are observable in the laser scan and the DKS-regime is dominated by less desirable multi-DKS states, showing the characteristic multi-step features [52];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' the single-DKS regime is marginal and only visible as a narrow step when the scan time is reduced to the millisecond level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' To explore the emergence of the breathing soliton at 5 a b c Comb repetition rate (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=') Kerr shiftts\\detuning (units of κ/2) −6 12 Time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=') Intracavity power (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=') comb fwd pump fwd comb bwd pump bwd laser detuning Kerr shift (bwd) Kerr shift (fwd) Kerr shift (hybrid modes) 0 Figure 4 | SIL-microcomb dynamics simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' a, The time- frequency spectrogram reconstructed from the numerical simula- tion during a wavelength-increasing scan of the pump laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' b, Corresponding intracavity powers for forward (beige) and back- ward (red) propagating microcomb modes (µ ̸= 0) as well as forward (orange) and backward (blue) propagating pump modes (µ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' c, Kerr-shifts of forward (orange) and backward (blue) propagating pump modes with respect to the ‘cold’ resonance frequency ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Detuning of pump laser (beige).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The dark lines indicate the effective hybrid modes frequencies (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' main text and SI for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' higher DFB currents (regime 3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3c,d), we per- form numeric simulation [53, 54], where we include forward and backward propagating waves (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Starting from a single DKS state at small detuning (within the well-known breather regime [46]), we integrate the cou- pled mode equations while imposing a slow laser scan to- wards longer wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The spectrogram of the repe- tition rate signal obtained from the numerical simulation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 4a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' it reproduces an abrupt appearance of sidebands in agreement with the experimental results of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The average power levels of the pump and comb (full spectrum without pump) in both forward- and backward-direction show rapid oscillations Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 4b, coin- ciding with the breathing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The phase-lag between comb and pump mode suggesting an oscillatory power flow between both, reminiscent of the detuning dependent in- termode breathing dynamics [55], observed in DKS in the presence of coupling between a DKS mode to a frequency degenerate higher-order transverse mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The latter is absent in our case, however, by design a strong coupling exists between forward and backward pump modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Al- though, up to a certain power level, the strongly coupled forward and backward modes are frequency degenerate nu- meric simulations suggest that this degeneracy is not gen- erally preserved (similar to CW symmetry breaking [56]) in the DKS regime and dependent on the state of opera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Based on the simulation, we derive the nonlinear fre- quency shifts of forward and backward propagating pump waves as well as those of the hybridized modes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' SI) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Strikingly, find that the regime of the breathing oscillation coincides with a situation where the differen- tial nonlinear shifts between forward and backward modes are small compared to the resonance width, suggesting forward-backward coupling as its origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Importantly, the phenomenon is restricted to a specific parameter regime, and can therefore easily be circumvented by choosing the operating parameters (notably pump power or detuning) accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Conclusion In conclusion, we have demonstrated microresonators with synthetic reflection for robust SIL and SIL-based micro- comb generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' By designing the synthetic reflection we enable deterministic low-threshold, single-DKS operation, highly-favorable for applications and address a major chal- lenge of this technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' For certain combinations of pump power and detuning we observe DKS breathing, which can readily be avoided by small changes in the operating pa- rameters, if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The presented results in conjunc- tion with the scalable, widely accessible fabrication pro- cess, the low-cost components, and, notably, its ease of operation meet important requirements of out-of-lab ap- plications and extension to other integrated photonic sys- tems, including normal dispersion combs [52, 57] and novel quantum light sources [58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Further research may ex- plore extending the presented results to combs in the back- ward direction (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' SI) [60], effectively blue-detuned DKS combs with potentially even higher conversion efficiency [49] or multiple pump wavelengths [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Materials Numerical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' To simulate the nonlinear dynamics we consider a system of coupled mode equations [53,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 54] for forward aµ and backward bµ mode amplitudes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' where µ denotes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='the relative (longitudinal) mode number with respect to the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='pump mode (m0 ↔ µ = 0): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='∂taµ = − (1 + iζµ)aµ + i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='µ′=ν+η−µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='aνaηa∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='µ′ + 2iaµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='|bη|2+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='+ iδµ0 2γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='κ bµ + fδµ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='∂tbµ = − (1 + iζµ)bµ + i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='µ′=ν+η−µ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='bνbηb∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='µ′ + 2ibµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='η ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='|aη|2+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='+ iδµ0 2γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='κ aµ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='where ζµ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='κ(ωµ − ωp − µD1) is a dimensionless detuning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='between the pump laser frequency ωp and cold resonance fre- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='quencies ωµ = ω0 + D1µ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='2D2µ2 (with D1/2π and D2/2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='being microresonator FSR and second-order dispersion respec- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The third term in each equation corresponds to the cross-phase modulation by the respective counter-propagating waves, while the fourth term represents the coupling between forward and backward propagating waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Instead of modeling the SIL dynamics by including laser rate equations, we numer- ically define the detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' This approach cannot describe the abrupt transition from the free-running laser to the SIL state, it remains however valid for the specified detuning and can qualitatively capture the features observed in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Simulation parameters similar to those of the experimental sys- tem are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Funding This project has received funding from the European Research Council (ERC) under the EU’s Horizon 2020 research and inno- vation program (grant agreement No 853564), from the EU’s Horizon 2020 research and innovation program (grant agree- ment No 965124) and through the Helmholtz Young Investiga- tors Group VH-NG-1404;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' the work was supported through the Maxwell computational resources operated at DESY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Data availability The datasets generated and analysed during the current study are available from the corresponding author on reasonable re- quest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Code availability Numeric simulation codes used in the current study are avail- able from the corresponding author on reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Disclosures The authors declare no competing interests, however disclose that J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} 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Physical Review Letters 124, 193601 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Lu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Bayesian tomography of high- dimensional on-chip biphoton frequency combs with randomized measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Nat Commun 13, 4338 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Fujii, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Effect on Kerr comb generation in a clockwise and counter-clockwise mode coupled mi- crocavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Express 25, 28969 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Chermoshentsev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Dual-laser self-injection locking to an integrated microresonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Optics Ex- press 30, 17094 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 9 Synthetic-self-injection locked microcombs - Supplemental Information Alexander E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Ulanov,1 Thibault Wildi,1 Nikolay G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Pavlov,2 John D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Jost,2 Maxim Karpov,2 Tobias Herr1,3,* 1Deutsches Elektronen-Synchrotron DESY, Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 85, 22607 Hamburg, Germany 2Enlightra Sarl, Rue de Lausanne 64, 1020 Renens, Switzerland 3Physics Department, Universit¨at Hamburg UHH, Luruper Chaussee 149, 22607 Hamburg, Germany ∗tobias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='herr@desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='de 1 Coupled mode equations and pump mode hybridization The normalized coupled mode equations (CMEs) for the pump in forward and backward directions read ∂ta0 = −(1 + iζ0)a0 + i|a0|2a0 + 2i|b0|2a0 + iβb0 + f (1) ∂tb0 = −(1 + iζ0)b0 + i|b0|2b0 + 2i|a0|2b0 + iβa0 (2) where for convenience the normalized coupling rate β = 2γ/κ ≥ 0 has been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Without loss of generality, f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The coefficient matrix of the system of equations (without the pump) M = �−(1 + iζ0) + i|a0|2 + 2i|b0|2 iβ iβ −(1 + iζ0) + i|b0|2 + 2i|a0|2 � (3) has the following Eigenvalues λ± = − � �1 + iζ0 − 3 2i(|a0|2 + |b0|2) ± i � β2 + �1 2(|a0|2 − |b0|2) �2 � � (4) = − � 1 + i(ζ0 − δζNL ± � β2 + δβ2 NL) � (5) and is diagonalized in the following Eigenbasis of hybridized forward-backward modes � � � 1 2(|a0|2 − |b0|2) ± � β2 + �1 2(|a0|2 − |b0|2) �2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' −β � � � (6) = � δβNL ± � β2 + δβ2 NL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' −β � (7) where δβNL = 1 2(|a0|2 − |b0|2) and δζNL = 3 2(|a0|2 + |b0|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The transformation matrices are: T = � δβNL + � β2 + δβ2 NL δβNL − � β2 + δβ2 NL −β −β � (8) and T −1 = � � � 1 2√ β2+δβ2 NL δβNL−√ β2+δβ2 NL 2β√ β2+δβ2 NL − 1 2√ β2+δβ2 NL −δβNL−√ β2+δβ2 NL 2β√ β2+δβ2 NL � � � (9) so that �a0 b0 � = T �a+ a− � (10) 10 and �a+ a− � = T −1 �a0 b0 � (11) where a± denote the (not specifically normalized) field amplitudes of the hybrid modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The steady state equations for the hybrid modes are 0 = − � 1 + i(ζ0 − δζNL + � β2 + δβ2 NL) � a+ + 1 2 � β2 + δβ2 NL f (12) 0 = − � 1 + i(ζ0 − δζNL − � β2 + δβ2 NL) � a− − 1 2 � β2 + δβ2 NL f (13) In non-normalized units,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' the effective resonance frequencies of the hybridized modes are ω±,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='eff = ω0 − κ 2 � δNL ± � β2 + δβ2 NL � (14) 2 Approximations for the forward pump mode under strong coupling In what follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' it is assumed that the coupling is strong β > 1 due to the strong coupling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' the power levels in forward and backward directions are approximately equal |a0|2 = |b0|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Note that due to symmetry breaking [56] this is only valid up to a certain power level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' We validated, by numeric integration of the CMEs 1 and 2, that this approximation is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' the detuning ζ0 is such that approximately only the lower frequency hybrid mode a− is driven, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' |a−| ≫ |a+|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Under these assumptions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='a0 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='δβNL − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='β2 + δβ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='NL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='a− ≈ −βa− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(15) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='b0 = −βa− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='and in consequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(1 + i(ζ0 − δζNL − β))a0 = f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Multiplying each side of the equation with its complex conjugate results in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(1 + (ζ0 − δζNL − β)2)|a0|2 = f 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='An immediate insight is that the strong coupling between forward and backward waves limits the power in the forward (or ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='backward) wave to values of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='|a0|2 ≤ f 2/4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(19) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Expressing δζNL via the field amplitudes gives ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(1 + (ζ0 − 3|a0|2 − β)2)|a0|2 = f 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='and for the detuning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='ζ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='± = β + 3|a0|2 ± � f 2 4|a0|2 − 1 (21) where ζ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='+ corresponds to an effective red-detuning and ζ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='− to an effective blue-detuning with regard to the lower-frequency hybrid mode a−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 11 3 Threshold condition and first oscillating sideband We consider two initially zero-power (except for vaccuum fluctuations) sidebands with mode number ±µ relative to the pumped mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Their CMEs are ∂ta+µ = −(1 + i(ζµ − 4|a0|2))a+µ + ia2 0a∗ −µ (22) ∂ta∗ −µ = −(1 − i(ζµ − 4|a0|2))a∗ −µ − ia∗2 0 a+µ (23) where again |a0|2 ≈ |b0|2 was assumed and ζµ = 2 κ(ω0 − ωp + 1 2D2µ2) = ζ0 + D2 κ µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' The Eigenvalues of this set of equations are λ± = −1 ± � |a0|4 − (ζµ − 4|a0|2)2 (24) The parametric gain experienced by the two sidebands therefore is G = κ � |a0|4 − � ζ0 + D2 κ µ2 − 4|a0|2 �2 (25) At least a intracavity power of |a0|2 = 1 is required to reach threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' With Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 19 it follows that for strong coupling the threshold pump power f 2 ≥ 4|a0| is at least four times the threshold power of a resonator without forward-backward coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='1 First oscillating sideband The phase mismatch between the pump wave and the resonator modes can be quantified via their effective (including nonlinear frequency shifts) detuning ζµ,eff from an equidistant D1-space frequency grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' A smaller ζµ,eff implies better phase matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' ζµ,eff = ζ0 + D2 κ µ2 − 4|a0|2 (26) = β − |a0|2 ± � f 2 4|a0|2 − 1 + D2 κ µ2 (27) For DKS the resonator is characterized by anomalous dispersion D2 > 0 (β ≫ D2/κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' It can therefore be guaranteed, that the first generated sideband pair (best phase matching) will be µ ± 1, if β − |a0|2 − � f 2 4|a0|2 − 1 > 0 (28) Assuming |a0|2 ≤ f 2/4 we find β > f 2/4 ⇔ γ/κ > f 2/8 (29) as a condition that guarantees that the first sideband pair will be generated at µ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='2 Threshold power The threshold power fth is the power level where the parametric threshold G > κ can be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 21 for the detuning into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 25, we obtain for the threshold condition |a0|4 − � β + 3|a0|2 ± � f 2 th 4|a0|2 − 1 + D2 κ µ2 − 4|a0|2 �2 = 1 (30) ⇔ |a0|4 − 1 = � β ± � f 2 th 4|a0|2 − 1 + D2 κ µ2 − |a0|2 �2 (31) Under the assumption that β − |a0|2 − � f2 th 4|a0|2 − 1 > 0 (condition for first oscillating sidebands µ = ±1), this results in f 2 th = 4|a0|2 + 4|a0|2 � β − |a0|2 + D2 κ µ2 − � |a0|4 − 1 �2 , (|a0|4 > 1) (32) This equation can be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' For example, β = 4 results in f 2 th ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='4 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Threshold power assuming zero-effective detuning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='A simplified threshold condition may be derived assuming that the threshold will be reached at zero effective detuning so that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='ζ0 = β + 3|a0|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(33) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='f 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='th = 4|a0|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(34) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='In this case the threshold condition is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='|a0|4 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='β + 3|a0|2 + D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='κ µ2 − 4|a0|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(35) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='⇔ f 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='th ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='16 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='β − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='4f 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='th + D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='κ µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(36) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='⇔ f 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='th = 2 + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='β + D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='κ µ2�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='β + D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='κ µ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(37) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='Assuming that β ≫ D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='κ µ2 this simplifies to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='f 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='th ≈ 2 + 2β2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='= 2β + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='⇔ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='f 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='th ≈ 4γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='κ + κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='(38) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='For β = 4 we find f 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='th = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='5, almost equal to what is obtained through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 4 Pump mode hybridization above threshold The derivations of Section 1 are only valid below threshold, where only the forward and backward pump mode are excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Above threshold, and in particular in presence of DKS, the effective frequencies of the hybrid mode resulting from the avoided mode crossing of the coupled forward and backward modes are ω±,eff = 1 2(ωa,eff + ωb,eff) + 1 2 � (ωa,eff − ωb,eff)2 + 4γ2 (39) where ωa,eff and ωb,eff are the effective (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' taking nonlinear frequency shifts into account) resonance frequencies of the forward and backward modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' ωa,eff = ω0 − κ 2 � Re( ˆF[|ψa(θ)|2ψa(θ)]µ=0/a0) + 2 � η |bη|2 � (40) ωb,eff = ω0 − κ 2 � Re( ˆF[|ψb(θ)|2ψb(θ)]µ=0/b0) + 2 � η |aη|2 � (41) (42) where ψa(θ) = ˆF −1[aµ] and ψb(θ) = ˆF −1[bµ] are the spatio-temporal field profiles and ˆF[ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' ]0 stands for the component corresponding to the pump mode ( ˆF denotes the Fourier transform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 5 Comb generation in the backward direction Due to the initially similar power-levels in forward and backward pump modes, combs may in principle not only be generated in the forward, but also in the backward direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Indeed, when the pump laser detuning is between the (effective) resonance frequencies of the hybridized modes, the backward-wave is usually stronger (despite the forward-pumping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' For the parameters considered in this work, we found that this range of detuning does not overlap with the soliton existence range and backward combs were not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' However, backward combs represent an interesting opportunity for additional research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' Aside from backward comb generation we note, that backward modulation instability, can trigger forward modulation instability (and comb generation) and vice versa, through a non-zero forward-backward coupling of the modulation instability sidebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' This can readily be included in the numeric model by introducing non-zero γ also for modes with µ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFPT4oBgHgl3EQfnzX3/content/2301.13132v1.pdf'} diff --git a/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf b/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8b8151d5518883f33ba0f20875f9d2958fd962d0 --- /dev/null +++ b/QNFRT4oBgHgl3EQf6zgQ/content/2301.13677v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1ec914d17ef1aafdac8e57f5a3881808fe65dd9abadb598c8ec5f98c1fa25275 +size 297422 diff --git a/QNFRT4oBgHgl3EQf6zgQ/vector_store/index.faiss b/QNFRT4oBgHgl3EQf6zgQ/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4195f8c77e0fa2a83e3473817dbf8b560af63ef2 --- /dev/null +++ b/QNFRT4oBgHgl3EQf6zgQ/vector_store/index.faiss 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+size 4849709 diff --git a/S9E4T4oBgHgl3EQfmA05/content/tmp_files/2301.05164v1.pdf.txt b/S9E4T4oBgHgl3EQfmA05/content/tmp_files/2301.05164v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e1f84f46afe0984344bf3666297ff6967d0475fc --- /dev/null +++ b/S9E4T4oBgHgl3EQfmA05/content/tmp_files/2301.05164v1.pdf.txt @@ -0,0 +1,928 @@ +ac-locking of thermally-induced sine-Gordon breathers +Duilio De Santis,1, ∗ Claudio Guarcello,2, 3 Bernardo Spagnolo,1, 4 Angelo Carollo,1 and Davide Valenti1 +1Dipartimento di Fisica e Chimica “E. Segr`e”, Group of Interdisciplinary Theoretical Physics, +Universit`a degli Studi di Palermo, I-90128 Palermo, Italy +2Dipartimento di Fisica “E. R. Caianiello”, Universit`a degli Studi di Salerno, I-84084 Fisciano, Salerno, Italy +3INFN, Sezione di Napoli, Gruppo Collegato di Salerno - Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy +4Radiophysics Department, Lobachevsky State University, 603950 Nizhniy Novgorod, Russia +(Dated: January 13, 2023) +A complete framework for exciting and detecting thermally-induced, stabilized sine-Gordon +breathers in ac-driven long Josephson junctions is developed. The formation of long-time stable +breathers locked to the ac source occurs for a sufficiently high temperature. The latter emerges +as a powerful control parameter, allowing for the remarkably stable localized modes to appear. +Nonmonotonic behaviors of both the breather generation probability and the energy spatial correla- +tions versus the thermal noise strength are found. The junction’s resistive switching characteristics +provides a clear experimental signature of the breather. +Introduction.—Owing to its simplicity and nonlinear +nature, the sine-Gordon (SG) equation [1] is universally +recognized as a fundamental modelling tool within the +scientific community [2]. +The SG framework, in fact, +provides a very accurate and intuitive viewpoint for a +large variety of phenomena occurring in, e.g., gravity and +black holes [2, 3], tectonic stress transfer [4], biology [5], +superconductivity and Josephson junctions (JJs) [2, 6], +Bose-Einstein condensates [7]. +A key feature of the SG equation is its rich spectrum +of solutions, which includes both kink-type and breather- +type solitons [1]. +The first are topological excitations +which can be visualized as 2π-twists in a mechanical +chain of linearly coupled pendula [8, 9]. A breather is +a space-localized, time-periodic bound state stemming +from the kink-antikink attraction [8, 9]. +The long Josephson junction (LJJ) is a (quasi) one- +dimensional, superconductor-based system whose elec- +trodynamics is reliably described by the SG model [1]. +Being the subject of many seminal experiments [10–12] +and striking applications [12–16], this device has played +an outstanding role in the spreading of the soliton con- +cept throughout natural and applied sciences [2, 8, 9]. In +LJJs, a kink represents a magnetic flux quantum Φ0 [1], +induced by a supercurrent loop, whose properties reflect +into the I -V characteristic of the junction [10–12]. +Due to its nontopological structure, mastering the +breather’s physics is a very tough challenge. In particu- +lar, experimental evidence of this oscillating state has +yet to be provided in LJJs, despite the numerous in- +vestigations on the matter [17–23], primarily due to its +friction-triggered radiative decay and its elusiveness with +respect to I -V measurements [20, 24]. The Josephson +breather’s detection would, therefore, solve a long-lasting +problem in nonlinear science, but it would also pave the +way for several applications in, e.g., information trans- +mission [25], quantum computation [26], generation of +THz radiation [27]. +Previous works (e.g., see Ref. [19]), analyzed the sta- +bilization of stationary SG breathers via ac-driving, with +specific ad-hoc initial conditions. Such a scenario, how- +ever, has remained experimentally unexplored. This is +presumably due to the practical difficulties in creating +persistent breather states, given the stabilization effect’s +crucial dependence on the initial condition. Moreover, +the phenomenon’s robustness against thermal fluctua- +tions has not been addressed so far. +On the other hand, the little discussed topic of +breathers in a noisy environment has recently gained +attention [21–23, 28, 29], and positive stochastically- +induced effects on both the generation and the dynamics +of these nonlinear waves have been demonstrated. The +present manuscript thus examines a lossy, ac-driven LJJ +in the presence of thermal noise. The emergence of long- +time stable breathers locked to the sinusoidal force is ob- +served for a sufficiently high temperature. +The latter +is, consequently, a powerful control parameter, allowing +for the localized modes to appear, while not endanger- +ing their persistence. The achievement of both the cre- +ation and the stabilization in a single effort should not +be overlooked, given the multistability of the SG system, +responsible for the possible emergence of kink-antikink +pairs. +As a result, both the probability of exciting solely +breathers and the energy spatial correlations are seen to +behave nonmonotonically versus the noise strength. Fur- +thermore, at fixed noise intensity, the excitation proba- +bility is evaluated in the ac frequency-amplitude space, +illustrating the reliability of the approach for different +breathing frequencies. A much-awaited, clear experimen- +tal signature of the stabilized bound state is finally found +in the junction’s resistive switching characteristics. +Note that, although the Josephson realm provides a +solid physical background for this letter, the formalism +is quite general, and an interdisciplinary flavor character- +izes the analysis. In other words, since many complex and +apparently different phenomena [2–7] can be understood +through the lens of the SG model, significant insights into +arXiv:2301.05164v1 [cond-mat.mes-hall] 12 Jan 2023 + +2 +its fundamental excitations have a wide scope within the +scientific community. The topic of SG breathers is indeed +of general interest: from DNA systems [30] and structural +geology [31] to high-Tc superconductivity [32]. +Other examples of breather-type states intensely +studied are: +polygonal breathers [33], +matter–wave +breathers [34], +breather wave molecules [35], +roto- +breathers in JJ ladders [36, 37]. Besides, in JJ parallel +arrays, the theoretically-predicted oscillobreathers, due +to their rapid pulsations, have eluded an experimental +verification for decades [38]. +Exploring the noisy, ac- +driven scenario in a fashion similar to that presented +here could lead to interesting developments even in the +discrete world [39, 40]. +The model.—Taking into account dissipation, an ac +current uniformly distributed in space, and thermal fluc- +tuations, the equation of motion for the LJJ reads [10, 41] +ϕxx − ϕtt − αϕt = sin ϕ − η sin(ωt) − γT (x, t), +(1) +with ϕ(x, t) indicating the phase difference between +the two superconducting wave functions (the notation +∂ϕ/∂x = ϕx is used throughout). The friction coefficient +α = G/ (ωpC) is defined in terms of the effective normal +conductance G, the capacitance per unit length C, and +the Josephson plasma frequency ωp = +� +2πJc/ (Φ0C), +with respect to which frequency is normalized in Eq. (1) +(Jc is the critical Josephson current density) [10]. The +spatial length scale is the Josephson penetration depth +λJ = +� +Φ0/ (2πJcLP ), where LP is the inductance per +unit length. Moreover, ω and η are, respectively, the nor- +malized frequency and amplitude of the external ac driv- +ing (η is given in units of Jc), and γT (x, t) is a Gaussian, +zero-average noise source with the correlation function +⟨γT (x1, t1)γT (x2, t2)⟩ = 2αΓδ(x1 − x2)δ(t1 − t2), +(2) +in which Γ = 2ekBT/ (ℏJcλJ) is the noise strength, pro- +portional to the absolute temperature T, e is the elec- +tron charge, kB is the Boltzmann constant, and ℏ is the +reduced Planck constant. Equation (1) is numerically in- +tegrated via an implicit finite-difference scheme, in the +spatio-temporal domain [−l/2, l/2] × [0, T ], with initial +conditions +ϕ(x, 0) = ϕt(x, 0) = 0, +(3) +and periodic boundary conditions +ϕ(−l/2, t) = ϕ(l/2, t), +(4) +the latter corresponding to an annular-geometry LJJ [11]. +More details, including the approximation of the stochas- +tic term, can be found in [42]. +In what follows, the +junction length is l = 50, the damping parameter is +α = 0.2 [14], and ω < 1, since below-plasma frequencies +are those natural to SG breathers [8, 9]. +FIG. +1. +Two +simulated +energy +density +profiles +ε(x, t) = (ϕ2 +t + ϕ2 +x)/2 + 1 − cos ϕ +[8, +9]. +In +panel +(a), +the spatio-temporal region [−21.5, −11.5] × [30, 130] is mag- +nified to better appreciate both the formation and the first +few oscillations of a single breather located at x ≈ −16.5. +In panel (b), +the inset focuses on [−22, 17] × [950, 975] +to illustrate the ac-locking of multiple nonlinear modes. +Parameter values: +T = 1000 (observation time), ω = 0.6, +η = 0.59, and Γ = 5 × 10−4. +Noise-induced, +stabilized +breathers.—Figure +1 +displays +two +simulated +energy +density +profiles +ε(x, t) = (ϕ2 +t + ϕ2 +x)/2 + 1 − cos ϕ [8, 9]. +Both pan- +els +demonstrate +that, +in +the +presence +of +thermal +fluctutations and ac forcing, remarkably stable breather +excitations can form in the junction. +In a purely +dissipative case, +breathers radiatively decay within +∼ 1/α = 5 [23], a lifetime which is surpassed by multiple +orders of magnitude here. Note also the stability of the +modes with respect to the position, i.e., their centers do +not drift away from the originary positions [x ≈ −16.5 +in Fig. 1(a)] over hundreds of oscillations, despite the +noise influence. These interesting features hold widely +among the different realizations. One or more breathers +typically appear in random spots within a few driving +cycles (t ≈ 50 in Fig. 1). +After a transient, a state +similar to that of Fig. 1, stable over very long times [43], +eventually sets in. +Further information regarding the stabilized oscilla- +tory modes is perhaps useful here: (i) their breathing +cycles are locked to the external ac force [Fig. 1(b), inset]; +(ii) they are strongly localized in space, over the charac- +teristic length λb (ω) = 1/ +√ +1 − ω2 [8, 9], i.e., the width of +an unperturbed breather at frequency ωb = ω; (iii) their +amplitude is ≳ Ab (ω) = 4 arctan +�√ +1 − ω2/ω +� +[8, 9], +which is that of an unperturbed breather at the driving’s +frequency ω [44]. +Keeping the parameter values ω = 0.6 and η = 0.59 as +in Fig. 1, the junction’s response versus the noise strength +Γ ∈ [10−5, 4 × 10−2] [14] is now explored, for T = 500 +and N = 1000 realizations. Specifically, simulating for a +time long enough to let the generation events to unravel, +the final state of each run is classified as follows: (a) no +excitations, if the phase profile is essentially flat over the +spatial domain; (b) breathers only, if the observed modes’ + +0123 +5 +0123456789 +6 +9 +1000 +750 +500 +七 +250 +0 +10 +20 +10 +20 +一10 +20 +—10 +0 +20 +b +a3 +FIG. 2. (a): Probability of having no excitations (Pa, blue), +breathers only (Pb, green), and at least a free kink-antikink +couple (Pc, red) versus Γ. (b): Energy-based coefficient of +spatial correlation, see Eq. (5), as a function of Γ. Parameter +values: T = 500, ω = 0.6, η = 0.59, and N = 1000. +amplitudes lie between Ab and 2π, the latter being the +phase value associated with kink-type structures [8, 9]; +(c) at least a free kink-antikink couple, if at least a 2π- +step excitation is present. +As illustrated in Fig. 2(a), for the lower Γ values, the +probability Pa of having no excitations is 1 (see the blue +circles). As the noise intensity is increased, a new sce- +nario soars, that of breather-only formation. Indeed, for +Γ roughly in [5 × 10−4, 10−2], the corresponding proba- +bility Pb is ≳ 0.9 (see the green circles). This provides a +rather wide range of working temperatures for the cur- +rent approach. The stochastic influence eventually be- +comes disruptive for the oscillatory bound state, and the +kink-antikink regime takes over for Γ > 10−2 (see the red +circles, Pc). The probability of exciting solely breathers +therefore exhibits a nonmonotonicity versus Γ, highlight- +ing the crucial role of the temperature as a control pa- +rameter in the setup. In this regard, the fact that ther- +mal noise can allow for the formation process, without +compromising the long-time stability of the breathers, is +noteworthy. +Furthermore, the energy spatial correlation evaluated +at the characteristic scale λb [42] +C¯ε(λb) ∝ +�� +¯ε(x)¯ε(x + λb)dx +� +�� +¯ε(x)dx +�2 +, +(5) +where ¯ε(x) is the time-averaged energy density, shows a +nonmonotonic behavior as a function of Γ [see Fig. 2(b)]. +Thus, an appropriate amount of environmental noise, in- +stead of degradation, enhances the junction’s sensitivity +to the external force, leading to nontrivial spatial cor- +relations—a somewhat counter-intuitive outcome. The +noise amplitude also impacts the typical timescale of the +generation events: for stronger fluctuations, they occur +earlier in the simulations. This aspect is quantitatively +addressed in [42]. +It is now important to examine, at fixed Γ > 0, the be- +havior of the breather-only generation probability Pb in +the frequency-amplitude parameter space [45]. To cope +with such a heavy computational task, N = 500 runs are +FIG. 3. +Probability of generating solely breathers in the +(ω, η) parameter space. +The red circle identifies the com- +bination ω = 0.6 and η = 0.59. Parameter values: T = 500, +Γ = 5 × 10−3, and N = 500. +performed for each (ω, η) pair, focusing on ω ∈ [0.5, 0.8] +and η ∈ [0.2, 0.8], with ∆ω = 0.02 and ∆η = 0.05. The +simulation time and noise amplitude are T = 500 and +Γ = 5 × 10−3, respectively. +Figure 3 shows that several high-Pb (ω, η) (green, yel- +low) areas exist for breather-only formation. Note that, +for the scenario of Fig. 1 to occur, the combined action +of noise and the deterministic force must provide an en- +ergy of the order of Eb (ω) = 16 +√ +1 − ω2 [8, 9], i.e., that +expected for a breather at frequency ω, without breaking +up any of the subsequent kink-antikink bonds. Two rea- +sons are behind the low-probability (purple) region. The +first one, for η ≳ 0.7 (see Fig. 3), is the kink-antikink (k- +ak) regime, associated to an excess of energy input. For +the remaining purple (ω, η) area, no excitations are ob- +served. One may notice that, at lower ω values, higher +amplitudes η are needed to excite the nonlinear breath- +ing states. This is qualitatively explained by the above +expression of Eb (ω), which implies that breathers with +lower frequencies require more energy. +Another topic worth discussing is the system’s topol- +ogy and its influence on the examined phenomenon. Due +to Eq. (4), the (initially null) topological charge is con- +served, thus no unpaired kinks/antikinks can arise. By +contrast, for Neumann-type boundary conditions, i.e., for +an overlap-geometry LJJ [21, 22], single kinks/antikinks +can emerge at the borders, usually forming bound states +with their virtual counterparts [19, 46]—what one may +call edge-breathers. The latter case was extensively ana- +lyzed as well (not shown here), and the overall picture is +not drastically altered. The difference is that in the peri- +odic framework, i.e., annular LJJs, there are no preferred +locations for the emergence of breather states, whereas +in the Neumann case, i.e., overlap LJJs, edge-breathers, +being essentially single-soliton modes, are more likely ob- +served since they provide an energetic advantage. +Detection.—The lowest dc current value to break up an +unperturbed breather into a kink-antikink pair crucially + +- +0.8 +0.8 +10.6 +0.6 +1 +0.4 +OD +0.4 +0.2 +0.2 +10-10 +6 +10-3 +10-5 +10-4 +10-2 +10-3 +10-2 +10-5 +10-4 +(a) +(b)1.0 +k-ak +0.9 +0.7 +0.8 +0.7 +0.6 +0.6 +~0.5 +0.5p +0.4 +0.4 +0.3 +no exc. +0.2 +0.3 +0.1 +0.2 +0.0 +0.5 +0.6 +0.7 +0.8 +34 +depends on its phase [17, 20]. Starting from this insight, +and taking full advantage of the developed setup, a much- +awaited, clear experimental signature of the oscillatory +bound state is provided. +The parameters ω = 0.6, η = 0.59, and Γ = 5 × 10−4 +are selected here to work with a highly favorable breather +formation scenario (see Figs. 2 and 3). +The physi- +cal idea behind the detection scheme is quite simple: +(i) excite stabilized breathers; (ii) embed their prop- +erties into the switching characteristics of the device +by destroying them at different stages of their oscilla- +tion cycle. More precisely, the ac-driven LJJ is first let +to evolve up to t = (t⋆ + τ), where t⋆ is a time much +greater than that typical for the occurrence of the gen- +eration events, and τ is an arbitrary (time) displace- +ment. With the chosen values of ω, η, and Γ, breathers +emerge roughly within t = 50 (see Fig. 1 and +[42]), +thus t⋆ = 250 is taken to allow the system to reach its +long-time stable configuration. +Next, the smooth cur- +rent bias γ {1 − exp[−0.1(t − t⋆ − τ)]} [20] is applied for +t > (t⋆ + τ), while the ac force is slowly turned off, and +one should record whether the junction switches to a re- +sistive state—namely, whether the kink-antikink splitting +is triggered and a measurable voltage drop appears. The +previous steps have then to be repeated a number of times +to obtain, for each different τ value, the minimal current +γsw leading to a significant switching probability over N +realizations, say, Psw ≥ 0.75. +A few relevant points underlying the above approach +should be mentioned. +Past proposals with a similar +goal [20] have encountered the serious issue of dissipation. +The modes’ stability for t ≤ (t⋆ + τ) practically solves +the problem here. Second, as previously mentioned, the +breather oscillations are locked to the ac-drive, ensuring +that breathers from all the repetitions at fixed τ arrive in +phase at t = (t⋆ + τ). This is crucial, since the whole idea +revolves around breaking up the solitonic bound states at +different stages of their oscillation cycle [47]. Note also +that the randomness in the number of breathers emerg- +ing in each realization does not harm the described se- +quence in any way. Lastly, the slow switch-off of the ac +driving for t > (t⋆ + τ) avoids the simultaneous action of +noise, the smooth current bias, and the ac source at full +strength. The latter situation, in fact, can potentially +lead to additional kink-antikink states that would pretty +much take over the switching dynamics of the junction. +The quantity γsw(τ) displays a peculiar oscillatory be- +havior (see Fig. 4). +A period approximately equal to +10 ≈ 2π/ω can be appreciated, which reflects the breath- +ing cycle. This outcome is markedly different from that +obtained both in the absence of excitations and in a +kink-antikink regime, where no sensitivity to the dis- +placement τ is exhibited. +Indeed, in the small-noise +case Γ = 10−5, where essentially no excitations appear +[Pa ≈ 1 in Fig. 2(a)], one gets Psw ≈ 0 for γ ∈ [0, 0.4], +independently of τ. +With Γ = 4 × 10−2 [Pc ≈ 1 in +FIG. 4. +Lowest current value γsw at which the resistive +state is triggered with probability Psw ≥ 0.75 as a func- +tion of the time displacement τ ∈ [0, 59]. +The ac driv- +ing’s slow switch-off consists in the time-dependent ampli- +tude η exp[−0.01(t − t⋆ − τ)] for t > (t⋆ + τ). Parameter val- +ues: T = 500, ω = 0.6, η = 0.59, Γ = 5 × 10−4, t⋆ = 250, and +N = 500. +Fig. 2(a), i.e., kink-antikink scenario] the minimal cur- +rent is γsw ≈ 0.17 ∀τ. +Conclusions.—This letter addresses the formation of +breathers stable over long times, for sufficiently high tem- +peratures, in ac-driven LJJs. Nonmonotonic behaviors of +both the probability of generating solely breathers and +the energy spatial correlations are obtained as a func- +tion of the noise strength, highlighting the latter’s criti- +cal role as a control parameter. The efficacy of the phe- +nomenon for different breathing frequencies is demon- +strated. +Lastly, the breather induces peculiar oscilla- +tions into the junction’s resistive switching characteris- +tics, which is exploitable to experimentally reveal it. +Preliminary simulations indicate that the results are +robust even to static disorder due, e.g., to impurities in +the device. It may also be interesting to design a setup +where preferred locations for the emergence of breathers +can be selected. This could be, reasonably, achieved by +locally heating the junction or by means of a spatially- +modulated ac force [19]. +The authors are very grateful to Prof. +A. Usti- +nov for suggesting the topic of breathers in Joseph- +son systems and for stimulating discussions. +DDS +gladly acknowledges fruitful discussions with Prof. +D. +Molteni. Most of the numerical runs were performed on +CINECA’s machine Galileo100 (Projects: IscrC NDJB +and IscrB 3DSBM). DDS, CG, BS, AC, DV acknowledge +the support of the Italian Ministry of University and Re- +search (MUR). BS also acknowledges the support of the +Government of the Russian Federation through Agree- +ment No. 074-02-2018-330 (2). +∗ duilio.desantis@unipa.it +[1] A. C. Scott, Encyclopedia of Nonlinear Science (Rout- + +0.35 +0.25 +SW +0.15 +10 +20 +30 +40 +50 +60 +T5 +ledge, 2006). +[2] J. Cuevas-Maraver, P. G. Kevrekidis, and F. Williams, +The sine-Gordon Model and its Applications (Springer, +2014). +[3] L. D. M. Villari, G. Marcucci, M. C. Braidotti, and +C. Conti, J. Phys. Commun. 2, 055016 (2018). +[4] V. G. Bykov, J. Seismol. 18, 497 (2014). +[5] V. G. Ivancevic and T. T. Ivancevic, J. Geom. 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No radiative decay was observed even for +higher T values. +[44] Furthermore, a test was run at Γ = 0, starting from an +exact breather at frequency ω, in the presence of the two +perturbations αϕt and η sin(ωt). The breather was seen +to adjust its amplitude to that observed for the same α, +ω, and η values, in the case of noise-induced formation. +[45] In the absence of thermal noise, no formation of nonlinear +modes occurs, regardless of the ω and η values. +[46] G. Costabile, +R. D. Parmentier, +B. Savo, +D. W. +McLaughlin, and A. C. Scott, Appl. Phys. Lett. 32, 587 +(1978). +[47] Each ‘stage’ corresponds to a displacement τ, and it has +to be replicated multiple times to evaluate Psw. + +Supplemental Material +ac-locking of thermally-induced sine-Gordon breathers +Duilio De Santis,1, ∗ Claudio Guarcello,2, 3 Bernardo Spagnolo,1, 4 Angelo Carollo,1 and Davide Valenti1 +1Dipartimento di Fisica e Chimica “E. Segr`e”, Group of Interdisciplinary Theoretical Physics, +Universit`a degli Studi di Palermo, I-90128 Palermo, Italy +2Dipartimento di Fisica “E. R. Caianiello”, Universit`a degli Studi di Salerno, I-84084 Fisciano, Salerno, Italy +3INFN, Sezione di Napoli, Gruppo Collegato di Salerno - Complesso Universitario di Monte S. Angelo, I-80126 Napoli, Italy +4Radiophysics Department, Lobachevsky State University, 603950 Nizhniy Novgorod, Russia +(Dated: January 13, 2023) +THE SINE-GORDON EQUATION +The sine-Gordon (SG) equation +ϕxx − ϕtt = sin ϕ, +(1) +which can be derived from the energy density ε(x, t) = (ϕ2 +t + ϕ2 +x)/2 + 1 − cos ϕ, admits topological soliton solutions +ϕ±(x, t) = 4 arctan +� +exp +� +± x − vt +√ +1 − v2 +�� +(2) +known as kinks (ϕ+) and antikinks (ϕ−). In Eq. (2), v < 1 is the constant velocity of the travelling wave, whose +relativistic behavior is due to the structure of Eq. (1). In the realm of long Josephson junctions (LJJs), the limiting +velocity is the propagation speed of electromagnetic signals in the device, commonly known as the Swihart velocity, +¯c = λJωp = 1/√LP C, with λJ being the Josephson penetration depth, ωp the Josephson plasma frequency, LP the +inductance per unit length, and C the capacitance per unit length [1]. +Breathers are space-localized, time-periodic solutions of Eq. (1) given by +ϕb(x, t) = 4 arctan +� +� +� +� +� +� +� +� +1 − ω2 +b +ωb +sin +� +ωb(t−vex) +√ +1−v2e +� +cosh +�√ +1−ω2 +b(x−vet) +√ +1−v2e +� +� +� +� +� +� +� +� +, +(3) +where 0 < ωb < 1 and ve < 1 are, respectively, the excitation’s proper frequency and the envelope velocity. Notably, +Eq. (3) can be obtained via analytic continuation from the profile describing the kink-antikink collision. If ve = 0—the +most relevant case for the present work—besides the duration of each breathing cycle Tb = 2π/ωb, the parameter +ωb yields the energy Eb = 16 +� +1 − ω2 +b, the amplitude Ab = 4 arctan +�� +1 − ω2 +b/ωb +� +, and the characteristic length +λb = 1/ +� +1 − ω2 +b of the nonlinear mode. Therefore, high (low) energy breathers possess low (high) frequencies and +high (low) oscillation amplitudes. +For more information concerning the SG equation, even in the presence of perturbation terms, see Refs. [2, 3] and +references therein. +NUMERICAL SOLUTION OF THE PERTURBED SINE-GORDON EQUATION +An implicit finite-difference scheme is employed to integrate the following perturbed SG equation +ϕxx − ϕtt − αϕt = sin ϕ − η sin(ωt), +(4) +where α is the dissipation coefficient and ω and η are, respectively, the frequency and amplitude of the monochromatic +force. More specifically, the spatial domain is divided into N cells of length ∆x = h and the temporal domain into +M intervals of duration ∆t = k. Within this framework, the restriction of ϕ(x, t) is indicated as ϕm +n = ϕ(nh, mk), for +arXiv:2301.05164v1 [cond-mat.mes-hall] 12 Jan 2023 + +2 +n = 1, ..., N and m = 1, ..., M. Then, one replaces the derivatives in Eq. (4) with [4] +ϕx = 1 +2h +� +ϕm +n+1 − ϕm +n−1 +� ++ O(h2), +ϕt = 1 +2k +� +ϕm+1 +n +− ϕm−1 +n +� ++ O(k2), +ϕxx = +1 +2h2 +� +ϕm+1 +n+1 − 2ϕm+1 +n ++ ϕm+1 +n−1 + ϕm−1 +n+1 − 2ϕm−1 +n ++ ϕm−1 +n−1 +� ++ O(h2 + k2), +ϕtt = 1 +k2 +� +ϕm+1 +n +− 2ϕm +n + ϕm−1 +n +� ++ O(k2). +(5) +If O(h2) and O(k2) terms are neglected, and both the initial and the boundary conditions are imposed, one gets a +system of equations, whose resolution determines the new (unknown) values ϕm+1 +n +, given the previous ones ϕm +n and +ϕm−1 +n +, with n = 1, ..., N. For periodic boundary conditions, the matrix representing the system is cyclic tridiagonal, +i.e., it has nonzero elements only on the diagonal, the subdiagonal, the superdiagonal, and in the corners. The solution +is therefore found by combining the Sherman-Morrison formula with Thomas’ algorithm (the latter being a simplified +form of Gaussian elimination) [5]. Moreover, Wilkinson’s iterative refinement method is used at each step to prevent +the accumulation of rounding errors [5]. +According to Ref. [6], the noisy perturbation γT (x, t), whose statistical properties are +⟨γT (x, t)⟩ = 0 +and +⟨γT (x1, t1)γT (x2, t2)⟩ = 2αΓδ(x1 − x2)δ(t1 − t2), +(6) +can be numerically handled as +√ +2αΓ W m +n +√ +hk +, +(7) +in which Γ is the noise strength and W m +n are independent normal random variables with zero mean and unit variance. +The computational scheme’s precision was tested by systematically varying the values of the space and time +steps, and by examining the discrepancy with a variety of analytical SG solutions, such as Eqs. (2) and (3), in +the α = η = Γ = 0 case. +Throughout the work, the chosen discretization steps are h = k = 0.01. +ENERGY-BASED ANALYSIS OF THE SPATIAL CORRELATIONS +The energy density ε(x, t), along with its spatial correlations, is a key physical quantity to characterize the emergence +of localized, coherent structures [7, 8]. In particular, considering the time average +¯ε(x) = 1 +T +� T +0 +ε(x, t)dt, +(8) +where T is the observation time, the following spatial correlation function is introduced +C¯ε(X) ∝ +�� +¯ε(x)¯ε(x + X)dx +� +�� +¯ε(x)dx +�2 +, +(9) +in which X is a space displacement, ⟨...⟩ denotes the ensemble average, and the integrals are performed over the whole +spatial domain. +To truly appreciate Eq. (9)’s significance, it is useful to discuss its behavior in distinct scenarios: (i) no excitations +(i.e., spatially-uniform condition); (ii) breathers stable both in amplitude and position over long times; (iii) turbulent +kink-antikink regime, with different excitations possibly appearing and wandering/annihilating over time. +In the +first, trivially-correlated case, the energy is equally distributed among all the system’s sites, therefore a flat C¯ε(X) +is obtained [C¯ε(X) = 1 under appropriate normalization in Eq. (9)]. When long-time stable breathers are present, +each one results in a prominent ¯ε(x) spike of width λb = 1/ +√ +1 − ω2, with λb being the characteristic length of an +unperturbed breather with frequency ωb = ω (ω is the frequency of the sinusoidal force, see above). Thus, C¯ε(X) +peaks at X = 0, and it decays to 1 with the typical scale λb as X is increased. +In the third situation, a nearly +spatially-homogeneous C¯ε(X) ≈ 1 is restored, but the underlying reason is rather different from that of (i). Many + +3 +FIG. 1. Energy-based coefficient of spatial correlation versus Γ ∈ [10−5, 4 × 10−2]. Parameter values: ∆x = ∆t = 0.01, l = 50, +T = 500, α = 0.2, ω = 0.6, η = 0.59, and N = 1000. +solitonic excitations are indeed observed temporarily in the system, but due to thermal agitation they do not leave +long-lasting marks on a few (random) spots. On average, see Eq. (8), the spatial domain is roughly explored in a +uniform way by these strongly fluctuation-driven transients. +In light of the above, it is natural to evaluate the system’s response in terms of C¯ε(X = λb), as a function of Γ. +As displayed in Fig. 1 [i.e., Fig. 2(b) in the main text, reported here for the reader’s convenience], such a quantity +behaves nonmonotonically versus Γ ∈ [10−5, 4 × 10−2]. The values of the remaining parameters are: ∆x = ∆t = 0.01, +l = 50 (system length), T = 500, α = 0.2, ω = 0.6, η = 0.59, and N = 1000 (number of realizations). At each in- +tegration step, the phase ϕ is regularized via moving averages before calculating ϕt and ϕx needed to evaluate +ε(x, t) = (ϕ2 +t + ϕ2 +x)/2 + 1 − cos ϕ. Clearly, the previously mentioned cases (i), (ii), and (iii) correspond to the phe- +nomenology observed at low, intermediate, and high noise strengths, respectively. Therefore, the trend in Fig. 2 is +well-understood on physical grounds. +In short, an appropriate amount of environmental noise can enhance the system’s sensitivity to the external force, +leading to nontrivial spatial correlations. +TYPICAL TIMESCALE OF THE SOLITONIC FORMATION EVENTS +The noise strength is reasonably expected to influence the typical timescale of the random emergence of the solitonic +states (breathers and kink-antikink couples). To examine this, focusing on the fraction of realizations where at least +one generation event takes place, one can record the time ˆt at which the first excitation with amplitude greater or +FIG. 2. Average time +�ˆt +� +and the corresponding standard deviation as a function of the noise amplitude Γ ∈ [10−5, 4 × 10−2]. +Parameter values: ∆x = ∆t = 0.01, l = 50, T = 500, α = 0.2, ω = 0.6, η = 0.59, and N = 1000. + +0.8 +0.6 +1 0.4 +X +0.2 +口 +0.01 +10-4 +-01 +10-2 +10-5 +I60 +40 +20 +10-4 +10-3 +10-2 +10-5 +I4 +equal to Ab = 4 arctan +�√ +1 − ω2/ω +� +is observed, with Ab being the amplitude of an unperturbed breather at frequency +ωb = ω. +Figure 2 shows the average time +�ˆt +� +and the corresponding standard deviation as a function of Γ ∈ [10−5, 4 × 10−2]. +The values of the remaining parameters are ∆x = ∆t = 0.01, l = 50, T = 500, α = 0.2, ω = 0.6, η = 0.59, and +N = 1000. +A clear result is found: for higher noise amplitudes, on average, nonlinear modes appear in earlier +stages of the simulations. +∗ duilio.desantis@unipa.it +[1] A. Barone and G. Patern`o, Physics and Applications of the Josephson Effect (Wiley, New York, 1982). +[2] A. C. Scott, Nonlinear Science: Emergence and Dynamics of Coherent Structures (Oxford, 2003). +[3] T. Dauxois and M. Peyrard, Physics of Solitons (Cambridge University Press, 2006). +[4] W. F. Ames, Numerical Methods for Partial Differential Equations (Academic press, 1977). +[5] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific +Computing (Cambridge University Press, 1992). +[6] H. C. Tuckwell, Wave Motion 65, 130 (2016). +[7] D. W. Brown, L. J. Bernstein, and K. Lindenberg, Phys. Rev. E 54, 3352 (1996). +[8] I. Daumont, T. Dauxois, and M. Peyrard, Nonlinearity 10, 617 (1997). + diff --git a/S9E4T4oBgHgl3EQfmA05/content/tmp_files/load_file.txt b/S9E4T4oBgHgl3EQfmA05/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..864af6a4164450373697f48a3f30b3ee3144b13c --- /dev/null +++ b/S9E4T4oBgHgl3EQfmA05/content/tmp_files/load_file.txt @@ -0,0 +1,747 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf,len=746 +page_content='ac-locking of thermally-induced sine-Gordon breathers Duilio De Santis,1, ∗ Claudio Guarcello,2, 3 Bernardo Spagnolo,1, 4 Angelo Carollo,1 and Davide Valenti1 1Dipartimento di Fisica e Chimica “E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Segr`e”, Group of Interdisciplinary Theoretical Physics, Universit`a degli Studi di Palermo, I-90128 Palermo, Italy 2Dipartimento di Fisica “E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Caianiello”, Universit`a degli Studi di Salerno, I-84084 Fisciano, Salerno, Italy 3INFN, Sezione di Napoli, Gruppo Collegato di Salerno - Complesso Universitario di Monte S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Angelo, I-80126 Napoli, Italy 4Radiophysics Department, Lobachevsky State University, 603950 Nizhniy Novgorod, Russia (Dated: January 13, 2023) A complete framework for exciting and detecting thermally-induced, stabilized sine-Gordon breathers in ac-driven long Josephson junctions is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The formation of long-time stable breathers locked to the ac source occurs for a sufficiently high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The latter emerges as a powerful control parameter, allowing for the remarkably stable localized modes to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Nonmonotonic behaviors of both the breather generation probability and the energy spatial correla- tions versus the thermal noise strength are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The junction’s resistive switching characteristics provides a clear experimental signature of the breather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='—Owing to its simplicity and nonlinear nature, the sine-Gordon (SG) equation [1] is universally recognized as a fundamental modelling tool within the scientific community [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The SG framework, in fact, provides a very accurate and intuitive viewpoint for a large variety of phenomena occurring in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', gravity and black holes [2, 3], tectonic stress transfer [4], biology [5], superconductivity and Josephson junctions (JJs) [2, 6], Bose-Einstein condensates [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' A key feature of the SG equation is its rich spectrum of solutions, which includes both kink-type and breather- type solitons [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The first are topological excitations which can be visualized as 2π-twists in a mechanical chain of linearly coupled pendula [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' A breather is a space-localized, time-periodic bound state stemming from the kink-antikink attraction [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The long Josephson junction (LJJ) is a (quasi) one- dimensional, superconductor-based system whose elec- trodynamics is reliably described by the SG model [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Being the subject of many seminal experiments [10–12] and striking applications [12–16], this device has played an outstanding role in the spreading of the soliton con- cept throughout natural and applied sciences [2, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In LJJs, a kink represents a magnetic flux quantum Φ0 [1], induced by a supercurrent loop, whose properties reflect into the I -V characteristic of the junction [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Due to its nontopological structure, mastering the breather’s physics is a very tough challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In particu- lar, experimental evidence of this oscillating state has yet to be provided in LJJs, despite the numerous in- vestigations on the matter [17–23], primarily due to its friction-triggered radiative decay and its elusiveness with respect to I -V measurements [20, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The Josephson breather’s detection would, therefore, solve a long-lasting problem in nonlinear science, but it would also pave the way for several applications in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', information trans- mission [25], quantum computation [26], generation of THz radiation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Previous works (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' [19]), analyzed the sta- bilization of stationary SG breathers via ac-driving, with specific ad-hoc initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Such a scenario, how- ever, has remained experimentally unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' This is presumably due to the practical difficulties in creating persistent breather states, given the stabilization effect’s crucial dependence on the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Moreover, the phenomenon’s robustness against thermal fluctua- tions has not been addressed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' On the other hand, the little discussed topic of breathers in a noisy environment has recently gained attention [21–23, 28, 29], and positive stochastically- induced effects on both the generation and the dynamics of these nonlinear waves have been demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The present manuscript thus examines a lossy, ac-driven LJJ in the presence of thermal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The emergence of long- time stable breathers locked to the sinusoidal force is ob- served for a sufficiently high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The latter is, consequently, a powerful control parameter, allowing for the localized modes to appear, while not endanger- ing their persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The achievement of both the cre- ation and the stabilization in a single effort should not be overlooked, given the multistability of the SG system, responsible for the possible emergence of kink-antikink pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' As a result, both the probability of exciting solely breathers and the energy spatial correlations are seen to behave nonmonotonically versus the noise strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Fur- thermore, at fixed noise intensity, the excitation proba- bility is evaluated in the ac frequency-amplitude space, illustrating the reliability of the approach for different breathing frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' A much-awaited, clear experimen- tal signature of the stabilized bound state is finally found in the junction’s resistive switching characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Note that, although the Josephson realm provides a solid physical background for this letter, the formalism is quite general, and an interdisciplinary flavor character- izes the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In other words, since many complex and apparently different phenomena [2–7] can be understood through the lens of the SG model, significant insights into arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='05164v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='mes-hall] 12 Jan 2023 2 its fundamental excitations have a wide scope within the scientific community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The topic of SG breathers is indeed of general interest: from DNA systems [30] and structural geology [31] to high-Tc superconductivity [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Other examples of breather-type states intensely studied are: polygonal breathers [33], matter–wave breathers [34], breather wave molecules [35], roto- breathers in JJ ladders [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Besides, in JJ parallel arrays, the theoretically-predicted oscillobreathers, due to their rapid pulsations, have eluded an experimental verification for decades [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Exploring the noisy, ac- driven scenario in a fashion similar to that presented here could lead to interesting developments even in the discrete world [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='—Taking into account dissipation, an ac current uniformly distributed in space, and thermal fluc- tuations, the equation of motion for the LJJ reads [10, 41] ϕxx − ϕtt − αϕt = sin ϕ − η sin(ωt) − γT (x, t), (1) with ϕ(x, t) indicating the phase difference between the two superconducting wave functions (the notation ∂ϕ/∂x = ϕx is used throughout).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The friction coefficient α = G/ (ωpC) is defined in terms of the effective normal conductance G, the capacitance per unit length C, and the Josephson plasma frequency ωp = � 2πJc/ (Φ0C), with respect to which frequency is normalized in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (1) (Jc is the critical Josephson current density) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The spatial length scale is the Josephson penetration depth λJ = � Φ0/ (2πJcLP ), where LP is the inductance per unit length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Moreover, ω and η are, respectively, the nor- malized frequency and amplitude of the external ac driv- ing (η is given in units of Jc), and γT (x, t) is a Gaussian, zero-average noise source with the correlation function ⟨γT (x1, t1)γT (x2, t2)⟩ = 2αΓδ(x1 − x2)δ(t1 − t2), (2) in which Γ = 2ekBT/ (ℏJcλJ) is the noise strength, pro- portional to the absolute temperature T, e is the elec- tron charge, kB is the Boltzmann constant, and ℏ is the reduced Planck constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Equation (1) is numerically in- tegrated via an implicit finite-difference scheme, in the spatio-temporal domain [−l/2, l/2] × [0, T ], with initial conditions ϕ(x, 0) = ϕt(x, 0) = 0, (3) and periodic boundary conditions ϕ(−l/2, t) = ϕ(l/2, t), (4) the latter corresponding to an annular-geometry LJJ [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' More details, including the approximation of the stochas- tic term, can be found in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In what follows, the junction length is l = 50, the damping parameter is α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='2 [14], and ω < 1, since below-plasma frequencies are those natural to SG breathers [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Two simulated energy density profiles ε(x, t) = (ϕ2 t + ϕ2 x)/2 + 1 − cos ϕ [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In panel (a), the spatio-temporal region [−21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='5, −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='5] × [30, 130] is mag- nified to better appreciate both the formation and the first few oscillations of a single breather located at x ≈ −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In panel (b), the inset focuses on [−22, 17] × [950, 975] to illustrate the ac-locking of multiple nonlinear modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Parameter values: T = 1000 (observation time), ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='59, and Γ = 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Noise-induced, stabilized breathers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='—Figure 1 displays two simulated energy density profiles ε(x, t) = (ϕ2 t + ϕ2 x)/2 + 1 − cos ϕ [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Both pan- els demonstrate that, in the presence of thermal fluctutations and ac forcing, remarkably stable breather excitations can form in the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In a purely dissipative case, breathers radiatively decay within ∼ 1/α = 5 [23], a lifetime which is surpassed by multiple orders of magnitude here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Note also the stability of the modes with respect to the position, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', their centers do not drift away from the originary positions [x ≈ −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='5 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 1(a)] over hundreds of oscillations, despite the noise influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' These interesting features hold widely among the different realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' One or more breathers typically appear in random spots within a few driving cycles (t ≈ 50 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' After a transient, a state similar to that of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 1, stable over very long times [43], eventually sets in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Further information regarding the stabilized oscilla- tory modes is perhaps useful here: (i) their breathing cycles are locked to the external ac force [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 1(b), inset];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (ii) they are strongly localized in space, over the charac- teristic length λb (ω) = 1/ √ 1 − ω2 [8, 9], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', the width of an unperturbed breather at frequency ωb = ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (iii) their amplitude is ≳ Ab (ω) = 4 arctan �√ 1 − ω2/ω � [8, 9], which is that of an unperturbed breather at the driving’s frequency ω [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Keeping the parameter values ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='59 as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 1, the junction’s response versus the noise strength Γ ∈ [10−5, 4 × 10−2] [14] is now explored, for T = 500 and N = 1000 realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Specifically, simulating for a time long enough to let the generation events to unravel, the final state of each run is classified as follows: (a) no excitations, if the phase profile is essentially flat over the spatial domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (b) breathers only, if the observed modes’ 0123 5 0123456789 6 9 1000 750 500 七 250 0 10 20 10 20 一10 20 —10 0 20 b a3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (a): Probability of having no excitations (Pa, blue), breathers only (Pb, green), and at least a free kink-antikink couple (Pc, red) versus Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (b): Energy-based coefficient of spatial correlation, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (5), as a function of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Parameter values: T = 500, ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='59, and N = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' amplitudes lie between Ab and 2π, the latter being the phase value associated with kink-type structures [8, 9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (c) at least a free kink-antikink couple, if at least a 2π- step excitation is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 2(a), for the lower Γ values, the probability Pa of having no excitations is 1 (see the blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' As the noise intensity is increased, a new sce- nario soars, that of breather-only formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Indeed, for Γ roughly in [5 × 10−4, 10−2], the corresponding proba- bility Pb is ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='9 (see the green circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' This provides a rather wide range of working temperatures for the cur- rent approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The stochastic influence eventually be- comes disruptive for the oscillatory bound state, and the kink-antikink regime takes over for Γ > 10−2 (see the red circles, Pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The probability of exciting solely breathers therefore exhibits a nonmonotonicity versus Γ, highlight- ing the crucial role of the temperature as a control pa- rameter in the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In this regard, the fact that ther- mal noise can allow for the formation process, without compromising the long-time stability of the breathers, is noteworthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Furthermore, the energy spatial correlation evaluated at the characteristic scale λb [42] C¯ε(λb) ∝ �� ¯ε(x)¯ε(x + λb)dx � �� ¯ε(x)dx �2 , (5) where ¯ε(x) is the time-averaged energy density, shows a nonmonotonic behavior as a function of Γ [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Thus, an appropriate amount of environmental noise, in- stead of degradation, enhances the junction’s sensitivity to the external force, leading to nontrivial spatial cor- relations—a somewhat counter-intuitive outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The noise amplitude also impacts the typical timescale of the generation events: for stronger fluctuations, they occur earlier in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' This aspect is quantitatively addressed in [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' It is now important to examine, at fixed Γ > 0, the be- havior of the breather-only generation probability Pb in the frequency-amplitude parameter space [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' To cope with such a heavy computational task, N = 500 runs are FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Probability of generating solely breathers in the (ω, η) parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The red circle identifies the com- bination ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6 and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Parameter values: T = 500, Γ = 5 × 10−3, and N = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' performed for each (ω, η) pair, focusing on ω ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='8] and η ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='8], with ∆ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='02 and ∆η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The simulation time and noise amplitude are T = 500 and Γ = 5 × 10−3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Figure 3 shows that several high-Pb (ω, η) (green, yel- low) areas exist for breather-only formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Note that, for the scenario of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 1 to occur, the combined action of noise and the deterministic force must provide an en- ergy of the order of Eb (ω) = 16 √ 1 − ω2 [8, 9], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', that expected for a breather at frequency ω, without breaking up any of the subsequent kink-antikink bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Two rea- sons are behind the low-probability (purple) region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The first one, for η ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='7 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 3), is the kink-antikink (k- ak) regime, associated to an excess of energy input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' For the remaining purple (ω, η) area, no excitations are ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' One may notice that, at lower ω values, higher amplitudes η are needed to excite the nonlinear breath- ing states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' This is qualitatively explained by the above expression of Eb (ω), which implies that breathers with lower frequencies require more energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Another topic worth discussing is the system’s topol- ogy and its influence on the examined phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Due to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (4), the (initially null) topological charge is con- served, thus no unpaired kinks/antikinks can arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' By contrast, for Neumann-type boundary conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', for an overlap-geometry LJJ [21, 22], single kinks/antikinks can emerge at the borders, usually forming bound states with their virtual counterparts [19, 46]—what one may call edge-breathers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The latter case was extensively ana- lyzed as well (not shown here), and the overall picture is not drastically altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The difference is that in the peri- odic framework, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', annular LJJs, there are no preferred locations for the emergence of breather states, whereas in the Neumann case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', overlap LJJs, edge-breathers, being essentially single-soliton modes, are more likely ob- served since they provide an energetic advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='—The lowest dc current value to break up an unperturbed breather into a kink-antikink pair crucially 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='4 OD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='2 10-10 6 10-3 10-5 10-4 10-2 10-3 10-2 10-5 10-4 (a) (b)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='0 k-ak 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6 ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='5p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='3 no exc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='8 34 depends on its phase [17, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Starting from this insight, and taking full advantage of the developed setup, a much- awaited, clear experimental signature of the oscillatory bound state is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The parameters ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='59, and Γ = 5 × 10−4 are selected here to work with a highly favorable breather formation scenario (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The physi- cal idea behind the detection scheme is quite simple: (i) excite stabilized breathers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (ii) embed their prop- erties into the switching characteristics of the device by destroying them at different stages of their oscilla- tion cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' More precisely, the ac-driven LJJ is first let to evolve up to t = (t⋆ + τ), where t⋆ is a time much greater than that typical for the occurrence of the gen- eration events, and τ is an arbitrary (time) displace- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' With the chosen values of ω, η, and Γ, breathers emerge roughly within t = 50 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 1 and [42]), thus t⋆ = 250 is taken to allow the system to reach its long-time stable configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Next, the smooth cur- rent bias γ {1 − exp[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='1(t − t⋆ − τ)]} [20] is applied for t > (t⋆ + τ), while the ac force is slowly turned off, and one should record whether the junction switches to a re- sistive state—namely, whether the kink-antikink splitting is triggered and a measurable voltage drop appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The previous steps have then to be repeated a number of times to obtain, for each different τ value, the minimal current γsw leading to a significant switching probability over N realizations, say, Psw ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' A few relevant points underlying the above approach should be mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Past proposals with a similar goal [20] have encountered the serious issue of dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The modes’ stability for t ≤ (t⋆ + τ) practically solves the problem here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Second, as previously mentioned, the breather oscillations are locked to the ac-drive, ensuring that breathers from all the repetitions at fixed τ arrive in phase at t = (t⋆ + τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' This is crucial, since the whole idea revolves around breaking up the solitonic bound states at different stages of their oscillation cycle [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Note also that the randomness in the number of breathers emerg- ing in each realization does not harm the described se- quence in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Lastly, the slow switch-off of the ac driving for t > (t⋆ + τ) avoids the simultaneous action of noise, the smooth current bias, and the ac source at full strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The latter situation, in fact, can potentially lead to additional kink-antikink states that would pretty much take over the switching dynamics of the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The quantity γsw(τ) displays a peculiar oscillatory be- havior (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' A period approximately equal to 10 ≈ 2π/ω can be appreciated, which reflects the breath- ing cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' This outcome is markedly different from that obtained both in the absence of excitations and in a kink-antikink regime, where no sensitivity to the dis- placement τ is exhibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Indeed, in the small-noise case Γ = 10−5, where essentially no excitations appear [Pa ≈ 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 2(a)], one gets Psw ≈ 0 for γ ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='4], independently of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' With Γ = 4 × 10−2 [Pc ≈ 1 in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Lowest current value γsw at which the resistive state is triggered with probability Psw ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='75 as a func- tion of the time displacement τ ∈ [0, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The ac driv- ing’s slow switch-off consists in the time-dependent ampli- tude η exp[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='01(t − t⋆ − τ)] for t > (t⋆ + τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Parameter val- ues: T = 500, ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='59, Γ = 5 × 10−4, t⋆ = 250, and N = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 2(a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', kink-antikink scenario] the minimal cur- rent is γsw ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='17 ∀τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='—This letter addresses the formation of breathers stable over long times, for sufficiently high tem- peratures, in ac-driven LJJs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Nonmonotonic behaviors of both the probability of generating solely breathers and the energy spatial correlations are obtained as a func- tion of the noise strength, highlighting the latter’s criti- cal role as a control parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The efficacy of the phe- nomenon for different breathing frequencies is demon- strated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Lastly, the breather induces peculiar oscilla- tions into the junction’s resistive switching characteris- tics, which is exploitable to experimentally reveal it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Preliminary simulations indicate that the results are robust even to static disorder due, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', to impurities in the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' It may also be interesting to design a setup where preferred locations for the emergence of breathers can be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' This could be, reasonably, achieved by locally heating the junction or by means of a spatially- modulated ac force [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The authors are very grateful to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Usti- nov for suggesting the topic of breathers in Joseph- son systems and for stimulating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' DDS gladly acknowledges fruitful discussions with Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Molteni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Most of the numerical runs were performed on CINECA’s machine Galileo100 (Projects: IscrC NDJB and IscrB 3DSBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' DDS, CG, BS, AC, DV acknowledge the support of the Italian Ministry of University and Re- search (MUR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' BS also acknowledges the support of the Government of the Russian Federation through Agree- ment No.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 102, 205505 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' [41] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Castellano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Torrioli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Cosmelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Costantini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Chiarello, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Carelli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Rotoli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Cirillo, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Kautz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' B 54, 15417 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' [42] See Supplemental Material for more information on the SG equation, the numerical techniques, the energy-based analysis of the spatial correlations, and a discussion of the typical timescale of the generation events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' [43] The choice T = 1000 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 1 was made for visualiza- tion purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' No radiative decay was observed even for higher T values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' [44] Furthermore, a test was run at Γ = 0, starting from an exact breather at frequency ω, in the presence of the two perturbations αϕt and η sin(ωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The breather was seen to adjust its amplitude to that observed for the same α, ω, and η values, in the case of noise-induced formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' [45] In the absence of thermal noise, no formation of nonlinear modes occurs, regardless of the ω and η values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' [46] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Costabile, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Parmentier, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Savo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' McLaughlin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Scott, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 32, 587 (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' [47] Each ‘stage’ corresponds to a displacement τ, and it has to be replicated multiple times to evaluate Psw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Supplemental Material ac-locking of thermally-induced sine-Gordon breathers Duilio De Santis,1, ∗ Claudio Guarcello,2, 3 Bernardo Spagnolo,1, 4 Angelo Carollo,1 and Davide Valenti1 1Dipartimento di Fisica e Chimica “E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Segr`e”, Group of Interdisciplinary Theoretical Physics, Universit`a degli Studi di Palermo, I-90128 Palermo, Italy 2Dipartimento di Fisica “E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Caianiello”, Universit`a degli Studi di Salerno, I-84084 Fisciano, Salerno, Italy 3INFN, Sezione di Napoli, Gruppo Collegato di Salerno - Complesso Universitario di Monte S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Angelo, I-80126 Napoli, Italy 4Radiophysics Department, Lobachevsky State University, 603950 Nizhniy Novgorod, Russia (Dated: January 13, 2023) THE SINE-GORDON EQUATION The sine-Gordon (SG) equation ϕxx − ϕtt = sin ϕ, (1) which can be derived from the energy density ε(x, t) = (ϕ2 t + ϕ2 x)/2 + 1 − cos ϕ, admits topological soliton solutions ϕ±(x, t) = 4 arctan � exp � ± x − vt √ 1 − v2 �� (2) known as kinks (ϕ+) and antikinks (ϕ−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (2), v < 1 is the constant velocity of the travelling wave, whose relativistic behavior is due to the structure of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In the realm of long Josephson junctions (LJJs), the limiting velocity is the propagation speed of electromagnetic signals in the device, commonly known as the Swihart velocity, ¯c = λJωp = 1/√LP C, with λJ being the Josephson penetration depth, ωp the Josephson plasma frequency, LP the inductance per unit length, and C the capacitance per unit length [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Breathers are space-localized, time-periodic solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (1) given by ϕb(x, t) = 4 arctan � � � � � � � � 1 − ω2 b ωb sin � ωb(t−vex) √ 1−v2e � cosh �√ 1−ω2 b(x−vet) √ 1−v2e � � � � � � � � , (3) where 0 < ωb < 1 and ve < 1 are, respectively, the excitation’s proper frequency and the envelope velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Notably, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (3) can be obtained via analytic continuation from the profile describing the kink-antikink collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' If ve = 0—the most relevant case for the present work—besides the duration of each breathing cycle Tb = 2π/ωb, the parameter ωb yields the energy Eb = 16 � 1 − ω2 b, the amplitude Ab = 4 arctan �� 1 − ω2 b/ωb � , and the characteristic length λb = 1/ � 1 − ω2 b of the nonlinear mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Therefore, high (low) energy breathers possess low (high) frequencies and high (low) oscillation amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' For more information concerning the SG equation, even in the presence of perturbation terms, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' [2, 3] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' NUMERICAL SOLUTION OF THE PERTURBED SINE-GORDON EQUATION An implicit finite-difference scheme is employed to integrate the following perturbed SG equation ϕxx − ϕtt − αϕt = sin ϕ − η sin(ωt), (4) where α is the dissipation coefficient and ω and η are, respectively, the frequency and amplitude of the monochromatic force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' More specifically, the spatial domain is divided into N cells of length ∆x = h and the temporal domain into M intervals of duration ∆t = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Within this framework, the restriction of ϕ(x, t) is indicated as ϕm n = ϕ(nh, mk), for arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='05164v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='mes-hall] 12 Jan 2023 2 n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', N and m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Then, one replaces the derivatives in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (4) with [4] ϕx = 1 2h � ϕm n+1 − ϕm n−1 � + O(h2), ϕt = 1 2k � ϕm+1 n − ϕm−1 n � + O(k2), ϕxx = 1 2h2 � ϕm+1 n+1 − 2ϕm+1 n + ϕm+1 n−1 + ϕm−1 n+1 − 2ϕm−1 n + ϕm−1 n−1 � + O(h2 + k2), ϕtt = 1 k2 � ϕm+1 n − 2ϕm n + ϕm−1 n � + O(k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (5) If O(h2) and O(k2) terms are neglected, and both the initial and the boundary conditions are imposed, one gets a system of equations, whose resolution determines the new (unknown) values ϕm+1 n , given the previous ones ϕm n and ϕm−1 n , with n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' For periodic boundary conditions, the matrix representing the system is cyclic tridiagonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', it has nonzero elements only on the diagonal, the subdiagonal, the superdiagonal, and in the corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The solution is therefore found by combining the Sherman-Morrison formula with Thomas’ algorithm (the latter being a simplified form of Gaussian elimination) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Moreover, Wilkinson’s iterative refinement method is used at each step to prevent the accumulation of rounding errors [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' According to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' [6], the noisy perturbation γT (x, t), whose statistical properties are ⟨γT (x, t)⟩ = 0 and ⟨γT (x1, t1)γT (x2, t2)⟩ = 2αΓδ(x1 − x2)δ(t1 − t2), (6) can be numerically handled as √ 2αΓ W m n √ hk , (7) in which Γ is the noise strength and W m n are independent normal random variables with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The computational scheme’s precision was tested by systematically varying the values of the space and time steps, and by examining the discrepancy with a variety of analytical SG solutions, such as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (2) and (3), in the α = η = Γ = 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Throughout the work, the chosen discretization steps are h = k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' ENERGY-BASED ANALYSIS OF THE SPATIAL CORRELATIONS The energy density ε(x, t), along with its spatial correlations, is a key physical quantity to characterize the emergence of localized, coherent structures [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In particular, considering the time average ¯ε(x) = 1 T � T 0 ε(x, t)dt, (8) where T is the observation time, the following spatial correlation function is introduced C¯ε(X) ∝ �� ¯ε(x)¯ε(x + X)dx � �� ¯ε(x)dx �2 , (9) in which X is a space displacement, ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='⟩ denotes the ensemble average, and the integrals are performed over the whole spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' To truly appreciate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (9)’s significance, it is useful to discuss its behavior in distinct scenarios: (i) no excitations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', spatially-uniform condition);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (ii) breathers stable both in amplitude and position over long times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (iii) turbulent kink-antikink regime, with different excitations possibly appearing and wandering/annihilating over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In the first, trivially-correlated case, the energy is equally distributed among all the system’s sites, therefore a flat C¯ε(X) is obtained [C¯ε(X) = 1 under appropriate normalization in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (9)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' When long-time stable breathers are present, each one results in a prominent ¯ε(x) spike of width λb = 1/ √ 1 − ω2, with λb being the characteristic length of an unperturbed breather with frequency ωb = ω (ω is the frequency of the sinusoidal force, see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Thus, C¯ε(X) peaks at X = 0, and it decays to 1 with the typical scale λb as X is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In the third situation, a nearly spatially-homogeneous C¯ε(X) ≈ 1 is restored, but the underlying reason is rather different from that of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Many 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Energy-based coefficient of spatial correlation versus Γ ∈ [10−5, 4 × 10−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Parameter values: ∆x = ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='01, l = 50, T = 500, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='2, ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='59, and N = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' solitonic excitations are indeed observed temporarily in the system, but due to thermal agitation they do not leave long-lasting marks on a few (random) spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' On average, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' (8), the spatial domain is roughly explored in a uniform way by these strongly fluctuation-driven transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In light of the above, it is natural to evaluate the system’s response in terms of C¯ε(X = λb), as a function of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' As displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 1 [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 2(b) in the main text, reported here for the reader’s convenience], such a quantity behaves nonmonotonically versus Γ ∈ [10−5, 4 × 10−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The values of the remaining parameters are: ∆x = ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='01, l = 50 (system length), T = 500, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='2, ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='59, and N = 1000 (number of realizations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' At each in- tegration step, the phase ϕ is regularized via moving averages before calculating ϕt and ϕx needed to evaluate ε(x, t) = (ϕ2 t + ϕ2 x)/2 + 1 − cos ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Clearly, the previously mentioned cases (i), (ii), and (iii) correspond to the phe- nomenology observed at low, intermediate, and high noise strengths, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Therefore, the trend in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 2 is well-understood on physical grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' In short, an appropriate amount of environmental noise can enhance the system’s sensitivity to the external force, leading to nontrivial spatial correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' TYPICAL TIMESCALE OF THE SOLITONIC FORMATION EVENTS The noise strength is reasonably expected to influence the typical timescale of the random emergence of the solitonic states (breathers and kink-antikink couples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' To examine this, focusing on the fraction of realizations where at least one generation event takes place, one can record the time ˆt at which the first excitation with amplitude greater or FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Average time �ˆt � and the corresponding standard deviation as a function of the noise amplitude Γ ∈ [10−5, 4 × 10−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Parameter values: ∆x = ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='01, l = 50, T = 500, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='2, ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='59, and N = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='4 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='2 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='01 10-4 01 10-2 10-5 I60 40 20 10-4 10-3 10-2 10-5 I4 equal to Ab = 4 arctan �√ 1 − ω2/ω � is observed, with Ab being the amplitude of an unperturbed breather at frequency ωb = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Figure 2 shows the average time �ˆt � and the corresponding standard deviation as a function of Γ ∈ [10−5, 4 × 10−2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' The values of the remaining parameters are ∆x = ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='01, l = 50, T = 500, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='2, ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='6, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='59, and N = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' A clear result is found: for higher noise amplitudes, on average, nonlinear modes appear in earlier stages of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' ∗ duilio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='desantis@unipa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content='it [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Barone and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Patern`o, Physics and Applications of the 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(Cambridge University Press, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' [4] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Ames, Numerical Methods for Partial Differential Equations (Academic press, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' [5] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Press, S.' metadata={'source': 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+page_content=' Dauxois, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} +page_content=' Peyrard, Nonlinearity 10, 617 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9E4T4oBgHgl3EQfmA05/content/2301.05164v1.pdf'} diff --git a/SNFRT4oBgHgl3EQf8zg4/content/tmp_files/2301.13685v1.pdf.txt b/SNFRT4oBgHgl3EQf8zg4/content/tmp_files/2301.13685v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..688be75408e5919cb0762c0fb9ffbfa9f79e6182 --- /dev/null +++ b/SNFRT4oBgHgl3EQf8zg4/content/tmp_files/2301.13685v1.pdf.txt @@ -0,0 +1,1111 @@ +The payload of the Lunar Gravitational-wave Antenna +J.V. van Heijningen,1 H.J.M. ter Brake,2 O. Gerberding,3 S. Chalathadka Subrahmanya,3 J. Harms,4 X. Bian,5 A. +Gatti,6 M. Zeoli,1 A. Bertolini,7 C. Collette,8 A. Perali,9, 10 N. Pinto,11 M. Sharma,11 F. Tavernier,6 and J. +Rezvani11 +1)Centre for Cosmology, Particle Physics and Phenomenology (CP3), UCLouvain, B-1348 Louvain-la-Neuve, +Belgium +2)Faculty of Science and Technology, University of Twente, 7522 NB Enschede, The Netherlands +3)Institut für Experimentalphysik, Universität Hamburg, 22761 Hamburg, Germany +4)Gran Sasso Science Institute (GSSI), I-67100 L’Aquila, Italy +5)Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China +6)ESAT-MICAS, Katholieke Universiteit Leuven, 3001 Leuven, Belgium +7)National institute of subatomic physics Nikhef, 1098 XG Amsterdam, The Netherlands +8)Precision Mechatronics Laboratory, Université de Liège, B-4000, Liège, Belgium +9)School of Pharmacy, Physics Unit, University of Camerino, I-62032 Camerino (MC), +Italy +10)INAF, I-62032 Camerino (MC), Italy +11)School of Science and Technology, Physics Division, University of Camerino, I-62032 Camerino (MC), +Italy +(Dated: 1 February 2023) +The toolbox to study the Universe grew on 14 September 2015 when the LIGO–Virgo collaboration heard a signal +from two colliding black holes between 30-250 Hz. Since then, many more gravitational waves have been detected +as detectors increased sensitivity. However, the current detector design sensitivity curves still have a lower cut-off +of 10 Hz. To detect even lower-frequency gravitational-wave signals, the Lunar Gravitational-wave Antenna will use +an array of seismic stations in a permanently shadowed crater. It aims to detect the differential between the elastic +response of the Moon and the suspended inertial sensor proof mass motion induced by gravitational waves. A cryogenic +superconducting inertial sensor is under development that aims for fm/√Hz sensitivity or better down to 1 Hz and is +planned to be deployed in seismic stations. Here, we describe the current state of research towards the inertial sensor, +its applications and additional auxiliary technologies in the payload of the lunar gravitational-wave detection mission. +The future of gravitational waves (GWs) is bright. After +the first detection of a binary black hole merger in 2015 1 +and a binary neutron star merger with electromagnetic coun- +terpart in 2017 2, the LIGO-Virgo-KAGRA collaboration has +detected more than 90 signals from black hole and/or neutron +star mergers in their first 3 observation runs 3 using the LIGO 4 +and Virgo 5 detectors. KAGRA 6, the first underground and +cryogenic detector, will join in the coming observation run. +All measured signals entered the LIGO/Virgo sensitive band +at around 30 Hz. Technical noise from many cross couplings +between angular and translational control, is the dominant +noise source below 30 Hz. By improving the low-frequency +performance, signals could be longer in-band and we could +have access to a population of BBH systems with a total mass +greater than 200 M⊙. +The Lunar GW Antenna (LGWA) 7 will detect GWs in the +decihertz region (0.1 – 1 Hz), giving access to even more mas- +sive BBH systems, white dwarf binaries and tidal disruption +events such as a star plunging into a black hole. LGWA uses +an array of extremely sensitive inertial sensors to probe di- +rectly the deformation of the lunar body as a result of the +passing GW. In summary, the lunar surface – and the rigidly +attached inertial sensor suspension frame – displaces accord- +ing to an elastic response determined by the stiffness of the +lunar body; the proof mass of the inertial sensor, however, dis- +places inertially and so the differential displacement between +proof mass and suspension frame holds the GW signal. More +details on this detection principle are found in ref. 8. +First, the mission concept is described in section I, focus- +ing on the heart of the antenna: the seismic station. In or- +der to achieve sufficient sensitivity to strain, we propose us- +ing an array of high-performance inertial sensors; section II +describes the development of such (sub-)fm/√Hz class iner- +tial sensors. A necessity to reach such sensitivity also down +to low frequency is the use of cryogenics which will lower +thermal noise and enable the use of high-Q superconducting +actuation and possibly sensing; sorption cooling and thermal +management is described in section III. High mechanical sen- +sitivity and low thermal noise are obtained by extremely soft +proof mass suspension. This sets strict requirements on the +leveling system, described in section IV. Finally, we detail the +synergy of LGWA inertial sensor development with the next- +generation terrestrial GW detector Einstein Telescope (ET) in +section V. +I. +MISSION CONCEPT AND SEISMIC STATIONS +The Lunar GW Antenna is a proposed kilometer-scale array +of four seismic stations deployed on the lunar surface. Each +station measures the horizontal surface displacement along +two orthogonal directions. The horizontal direction is chosen +to be able to build softer proof-mass suspensions, which ben- +efits the instrument sensitivity (see following sections). The +LGWA deployment site is one of the permanently shadowed +regions inside a crater at the lunar north or south pole. With- +out direct sunlight, alternatives to solar panels on our stations +are investigated. One of the possible power system for LGWA +arXiv:2301.13685v1 [gr-qc] 31 Jan 2023 + +2 +laser-power beaming system using solar panels on the crater +edge 9. +While each seismometer has the capability to observe a GW +signal, the array is proposed as a tool for the reduction of the +seismic background in LGWA data. The models of the seis- +mic background still need to be improved, but the prelimi- +nary results indicate that a background limitation of GW mea- +surements with LGWA should be expected above 0.1 Hz7,8,10. +Work is underway to generalize noise-cancellation methods +developed for current GW detectors11 to be applicable to +LGWA. The star-like array configuration shown in figure 1 +is proposed with the idea to achieve best noise cancellation in +the central sensor. +FIG. 1. Lunar mosaic of about 1500 Clementine images of the lunar +south polar region. The projection is orthographic, centred on the +south pole out to 70o S. The Schrödinger Basin (320 km in diameter) +is located in the lower right. The inset shows an example crater near +the south pole with a star-like deployment configuration of a lander +and four seismic stations in a kilometer-scale array of seismic station +containing cryogenic inertial sensors. Adapted from ref. 12. +Crucial for the success of LGWA is the excellent quality +of the Moon as ultra-quiet elastic body responding to the +extremely weak spacetime fluctuations. +The lunar seismic +background from meteoroid impacts is predicted to be sev- +eral orders of magnitude quieter than the terrestrial seismic +background10. Other sources of surface displacement must +generally be considered. Albeit higher in magnitude when +compared to other types of moonquakes, shallow moonquakes +are rare and not expected to significantly reduce observation +time of lunar GW detectors. Deep moonquakes are more fre- +quent, but the corresponding background noise is expected to +lie below the one from meteoroid impacts. Also thermal ef- +fects can lead to seismic events. The so-called thermal moon- +quakes were observed in large numbers with the Apollo seis- +mic stations around sunset and sunrise13. It is also to be ex- +pected that temperature changes lead to ground tilts and defor- +mations of payload and lander causing additional disturbances +of seismic measurements14. +In order to avoid performance limitations from thermal ef- +fects, it was proposed to deploy LGWA inside a permanently +shadowed region (PSR). The PSRs are formed by craters at +the lunar poles. They can have temperatures continuously be- +low 40 K and be thermally stable with temperature fluctua- +tions driven by heat flow from the lunar interior, infrared light +emitted by sunlit parts of the lunar surface, and by scattered +sunlight15. The cold temperatures of a PSR will have the addi- +tional benefit to act as a natural cryo-cooler of the proof mass, +which lowers thermal noise and enables a sorption-based tech- +nology to cool the LGWA proof masses to 4 K (see section +III). A concept drawing of an LGWA seismic station contain- +ing the inertial sensor, a sorption cooler and levelling systems +is shown in figure 2. +FIG. 2. Conceptual overview of a seismic station on a tilted surface +on the lunar regolith. Roughness and tilt of lunar surface exagger- +ated for illustrative purposes. Several subsystems vital to successful +operation are depicted and further detailed in the text. Subsystems +are not shown to scale here. +Since it is important to have reliable models of the seis- +mic background for the planning of LGWA, it was proposed +to deploy a geophysical explorer mission inside a PSR called +LGWA Soundcheck 16. The sensitivity target is less ambitious +(picometer resolution in the decihertz band), but nevertheless, +it will mark a major step forward in lunar seismometer tech- +nology and beat the sensitivity of Apollo seismometers by 2 – +3 orders of magnitude below 1 Hz. LGWA Soundcheck will +allow us to make a greatly improved prediction of the seis- +mic background spectrum based on the observed distribution +of seismic events inside a PSR. +II. +INERTIAL SENSOR DEVELOPMENT +An LGWA inertial sensor has stringent requirements such +as fm/√Hz sensitivity down to 1 Hz, deployablility, low heat +dissipation and favourable electronic characteristics. While +still under development, we describe the current R&D efforts + +lander +seismic +stationC/He +W +90 K radi +sorption +coolers +cryostat 15 K +4K +platform +levelling +system3 +here. The proof mass will be suspended by means of a folded +Watt’s linkage, a common way17 to achieve a compact, low- +resonance-frequency device. To achieve low thermal noise, +the target proof mass will be 10 kg. By using niobium, which +has a 8.4 g/cm3 density, such device with all auxiliary sensing +and actuation system can fit in a volume 200×200×100 mm3. +The readout of the proof mass motion, and therefore ul- +timately the differential signal between the elastic response +of the Moon to passing GWs and the inertial proof mass +which holds the GW signal, is a cm-scale interferometer. +An example of such opto-mechanical device is a room tem- +perature version of an interferometrically Watt’s linkage that +reached 8 fm/√Hz from 30 Hz 18. The used interferometric +readout, based on ref. 19, reached 4 fm/√Hz from 4 Hz on- +wards 20. This readout needs feedback to keep the working +point halfway up the fringe (the linear part of the sinusoid) as +any deviation makes the output non-linear and degrades the +subtraction of common mode noise between the two interfer- +ometer output ports. Without feedback the typical micrometer +motion on Earth of the sensor frame would cause the sinu- +soidal error signal to move between fringes. +The feedback is provided by an actuator that locks the proof +mass to the suspension frame. The signal sent to the actuator +is then proportional to force and acceleration and serves as +the sensor output. Often, a coil-magnet actuator is used in +force-feedback inertial sensors. However, in the previously +discussed 8 fm/√Hz results, thermal noise was expected to be +dominant below 10 Hz. While the used Watt’s linkages can +have mechanical quality factors above 5000, the permanent +magnet and its eddy current damping of the moving metal +pieces had degraded the Q to below 100 20. LGWA requires +lower-frequency fm/√Hz sensitivity which can only be ob- +tained by lowering thermal noise which goes as 21 +x2 +th = +4kBTω2 +0φ +mω +� +(ω2 +0 −ω2)2 +ω4 +0φ 2�, +(1) +where xth denotes the thermal noise displacement amplitude +spectral density (ASD), kB Boltzmann’s constant, T the tem- +perature, ω0 the angular resonance frequency, φ(= 1/Q for +structurally damped suspensions) the loss angle and ω the an- +gular frequency. Low temperatures and increased mass will +obviously help, but different actuators that will not (domi- +nantly) damp the Watt’s linkage are necessary. +Therefore, +superconducting actuators that use the Meissner effect rather +than a magnet to exert a force on the proof mass are inves- +tigated 22,23. The superconducting thin film coils and super- +conducting surface (depicted by orange rectangles) can, de- +pending on the achieved cooling level or other application, be +manufactured from niobium (Tc = 9.2 K), MgB2 (Tc = 40 K) +or YBCO (Tc = 93 K). To be in the necessary full magnetic +expulsion state, temperatures around 60% of Tc or lower is +needed. +The current design follows from an initial cryogenic iner- +tial sensor concept first proposed in ref. 24, which was subse- +quently updated 23. Currently, we investigate what is depicted +in figure 3. The resonance frequency of the Watt’s linkage +can be coarsely set by the sliding tuning mass, which changes +mass distribution between inverted and regular pendulum. Af- +ter cooldown of the mechanics, the resonance frequency may +have changed. A DC current on one of the tuning coils can ef- +fectively change the mass distribution thereby tuning the res- +onance frequency. +FIG. 3. A cryogenic superconducting monolithic inertial sensor. The +proof mass is suspended from the frame by a regular pendulum and +inverted pendulum. +This monolithic configuration is known as a +Watt’s linkage and allows for an arbitrarily low natural frequency, +which increases the mechanical sensitivity. The proof mass motion +is monitored by an interferometric readout and the custom cryo-chip +is under development using 65 nm CMOS technology. More details +are found in the text. +The estimate of sensitivity is made by modelling the dis- +placement noises of mechanical and interferometric nature. +Most models for these noises are described in refs. 18,24. The +actuator noise model is a simple current driver model 20. We +use the parameters in table I and arrive at the noise budget +shown in figure 6(a). This noise budget is roughly the same as +the "opto-mechanical" trace in figures 2, 3, 4 and 5 of ref. 7. +The sensitivity of the four-sensor array is a factor 2 lower. +LGWA sensitivity is obtained by dividing out the Moon’s re- +sponse, i.e. the expected surface motion per unit strain. An ex- +ample of such modeled response is found in figure 1 of ref. 7. +The used readout scheme is an example femtometer-class +interferometer. There are other options to realise an optical +readout with similar or even lower predicted sensitivity. The +trade-off between displacement readout schemes relies heav- +ily on the required dynamic range and the ability, and corre- +sponding benefits, of operating at a specific or a random op- +erating point. So-called multi-fringe interferometric sensors +implement phasemeters to read out the phase at any operating +point and with large, mostly multi-fringe, dynamic range26. +These types of interferometers are limited to femtometer-level +sensitivities by effective technical-fundamental limitations in +their readout, especially by digitisation noise and to provide +linear sensing over a wide range they typically do not employ +optical resonators to enhance the signals27. +The best space-based demonstration of such displacement +sensors is the multi-fringe heterodyne interferometry realised +in LISA pathfinder28, which achieved a displacement mea- +surement noise floor of 30 fm/√Hz around 1 Hz, mostly lim- +ited by ADC quantisation noise in the digital phasemeter. A +lower digitisation noise floor could be realised with commer- + +(not in vacuum) +VVVVV +laser +gain calibration +actuator +22 +PBS + pendulum +PD2 c +pendulum +tuning +mass +inverted +PD1 +ctrl +piezo +custom +cryo-chip +tuning coils +actuator 2 +calibration4 +TABLE I. Mechanical, readout and electronics parameters for both +the interferometrically and SQUID read out Watt’s linkage. +Parameter +Value +Unit +Proof mass +10 +kg +Natural frequency +0.25 +Hz +Temperature +5 +K +Coil-superconductor gap +0.1 +mm +Actuator strength +50 +µN/A +Niobium with interferometric readout +Watt’s linkage material +Nb +- +Quality factor +1·104 +- +Frequency noisea +500 · f −1/2 Hz/√Hz +Static differential arm length +0.5 +mm +Injected laser power +10 +mW +Wavelength +1550 +nm +TIA feedback resistor +20 +kΩ +Silicon with SQUID readout +Watt’s linkage material +Si +- +Quality factor +1·106 +- +SQUID energy resolution EA +2500 ¯h +J/Hz +signal to SQUID coupling efficiency ηβ +0.25 +- +1/√f corner frequency fc +0.1 +Hz +a Typical value for high-end lasers e.g. The RockTM from NP Photonics 25. +cially available ADCs. A critical part of the low-frequency +noise floor that has to be evaluated for LGWA is the achievable +temperature stability and the corresponding thermally driven +couplings, namely thermoelastic and thermorefractive noise, +which were suppressed in LISA Pathfinder by the exceptional +temperature stability29. These thermally driven noise sources +will be critical for any interferometric readout scheme and +need to be studied with respect to the cryogenic environment +of the proof mass. Thermal compensation strategies can be +employed, but are complicated, in design and in testing, by the +cryogenic operating temperatures. These noise source are also +critical for any opto-mechanical laser frequency reference, be +it a proper 2nd, equally long, arm in the local interferometer +topology or some external, disjoint reference. +For the LGWA and especially LGWA Soundcheck the +power consumption of the payload might be a critical fac- +tor, with the laser sources being a significant driver of such +a budget. Accordingly, the power consumption of any given +interferometric readout has to be taken into account, as well as +their influence on the potentially reduced power consumption +in the active feedback to control the proof mass. This might +benefit interferometric readout schemes that require little or +no opto-electronic elements, slow signal digitisation and lit- +tle signal post-processing. In addition to the readout scheme +shown in figure 3, a higher dynamic range option that can +achieve femtometer-level displacement noise with no addi- +tional active components is quadrature homodyne interferom- +etry, which has already been used to demonstrate compact in- +terferometric readout of inertial sensors 30 and demonstrated +a noise floor of 20 fm/√Hz 31. Depending on the dynamic +range and the optical design, especially with regards to ghost +beams and polarisation contamination32, such a readout might +require additional digital signal processing with a Lissajous fit +to suppress periodic non-linearity, which again might limits its +advantage in terms of power consumption. +Finally, optical resonators can be employed in compact dis- +placement sensors to achieve sub-femtometer displacement +readout noise floors at the cost of readout range and linear- +ity, for example using fiber-based implementations, as demon- +strated in ref. 33. Combining optical cavities with operation- +point independent, wider-range readout is, however, non- +trivial. Using a strong frequency-modulated laser with an op- +tical resonator promises noise floors of 10−16 m/√Hz27, but +might require too much effort with respect to opto-electronics +and signal processing for the readout of only two displace- +ments in a single LGWA station. A more relevant approach +might be to lock one laser to an optical cavity between the +proof mass and an external mirror and to measure its fre- +quency variations with changing length. Such a scheme re- +quires a second ultra-stable laser to generate a beat note, +but, combined with a corresponding real-time digital signal- +processing system, this scheme can also realise the locking of +both lasers to their respective optical resonators34, as depicted +in figure 4. +FIG. 4. Heterodyne cavity-tracking readout scheme with co-located +ultra-stable optical cavity. Tracking the motion of the proof mass +requires a high-dynamic range phase readout system. Cavity length +L, wavelength λ and phase readout bandwidth BW determine the +maximum one-way displacement tracking range ∆Lmax = λ/2 · +BW/(c/(2L)) = λ/2·BW/FSR. +This readout senses one degree-of-freedom, adding another +axis demands an additional laser that is locked to the cor- +responding cavity. +Hence, in order to measure the hori- +zontal surface displacement along two orthogonal directions, +each seismic station requires in total three laser sources. For +resonator lengths of 5 cm the beat frequency will shift by +3 GHz for a displacement of λ/2, a frequency shift that +could be tracked with a high-bandwidth, frequency-tracking +phasemeter 35 with negligible frequency tracking noise. Field- +programmable gate arrays with integrated high-speed data +converters are available to implement such tracking systems +with several GHz of bandwidth. A heterodyne cavity-tracking +readout scheme can, in principle, achieve readout noise levels +of 10−17 m/√Hz with reasonable levels of cavity Finesse, be- +cause they are not directly limited by digitisation noise and the +influence of shot-noise is suppressed by the optical enhance- +ment. In practise this readout will be limited by the stability +of the available frequency reference, which could be a sepa- + +Cryostat +proof mass readout +spacer +L = 5 cm +Ultra-stable +Optical Cavity5 +rate cavity as developed for space-based optical clocks or fun- +damental physics experiments36 that is co-located within the +cryostat to reduce thermal effects like coating thermal noise, +as shown in figure 4. If available, the ultra-stable laser can +also be a fully separate device connected only via fiber. The +lasers, the phase readout system and the fiber-optics do, to +first order, not have stringent environmental noise couplings +and can be placed outside the cryostat. The additional com- +plexities and power consumption of a heterodyne cavity lock- +ing scheme make it unsuited for LGWA Soundcheck, but the +promise of mid-range dynamic range and extremely low read- +out noise floor make it a promising candidate for the full +LGWA readout, assuming other noise sources can be brought +to sufficiently low levels, at least at the higher readout fre- +quencies. Detailed studies of amplitude noise37 and of tilt- +to-length coupling38 will have to be done for any design and +readout scheme. +Besides the different interferometric readout strategies de- +scribed above, superconductivity can be used to read out the +proof mass position with high precision. If a superconductor +moves with respect to a superconducting coil carrying a per- +sistent current, the inductance of coil-superconductor system +changes. The current in the coil will change correspondingly +to keep the flux in the system conserved. Due to flux con- +servation in superconducting loops, the current change can be +converted to magnetic field change simply by connecting an- +other coil. This changing magnetic field can subsequently be +picked up by a Superconducting QUantum Interference De- +vice (SQUID), which is known for its extreme sensitivity to +changing magnetic fields. This readout strategy has been sug- +gested, e.g., in ref. 39 for gravity gradiometry. Using two sens- +ing coils in parallel and sandwiching a superconductor, and a +third one to convert the current signal into magnetic signal, +the motion of the superconductor can be read out with sub- +femtometer precision. On the right side of figure 5(a) such +dual coil sandwich configuration is shown. +The superconducting readout provides an error signal for +a feedback loop with a superconducting actuator, which can +also employ a dual coil sandwich architecture. The supercon- +ducting coils can be loaded with a persistent current as shown +in figure 5(b). By sending an actuation current running in par- +allel in the two coils, we can increase the current, and corre- +sponding magnetic force, on one side and reduce the magnetic +force on the other side, generating a net (feedback) force on +the superconductor. The magnetic force between a coil and +a superconducting surface is proportional to the square of the +current in the coil (Ipers +Iact)2 = I2 +pers +2IpersIact +I2 +act. Large +persistent currents (> 1 A currents are common 40) will give +the largest coupling to the signal current. However, because +the persistent currents in the coils push from either side there +is a positive stiffness roughly equal to the DC force from each +coil, divided by the coil-surface gap. This added stiffness can +be corrected for using the tuning mass and coils. The main +advantage of this strategy is that only small currents (< 100 +µA) will have to be generated by the on-chip current driver. +Moreover, the dual coil architecture linearizes the relation be- +tween the actuation current and the feedback force which will +simplify the control and data analysis. +(a) +(b) +(c) +FIG. 5. (a) a silicon Watt’s linkage with superconductive readout and +actuation, (b) dual coil sandwich configuration used for sensing and +actuation. More details found in text and (c) a zoom of the monolithic +niobium and quasi-monolithic silicon flexures +To decrease the thermal noise even further, a silicon Watt’s +linkage is proposed. Silicon is a crystalline material exhibiting +low mechanical loss at cryogenic temperatures, with a bulk +Q of 108 41. The thin flexures allowing for their low stiff- +ness of metallic Watt’s linkages have historically been fab- +ricated using electro-discharge machining (EDM) techniques +as shown in figure 5(c). A more difficult hybrid procedure +for silicon must be followed as using EDM to cut the delicate +flexures is expected to result in surface damage and thus lossy +flexures. The frame and proof mass are manufactured from +highly doped silicon, which can be cut using EDM. The legs +including the flexures are (laser assisted plasma) etched out +of a thick 500 µm wafer and hydro catalysis bonded (HCB) +to the frame and proof mass. HCB is famous for producing +quasi-monolithic bonds in mirror suspensions of the current +interferometric GW detectors 42. Figure 2 in ref. 22 shows a +possible HCB assembly procedure for a silicon Watt’s link- +age. The quasi-monolithic silicon Watt’s linkage is expected +to have a Q of 106, thereby lowering the thermal noise by an +order of magnitude with respect to the niobium variant. + +superconducting +superconducting +actuator +sensor +pendulum +pendulum +tuning +inverted +mass +custom +nn cryo-chip +proof mass +output +tuning coils +levelling screwsactuation/signal current +persistent current6 +(a) +(b) +FIG. 6. Minimum detectable inertial displacement for a structurally +damped accelerometer with (a) niobium mechanics and interferomet- +ric readout and (b) silicon mechanics and SQUID readout. +The SQUID readout has a sub-fm/√Hz sensitivity cor- +rected for the sensor mechanics as 43 +x2 +squid = 2EA(1+ fc/f) +mω0ηβ +(ω2 −ω2 +0)2 +ω2 +0/Q +ω4 +, +(2) +where most symbols have been denoted in table I. The SQUID +has a 1/f characteristic below fc in its power spectral density. +The same actuator noise model as the niobium version and the +silicon proof mass suspension thermal noise model complete +the noise budget as presented in figure 6(b). +III. +SORPTION COOLING AND THERMAL +MANAGEMENT +Cryogenic cooling of the inertial sensor will be established +by combining two vibration-free cooling technologies; High- +emissivity radiator panels will be used to provide heat-sink +platforms at temperature levels of about 50 K and 90 K. Next, +a two-stage sorption-based Joule-Thomson cooler will be heat +sunk to these platforms and will cool further down to 14.5 K +and 4.5 K. This sorption-based cooling technology has been +developed at the University of Twente in the past two decades. +It operates with a thermal compressor rather than a mechan- +ical compressor as conventional cryogenic coolers do. Apart +from a few passive valves it has no mechanical moving parts +and, therefore, offers operation at an extremely low level of +emitted vibrations and a long lifetime because of the absence +of wear. Both aspects are obviously attractive in space appli- +cations. The operation of a sorption compressor is based on +the cyclic adsorption and desorption of a working gas at a sor- +ber material such as, in our case, activated carbon. Activated +carbon is a material that by its highly porous structure has a +very large internal surface so that it can adsorb large quan- +tities of gas. By heating the sorber, the gas is desorbed and +a high pressure can be established. By expanding this high- +pressure gas in a Joule-Thomson (JT) cold stage, cooling can +be obtained. The operating principles and the thermodynam- +ics involved, are discussed in many papers 44–48. +The baseline cooler chain of the LGWA project is schemat- +ically depicted in figure 7 and resembles the Darwin cooler +that was developed in an earlier ESA-TRP project 45. The +first stage of the LGWA sorption cooler operates with hydro- +gen gas and realizes a temperature of 15 K. The second-stage +sorption cooler operates with helium gas and, precooled by the +hydrogen stage, it reaches 4.5 K. The hydrogen compressor is +thermally linked to the 90 K radiator heat sink. The hydrogen +gas is precooled by a 50 K radiator that also serves as the heat +sink for the helium compressor. Based on the performance of +the two stages of the Darwin cooler, the gross cooling pow- +ers at both stages in the LGWA project are expected to be +36 mW at the 15K stage (of which 6 mW are used to precool +the helium gas in the second stage), and 4.5 mW at the 4.5 K +stage. The total electric input power to the coolers is slightly +more than 6 W; 4.2 W in the compressor of the hydrogen stage +and 1.9 W in that of the helium stage. This input power, plus +the power taken from the cold interfaces is emitted to deep +space at the two radiator panels. In previous work, the radia- +tor temperatures were optimized aiming at minimum radiator +size, resulting in actual temperatures of 87 K and 51 K. The +required radiator panel areas are 1.6 m2 and 8.2 m2, respec- +tively 49. This setup is schematically depicted in figure 7. The +cooler mass is expected to be 10 kg of which both stages are +about half of that 45,49–51. The cooling powers as indicated in +figure 7 are not fully available as net cooling power. Part of it +is used to take up parasitic heat loads due to conduction and +radiation. The heat load budgets are listed in table III. In or- +der to withstand launch loads, all frames will be mechanically +fixed. Once positioned on the moon surface, these launch-load +connections will be disconnected allowing for the 15 K frame +to be leveled with respect to the moon surface, as illustrated in +figure 2. The remaining support structures are anticipated to +be G10 struts between leveling platform and 15 K frame, and +Kevlar straps between 15 K frame and 4.5 K cold mass. +A sorption-based Joule-Thomson cooler has been launched +and successfully operated in space in the ESA-Planck mission +(2009-2013) 52,53. It provided cooling power of 1 W at about + +10-10 +suspension thermal +frequency +relative intensity +10-11 +shot +displacement [m/√Hz] +actuator driver +electronic +- Total +10-15 +10-16 +10-2 +10-1 +100 +101 +frequency [Hz]10-10 +suspension thermal +-SQUID readout +actuator driver +10-11 +-total +10-14 +10-15 +10-16 +10-2 +10-1 +100 +10 +frequency [Hz]7 +15 K +Total gross cooling power +36 mW +Precooling He stage +6 mW +Radiation from 50 K environment +20 mW +Conductive load through support (G10 struts) +9 mW +Conductive load via cooler tubing +1 mW +Emissivity +0.1 +4.5 K +Total gross cooling power +4.5 mW +Radiation from 15 K environment +0.1 mW +Conductive load through support (Kevlar straps) +0.9 mW +Conductive load via cooler tubing +0.1 mW +Dissipation and conductive load of sensor + electronics 3.4 mW +Emissivity +0.1 +TABLE II. Heat load budgets at the 15 K and 4.5 K cold-tip inter- +faces. +20 K using hydrogen as the working fluid. However, the com- +pressor sorber material was a metal hydride which is a chem- +ical absorber whereas in our compressor technology activated +carbon is applied which is a physical adsorber. The big differ- +ence is that a chemical absorber degenerates over time limit- +ing the lifetime of the cooler in mission (in Planck 2 years), +whereas the adsorption process with carbon is fully reversible +and does not limit the lifetime of the cooler. Our carbon based +sorption compressor technology was qualified at TRL5 (sur- +viving launch vibrations) in one of the recent ESA projects 54. +IV. +SEISMOMETER LEVELING SYSTEM +A leveling system is needed to achieve an initial alignment +of the seismometer platform to compensate ground slope and +then to keep it aligned within a few microradians. The require- +ment of the alignment accuracy is set by the softness of the +proof-mass suspension through the tilt-to-horizontal coupling +dpm = gθ/ω2 +0. The critical dimension in figure 3 and 5(a) is +the 100 µm gap between coils and superconductor in the actu- +ator. The leveling system should be more precise than 30 µm +in proof-mass positioning to ensure that the superconductor +does not make contact with the sandwiched coils. +A platform meeting similar requirements was developed for +the SEIS experiment of the Mars InSight mission55–57. This +system features a MEMS-based rough alignment to compen- +sate for up to 15◦ of ground slope, and a precision alignment +system that reaches a few microradians using high-precision +tiltmeters. An important new requirement for the LGWA plat- +form is that it must be compatible with the cold environment +of a PSR, which constraints above all the technologies that +can be used for the high-precision tiltmeters. +An alternative to using high-precision tiltmeters might +be to realize the LGWA seismic sensors with a high dy- +namic range laser-interferometric readout of the proof-mass +displacement58. Exploiting the tilt-to-horizontal coupling, tilt +can be measured and compensated by observing the move- +FIG. 7. Schematic diagram of the two-stage sorption-based cooler +with a cooling power of 30 mW at 15 K and 4.5 W at 4.5 K; Electric +input power is indicated in blue; heat flows in red. +ment of the proof mass. With the rough tilt alignment stage, +one can assess what sign the high-precision adjustment must +have, i.e., with which side of its frame the proof mass makes +contact before the fine-alignment is engaged. +V. +SYNERGY WITH EINSTEIN TELESCOPE +On Earth, ET features an underground and cryogenic de- +sign and aims to be sensitive to GWs down to 3 Hz. Meth- +ods to apply low-vibration cryogenic cooling of the mirrors +in a cryostat to lower thermal noise are currently investigated +in research facilities 59–61. Close to the mirror spurious vibra- +tions could be injected by the application of cooling power. To +ensure the lower cryogenic stages are indeed at low enough vi- +bration levels, new inertial sensors such as described here are +necessary. +ET aims to be 10 times more sensitive than current detec- +tors above 10 Hz and stretch its lower bandwidth limit down +to 3 Hz. Cooling down of the input and end mirrors down +to around 10 K is needed to reduce the dominant noise at +low frequency: thermal noise. To extract heat, the penulti- +mate mass above the mirror shown in figure 8 (right) operates +at about 5 K. Cooling the penultimate mass cannot be done +radiatively due to the low temperature and required power +(several 100 mW) and therefore some physical connection be- +tween cryocoolers and the suspension final stages is required. + +4.2 W +H2/ +4.2 W +90 K +carbon +1.6 m² +1.9 Wel +1.9 W +He/ +50 K +carbon +25 mW +8.2 m2 +6 mW +15.K +30 mW +4.5 K +4.5 mW8 +The cooling power is applied by low-vibration cryocoolers +and using flexible heat links. However, there is still a risk that +unwanted vibrations end up in the penultimate stages, close +to the mirrors where extremely tiny displacements in the de- +tection bandwidth are required. The cryogenic temperatures +provide opportunities for new, superconductive actuators and +(inertial) sensors. The use of superconductive coils reduces +the cooling power (and therefore vibrations) otherwise needed +for dissipative elements, such as the resistive copper actuator +coils in figure 8 (left). Extremely sensitive inertial sensors, +such as presented here, are needed to monitor the platform +motion. +FIG. 8. The final stages of (left) current room-temperature mirror +suspensions and (right) future cryogenic mirror suspensions, where +the low temperatures provide opportunities for new actuators and (in- +ertial) sensors. Ultimate configuration for ET may differ; however, +similar sensing and actuation solutions will be necessary. +In GW detector suspensions, actuators are used in an hierar- +chical way in terms of strength and range; the most low-noise, +short-range actuators are needed close to the mirror where +residual acceleration is extremely small. At the top of the sus- +pension chain, actuation noise requirements are less stringent, +but those actuators will have to operate over a larger range. +Most actuators used in today’s GW detectors are (some form +of) coil-magnet actuator as these are easy develop, install and +use. The use of permanent magnets close to moving metals +can cause harmful eddy currents and stray magnetic field can +exert unwanted forces on the suspended objects. The former +is largely solved by using plastics (e.g. PEEK) near the mag- +nets and the latter is often solved by placing the magnets on +the same object in opposite polarity. +The cryogenic GW detector KAGRA operates 23 kg mir- +rors dissipating 0.5 mW 62 at the actuators and ET mirrors are +10 times as massive 63, thus dissipating >10 mW if old re- +sistive actuators are used. This is of order 10% compared to +the expected absorption of laser light and thermal radiation +of mirror and payload, respectively. Lastly, the sub-fm/√Hz +dual coil position sensor with SQUID readout can be used as +differential sensors between cage and (pen)ultimate stage(s). +CONCLUSION AND FUTURE WORK +To open up GW science in the decihertz range, there have +been space-borne proposals, such as DECIGO 64 and BBO 65. +While they promise higher sensitivity than LGWA, many tech- +nological challenges remain and a longer timeline is expected. +Here, we have presented several technologies that make up the +payload and detail several different options in the inertial sen- +sor design. +While the niobium Watt’s linkage fabrication processes and +interferometric readout technology is more mature, the silicon +Watt’s linkage with SQUID readout may result in roughly one +order of magnitude lower thermal and readout noise. Note +that a tenfold sensitivity improvement will lead to larger range +and thus an expected factor thousand more GW signals. In +both designs we propose actuators with superconducting coils +which are also necessary for the sensing part in the SQUID +readout. The development of the inertial sensor as well as the +sensing and actuation technology shows strong synergy with +future cryogenic GW detector ET. +The inertial sensors with extreme sensitivity have to be +tested in extremely quiet and cold environments. Such test +facilities in the form of actively isolated platforms inspired +by the LIGO HAM table designs 66 are being developed as +part of the E-TEST effort in Belgium 59,61 and the GEM- +INI facility in the underground National Laboratories of Gran +Sasso 67. The aimed-for sensitivity at 1 Hz is about 5 orders of +magnitude smaller than the Earth’s seismic motion at that fre- +quency. Placing two or three identical sensors on the isolated +platform allows for subtraction of common mode noise using +the Wiener filter 68 or three-channel correlation techniques 69 +resulting in a sensor self-noise measurement. +The technology necessary for LGWA will either be spe- +cific development of existing space technology (levelling sys- +tem, sorption cooler, thermal management systems etc.) or +in parallel with terrestrial GW instrumentation R&D in iner- +tial sensing and active isolation. Future terrestrial GW de- +tector isolation has to stretch to lower frequencies and needs +better low-frequency inertial sensors and active isolation per- +formance for that. For a space application as LGWA, how- +ever, there will be extra (space) engineering necessary. Be- +fore LGWA will fly, the aforementioned LGWA Soundcheck +also requires some technology development. Its strategy is to +combine technologies that have already flown in space. For in- +stance, elements of the interferometer topology developed for +LISA (Pathfinder) can be adopted for the readout of Sound- +check. R&D for LISA and other space missions will also +have overlap with the technologies presented here. In this +context, payload technology development continues towards +cryogenic, (sub-)fm/√Hz inertial sensing on the lunar surface +for GW detection and lunar geophysics. + +flexible +heat links +from cryo-cooler +cryo-shields +new (inertial) +sensors +new actuators9 +ACKNOWLEDGEMENTS +Oliver Gerberding and Shreevathsa Chalathadka Sub- +rahmanya are funded by the Deutsche Forschungsgemein- +schaft (DFG, German Research Foundation) under Germany’s +Excellence Strategy—EXC 2121 “Quantum Universe”— +390833306. +Filip Tavernier and Alberto Gatti are funded +by internal KU Leuven funds (iBOF-21-084). +Filip Tav- +ernier, Alberto Gatti, Christophe Collette, Joris van Heijnin- +gen and this research are partially funded by Interreg V-A Eu- +regio Maas-Rijn under the E-TEST project (EMR113). Mor- +gane Zeoli is funded by the Fonds National de la Recherche +Scientifique (FNRS) under projet de recherche STELLAR +(T.0022.22). +1B.P. Abbott et al. 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ET design report update. Technical report, 2020. +ET-0007B-20. +64S. Kawamura et al. Current status of space gravitational wave antenna DE- +CIGO and B-DECIGO. Progress of Theoretical and Experimental Physics, +2021(5), 02 2021. 05A105. +65Jeff Crowder and Neil J. Cornish. Beyond LISA: Exploring future gravita- +tional wave missions. Phys. Rev. D, 72:083005, Oct 2005. +66F Matichard et al. Seismic isolation of advanced ligo: Review of strat- +egy, instrumentation and performance. Classical and Quantum Gravity, +32(18):185003, aug 2015. +67Jan Harms. +GEMINI: A new underground seismic-isolation facility at +LNGS. +68J. Harms, E. L. Bonilla, M. W. Coughlin, J. Driggers, S. E. Dwyer, D. J. +McManus, M. P. Ross, B. J. J. Slagmolen, and K. Venkateswara. Observa- +tion of a potential future sensitivity limitation from ground motion at ligo +hanford. Phys. Rev. D, 101:102002, May 2020. +69Reinoud Sleeman, Arie van Wettum, and Jeannot Trampert. 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Bulletin of the Seismological Society of +America, 96(1):258–271, 02 2006. + diff --git a/SNFRT4oBgHgl3EQf8zg4/content/tmp_files/load_file.txt b/SNFRT4oBgHgl3EQf8zg4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5d02c5780dbaabf732cf3cc662ce72bb27eb88d5 --- /dev/null +++ b/SNFRT4oBgHgl3EQf8zg4/content/tmp_files/load_file.txt @@ -0,0 +1,863 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf,len=862 +page_content='The payload of the Lunar Gravitational-wave Antenna J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' van Heijningen,1 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' University of Twente,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 7522 NB Enschede,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The Netherlands 3)Institut für Experimentalphysik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Universität Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 22761 Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Germany 4)Gran Sasso Science Institute (GSSI),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' I-67100 L’Aquila,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Italy 5)Institute of Mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' China 6)ESAT-MICAS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Katholieke Universiteit Leuven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 3001 Leuven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Belgium 7)National institute of subatomic physics Nikhef,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 1098 XG Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The Netherlands 8)Precision Mechatronics Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Université de Liège,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' B-4000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Liège,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Belgium 9)School of Pharmacy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Physics Unit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' University of Camerino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' I-62032 Camerino (MC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Italy 10)INAF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' I-62032 Camerino (MC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Italy 11)School of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Physics Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' University of Camerino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' I-62032 Camerino (MC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Italy (Dated: 1 February 2023) The toolbox to study the Universe grew on 14 September 2015 when the LIGO–Virgo collaboration heard a signal from two colliding black holes between 30-250 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Since then, many more gravitational waves have been detected as detectors increased sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' However, the current detector design sensitivity curves still have a lower cut-off of 10 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' To detect even lower-frequency gravitational-wave signals, the Lunar Gravitational-wave Antenna will use an array of seismic stations in a permanently shadowed crater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' It aims to detect the differential between the elastic response of the Moon and the suspended inertial sensor proof mass motion induced by gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' A cryogenic superconducting inertial sensor is under development that aims for fm/√Hz sensitivity or better down to 1 Hz and is planned to be deployed in seismic stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Here, we describe the current state of research towards the inertial sensor, its applications and additional auxiliary technologies in the payload of the lunar gravitational-wave detection mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The future of gravitational waves (GWs) is bright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' After the first detection of a binary black hole merger in 2015 1 and a binary neutron star merger with electromagnetic coun- terpart in 2017 2, the LIGO-Virgo-KAGRA collaboration has detected more than 90 signals from black hole and/or neutron star mergers in their first 3 observation runs 3 using the LIGO 4 and Virgo 5 detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' KAGRA 6, the first underground and cryogenic detector, will join in the coming observation run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' All measured signals entered the LIGO/Virgo sensitive band at around 30 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Technical noise from many cross couplings between angular and translational control, is the dominant noise source below 30 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' By improving the low-frequency performance, signals could be longer in-band and we could have access to a population of BBH systems with a total mass greater than 200 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The Lunar GW Antenna (LGWA) 7 will detect GWs in the decihertz region (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='1 – 1 Hz), giving access to even more mas- sive BBH systems, white dwarf binaries and tidal disruption events such as a star plunging into a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' LGWA uses an array of extremely sensitive inertial sensors to probe di- rectly the deformation of the lunar body as a result of the passing GW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' In summary, the lunar surface – and the rigidly attached inertial sensor suspension frame – displaces accord- ing to an elastic response determined by the stiffness of the lunar body;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' the proof mass of the inertial sensor, however, dis- places inertially and so the differential displacement between proof mass and suspension frame holds the GW signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' More details on this detection principle are found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' First, the mission concept is described in section I, focus- ing on the heart of the antenna: the seismic station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' In or- der to achieve sufficient sensitivity to strain, we propose us- ing an array of high-performance inertial sensors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' section II describes the development of such (sub-)fm/√Hz class iner- tial sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' A necessity to reach such sensitivity also down to low frequency is the use of cryogenics which will lower thermal noise and enable the use of high-Q superconducting actuation and possibly sensing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' sorption cooling and thermal management is described in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' High mechanical sen- sitivity and low thermal noise are obtained by extremely soft proof mass suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This sets strict requirements on the leveling system, described in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Finally, we detail the synergy of LGWA inertial sensor development with the next- generation terrestrial GW detector Einstein Telescope (ET) in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' MISSION CONCEPT AND SEISMIC STATIONS The Lunar GW Antenna is a proposed kilometer-scale array of four seismic stations deployed on the lunar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Each station measures the horizontal surface displacement along two orthogonal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The horizontal direction is chosen to be able to build softer proof-mass suspensions, which ben- efits the instrument sensitivity (see following sections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The LGWA deployment site is one of the permanently shadowed regions inside a crater at the lunar north or south pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' With- out direct sunlight, alternatives to solar panels on our stations are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' One of the possible power system for LGWA arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='13685v1 [gr-qc] 31 Jan 2023 2 laser-power beaming system using solar panels on the crater edge 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' While each seismometer has the capability to observe a GW signal, the array is proposed as a tool for the reduction of the seismic background in LGWA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The models of the seis- mic background still need to be improved, but the prelimi- nary results indicate that a background limitation of GW mea- surements with LGWA should be expected above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='1 Hz7,8,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Work is underway to generalize noise-cancellation methods developed for current GW detectors11 to be applicable to LGWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The star-like array configuration shown in figure 1 is proposed with the idea to achieve best noise cancellation in the central sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Lunar mosaic of about 1500 Clementine images of the lunar south polar region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The projection is orthographic, centred on the south pole out to 70o S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The Schrödinger Basin (320 km in diameter) is located in the lower right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The inset shows an example crater near the south pole with a star-like deployment configuration of a lander and four seismic stations in a kilometer-scale array of seismic station containing cryogenic inertial sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Adapted from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Crucial for the success of LGWA is the excellent quality of the Moon as ultra-quiet elastic body responding to the extremely weak spacetime fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The lunar seismic background from meteoroid impacts is predicted to be sev- eral orders of magnitude quieter than the terrestrial seismic background10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Other sources of surface displacement must generally be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Albeit higher in magnitude when compared to other types of moonquakes, shallow moonquakes are rare and not expected to significantly reduce observation time of lunar GW detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Deep moonquakes are more fre- quent, but the corresponding background noise is expected to lie below the one from meteoroid impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Also thermal ef- fects can lead to seismic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The so-called thermal moon- quakes were observed in large numbers with the Apollo seis- mic stations around sunset and sunrise13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' It is also to be ex- pected that temperature changes lead to ground tilts and defor- mations of payload and lander causing additional disturbances of seismic measurements14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' In order to avoid performance limitations from thermal ef- fects, it was proposed to deploy LGWA inside a permanently shadowed region (PSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The PSRs are formed by craters at the lunar poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' They can have temperatures continuously be- low 40 K and be thermally stable with temperature fluctua- tions driven by heat flow from the lunar interior, infrared light emitted by sunlit parts of the lunar surface, and by scattered sunlight15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The cold temperatures of a PSR will have the addi- tional benefit to act as a natural cryo-cooler of the proof mass, which lowers thermal noise and enables a sorption-based tech- nology to cool the LGWA proof masses to 4 K (see section III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' A concept drawing of an LGWA seismic station contain- ing the inertial sensor, a sorption cooler and levelling systems is shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Conceptual overview of a seismic station on a tilted surface on the lunar regolith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Roughness and tilt of lunar surface exagger- ated for illustrative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Several subsystems vital to successful operation are depicted and further detailed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Subsystems are not shown to scale here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Since it is important to have reliable models of the seis- mic background for the planning of LGWA, it was proposed to deploy a geophysical explorer mission inside a PSR called LGWA Soundcheck 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The sensitivity target is less ambitious (picometer resolution in the decihertz band), but nevertheless, it will mark a major step forward in lunar seismometer tech- nology and beat the sensitivity of Apollo seismometers by 2 – 3 orders of magnitude below 1 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' LGWA Soundcheck will allow us to make a greatly improved prediction of the seis- mic background spectrum based on the observed distribution of seismic events inside a PSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' INERTIAL SENSOR DEVELOPMENT An LGWA inertial sensor has stringent requirements such as fm/√Hz sensitivity down to 1 Hz, deployablility, low heat dissipation and favourable electronic characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' While still under development, we describe the current R&D efforts lander seismic stationC/He W 90 K radi sorption coolers cryostat 15 K 4K platform levelling system3 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The proof mass will be suspended by means of a folded Watt’s linkage, a common way17 to achieve a compact, low- resonance-frequency device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' To achieve low thermal noise, the target proof mass will be 10 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' By using niobium, which has a 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='4 g/cm3 density, such device with all auxiliary sensing and actuation system can fit in a volume 200×200×100 mm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The readout of the proof mass motion, and therefore ul- timately the differential signal between the elastic response of the Moon to passing GWs and the inertial proof mass which holds the GW signal, is a cm-scale interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' An example of such opto-mechanical device is a room tem- perature version of an interferometrically Watt’s linkage that reached 8 fm/√Hz from 30 Hz 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The used interferometric readout, based on ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 19, reached 4 fm/√Hz from 4 Hz on- wards 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This readout needs feedback to keep the working point halfway up the fringe (the linear part of the sinusoid) as any deviation makes the output non-linear and degrades the subtraction of common mode noise between the two interfer- ometer output ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Without feedback the typical micrometer motion on Earth of the sensor frame would cause the sinu- soidal error signal to move between fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The feedback is provided by an actuator that locks the proof mass to the suspension frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The signal sent to the actuator is then proportional to force and acceleration and serves as the sensor output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Often, a coil-magnet actuator is used in force-feedback inertial sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' However, in the previously discussed 8 fm/√Hz results, thermal noise was expected to be dominant below 10 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' While the used Watt’s linkages can have mechanical quality factors above 5000, the permanent magnet and its eddy current damping of the moving metal pieces had degraded the Q to below 100 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' LGWA requires lower-frequency fm/√Hz sensitivity which can only be ob- tained by lowering thermal noise which goes as 21 x2 th = 4kBTω2 0φ mω � (ω2 0 −ω2)2 +ω4 0φ 2�, (1) where xth denotes the thermal noise displacement amplitude spectral density (ASD), kB Boltzmann’s constant, T the tem- perature, ω0 the angular resonance frequency, φ(= 1/Q for structurally damped suspensions) the loss angle and ω the an- gular frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Low temperatures and increased mass will obviously help, but different actuators that will not (domi- nantly) damp the Watt’s linkage are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Therefore, superconducting actuators that use the Meissner effect rather than a magnet to exert a force on the proof mass are inves- tigated 22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The superconducting thin film coils and super- conducting surface (depicted by orange rectangles) can, de- pending on the achieved cooling level or other application, be manufactured from niobium (Tc = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='2 K), MgB2 (Tc = 40 K) or YBCO (Tc = 93 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' To be in the necessary full magnetic expulsion state, temperatures around 60% of Tc or lower is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The current design follows from an initial cryogenic iner- tial sensor concept first proposed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 24, which was subse- quently updated 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Currently, we investigate what is depicted in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The resonance frequency of the Watt’s linkage can be coarsely set by the sliding tuning mass, which changes mass distribution between inverted and regular pendulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Af- ter cooldown of the mechanics, the resonance frequency may have changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' A DC current on one of the tuning coils can ef- fectively change the mass distribution thereby tuning the res- onance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' A cryogenic superconducting monolithic inertial sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The proof mass is suspended from the frame by a regular pendulum and inverted pendulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This monolithic configuration is known as a Watt’s linkage and allows for an arbitrarily low natural frequency, which increases the mechanical sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The proof mass motion is monitored by an interferometric readout and the custom cryo-chip is under development using 65 nm CMOS technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' More details are found in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The estimate of sensitivity is made by modelling the dis- placement noises of mechanical and interferometric nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Most models for these noises are described in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 18,24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The actuator noise model is a simple current driver model 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' We use the parameters in table I and arrive at the noise budget shown in figure 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This noise budget is roughly the same as the "opto-mechanical" trace in figures 2, 3, 4 and 5 of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The sensitivity of the four-sensor array is a factor 2 lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' LGWA sensitivity is obtained by dividing out the Moon’s re- sponse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' the expected surface motion per unit strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' An ex- ample of such modeled response is found in figure 1 of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The used readout scheme is an example femtometer-class interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' There are other options to realise an optical readout with similar or even lower predicted sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The trade-off between displacement readout schemes relies heav- ily on the required dynamic range and the ability, and corre- sponding benefits, of operating at a specific or a random op- erating point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' So-called multi-fringe interferometric sensors implement phasemeters to read out the phase at any operating point and with large, mostly multi-fringe, dynamic range26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' These types of interferometers are limited to femtometer-level sensitivities by effective technical-fundamental limitations in their readout, especially by digitisation noise and to provide linear sensing over a wide range they typically do not employ optical resonators to enhance the signals27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The best space-based demonstration of such displacement sensors is the multi-fringe heterodyne interferometry realised in LISA pathfinder28, which achieved a displacement mea- surement noise floor of 30 fm/√Hz around 1 Hz, mostly lim- ited by ADC quantisation noise in the digital phasemeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' A lower digitisation noise floor could be realised with commer- (not in vacuum) VVVVV laser gain calibration actuator 22 PBS pendulum PD2 c pendulum tuning mass inverted PD1 ctrl piezo custom cryo-chip tuning coils actuator 2 calibration4 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Mechanical, readout and electronics parameters for both the interferometrically and SQUID read out Watt’s linkage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Parameter Value Unit Proof mass 10 kg Natural frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='25 Hz Temperature 5 K Coil-superconductor gap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='1 mm Actuator strength 50 µN/A Niobium with interferometric readout Watt’s linkage material Nb Quality factor 1·104 Frequency noisea 500 · f −1/2 Hz/√Hz Static differential arm length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 mm Injected laser power 10 mW Wavelength 1550 nm TIA feedback resistor 20 kΩ Silicon with SQUID readout Watt’s linkage material Si Quality factor 1·106 SQUID energy resolution EA 2500 ¯h J/Hz signal to SQUID coupling efficiency ηβ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='25 1/√f corner frequency fc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='1 Hz a Typical value for high-end lasers e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The RockTM from NP Photonics 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' cially available ADCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' A critical part of the low-frequency noise floor that has to be evaluated for LGWA is the achievable temperature stability and the corresponding thermally driven couplings, namely thermoelastic and thermorefractive noise, which were suppressed in LISA Pathfinder by the exceptional temperature stability29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' These thermally driven noise sources will be critical for any interferometric readout scheme and need to be studied with respect to the cryogenic environment of the proof mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Thermal compensation strategies can be employed, but are complicated, in design and in testing, by the cryogenic operating temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' These noise source are also critical for any opto-mechanical laser frequency reference, be it a proper 2nd, equally long, arm in the local interferometer topology or some external, disjoint reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' For the LGWA and especially LGWA Soundcheck the power consumption of the payload might be a critical fac- tor, with the laser sources being a significant driver of such a budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Accordingly, the power consumption of any given interferometric readout has to be taken into account, as well as their influence on the potentially reduced power consumption in the active feedback to control the proof mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This might benefit interferometric readout schemes that require little or no opto-electronic elements, slow signal digitisation and lit- tle signal post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' In addition to the readout scheme shown in figure 3, a higher dynamic range option that can achieve femtometer-level displacement noise with no addi- tional active components is quadrature homodyne interferom- etry, which has already been used to demonstrate compact in- terferometric readout of inertial sensors 30 and demonstrated a noise floor of 20 fm/√Hz 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Depending on the dynamic range and the optical design, especially with regards to ghost beams and polarisation contamination32, such a readout might require additional digital signal processing with a Lissajous fit to suppress periodic non-linearity, which again might limits its advantage in terms of power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Finally, optical resonators can be employed in compact dis- placement sensors to achieve sub-femtometer displacement readout noise floors at the cost of readout range and linear- ity, for example using fiber-based implementations, as demon- strated in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Combining optical cavities with operation- point independent, wider-range readout is, however, non- trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Using a strong frequency-modulated laser with an op- tical resonator promises noise floors of 10−16 m/√Hz27, but might require too much effort with respect to opto-electronics and signal processing for the readout of only two displace- ments in a single LGWA station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' A more relevant approach might be to lock one laser to an optical cavity between the proof mass and an external mirror and to measure its fre- quency variations with changing length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Such a scheme re- quires a second ultra-stable laser to generate a beat note, but, combined with a corresponding real-time digital signal- processing system, this scheme can also realise the locking of both lasers to their respective optical resonators34, as depicted in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Heterodyne cavity-tracking readout scheme with co-located ultra-stable optical cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Tracking the motion of the proof mass requires a high-dynamic range phase readout system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Cavity length L, wavelength λ and phase readout bandwidth BW determine the maximum one-way displacement tracking range ∆Lmax = λ/2 · BW/(c/(2L)) = λ/2·BW/FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This readout senses one degree-of-freedom, adding another axis demands an additional laser that is locked to the cor- responding cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Hence, in order to measure the hori- zontal surface displacement along two orthogonal directions, each seismic station requires in total three laser sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' For resonator lengths of 5 cm the beat frequency will shift by 3 GHz for a displacement of λ/2, a frequency shift that could be tracked with a high-bandwidth, frequency-tracking phasemeter 35 with negligible frequency tracking noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Field- programmable gate arrays with integrated high-speed data converters are available to implement such tracking systems with several GHz of bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' A heterodyne cavity-tracking readout scheme can, in principle, achieve readout noise levels of 10−17 m/√Hz with reasonable levels of cavity Finesse, be- cause they are not directly limited by digitisation noise and the influence of shot-noise is suppressed by the optical enhance- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' In practise this readout will be limited by the stability of the available frequency reference, which could be a sepa- Cryostat proof mass readout spacer L = 5 cm Ultra-stable Optical Cavity5 rate cavity as developed for space-based optical clocks or fun- damental physics experiments36 that is co-located within the cryostat to reduce thermal effects like coating thermal noise, as shown in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' If available, the ultra-stable laser can also be a fully separate device connected only via fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The lasers, the phase readout system and the fiber-optics do, to first order, not have stringent environmental noise couplings and can be placed outside the cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The additional com- plexities and power consumption of a heterodyne cavity lock- ing scheme make it unsuited for LGWA Soundcheck, but the promise of mid-range dynamic range and extremely low read- out noise floor make it a promising candidate for the full LGWA readout, assuming other noise sources can be brought to sufficiently low levels, at least at the higher readout fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Detailed studies of amplitude noise37 and of tilt- to-length coupling38 will have to be done for any design and readout scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Besides the different interferometric readout strategies de- scribed above, superconductivity can be used to read out the proof mass position with high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' If a superconductor moves with respect to a superconducting coil carrying a per- sistent current, the inductance of coil-superconductor system changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The current in the coil will change correspondingly to keep the flux in the system conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Due to flux con- servation in superconducting loops, the current change can be converted to magnetic field change simply by connecting an- other coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This changing magnetic field can subsequently be picked up by a Superconducting QUantum Interference De- vice (SQUID), which is known for its extreme sensitivity to changing magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This readout strategy has been sug- gested, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=', in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 39 for gravity gradiometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Using two sens- ing coils in parallel and sandwiching a superconductor, and a third one to convert the current signal into magnetic signal, the motion of the superconductor can be read out with sub- femtometer precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' On the right side of figure 5(a) such dual coil sandwich configuration is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The superconducting readout provides an error signal for a feedback loop with a superconducting actuator, which can also employ a dual coil sandwich architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The supercon- ducting coils can be loaded with a persistent current as shown in figure 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' By sending an actuation current running in par- allel in the two coils, we can increase the current, and corre- sponding magnetic force, on one side and reduce the magnetic force on the other side, generating a net (feedback) force on the superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The magnetic force between a coil and a superconducting surface is proportional to the square of the current in the coil (Ipers +Iact)2 = I2 pers +2IpersIact +I2 act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Large persistent currents (> 1 A currents are common 40) will give the largest coupling to the signal current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' However, because the persistent currents in the coils push from either side there is a positive stiffness roughly equal to the DC force from each coil, divided by the coil-surface gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This added stiffness can be corrected for using the tuning mass and coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The main advantage of this strategy is that only small currents (< 100 µA) will have to be generated by the on-chip current driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Moreover, the dual coil architecture linearizes the relation be- tween the actuation current and the feedback force which will simplify the control and data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' (a) a silicon Watt’s linkage with superconductive readout and actuation, (b) dual coil sandwich configuration used for sensing and actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' More details found in text and (c) a zoom of the monolithic niobium and quasi-monolithic silicon flexures To decrease the thermal noise even further, a silicon Watt’s linkage is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Silicon is a crystalline material exhibiting low mechanical loss at cryogenic temperatures, with a bulk Q of 108 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The thin flexures allowing for their low stiff- ness of metallic Watt’s linkages have historically been fab- ricated using electro-discharge machining (EDM) techniques as shown in figure 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' A more difficult hybrid procedure for silicon must be followed as using EDM to cut the delicate flexures is expected to result in surface damage and thus lossy flexures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The frame and proof mass are manufactured from highly doped silicon, which can be cut using EDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The legs including the flexures are (laser assisted plasma) etched out of a thick 500 µm wafer and hydro catalysis bonded (HCB) to the frame and proof mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' HCB is famous for producing quasi-monolithic bonds in mirror suspensions of the current interferometric GW detectors 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Figure 2 in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 22 shows a possible HCB assembly procedure for a silicon Watt’s link- age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The quasi-monolithic silicon Watt’s linkage is expected to have a Q of 106, thereby lowering the thermal noise by an order of magnitude with respect to the niobium variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' superconducting superconducting actuator sensor pendulum pendulum tuning inverted mass custom nn cryo-chip proof mass output tuning coils levelling screwsactuation/signal current persistent current6 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Minimum detectable inertial displacement for a structurally damped accelerometer with (a) niobium mechanics and interferomet- ric readout and (b) silicon mechanics and SQUID readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The SQUID readout has a sub-fm/√Hz sensitivity cor- rected for the sensor mechanics as 43 x2 squid = 2EA(1+ fc/f) mω0ηβ (ω2 −ω2 0)2 +ω2 0/Q ω4 , (2) where most symbols have been denoted in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The SQUID has a 1/f characteristic below fc in its power spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The same actuator noise model as the niobium version and the silicon proof mass suspension thermal noise model complete the noise budget as presented in figure 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' SORPTION COOLING AND THERMAL MANAGEMENT Cryogenic cooling of the inertial sensor will be established by combining two vibration-free cooling technologies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' High- emissivity radiator panels will be used to provide heat-sink platforms at temperature levels of about 50 K and 90 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Next, a two-stage sorption-based Joule-Thomson cooler will be heat sunk to these platforms and will cool further down to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 K and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This sorption-based cooling technology has been developed at the University of Twente in the past two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' It operates with a thermal compressor rather than a mechan- ical compressor as conventional cryogenic coolers do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Apart from a few passive valves it has no mechanical moving parts and, therefore, offers operation at an extremely low level of emitted vibrations and a long lifetime because of the absence of wear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Both aspects are obviously attractive in space appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The operation of a sorption compressor is based on the cyclic adsorption and desorption of a working gas at a sor- ber material such as, in our case, activated carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Activated carbon is a material that by its highly porous structure has a very large internal surface so that it can adsorb large quan- tities of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' By heating the sorber, the gas is desorbed and a high pressure can be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' By expanding this high- pressure gas in a Joule-Thomson (JT) cold stage, cooling can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The operating principles and the thermodynam- ics involved, are discussed in many papers 44–48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The baseline cooler chain of the LGWA project is schemat- ically depicted in figure 7 and resembles the Darwin cooler that was developed in an earlier ESA-TRP project 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The first stage of the LGWA sorption cooler operates with hydro- gen gas and realizes a temperature of 15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The second-stage sorption cooler operates with helium gas and, precooled by the hydrogen stage, it reaches 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The hydrogen compressor is thermally linked to the 90 K radiator heat sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The hydrogen gas is precooled by a 50 K radiator that also serves as the heat sink for the helium compressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Based on the performance of the two stages of the Darwin cooler, the gross cooling pow- ers at both stages in the LGWA project are expected to be 36 mW at the 15K stage (of which 6 mW are used to precool the helium gas in the second stage), and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 mW at the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 K stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The total electric input power to the coolers is slightly more than 6 W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='2 W in the compressor of the hydrogen stage and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='9 W in that of the helium stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This input power, plus the power taken from the cold interfaces is emitted to deep space at the two radiator panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' In previous work, the radia- tor temperatures were optimized aiming at minimum radiator size, resulting in actual temperatures of 87 K and 51 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The required radiator panel areas are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='6 m2 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='2 m2, respec- tively 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This setup is schematically depicted in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The cooler mass is expected to be 10 kg of which both stages are about half of that 45,49–51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The cooling powers as indicated in figure 7 are not fully available as net cooling power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Part of it is used to take up parasitic heat loads due to conduction and radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The heat load budgets are listed in table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' In or- der to withstand launch loads, all frames will be mechanically fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Once positioned on the moon surface, these launch-load connections will be disconnected allowing for the 15 K frame to be leveled with respect to the moon surface, as illustrated in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The remaining support structures are anticipated to be G10 struts between leveling platform and 15 K frame, and Kevlar straps between 15 K frame and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 K cold mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' A sorption-based Joule-Thomson cooler has been launched and successfully operated in space in the ESA-Planck mission (2009-2013) 52,53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' It provided cooling power of 1 W at about ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='suspension thermal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='frequency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='relative intensity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10-11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='shot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='displacement [m/√Hz] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='actuator driver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='electronic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='Total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10-15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10-16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='frequency [Hz]10-10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='suspension thermal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='SQUID readout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='actuator driver ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10-11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='total ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10-14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10-15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10-16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='frequency [Hz]7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='15 K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='Total gross cooling power ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='36 mW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='Precooling He stage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='6 mW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='Radiation from 50 K environment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='20 mW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='Conductive load through support (G10 struts) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='9 mW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='Conductive load via cooler tubing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='1 mW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='Emissivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 K Total gross cooling power 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 mW Radiation from 15 K environment 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='1 mW Conductive load through support (Kevlar straps) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='9 mW Conductive load via cooler tubing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='1 mW Dissipation and conductive load of sensor + electronics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='4 mW Emissivity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='1 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Heat load budgets at the 15 K and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 K cold-tip inter- faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 20 K using hydrogen as the working fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' However, the com- pressor sorber material was a metal hydride which is a chem- ical absorber whereas in our compressor technology activated carbon is applied which is a physical adsorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The big differ- ence is that a chemical absorber degenerates over time limit- ing the lifetime of the cooler in mission (in Planck 2 years), whereas the adsorption process with carbon is fully reversible and does not limit the lifetime of the cooler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Our carbon based sorption compressor technology was qualified at TRL5 (sur- viving launch vibrations) in one of the recent ESA projects 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' SEISMOMETER LEVELING SYSTEM A leveling system is needed to achieve an initial alignment of the seismometer platform to compensate ground slope and then to keep it aligned within a few microradians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The require- ment of the alignment accuracy is set by the softness of the proof-mass suspension through the tilt-to-horizontal coupling dpm = gθ/ω2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The critical dimension in figure 3 and 5(a) is the 100 µm gap between coils and superconductor in the actu- ator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The leveling system should be more precise than 30 µm in proof-mass positioning to ensure that the superconductor does not make contact with the sandwiched coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' A platform meeting similar requirements was developed for the SEIS experiment of the Mars InSight mission55–57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This system features a MEMS-based rough alignment to compen- sate for up to 15◦ of ground slope, and a precision alignment system that reaches a few microradians using high-precision tiltmeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' An important new requirement for the LGWA plat- form is that it must be compatible with the cold environment of a PSR, which constraints above all the technologies that can be used for the high-precision tiltmeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' An alternative to using high-precision tiltmeters might be to realize the LGWA seismic sensors with a high dy- namic range laser-interferometric readout of the proof-mass displacement58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Exploiting the tilt-to-horizontal coupling, tilt can be measured and compensated by observing the move- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Schematic diagram of the two-stage sorption-based cooler with a cooling power of 30 mW at 15 K and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 W at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Electric input power is indicated in blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' heat flows in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' ment of the proof mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' With the rough tilt alignment stage, one can assess what sign the high-precision adjustment must have, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=', with which side of its frame the proof mass makes contact before the fine-alignment is engaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' SYNERGY WITH EINSTEIN TELESCOPE On Earth, ET features an underground and cryogenic de- sign and aims to be sensitive to GWs down to 3 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Meth- ods to apply low-vibration cryogenic cooling of the mirrors in a cryostat to lower thermal noise are currently investigated in research facilities 59–61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Close to the mirror spurious vibra- tions could be injected by the application of cooling power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' To ensure the lower cryogenic stages are indeed at low enough vi- bration levels, new inertial sensors such as described here are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' ET aims to be 10 times more sensitive than current detec- tors above 10 Hz and stretch its lower bandwidth limit down to 3 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Cooling down of the input and end mirrors down to around 10 K is needed to reduce the dominant noise at low frequency: thermal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' To extract heat, the penulti- mate mass above the mirror shown in figure 8 (right) operates at about 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Cooling the penultimate mass cannot be done radiatively due to the low temperature and required power (several 100 mW) and therefore some physical connection be- tween cryocoolers and the suspension final stages is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='2 W H2/ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='2 W 90 K carbon 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='6 m² 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='9 Wel 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='9 W He/ 50 K carbon 25 mW 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='2 m2 6 mW 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='K 30 mW 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 K 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 mW8 The cooling power is applied by low-vibration cryocoolers and using flexible heat links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' However, there is still a risk that unwanted vibrations end up in the penultimate stages, close to the mirrors where extremely tiny displacements in the de- tection bandwidth are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The cryogenic temperatures provide opportunities for new, superconductive actuators and (inertial) sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The use of superconductive coils reduces the cooling power (and therefore vibrations) otherwise needed for dissipative elements, such as the resistive copper actuator coils in figure 8 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Extremely sensitive inertial sensors, such as presented here, are needed to monitor the platform motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The final stages of (left) current room-temperature mirror suspensions and (right) future cryogenic mirror suspensions, where the low temperatures provide opportunities for new actuators and (in- ertial) sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Ultimate configuration for ET may differ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' however, similar sensing and actuation solutions will be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' In GW detector suspensions, actuators are used in an hierar- chical way in terms of strength and range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' the most low-noise, short-range actuators are needed close to the mirror where residual acceleration is extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' At the top of the sus- pension chain, actuation noise requirements are less stringent, but those actuators will have to operate over a larger range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Most actuators used in today’s GW detectors are (some form of) coil-magnet actuator as these are easy develop, install and use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The use of permanent magnets close to moving metals can cause harmful eddy currents and stray magnetic field can exert unwanted forces on the suspended objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The former is largely solved by using plastics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' PEEK) near the mag- nets and the latter is often solved by placing the magnets on the same object in opposite polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The cryogenic GW detector KAGRA operates 23 kg mir- rors dissipating 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='5 mW 62 at the actuators and ET mirrors are 10 times as massive 63, thus dissipating >10 mW if old re- sistive actuators are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' This is of order 10% compared to the expected absorption of laser light and thermal radiation of mirror and payload, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Lastly, the sub-fm/√Hz dual coil position sensor with SQUID readout can be used as differential sensors between cage and (pen)ultimate stage(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK To open up GW science in the decihertz range, there have been space-borne proposals, such as DECIGO 64 and BBO 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' While they promise higher sensitivity than LGWA, many tech- nological challenges remain and a longer timeline is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Here, we have presented several technologies that make up the payload and detail several different options in the inertial sen- sor design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' While the niobium Watt’s linkage fabrication processes and interferometric readout technology is more mature, the silicon Watt’s linkage with SQUID readout may result in roughly one order of magnitude lower thermal and readout noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Note that a tenfold sensitivity improvement will lead to larger range and thus an expected factor thousand more GW signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' In both designs we propose actuators with superconducting coils which are also necessary for the sensing part in the SQUID readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The development of the inertial sensor as well as the sensing and actuation technology shows strong synergy with future cryogenic GW detector ET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The inertial sensors with extreme sensitivity have to be tested in extremely quiet and cold environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Such test facilities in the form of actively isolated platforms inspired by the LIGO HAM table designs 66 are being developed as part of the E-TEST effort in Belgium 59,61 and the GEM- INI facility in the underground National Laboratories of Gran Sasso 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The aimed-for sensitivity at 1 Hz is about 5 orders of magnitude smaller than the Earth’s seismic motion at that fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Placing two or three identical sensors on the isolated platform allows for subtraction of common mode noise using the Wiener filter 68 or three-channel correlation techniques 69 resulting in a sensor self-noise measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' The technology necessary for LGWA will either be spe- cific development of existing space technology (levelling sys- tem, sorption cooler, thermal management systems etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=') or in parallel with terrestrial GW instrumentation R&D in iner- tial sensing and active isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Future terrestrial GW de- tector isolation has to stretch to lower frequencies and needs better low-frequency inertial sensors and active isolation per- formance for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' For a space application as LGWA, how- ever, there will be extra (space) engineering necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Be- fore LGWA will fly, the aforementioned LGWA Soundcheck also requires some technology development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Its strategy is to combine technologies that have already flown in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' For in- stance, elements of the interferometer topology developed for LISA (Pathfinder) can be adopted for the readout of Sound- check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' R&D for LISA and other space missions will also have overlap with the technologies presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' In this context, payload technology development continues towards cryogenic, (sub-)fm/√Hz inertial sensing on the lunar surface for GW detection and lunar geophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' flexible heat links from cryo-cooler cryo-shields new (inertial) sensors new actuators9 ACKNOWLEDGEMENTS Oliver Gerberding and Shreevathsa Chalathadka Sub- rahmanya are funded by the Deutsche Forschungsgemein- schaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC 2121 “Quantum Universe”— 390833306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Filip Tavernier and Alberto Gatti are funded by internal KU Leuven funds (iBOF-21-084).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Filip Tav- ernier, Alberto Gatti, Christophe Collette, Joris van Heijnin- gen and this research are partially funded by Interreg V-A Eu- regio Maas-Rijn under the E-TEST project (EMR113).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' Mor- gane Zeoli is funded by the Fonds National de la Recherche Scientifique (FNRS) under projet de recherche STELLAR (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='0022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNFRT4oBgHgl3EQf8zg4/content/2301.13685v1.pdf'} +page_content=' 1B.' metadata={'source': 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b/WdFOT4oBgHgl3EQf7jS_/content/tmp_files/2301.12963v1.pdf.txt @@ -0,0 +1,788 @@ +arXiv:2301.12963v1 [math.DS] 30 Jan 2023 +On Sharp Bounds for the Dynamic +Asymptotic Dimension +Samantha Pilgrim +January 31, 2023 +Abstract: +We prove the dynamic asymptotic dimension of a free isometric +action on a space of finite doubling dimension is either infinite or equal to the +asymptotic dimension of the acting group; and give a full description of the +dynamic asymptotic dimension of translation actions on compact Lie groups +in terms of the amenability and asymptotic dimension of the acting group. +1 +Introduction +The dynamic asymptotic dimension (DAD) of a group action Γ ↷ X was +first introduced by Guentner, Willett, and Yu [9], and was shown to relate +to conditions used by Bartels, L¨uck, and Reich in work on the Farrell Jones +conjecture on manifold topology [2] [3], as well as to the nuclear dimension +of the crossed product [17] and calculations of its K-theory [8]. The DAD is +also related to other dynamical dimension theories [10] known to take into +account both the asymptotic dimension of Γ and the topological dimension +of X. However, it has been suspected since its introduction that the dynamic +asymptotic dimension may actually coincide with the asymptotic dimension +of Γ whenever it is finite (it is possible to have DAD(Γ ↷ X) = ∞ while +asdimΓ < ∞). The case where Γ is virtually cyclic is proved in [1], providing +an important test-case for this conjecture. More generally, the work of [15] +gives an upper bound of asdimΓ + dim X for isometric actions by finitely +generated groups on manifolds; and [5] shows that many actions on Cantor +sets have DAD = asdimΓ. +1 + +This note documents progress on a question [15, 8.6] originally posed +by Willett by giving the first sharp bounds for the dynamic asymptotic di- +mension of free isometric actions on spaces of dimension greater than (or +equal to) zero, modulo the fairly mild assumption that X has finite dou- +bling dimension. In this case we show the dimension is always either ∞ or +asdimΓ. We accomplish this by introducing a new way of formulating the +doubling dimension of a space, which we call Cantor decomposability. This +property, together with the theory of residually finite group actions, allows +many isometric actions to be modeled by several sequences of partial dynam- +ical systems on discrete spaces which asymptotically resemble the structure +of Γ. This property is reminiscent of box spaces of residually finite groups, +and our investigations are partly inspired by past work on the asymptotic +dimension of such spaces [6], as well as by the relation between box spaces +and odometers [13]. In line with [6], which describes among other things +the asymptotic dimension of box spaces of residually finite groups; we use +the theory of residually finite group actions introduced in [11], together with +Cantor decomposability, to reduce the problem to geometry. +We apply the main result in order to calculate the dimension of translation +actions on compact Lie groups in terms of the amenability and asymptotic +dimension of the acting group. +We also compute the dimension of many +isometric actions by Zn. +2 +DAD of partial dynamical systems +We begin by establishing some notation and terminology. We will assume +throughout that the group Γ is finitely generated with finite generating set +F, that F = F −1 is symmetric, and that e ∈ F (where e denotes the identity +in Γ). The following definition comes from [7, 2.1]. See [7, Part I] for a more +complete treatment of partial actions. +Definition 2.1. A topological partial action of Γ on a topological space X +is a pair ({Dγ}γ∈Γ, {θγ}γ∈Γ) consisting of a collection {Dγ}γ∈Γ of subsets of +X, and a collection {θγ}γ∈Γ of homeomorphisms, θγ : Dγ → Dγ−1 such that +(i) De = X, and θe is the identity map. +(ii) θγ ◦ θλ ⊆ θγλ, for all γ and λ in Γ. +2 + +Here, the composition θγ ◦ θλ denotes the map whose domain is the set of +all x ∈ X for which θγ(θλ(x)) makes sense. In other words, this is the set +θ−1 +λ (Dγ) = θ−1 +λ (Dλ−1 ∩ Dγ). The symbol “⊆” means that the function on the +right-hand-side is an extension of the function on the left-hand-side. Notice +also that θγ−1 = θ−1 +γ . +A partial dynamical system is a quadruple (X, Γ, {Dγ}γ∈F, {θγ}γ∈Γ). An +action Γ ↷ X is then a partial dynamical system with Dγ = X for all γ ∈ Γ. +A partial dynamical system is free if θγ(x) = θλ(x) iff γ = λ. +We may +sometimes still write γ · x for θγ(x) when it is not ambiguous to do so. +Some partial dynamical systems we will deal with later are restrictions of +actions or partial actions, in which case we will write Γ ↷ A for a subset A ⊂ +X to denote the partial dynamical system (A, Γ, {A ∩ h−1 +γ (A)}γ∈F, {hγ}γ∈F) +where hγ comes from Γ ↷ X or some partial action on X. +Definition 2.2. Let (X, Γ, {Dγ}γ∈F, {θγ}γ∈Γ) and (Y, Γ, {Eγ}γ∈F, {ργ}γ∈Γ) +be topological partial dynamical systems. By a conjugacy, we mean a home- +omorphism f : X → Y such that f(θγ(x)) = ργ(f(x)) whenever θγ(x) is +defined (so ργ(f(x)) must be defined in this case). +Definition 2.3. Let (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) be a partial dynamical system +and S ⊂ Γ a finite subset with S = S−1 and e ∈ S, and A ⊂ X. An S- +chain in A is a finite sequence x0, . . . , xn of points in A such that for all +0 ≤ i ≤ n − 1, θγ(xi) = xi+1 for some γ ∈ S. Two points in A are in the +same S-component if they are connected by an S-chain in A. We say a cover +V = {Vj}d +j=0 is a (d, S, M)-cover for Γ ↷ X (or just X if unambiguous) if +all S-components of each Vi have cardinality at most M. +Definition 2.4. The dynamic asymptotic dimension of a free partial dy- +namical system is the smallest integer d such that for every finite subset +S ⊂ Γ with S = S−1 and e ∈ S, there is M > 0 and a (d, S, M)-cover for +(X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ). If no such d exists, the dimension is defined to be +∞. As we are assuming Γ is finitely generated with finite generating set F, +we can assume S has the form F r for some r > 0. A d-dimensional control +function for (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) (with Γ finitely generated) is a function +D : N → N such that for every integer r > 0, there is a (d, F r, D(r))-cover +for (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ). +We will use the notation described in the last two definitions (which +relates to the definition of DAD for free actions) even for actions which are +not free. +3 + +Notice that DAD appears a priori to be sensitive to the topology of X as +the cover U is required to be open. +Definition 2.5. If (Xn, Γ, {Dn +γ}γ∈Γ, {θn +γ}γ∈Γ) is a sequence of partial dy- +namical systems, we say DAD(Γ ↷ Xn)n ≤ d uniformly if there is a single +d dimensional control function for all n. +Definition 2.6. Suppose CF(Γ) is a Cayley graph of a finitely generated +group Γ with F being a generating set. By this we mean the set Γ with the +metric given by d(x, y) = |yx−1|F where |w|S denotes the minimal length of +a word in S which equals w as an element of Γ. Notice that this is invariant +under right multiplication. If r > 0, an r-chain in A ⊂ CF(Γ) is a finite +sequence x0, . . . , xn of points in A such that d(xi, xi+1) ≤ r for 0 ≤ i ≤ n−1. +Two points g, h ∈ A ⊂ CF(Γ) are in the same r-component of A if they +are connected by an r-chain in A. +We say a cover U = {U0, . . . , Ud} of +CF(Γ) such that the r-components of each Ui have diameter at most Mr is +a (d, r, Mr)-cover for CF(Γ). The asymptotic dimension of CF(Γ), which we +write as asdimCF(Γ), is the least integer d such that for all r > 0 there is +a (d, r, Mr)-cover for CF(Γ). As asdimCF(Γ) is independent of F, we may +simply write asdimΓ. When referring to the action of Γ on itself, we mean +the action by left multiplication. +Lemma 2.7. Suppose Γ ↷ X is a free action by a finitely generated group. +Then DAD(Γ ↷ X) ≥ asdimΓ. In fact, this inequality holds even if one +modifies the definition of DAD to allow covers by Borel sets, or even arbitrary +sets. +Proof. Let F be a finite generating set for Γ as usual. Suppose DAD(Γ ↷ +X) ≤ d. Let r > 0. Then there is a (d, F r, Mr)-cover V for Γ ↷ X. Let O(x) +be the orbit of some x ∈ X. Identify O(x) with CF(Γ) by γ · x �→ γ. Then V +gives a cover V′ of CF(Γ). Moreover, two points y, y′ ∈ O(x) are connected +by an F r chain iff their corresponding elements in CF(Γ) are connected by +an r-chain. Therefore, the r-components of each V ∈ V′ are Mr-bounded, so +V′ is a (d, r, Mr)-cover for CF(Γ). +Saying anything about the reverse inequality will require considerably +more work. +4 + +3 +Union theorem +Union theorems are common in dimension theories. The intuition is that +a finite union of objects with dimension ≤ d should still have dimension +≤ d. Our method of showing this for DAD comes from [4, Lemma 3.5 and +Corollary 3.6]. This section also includes an application of the union theorem +which will be used later, as well as some other technical lemmas. +Lemma 3.1. Suppose Γ ↷ X is free and A, B ⊂ X such that the F r- +components of B have cardinality at most R and the F r(R+2)-components +of A have cardinality at most D. Then the F r-components of A ∪ B have +cardinality at most #B(D−1)(R+1)+1 +e +(CF r(Γ)). +Proof. Consider an F r-chain in A ∪ B: x0, . . . , xn and assume this chain has +no repeated points. Suppose xi and xj are two consecutive elements of A, so +xi+1, . . . , xj−1 are all points in B. Then these points form an F r-chain in B +which therefore has cardinality at most R, and therefore length at most R +(since there are no repeated points). But then xi+1 and xj−1 are connected +by something in F rR, and so xi and xj are in the same F r(R+2)-component +of A. By similar reasoning, all elements of A in the original chain are in the +same F r(R+2)-component of A. The subchain x0, . . . , xn of points contained +in A then has cardinality at most D, hence length at most D. Since each +chain between each element of A has length at most R, the length of the +original chain is at most (D − 1)(R + 1) + 1. The F r-components of A ∪ B +therefore have cardinality at most #B(D−1)(R+1)+1 +e +(CF r(Γ)). +Theorem 3.2. (Finite Union Theorem) Suppose Γ ↷ X and {Ai}K +i=0 are +open subsets of X. Suppose fi is a d-dimensional control function for Γ ↷ Ai. +Let r > 0. Define ri and Ri inductively starting with r0 = r, R0 = f0(r) and +then for 1 ≤ i ≤ K defining ri = r(Ri−1 + 2), and +Ri = #B(Ri−1−1)(fi(ri)+1)+1 +e +(CF r(Γ)). +Then there is a (d, F r, RK)-cover for Γ ↷ ∪iAi. Notice that RK depends +only on the fi and K. +Proof. Take a (d, F ri, fi(ri))-cover V(i) = {V (i) +j }d +j=0 of Ai for each i. Form a +cover V of ∪K +i=0Ai by putting Vj = ∪K +i=0V (i) +j . Then use the previous lemma +and induction. At the s-th step of the induction (a union of s + 1 sets), we +have r = r and rR + 2r = rRs−1 + 2r = rs (s = 1, . . . , K). +5 + +Definition 3.3. Suppose Γ has finite generating set F and (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) +is a partial dynamical system. For P ∈ N, we define +BP +x (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) := {y ∈ X | y = θγ(x) for some γ ∈ BP +e (CF(Γ))}. +We may also just write BP +x or BP +x (X) when unambiguous. +Lemma 3.4. Suppose (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) is a partial dynamical system +such that θγ(x) = θγ′(x) for γ, γ′ ∈ BP +e (CF(Γ)) if and only if γ = γ′. Then +this system restricted to any BP +x is conjugate to a restricted action Γ ↷ A ⊂ +BP +e (CF(Γ)) by left multiplication. +Proof. Consider the subset BP +x = {yγ | yγ = θγ(x) with γ ∈ BP +e (CF(Γ))}. +Identify BP +x with a subset A of BP +e (CF(Γ)) by the map f(yγ) := γ. Then if +yδ, yδ′ ∈ BP +x are such that θγ(yδ) = yδ′, we have yδ′ = θγ(yδ) = θγ ◦ θδ(x) = +θγδ(x) = yγδ implying γδ = δ′. Hence, f(θγ(yδ)) = f(yδ′) = δ′ = γδ = γ · δ, +so f is equivariant. +Lemma 3.5. Suppose Γ is a finitely generated group with generating set F +and CF(Γ) is a Cayley graph. Let (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) be a free par- +tial dynamical system. +Then a (d, r, M)-cover for CF(Γ) gives rise to a +(d, F r, #BM +e (CF(Γ))-cover for (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ). +Proof. For each orbit of (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ), identify the points in X +with a subset of CF(Γ) by choosing any x0 to correspond to the identity and +then identifying θγ(x) with γ. Then notice that two points are connected by +an F r chain iff their corresponding elements in CF(Γ) are connected by an +r-chain. +Next, we want to establish some lemmas about the DAD of sequences of +actions. The content of the following lemma is essentially that of [6, 3.1] and +mirrors the theory of box spaces of residually finite groups. In the proof of +the main theorem, this is part of what allows us to infer a tight bound on +the dimension whenever the dimension is finite. +Lemma 3.6. Suppose Γ is finitely generated with generating set F and +(Gn, Γ, {Dn +γ}γ∈Γ, {θn +γ}γ∈Γ)n is a sequence of partial dynamical systems with +Gn finite such that for all r > 0 there is N > 0 such that there is a +(d, F r, Mr)-cover for (Gn, Γ, {Dn +γ}γ∈Γ, {θn +γ}γ∈Γ) for all n ≥ N (Mr does not +depend on n). Suppose also that for all P > 0 there is N′ such that for +6 + +n ≥ N′ and any r′ > 0, the system (Gn, Γ, {Dn +γ}γ∈Γ, {θn +γ}γ∈Γ)n restricted +to any BP +x (x ∈ Gn) has a (d′, F r′, M′ +r′)-cover with M′ depending only on +r′ and Γ. +Then for all r > 0, there is N′′ and a (d′, F r, M′′ +r )-cover for +(Gn, Γ, {Dn +γ}γ∈Γ, {θn +γ}γ∈Γ)n for all n ≥ N′′. Moreover, M′′ +r depends only on +d and the function r �→ M′ +r. +Proof. Fix r > 0. For each 0 ≤ i ≤ d, define fi(r) = M′ +r for all i. Define ri +and Ri inductively as in 3.2. By hypothesis, there is N such that for n ≥ N, +there exists a (d, F RK, MRK)-cover V = {Vi}d +i=0 for (Gn, Γ, {Dn +γ}γ∈Γ, {θn +γ}γ∈Γ). +Then find N′ ≥ N so that (Gn, Γ, {Dn +γ}γ∈Γ, {θn +γ}γ∈Γ)n restricted to any +B +MRK +1 +x +has a (d′, F ri, fi(ri))-cover. Find such a cover Ui = {Ui +k}d′ +k=0 for +each F RK-component of Vi. For each 0 ≤ i ≤ d, form a (d′, F ri, fi(ri))-cover +U = {Uk}d′ +k=0 of Vi by putting Uk = � +j Uj +k. It follows from 3.2 that there is +a (d′, F r, gd(r))-cover for each (Gn, Γ, {Dn +γ}γ∈Γ, {θn +γ}γ∈Γ) with n ≥ N′. +Definition 3.7. If Γ is a group and CF(Γ) is a Cayley graph (F is a finite +generating set), we say Γ has polynomial growth of degree at most d if there +are d > 0 and C > 0 such that if G(r) = #Br +e(CF(Γ)), then Gn(r) ≤ Crd. A +sequence of partial dynamical systems (Gn, Γ, {Dn +γ}γ∈Γ, {θn +γ}γ∈Γ)n has poly- +nomial growth of degree at most d if there are d > 0 and C > 0 such that if +Gn(r) = supx∈Gn #{y : y = γ · x for γ ∈ Br +e(CF(Γ))}, then Gn(r) ≤ Crd for +all n. +We will make use of a standard ‘greedy algorithm’ in this section and +again later, so we formalize such an algorithm with a lemma. +Lemma 3.8. Suppose S is a finite set and ∼ is a reflexive, symmetric rela- +tion on S such that #{t ∈ S | t ̸= s and s ∼ t} ≤ D for all s ∈ S. Then +there is a cover C = {Ci}D +i=0 such that s ≁ t for all s ̸= t in Ui for all i. +Proof. Assign elements to the sets in C in the following way. Start by picking +any element s0 ∈ S. Then there are at most D other elements of S which +are related to s0. Assign s0 ∈ C0 and those other elements to C1 through CD +so that at most one belongs to each. Then consider one of the elements just +discussed other than s0. This element is related to at most D other elements +of S including s0, so we can repeat the process. Continue in this way until +all elements of S are assigned to some set in C. +7 + +Lemma 3.9. Suppose Γ is finitely generated and (Gn, Γ, {Dn +γ}γ∈Γ, {θn +γ}γ∈Γ)n +is a sequence of partial dynamical systems on finite sets with polynomial +growth of degree at most d. Then DAD(Gn, Γ, {Dn +γ}γ∈Γ, {θn +γ}γ∈Γ)n ≤ K = +4d + 1. +Proof. The proof is essentially the same as that of [6, 3.3]. +Fix F ⊂ Γ a finite generating set. +For a partial dynamical system +(G, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) on a discrete space, the following condition implies +its dynamic asymptotic dimension is at most K − 1: for all R > 0 there is +M > 0 and a cover V of G by sets of cardinality at most some M such that +if x ∈ V ∈ V, there are at most K − 1 other elements V ′ ∈ V such that +γ · x ∈ V ′ for some γ ∈ F R. To see this, notice that one can use a greedy +algorithm as in 3.8 to sort the elements of V into K subcollections such that +taking the union over each subcollection yields a (d, F r, M)-cover. +By assumption, we have that #BR +x (Gn) ≤ CRd for all R > 0, x ∈ Gn +for any n. +Let K = 4d + 1, R > 1 and S0 = 4m+1R where m is such +that (K/4d)m ≥ CRd. We claim that for every n there exists Rn such that +R ≤ Rn ≤ S0 +4 and |B4Rn +x +(Gn)| ≤ K|BRn +x (Gn)| (where x ∈ Gn is arbitrary). +If this were not the case then Ki|BR +x (Gn)| < |B4iR +x +(Gn)| ≤ C4idRd for every +i with 4iR ≤ S0, so setting i = m would give C4mdRd > Km|BR +x (Gn)| ≥ +C4mdRd|BR +x (Gn)|. But then we would have 1 > |BR +x (Gn)|, a contradiction. +Now, take Xn maximal in Gn such that BRn +x (Gn) and BRn +y (Gn) are disjoint +for all x, y ∈ Xn. Consider the collection V = {B2Rn +x +(Gn) | x ∈ Xn}. If z ∈ +Gn, maximality of Xn implies there is x ∈ Xm such that BRn +z (Gn)∩BRn +x (Gn) +and so z ∈ B2Rn +x +(Gn), and so V covers Gn. +Finally, we check that if x ∈ Gn and x ∈ V ∈ V, there are at most K +other elements V ′ ∈ V such that γ · x ∈ V ′ for some γ ∈ F Rn. Let z ∈ Gn. +For every B2Rn +x +(Gn) ∈ V which has an element at a distance at most Rn to z, +we have that x ∈ B2Rn+Rn +z +(Gn) ⊂ B3Rn +z +(Gn). Now consider B3Rn +z +(Gn) ∩ Xn. +Since BRn +x (Gn) and BRn +y (Gn) are disjoint for any x and y in Xm, we have +that #(B3Rn +z +(Gn) ∩ Xn) ≤ #B4Rn +z +(Gn) +#BRn +z +(Gn) ≤ 4d + 1. +If V ′ ∈ V is such that there is γ ∈ F Rn with γ · V ∩ V ′ ̸= ∅, then there +is z ∈ V ′ with z ∈ B3Rn +z +(Gn) ∩ Xn. Therefore, there are at most 4d + 1 such +V ′ ∈ V. This completes the proof. +Lemma 3.10. Suppose Γ has polynomial growth and (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) +is a partial dynamical system. Then (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) has polynomial +8 + +growth bounded by that of Γ. That is, a d > 0 and C > 0 that work for Γ in +the sense of 3.7 also work for the whole sequence of partial actions. +4 +Residual finiteness +We will also make use of a property which allows group actions to be ap- +proximated by actions on finite sets. +The definition below was originally +introduced by Kerr and Nowak in [11]. +Definition 4.1. An action Γ ↷ X on a metric space by homeomorphisms +is residually finite if for all F ⊂ Γ finite and ǫ > 0 there is a finite set +E, an action Γ ↷ E, and a map ζ : E → X such that ζ(E) is ǫ-dense +in X (meaning every ǫ-ball in X intersects E nontrivially) and that d(ζ(γ · +e), γ ·ζ(e)) < ǫ for all γ ∈ F. Such an action is called a (F, ǫ)-approximating +action for Γ ↷ X. If X has no isolated points, a perturbation argument shows +we can assume ζ to be an inclusion. If (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) is a partial +dynamical system, E is a finite set and ζ : E → X is a map, we similarly say +a partial dynamical system (E, Γ, {DE +γ }γ∈Γ, {θE +γ }γ∈Γ) is (F, ǫ)-approximating +if E is ǫ-dense in X and if, whenever θγ(ζ(x)) is defined for γ ∈ F and +x ∈ E, we have that θE +γ (x) is defined and d(θγ(ζ(x)), ζ(θE +γ (x))) < ǫ. +As with the theory of residually finite groups, there is a close relationship +between isometry and residual finiteness. We can make use of residual finite- +ness of actions later in 5.6 without loss of generality in the main theorem +thanks to the following result. +Theorem 4.2 ([14, Theorem 3.6]). Every faithful, isometric action by a +finitely generated, amenable group is residually finite. +The work of [6] shows that the asymptotic dimension of box spaces of +residually finite groups is closely related to the asymptotic dimension of +the group itself. As the dynamic asymptotic dimension behaves somewhat +like the asymptotic dimension of a box space (both are infinite in the non- +amenable case for example), we expect that residual finiteness of actions may +be helpful to relating their DAD to the asymptotic dimension of the acting +group. +9 + +5 +Cantor-like decompositions +A key ingredient in the proof of the main theorem is a technical condition on +X which allows an isometric, residually finite action to be modeled by several +sequences of partial dynamical systems on finite spaces, thereby allowing us +to apply the work of the previous section. We will see that many spaces +satisfy this property. +Isometric actions on Cantor sets can be described as inverse limits of +finite actions [14, 2.3], which allows their DAD to be studied more easily. We +therefore would like to cover X by Cantor sets and use the union theorem +from section 3. Instead of doing exactly this, we cover X by sets which (as +ǫ → 0) resemble the sets one might intersect in constructing a Cantor set-like +object. +Definition 5.1. Let X be a compact metric space and let ǫ, δ > 0. We say X +is (K, ǫ, δ)-Cantor decomposable if there is an open cover U = U0 ⊔ . . . ⊔ UK +of X with diam(U) < ǫ for all U ∈ Ui and all i, and d(U, V ) > δ for all +distinct U, V ∈ Ui and all i. We call such a cover a (K, ǫ, δ)-decomposition. +This condition comes from trying to force an argument like [14, 2.3] to +work by dividing X into different subspaces. Notice that the definition is +somewhat reminiscent of asymptotic dimension, but small-scale. We there- +fore expect K to be related to some notion of metric dimension for X. +Definition 5.2. A metric space (X, d) is M-doubling if for all r > 0 and +x ∈ X, the closed ball B2r +x can be covered by at most M closed balls of radius +r. Further define dimd(X) := log2 M, which we call the doubling dimension +of X. +The origins of this concept are not precisely clear and it may have been +rediscovered independently multiple times. It can be reasonably attributed to +Assouad, though his is a different, equivalent definition. See the introduction +of [12] for additional exposition. +Example 5.3. The space R is 2-doubling and therefore has doubling dimen- +sion 1. +Working out M exactly for higher-dimension Euclidean spaces is +difficult, but it is more straightforward to see that Rd is M-doubling for some +M ≤ 4 · 2d. This follows since a d-cube with side length l can be covered +by 4 d-balls of diameter l, and it takes 2d d-cubes with side length l/2 to +cover one with side length l. More generally, a Riemannian manifold admits +an isometric embedding into some Euclidean space by the Nash embedding +theorem, and therefore has finite doubling dimension. +10 + +As alluded to earlier, we will now see that Cantor decomposability relates +quantitatively to the doubling dimension. +Lemma 5.4. Suppose (X, d) is compact and M-doubling. Then (X, d) has +a (M⌈2+log2(k+3)⌉, ǫ, kǫ)-decomposition for all ǫ > 0 and k ∈ N. If X has a +(K, ǫ, 2ǫ)-decomposition for all ǫ > 0, then X is K-doubling. In particular, +this implies X has finite doubling dimension iff for all k there exists K such +that X is (K, ǫ, kǫ)-Cantor decomposable for all ǫ > 0. +Proof. For the first claim, fix ǫ > 0. Let C be a cover of X by (closed) ǫ/2- +balls using as few balls as possible. If x ∈ X and U = {B ∈ C|B∩B(k+2)ǫ +x +̸= ∅} +then U is a cover of � +B∈U B using as few ǫ/2-balls as possible: if � +B∈U B +had a cover U′ by fewer balls, then (C \ U) ∪ U′ would be a cover of X by +fewer balls. Therefore, since � +B∈U B ⊂ B(k+3)ǫ +x +, #U ≤ M⌈2+log2(k+3)⌉ (where +⌈·⌉ denotes the ceiling function). +It is therefore possible to color the balls using #U ≤ M⌈2+log2(k+3)⌉ colors +such that any two balls of the same color are distance at least (k + 1)ǫ apart. +This can be shown via a standard ‘greedy algorithm’ (see 3.8). Finally, we +can replace each ball with an open neighborhood of itself while keeping the +diameter of each < ǫ and keeping balls in the same collection > kǫ apart. +For the second claim, suppose X is (K, ǫ, 2ǫ)-Cantor decomposable for +all ǫ > 0. Given a ball B ⊂ X of radius 2ǫ, find a cover of X witnessing +its (K, ǫ, 2ǫ)-Cantor decomposability. Then the subcollection of sets in this +cover which intersect B is a cover of B. Moreover, any set intersecting B is +within 2ǫ of any other such set, so this cover of B contains at most K sets +of diameter < ǫ. +The existence of (K, ǫ, δ)-decompositions for δ large compared to ǫ really +does require finite doubling dimension rather than finiteness of other types +of dimension (e.g. Lebesgue covering dimension). We give an example to +show this. +Example 5.5. There exists a Cantor set with infinite doubling dimension. +Proof. Let C be the space of sequences (an)∞ +n=1 where an ∈ {1, . . . , n}. Give +C the metric d((an), (bn)) = � +n +|an−bn| +n2n . Then a ball of radius +1 +2k centered at +(an) consists of sequences which agree with (an) up to at least index k. Such +a ball therefore requires k + 1 balls of radius +1 +2k+1 to cover it. +11 + +We conclude this section by showing how Cantor decompositions can be +used to model certain isometric actions by partial dynamical systems acting +on decompositions of X. +Lemma 5.6. Suppose Γ ↷ X (denoted by γ · x) is isometric and residually +finite, that Γ has finite generating set F, and that X has no isolated points +and finite doubling dimension. Let r > 0 and P > 0. Then there is ǫ0 such +that if ǫ < ǫ0 and Uǫ = Uǫ +0 ⊔· · ·⊔Uǫ +K is a (K, ǫ, 15ǫ)-decomposition of X such +that, for all i and U ∈ Ui, U contains a ball of radius ǫ/2 (this can be done +by first taking a (K, ǫ/2, 16ǫ)-decomposition for each ǫ and then replacing +each set by its ǫ/2-neighborhood); then there is a partial dynamical system +(Uǫ +i , Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) such that +(i) If γ ∈ F r and γ · U ∩ V ̸= ∅ for U, V ∈ Uǫ +i , then θγ(U) = V . +(ii) If w = fj · · · f1 and w′ = f ′ +p · · · f ′ +1 are words in F r with length ≤ P + 1 +which are not equal as elements of Γ, then θfj ◦ · · · ◦ θf1(x) ̸= θf′p ◦ · · · ◦ +θf′ +1(x) for all x where both sides are defined. +Proof. Start by letting ǫ be sufficiently small that translates (according to the +action Γ ↷ X) of subsets of diameter < ǫ by elements of F (P +1)r are disjoint +(using that Γ ↷ X is free). This will later be used to ensure condition (ii) +holds. +Let Γ ↷ E ⊂ X be a (F r, ǫ/2)-approximating action for Γ ↷ X. Use +δγ(x) to denote the action Γ ↷ E. +Now, for each i, we can restrict the action Γ ↷ E to a partial action +Γ ↷ E ∩ N2ǫ(� +U∈Uǫ +i U) (where N2ǫ(·) denotes the open 2ǫ-neighborhood). +Each x ∈ E ∩ N2ǫ(� +U∈Uǫ +i U) can be associated to an ˜x ∈ � +U∈Uǫ +i U in such +a way that ˜x = x if x ∈ � +U∈Uǫ +i U, d(x, ˜x) < 3ǫ, and ˜x ̸= ˜y for x ̸= y +(since X has no isolated points). Let ˜E = {˜x +| x ∈ E ∩ N2ǫ(� +U∈Uǫ +i U)} +and conjugate by the correspondence x �→ ˜x to obtain a partial dynamical +system ( ˜E, Γ, {αγ}γ∈Γ, {Cγ}γ∈Γ). +Then, if there is γ ∈ F r and γ · U ∩ V ̸= ∅ for some V ∈ Uǫ +i ; then (as E is +ǫ/2-dense and as U contains an ǫ/2-ball) there is x ∈ U with x ∈ E so that +˜x = x and d(δγ(x), V ) < 2ǫ, and so � +δγ(x) = αγ(x) ∈ V . This will guarantee +condition (i). Also notice that d(γ · ˜x, αγ(˜x)) < 7ǫ for any ˜x ∈ ˜E. +12 + +Then the partial dynamical system ( ˜E, Γ, {αγ}γ∈Γ, {Cγ}γ∈Γ) is (F r, 7ǫ)- +approximating for the restricted partial action Γ ↷ � +U∈Uǫ +i U and every U ∈ +Uǫ +i contains at least one element of ˜E (since the set E was ǫ/2-dense). +Moreover, if x ∈ U with x ∈ ˜E and αγ(x) ∈ V ∈ Uǫ +i , and y ∈ U with +y ∈ ˜E; then if αγ(y) ∈ � +U∈Uǫ +i U, d(αγ(x), αγ(y)) < 15ǫ, and so αγ(y) ∈ +V . +For γ ∈ Γ, restrict αγ to the points x ∈ ˜E such that there exists a +word fm · · · f1 in F r which is equal to γ in Γ and such that the composition +αfm ◦· · ·◦αf1(x) is defined (recall that since we are restricting a partial action +of Γ, such compositions must agree with αγ whenever they are defined). +Having been restricted this way, each partial bijection αγ for γ ∈ Γ de- +scends to a partial bijection βγ of Uǫ +i , and form the partial dynamical system +(Uǫ +i , Γ, {Dγ}γ∈Γ, {βγ}γ∈Γ). We have that βγ ◦ βλ ⊆ βγλ since βγ is induced +from αγ on a quotient space. Explicitly, this definition amounts to saying +βγ(U) = V iff there exists ˜x ∈ ˜E∩U and a sequence f1, . . . , fn ∈ F r such that +αfm◦· · ·◦αf1(x) is defined and in V . This does not depend on ˜x by the preced- +ing paragraph, and does not depend on f1, . . . , fm since αfm ◦ · · · ◦ αf1 = αγ +whenever the left hand side is defined. The descended self maps are still +partial bijections since non-injectivity would imply d(γ · x, γ · y) < 15ǫ for +x ∈ U ∈ Uǫ +i , y ∈ V ∈ Uǫ +i , and U ̸= V ; and this would contradict that the +original action is isometric. As noted in the third paragraph, these systems +satisfy (i). +Now we make a further restriction. For each γ ∈ Γ, replace Dγ with +the subset of itself consisting of those U ∈ Uǫ +i such that there exists a word +fm · · · f1 in F r which is equal to γ in Γ and such that for all m0 ∈ {1, . . . , m} +there exists V ∈ Uǫ +i such that γm0 · · · γ1·U∩V ̸= ∅ and let θγ be the restriction +of βγ to this new Dγ. Notice that this preserves (i) while now ensuring (ii) +by the choice of ǫ we made at the beginning. +Condition (ii) is needed since the partial dynamical systems constructed +here are not free, so it is important that they be asymptotically free in an +appropriate sense. This condition is also used with 3.4 to show these partial +dynamical systems locally resemble Γ ↷ Γ. Condition (i) ensures a (F r, M)- +cover for the system on Uǫ +i gives rise to a (F r, M)-cover for the restricted +action Γ ↷ � +U∈Uǫ +i U. +The sequences of actions constructed above are therefore similar to box +spaces of residually finite groups. Indeed, we are motivated by past results +relating dynamic asymptotic dimension to the asymptotic dimension of box +spaces [16]. +13 + +6 +Application to DAD +Lemma 6.1. Let Γ ↷ X and suppose there is a (d, F k, D)-cover for Γ ↷ X. +Then there is δ > 0 such that for Uǫ as in 5.6 with ǫ < δ, r = k, and +p = D +1, there is a (d, F r, D)-cover for (Uǫ +i , Γ, {Di,ǫ +γ }γ∈Γ, {θi,ǫ +γ }γ∈Γ) for each +i. +Proof. Fix r > 0 and let V = {Vm}d +m=0 be a (d, F r, D)-cover of X. This cover +has some Lebesgue number λ. For ǫ < λ/4 each element of Uǫ +i is entirely +contained in at least one Vm, so we can form a cover of Ui by d + 1 sets (by +making some choices). Moreover, for each x ∈ U ∈ Ui with U ⊂ Vm for some +m, any F r-chain x = x0, . . . , xn with cardinality more than D must have +some xj /∈ Vm. Replace each Vm by its λ/3-interior, which can be done for +all i while maintaining that the Vm cover X and that each U ∈ Ui is entirely +contained in some Vm). Now any F r-chain as before with cardinality more +than D must have some xj /∈ Nλ/3(Vm). Then by taking ǫ < +λ +3(D+1), any +F r-chain in Ui with more than D elements beginning at U ⊂ Vm will contain +some U′ entirely outside of Vm. +Theorem 6.2. Suppose Γ is finitely generated, that Γ ↷ X is free and iso- +metric, and that X has finite doubling dimension and no isolated points. +Suppose further that DAD(Γ ↷ X) ≤ d for some d. +Then DAD(Γ ↷ +X) ≤ asdimΓ. More precisely, if F ⊂ Γ, then for every r > 0 there is a +(asdimΓ, F r, D)-cover for Γ ↷ X and D depends only on K, r, CF(Γ), and +d. +Proof. As always, assume Γ has finite generating set F. If Γ is not amenable, +then DAD(Γ ↷ X) = ∞. This follows from [9, Corollary 8.27], which shows +that a groupoid with finite dynamic asymptotic dimension must be amenable. +Since amenable measure-preserving actions must be by amenable groups (and +isometric actions fix measures), this means the dimension is infinite when Γ +is non-amenable. If Γ is amenable, Γ ↷ X is residually finite by 4.2, and we +can therefore apply 5.6. +For each 0 ≤ i ≤ K, let n = 1, 2, . . ., and form a sequence (Uǫn +i , Γ, {Di,ǫn +γ +}γ∈Γ, {θi,ǫn +γ +}γ∈Γ)n +of partial dynamical system with the properties from 5.6 with r = n, P = n +and ǫn < 1/n. +Since Γ ↷ X is finite dimensional, 6.1 shows we have, for any r > 0, a +(d, F r, Mr)-cover for these systems for n sufficiently large and all i. Then by +14 + +5.6(ii), 3.4, 3.5. and 3.6, we have a (asdimΓ, F r, M′ +r)-cover for each of these +systems and M′ +r depends only on r, CF(Γ), and d. +Then for n ≥ r, by 5.6(i), this gives a (asdimΓ, F r, M′ +r)-cover for the +restricted action Γ ↷ � +U∈Ui U for each i. +Define fi(r) := M′ +r for all i and let ri and Ri be as in 3.2. Then, letting +n ≥ RK, we can apply 3.2 to get a (asdimΓ, F r, RK)-cover for Γ ↷ X, while +also ensuring n is sufficiently large that 5.6(ii) holds with P = RK. Since +RK depends only on K, r, CF r(Γ), and d, we are done. +Corollary 6.3. Suppose Γ ↷ X is free and isometric and that X has finite +doubling dimension and no isolated points (e.g. is a Riemannian manifld). +Then either DAD(Γ ↷ X) = ∞ or DAD(Γ ↷ X) = asdimΓ. +Proof. This follows from 6.2 and 2.7. +7 +Further discussion +It is now possible to give a complete description of the asymptotic dimension +of translation actions on compact Lie groups in terms of the amenability and +asymptotic dimension of the acting group. +Theorem 7.1. Let Γ ↷ G be a translation action by a finitely generated +subgroup of a compact Lie group. Then DAD(Γ ↷ G) = asdimΓ if Γ is +amenable and DAD(Γ ↷ G) = ∞ otherwise. +Proof. If Γ is not amenable, then DAD(Γ ↷ G) = ∞ by the same argument +as in 6.2, as this action is Haar measure-preserving. Assume therefore that +Γ is amenable. +Representation theory shows G ≤ Un is a subgroup of the n × n unitaries +for some n. Since DAD does not increase when passing to a closed, invariant +subset, we can assume G = Un. +Fix a finite subset F ⊂ Γ. The group Γ is a finitely generated, amenable +subgroup of a compact Lie group, and is therefore virtually abelian. This +follows from a combination of the Tits alternative, Lie’s theorem, and Engel’s +theorem (see, for instance, [18] for a proof). +In particular, this means Γ is virtually nilpotent, hence polynomial growth. +Moreover, Un ⊂ Mn(C) ∼= Cn2 has finite doubling dimension. +15 + +The sequence of partial dynamical systems constructed in 5.6 have uni- +form polynomial growth by 3.10 and so have asymptotic dimension uniformly +≤ d for some d. We can therefore apply 6.3. +We can similarly compute the DAD of many actions by Zd. +Theorem 7.2. Suppose Zd ↷ X is a free, isometric action and X has finite +doubling dimension (e.g. +is a Riemannian manifold). +Then DAD(Zd ↷ +X) = d. +Proof. Apply 6.2 and 3.10 as in the proof above. +Although previously suspected, it remains somewhat surprising that the +dynamic asymptotic dimension does not depend on the topological complex- +ity of X. Said another way, 6.3 shows that dynamic asymptotic dimension +can be equivalently defined using covers by Borel sets or even arbitrary sets +(at least for actions satisfying the hypotheses of 6.3). This is spiritually sim- +ilar to the results about Borel asymptotic dimension found in [5, Theorems +4.2 and 6.4]. +We can also obtain more precise bounds for other dynamical dimensions. +As mentioned earlier, the dynamic asymptotic dimension is related to other +dimension theories for group actions by [10, 5.1.4]. This and corollary 6.3 +show together that if dim(Γ ↷ X) is the amenability dimension (or the tower +dimension), then either dim(Γ ↷ X) = ∞ or dim(Γ ↷ X) ≤ (asdimΓ + +1)(dim X + 1) − 1 where dim X is the covering dimension of X. +References +[1] Massoud Amini, Kang Li, Damian Sawicki, and Ali Shakibazadeh, Dynamic asymp- +totic dimension for actions of virtually cyclic groups, Proc. Edinb. Math. Soc. 64 +(2021), no. 2, 364–372. +[2] Arthur Bartels, On proofs of the Farrell-Jones conjecture, Topology and geometric +group theory, 2016, pp. 1–31. MR3598160 +[3] Arthur Bartels, Wolfgang L¨uck, and Holger Reich, Equivariant covers for hyperbolic +groups, Geom. Topol. 12 (200610). +[4] Nikolay Brodskiy, Jerzy Dydak, Michael Levin, and Atish Mitra, A Hurewicz theorem +for the Assouad–Nagata dimension, J. London Math. Soc. 77 (2006), 741–756. +[5] Clinton Conley, Steve Jackson, Andrew Marks, Brandon Seward, and Robin +Tucker-Drob, Borel asymptotic dimension and hyperfinite equivalence relations, +arXiv:2009.06721[math.DS] (2020). +16 + +[6] Thiebout Delabie and Matthew Tointon, The asymptotic dimension of box spaces of +virtually nilpotent groups, Discrete Math. 341 (2018), 1036–1040. +[7] Ruy Exel, Partial dynamical systems, Fell bundles and applications, arXiv: Operator +Algebras (2015). +[8] E. Guentner, R. Willett, and Guoliang Yu, Dynamic asymptotic dimension and con- +trolled operator K-theory, arXiv: K-Theory and Homology (2016). +[9] Erik Guentner, Rufus Willett, and Guoliang Yu, Dynamic asymptotic dimension: +relation to dynamics, topology, coarse geometry, and C∗-algebras, Math. Ann. 367 +(201510). +[10] David Kerr, Dimension, comparison, and almost finiteness, J. Eur. Math. Soc. 22 +(2020), no. 11, 3697–3745. +[11] David Kerr and Piotr Nowak, Residually finite actions and crossed products, Ergod. +Theory Dyn. Syst. 32 (2011). +[12] Jouni Luukkainen and Eero Saksman, Every complete doubling metric space carries +a doubling measure, Proceedings of the American Mathematical Society 126 (1998), +no. 2, 531–534. +[13] Samantha Pilgrim, Asymptotic dimension of box spaces and odometers of elementary +amenable groups, 2022. +[14] S.J. Pilgrim, Isometric actions and finite approximations, Ergod. Theory Dyn. Syst. +(2022), 1–7. +[15] Damian Sawicki and Dawid Kielak, Warped cones, (non-)rigidity, and piecewise prop- +erties, Proc. London Math. Soc. 118 (201809). +[16] G´abor Szab´o, Jianchao Wu, and Joachim Zacharias, Rokhlin dimension for actions +of residually finite groups, Ergodic Theory and Dynamical Systems 39 (2019), no. 8, +2248–2304. +[17] Wilhelm Winter and Joachim Zacharias, The nuclear dimension of C∗-algebras, Ad- +vances in Mathematics 224 (2010), 461–498, available at 0903.4914. +[18] Qiaochu Yuan, Response to stack exchange question, 2020. +17 + diff --git a/WdFOT4oBgHgl3EQf7jS_/content/tmp_files/load_file.txt b/WdFOT4oBgHgl3EQf7jS_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..549aafaf0221234aae55ab505b1fb526bc03dcdd --- /dev/null +++ b/WdFOT4oBgHgl3EQf7jS_/content/tmp_files/load_file.txt @@ -0,0 +1,528 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf,len=527 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='12963v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='DS] 30 Jan 2023 On Sharp Bounds for the Dynamic Asymptotic Dimension Samantha Pilgrim January 31, 2023 Abstract: We prove the dynamic asymptotic dimension of a free isometric action on a space of finite doubling dimension is either infinite or equal to the asymptotic dimension of the acting group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' and give a full description of the dynamic asymptotic dimension of translation actions on compact Lie groups in terms of the amenability and asymptotic dimension of the acting group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 1 Introduction The dynamic asymptotic dimension (DAD) of a group action Γ ↷ X was first introduced by Guentner, Willett, and Yu [9], and was shown to relate to conditions used by Bartels, L¨uck, and Reich in work on the Farrell Jones conjecture on manifold topology [2] [3], as well as to the nuclear dimension of the crossed product [17] and calculations of its K-theory [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The DAD is also related to other dynamical dimension theories [10] known to take into account both the asymptotic dimension of Γ and the topological dimension of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' However, it has been suspected since its introduction that the dynamic asymptotic dimension may actually coincide with the asymptotic dimension of Γ whenever it is finite (it is possible to have DAD(Γ ↷ X) = ∞ while asdimΓ < ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The case where Γ is virtually cyclic is proved in [1], providing an important test-case for this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' More generally, the work of [15] gives an upper bound of asdimΓ + dim X for isometric actions by finitely generated groups on manifolds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' and [5] shows that many actions on Cantor sets have DAD = asdimΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 1 This note documents progress on a question [15, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6] originally posed by Willett by giving the first sharp bounds for the dynamic asymptotic di- mension of free isometric actions on spaces of dimension greater than (or equal to) zero, modulo the fairly mild assumption that X has finite dou- bling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' In this case we show the dimension is always either ∞ or asdimΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We accomplish this by introducing a new way of formulating the doubling dimension of a space, which we call Cantor decomposability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This property, together with the theory of residually finite group actions, allows many isometric actions to be modeled by several sequences of partial dynam- ical systems on discrete spaces which asymptotically resemble the structure of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This property is reminiscent of box spaces of residually finite groups, and our investigations are partly inspired by past work on the asymptotic dimension of such spaces [6], as well as by the relation between box spaces and odometers [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' In line with [6], which describes among other things the asymptotic dimension of box spaces of residually finite groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' we use the theory of residually finite group actions introduced in [11], together with Cantor decomposability, to reduce the problem to geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We apply the main result in order to calculate the dimension of translation actions on compact Lie groups in terms of the amenability and asymptotic dimension of the acting group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We also compute the dimension of many isometric actions by Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 2 DAD of partial dynamical systems We begin by establishing some notation and terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We will assume throughout that the group Γ is finitely generated with finite generating set F, that F = F −1 is symmetric, and that e ∈ F (where e denotes the identity in Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The following definition comes from [7, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' See [7, Part I] for a more complete treatment of partial actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' A topological partial action of Γ on a topological space X is a pair ({Dγ}γ∈Γ, {θγ}γ∈Γ) consisting of a collection {Dγ}γ∈Γ of subsets of X, and a collection {θγ}γ∈Γ of homeomorphisms, θγ : Dγ → Dγ−1 such that (i) De = X, and θe is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' (ii) θγ ◦ θλ ⊆ θγλ, for all γ and λ in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 2 Here, the composition θγ ◦ θλ denotes the map whose domain is the set of all x ∈ X for which θγ(θλ(x)) makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' In other words, this is the set θ−1 λ (Dγ) = θ−1 λ (Dλ−1 ∩ Dγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The symbol “⊆” means that the function on the right-hand-side is an extension of the function on the left-hand-side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Notice also that θγ−1 = θ−1 γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' A partial dynamical system is a quadruple (X, Γ, {Dγ}γ∈F, {θγ}γ∈Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' An action Γ ↷ X is then a partial dynamical system with Dγ = X for all γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' A partial dynamical system is free if θγ(x) = θλ(x) iff γ = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We may sometimes still write γ · x for θγ(x) when it is not ambiguous to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Some partial dynamical systems we will deal with later are restrictions of actions or partial actions, in which case we will write Γ ↷ A for a subset A ⊂ X to denote the partial dynamical system (A, Γ, {A ∩ h−1 γ (A)}γ∈F, {hγ}γ∈F) where hγ comes from Γ ↷ X or some partial action on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let (X, Γ, {Dγ}γ∈F, {θγ}γ∈Γ) and (Y, Γ, {Eγ}γ∈F, {ργ}γ∈Γ) be topological partial dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' By a conjugacy, we mean a home- omorphism f : X → Y such that f(θγ(x)) = ργ(f(x)) whenever θγ(x) is defined (so ργ(f(x)) must be defined in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) be a partial dynamical system and S ⊂ Γ a finite subset with S = S−1 and e ∈ S, and A ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' An S- chain in A is a finite sequence x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' , xn of points in A such that for all 0 ≤ i ≤ n − 1, θγ(xi) = xi+1 for some γ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Two points in A are in the same S-component if they are connected by an S-chain in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We say a cover V = {Vj}d j=0 is a (d, S, M)-cover for Γ ↷ X (or just X if unambiguous) if all S-components of each Vi have cardinality at most M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The dynamic asymptotic dimension of a free partial dy- namical system is the smallest integer d such that for every finite subset S ⊂ Γ with S = S−1 and e ∈ S, there is M > 0 and a (d, S, M)-cover for (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If no such d exists, the dimension is defined to be ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' As we are assuming Γ is finitely generated with finite generating set F, we can assume S has the form F r for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' A d-dimensional control function for (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) (with Γ finitely generated) is a function D : N → N such that for every integer r > 0, there is a (d, F r, D(r))-cover for (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We will use the notation described in the last two definitions (which relates to the definition of DAD for free actions) even for actions which are not free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 3 Notice that DAD appears a priori to be sensitive to the topology of X as the cover U is required to be open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If (Xn, Γ, {Dn γ}γ∈Γ, {θn γ}γ∈Γ) is a sequence of partial dy- namical systems, we say DAD(Γ ↷ Xn)n ≤ d uniformly if there is a single d dimensional control function for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose CF(Γ) is a Cayley graph of a finitely generated group Γ with F being a generating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' By this we mean the set Γ with the metric given by d(x, y) = |yx−1|F where |w|S denotes the minimal length of a word in S which equals w as an element of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Notice that this is invariant under right multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If r > 0, an r-chain in A ⊂ CF(Γ) is a finite sequence x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' , xn of points in A such that d(xi, xi+1) ≤ r for 0 ≤ i ≤ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Two points g, h ∈ A ⊂ CF(Γ) are in the same r-component of A if they are connected by an r-chain in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We say a cover U = {U0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' , Ud} of CF(Γ) such that the r-components of each Ui have diameter at most Mr is a (d, r, Mr)-cover for CF(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The asymptotic dimension of CF(Γ), which we write as asdimCF(Γ), is the least integer d such that for all r > 0 there is a (d, r, Mr)-cover for CF(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' As asdimCF(Γ) is independent of F, we may simply write asdimΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' When referring to the action of Γ on itself, we mean the action by left multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose Γ ↷ X is a free action by a finitely generated group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then DAD(Γ ↷ X) ≥ asdimΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' In fact, this inequality holds even if one modifies the definition of DAD to allow covers by Borel sets, or even arbitrary sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let F be a finite generating set for Γ as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose DAD(Γ ↷ X) ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then there is a (d, F r, Mr)-cover V for Γ ↷ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let O(x) be the orbit of some x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Identify O(x) with CF(Γ) by γ · x �→ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then V gives a cover V′ of CF(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Moreover, two points y, y′ ∈ O(x) are connected by an F r chain iff their corresponding elements in CF(Γ) are connected by an r-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Therefore, the r-components of each V ∈ V′ are Mr-bounded, so V′ is a (d, r, Mr)-cover for CF(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Saying anything about the reverse inequality will require considerably more work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 4 3 Union theorem Union theorems are common in dimension theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The intuition is that a finite union of objects with dimension ≤ d should still have dimension ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Our method of showing this for DAD comes from [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='5 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This section also includes an application of the union theorem which will be used later, as well as some other technical lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose Γ ↷ X is free and A, B ⊂ X such that the F r- components of B have cardinality at most R and the F r(R+2)-components of A have cardinality at most D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then the F r-components of A ∪ B have cardinality at most #B(D−1)(R+1)+1 e (CF r(Γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Consider an F r-chain in A ∪ B: x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' , xn and assume this chain has no repeated points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose xi and xj are two consecutive elements of A, so xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' , xj−1 are all points in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then these points form an F r-chain in B which therefore has cardinality at most R, and therefore length at most R (since there are no repeated points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' But then xi+1 and xj−1 are connected by something in F rR, and so xi and xj are in the same F r(R+2)-component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' By similar reasoning, all elements of A in the original chain are in the same F r(R+2)-component of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The subchain x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' , xn of points contained in A then has cardinality at most D, hence length at most D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Since each chain between each element of A has length at most R, the length of the original chain is at most (D − 1)(R + 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The F r-components of A ∪ B therefore have cardinality at most #B(D−1)(R+1)+1 e (CF r(Γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' (Finite Union Theorem) Suppose Γ ↷ X and {Ai}K i=0 are open subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose fi is a d-dimensional control function for Γ ↷ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Define ri and Ri inductively starting with r0 = r, R0 = f0(r) and then for 1 ≤ i ≤ K defining ri = r(Ri−1 + 2), and Ri = #B(Ri−1−1)(fi(ri)+1)+1 e (CF r(Γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then there is a (d, F r, RK)-cover for Γ ↷ ∪iAi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Notice that RK depends only on the fi and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Take a (d, F ri, fi(ri))-cover V(i) = {V (i) j }d j=0 of Ai for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Form a cover V of ∪K i=0Ai by putting Vj = ∪K i=0V (i) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then use the previous lemma and induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' At the s-th step of the induction (a union of s + 1 sets), we have r = r and rR + 2r = rRs−1 + 2r = rs (s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' , K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 5 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose Γ has finite generating set F and (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) is a partial dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' For P ∈ N, we define BP x (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) := {y ∈ X | y = θγ(x) for some γ ∈ BP e (CF(Γ))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We may also just write BP x or BP x (X) when unambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) is a partial dynamical system such that θγ(x) = θγ′(x) for γ, γ′ ∈ BP e (CF(Γ)) if and only if γ = γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then this system restricted to any BP x is conjugate to a restricted action Γ ↷ A ⊂ BP e (CF(Γ)) by left multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Consider the subset BP x = {yγ | yγ = θγ(x) with γ ∈ BP e (CF(Γ))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Identify BP x with a subset A of BP e (CF(Γ)) by the map f(yγ) := γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then if yδ, yδ′ ∈ BP x are such that θγ(yδ) = yδ′, we have yδ′ = θγ(yδ) = θγ ◦ θδ(x) = θγδ(x) = yγδ implying γδ = δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Hence, f(θγ(yδ)) = f(yδ′) = δ′ = γδ = γ · δ, so f is equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose Γ is a finitely generated group with generating set F and CF(Γ) is a Cayley graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) be a free par- tial dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then a (d, r, M)-cover for CF(Γ) gives rise to a (d, F r, #BM e (CF(Γ))-cover for (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' For each orbit of (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ), identify the points in X with a subset of CF(Γ) by choosing any x0 to correspond to the identity and then identifying θγ(x) with γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then notice that two points are connected by an F r chain iff their corresponding elements in CF(Γ) are connected by an r-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Next, we want to establish some lemmas about the DAD of sequences of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The content of the following lemma is essentially that of [6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='1] and mirrors the theory of box spaces of residually finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' In the proof of the main theorem, this is part of what allows us to infer a tight bound on the dimension whenever the dimension is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose Γ is finitely generated with generating set F and (Gn, Γ, {Dn γ}γ∈Γ, {θn γ}γ∈Γ)n is a sequence of partial dynamical systems with Gn finite such that for all r > 0 there is N > 0 such that there is a (d, F r, Mr)-cover for (Gn, Γ, {Dn γ}γ∈Γ, {θn γ}γ∈Γ) for all n ≥ N (Mr does not depend on n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose also that for all P > 0 there is N′ such that for 6 n ≥ N′ and any r′ > 0, the system (Gn, Γ, {Dn γ}γ∈Γ, {θn γ}γ∈Γ)n restricted to any BP x (x ∈ Gn) has a (d′, F r′, M′ r′)-cover with M′ depending only on r′ and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then for all r > 0, there is N′′ and a (d′, F r, M′′ r )-cover for (Gn, Γ, {Dn γ}γ∈Γ, {θn γ}γ∈Γ)n for all n ≥ N′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Moreover, M′′ r depends only on d and the function r �→ M′ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Fix r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' For each 0 ≤ i ≤ d, define fi(r) = M′ r for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Define ri and Ri inductively as in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' By hypothesis, there is N such that for n ≥ N, there exists a (d, F RK, MRK)-cover V = {Vi}d i=0 for (Gn, Γ, {Dn γ}γ∈Γ, {θn γ}γ∈Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then find N′ ≥ N so that (Gn, Γ, {Dn γ}γ∈Γ, {θn γ}γ∈Γ)n restricted to any B MRK +1 x has a (d′, F ri, fi(ri))-cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Find such a cover Ui = {Ui k}d′ k=0 for each F RK-component of Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' For each 0 ≤ i ≤ d, form a (d′, F ri, fi(ri))-cover U = {Uk}d′ k=0 of Vi by putting Uk = � j Uj k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' It follows from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2 that there is a (d′, F r, gd(r))-cover for each (Gn, Γ, {Dn γ}γ∈Γ, {θn γ}γ∈Γ) with n ≥ N′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If Γ is a group and CF(Γ) is a Cayley graph (F is a finite generating set), we say Γ has polynomial growth of degree at most d if there are d > 0 and C > 0 such that if G(r) = #Br e(CF(Γ)), then Gn(r) ≤ Crd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' A sequence of partial dynamical systems (Gn, Γ, {Dn γ}γ∈Γ, {θn γ}γ∈Γ)n has poly- nomial growth of degree at most d if there are d > 0 and C > 0 such that if Gn(r) = supx∈Gn #{y : y = γ · x for γ ∈ Br e(CF(Γ))}, then Gn(r) ≤ Crd for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We will make use of a standard ‘greedy algorithm’ in this section and again later, so we formalize such an algorithm with a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose S is a finite set and ∼ is a reflexive, symmetric rela- tion on S such that #{t ∈ S | t ̸= s and s ∼ t} ≤ D for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then there is a cover C = {Ci}D i=0 such that s ≁ t for all s ̸= t in Ui for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Assign elements to the sets in C in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Start by picking any element s0 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then there are at most D other elements of S which are related to s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Assign s0 ∈ C0 and those other elements to C1 through CD so that at most one belongs to each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then consider one of the elements just discussed other than s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This element is related to at most D other elements of S including s0, so we can repeat the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Continue in this way until all elements of S are assigned to some set in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 7 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose Γ is finitely generated and (Gn, Γ, {Dn γ}γ∈Γ, {θn γ}γ∈Γ)n is a sequence of partial dynamical systems on finite sets with polynomial growth of degree at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then DAD(Gn, Γ, {Dn γ}γ∈Γ, {θn γ}γ∈Γ)n ≤ K = 4d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The proof is essentially the same as that of [6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Fix F ⊂ Γ a finite generating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' For a partial dynamical system (G, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) on a discrete space, the following condition implies its dynamic asymptotic dimension is at most K − 1: for all R > 0 there is M > 0 and a cover V of G by sets of cardinality at most some M such that if x ∈ V ∈ V, there are at most K − 1 other elements V ′ ∈ V such that γ · x ∈ V ′ for some γ ∈ F R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' To see this, notice that one can use a greedy algorithm as in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='8 to sort the elements of V into K subcollections such that taking the union over each subcollection yields a (d, F r, M)-cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' By assumption, we have that #BR x (Gn) ≤ CRd for all R > 0, x ∈ Gn for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let K = 4d + 1, R > 1 and S0 = 4m+1R where m is such that (K/4d)m ≥ CRd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We claim that for every n there exists Rn such that R ≤ Rn ≤ S0 4 and |B4Rn x (Gn)| ≤ K|BRn x (Gn)| (where x ∈ Gn is arbitrary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If this were not the case then Ki|BR x (Gn)| < |B4iR x (Gn)| ≤ C4idRd for every i with 4iR ≤ S0, so setting i = m would give C4mdRd > Km|BR x (Gn)| ≥ C4mdRd|BR x (Gn)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' But then we would have 1 > |BR x (Gn)|, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Now, take Xn maximal in Gn such that BRn x (Gn) and BRn y (Gn) are disjoint for all x, y ∈ Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Consider the collection V = {B2Rn x (Gn) | x ∈ Xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If z ∈ Gn, maximality of Xn implies there is x ∈ Xm such that BRn z (Gn)∩BRn x (Gn) and so z ∈ B2Rn x (Gn), and so V covers Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Finally, we check that if x ∈ Gn and x ∈ V ∈ V, there are at most K other elements V ′ ∈ V such that γ · x ∈ V ′ for some γ ∈ F Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let z ∈ Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' For every B2Rn x (Gn) ∈ V which has an element at a distance at most Rn to z, we have that x ∈ B2Rn+Rn z (Gn) ⊂ B3Rn z (Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Now consider B3Rn z (Gn) ∩ Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Since BRn x (Gn) and BRn y (Gn) are disjoint for any x and y in Xm, we have that #(B3Rn z (Gn) ∩ Xn) ≤ #B4Rn z (Gn) #BRn z (Gn) ≤ 4d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If V ′ ∈ V is such that there is γ ∈ F Rn with γ · V ∩ V ′ ̸= ∅, then there is z ∈ V ′ with z ∈ B3Rn z (Gn) ∩ Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Therefore, there are at most 4d + 1 such V ′ ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose Γ has polynomial growth and (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) is a partial dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) has polynomial 8 growth bounded by that of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' That is, a d > 0 and C > 0 that work for Γ in the sense of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='7 also work for the whole sequence of partial actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 4 Residual finiteness We will also make use of a property which allows group actions to be ap- proximated by actions on finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The definition below was originally introduced by Kerr and Nowak in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' An action Γ ↷ X on a metric space by homeomorphisms is residually finite if for all F ⊂ Γ finite and ǫ > 0 there is a finite set E, an action Γ ↷ E, and a map ζ : E → X such that ζ(E) is ǫ-dense in X (meaning every ǫ-ball in X intersects E nontrivially) and that d(ζ(γ · e), γ ·ζ(e)) < ǫ for all γ ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Such an action is called a (F, ǫ)-approximating action for Γ ↷ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If X has no isolated points, a perturbation argument shows we can assume ζ to be an inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If (X, Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) is a partial dynamical system, E is a finite set and ζ : E → X is a map, we similarly say a partial dynamical system (E, Γ, {DE γ }γ∈Γ, {θE γ }γ∈Γ) is (F, ǫ)-approximating if E is ǫ-dense in X and if, whenever θγ(ζ(x)) is defined for γ ∈ F and x ∈ E, we have that θE γ (x) is defined and d(θγ(ζ(x)), ζ(θE γ (x))) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' As with the theory of residually finite groups, there is a close relationship between isometry and residual finiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We can make use of residual finite- ness of actions later in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6 without loss of generality in the main theorem thanks to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2 ([14, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Every faithful, isometric action by a finitely generated, amenable group is residually finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The work of [6] shows that the asymptotic dimension of box spaces of residually finite groups is closely related to the asymptotic dimension of the group itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' As the dynamic asymptotic dimension behaves somewhat like the asymptotic dimension of a box space (both are infinite in the non- amenable case for example), we expect that residual finiteness of actions may be helpful to relating their DAD to the asymptotic dimension of the acting group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 9 5 Cantor-like decompositions A key ingredient in the proof of the main theorem is a technical condition on X which allows an isometric, residually finite action to be modeled by several sequences of partial dynamical systems on finite spaces, thereby allowing us to apply the work of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We will see that many spaces satisfy this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Isometric actions on Cantor sets can be described as inverse limits of finite actions [14, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='3], which allows their DAD to be studied more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We therefore would like to cover X by Cantor sets and use the union theorem from section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Instead of doing exactly this, we cover X by sets which (as ǫ → 0) resemble the sets one might intersect in constructing a Cantor set-like object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let X be a compact metric space and let ǫ, δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We say X is (K, ǫ, δ)-Cantor decomposable if there is an open cover U = U0 ⊔ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' ⊔ UK of X with diam(U) < ǫ for all U ∈ Ui and all i, and d(U, V ) > δ for all distinct U, V ∈ Ui and all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We call such a cover a (K, ǫ, δ)-decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This condition comes from trying to force an argument like [14, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='3] to work by dividing X into different subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Notice that the definition is somewhat reminiscent of asymptotic dimension, but small-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We there- fore expect K to be related to some notion of metric dimension for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' A metric space (X, d) is M-doubling if for all r > 0 and x ∈ X, the closed ball B2r x can be covered by at most M closed balls of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Further define dimd(X) := log2 M, which we call the doubling dimension of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The origins of this concept are not precisely clear and it may have been rediscovered independently multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' It can be reasonably attributed to Assouad, though his is a different, equivalent definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' See the introduction of [12] for additional exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The space R is 2-doubling and therefore has doubling dimen- sion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Working out M exactly for higher-dimension Euclidean spaces is difficult, but it is more straightforward to see that Rd is M-doubling for some M ≤ 4 · 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This follows since a d-cube with side length l can be covered by 4 d-balls of diameter l, and it takes 2d d-cubes with side length l/2 to cover one with side length l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' More generally, a Riemannian manifold admits an isometric embedding into some Euclidean space by the Nash embedding theorem, and therefore has finite doubling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 10 As alluded to earlier, we will now see that Cantor decomposability relates quantitatively to the doubling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose (X, d) is compact and M-doubling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then (X, d) has a (M⌈2+log2(k+3)⌉, ǫ, kǫ)-decomposition for all ǫ > 0 and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If X has a (K, ǫ, 2ǫ)-decomposition for all ǫ > 0, then X is K-doubling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' In particular, this implies X has finite doubling dimension iff for all k there exists K such that X is (K, ǫ, kǫ)-Cantor decomposable for all ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' For the first claim, fix ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let C be a cover of X by (closed) ǫ/2- balls using as few balls as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If x ∈ X and U = {B ∈ C|B∩B(k+2)ǫ x ̸= ∅} then U is a cover of � B∈U B using as few ǫ/2-balls as possible: if � B∈U B had a cover U′ by fewer balls, then (C \\ U) ∪ U′ would be a cover of X by fewer balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Therefore, since � B∈U B ⊂ B(k+3)ǫ x , #U ≤ M⌈2+log2(k+3)⌉ (where ⌈·⌉ denotes the ceiling function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' It is therefore possible to color the balls using #U ≤ M⌈2+log2(k+3)⌉ colors such that any two balls of the same color are distance at least (k + 1)ǫ apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This can be shown via a standard ‘greedy algorithm’ (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Finally, we can replace each ball with an open neighborhood of itself while keeping the diameter of each < ǫ and keeping balls in the same collection > kǫ apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' For the second claim, suppose X is (K, ǫ, 2ǫ)-Cantor decomposable for all ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Given a ball B ⊂ X of radius 2ǫ, find a cover of X witnessing its (K, ǫ, 2ǫ)-Cantor decomposability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then the subcollection of sets in this cover which intersect B is a cover of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Moreover, any set intersecting B is within 2ǫ of any other such set, so this cover of B contains at most K sets of diameter < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The existence of (K, ǫ, δ)-decompositions for δ large compared to ǫ really does require finite doubling dimension rather than finiteness of other types of dimension (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Lebesgue covering dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We give an example to show this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' There exists a Cantor set with infinite doubling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let C be the space of sequences (an)∞ n=1 where an ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Give C the metric d((an), (bn)) = � n |an−bn| n2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then a ball of radius 1 2k centered at (an) consists of sequences which agree with (an) up to at least index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Such a ball therefore requires k + 1 balls of radius 1 2k+1 to cover it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 11 We conclude this section by showing how Cantor decompositions can be used to model certain isometric actions by partial dynamical systems acting on decompositions of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose Γ ↷ X (denoted by γ · x) is isometric and residually finite, that Γ has finite generating set F, and that X has no isolated points and finite doubling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let r > 0 and P > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then there is ǫ0 such that if ǫ < ǫ0 and Uǫ = Uǫ 0 ⊔· · ·⊔Uǫ K is a (K, ǫ, 15ǫ)-decomposition of X such that, for all i and U ∈ Ui, U contains a ball of radius ǫ/2 (this can be done by first taking a (K, ǫ/2, 16ǫ)-decomposition for each ǫ and then replacing each set by its ǫ/2-neighborhood);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' then there is a partial dynamical system (Uǫ i , Γ, {Dγ}γ∈Γ, {θγ}γ∈Γ) such that (i) If γ ∈ F r and γ · U ∩ V ̸= ∅ for U, V ∈ Uǫ i , then θγ(U) = V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' (ii) If w = fj · · · f1 and w′ = f ′ p · · · f ′ 1 are words in F r with length ≤ P + 1 which are not equal as elements of Γ, then θfj ◦ · · · ◦ θf1(x) ̸= θf′p ◦ · · · ◦ θf′ 1(x) for all x where both sides are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Start by letting ǫ be sufficiently small that translates (according to the action Γ ↷ X) of subsets of diameter < ǫ by elements of F (P +1)r are disjoint (using that Γ ↷ X is free).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This will later be used to ensure condition (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let Γ ↷ E ⊂ X be a (F r, ǫ/2)-approximating action for Γ ↷ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Use δγ(x) to denote the action Γ ↷ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Now, for each i, we can restrict the action Γ ↷ E to a partial action Γ ↷ E ∩ N2ǫ(� U∈Uǫ i U) (where N2ǫ(·) denotes the open 2ǫ-neighborhood).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Each x ∈ E ∩ N2ǫ(� U∈Uǫ i U) can be associated to an ˜x ∈ � U∈Uǫ i U in such a way that ˜x = x if x ∈ � U∈Uǫ i U, d(x, ˜x) < 3ǫ, and ˜x ̸= ˜y for x ̸= y (since X has no isolated points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let ˜E = {˜x | x ∈ E ∩ N2ǫ(� U∈Uǫ i U)} and conjugate by the correspondence x �→ ˜x to obtain a partial dynamical system ( ˜E, Γ, {αγ}γ∈Γ, {Cγ}γ∈Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then, if there is γ ∈ F r and γ · U ∩ V ̸= ∅ for some V ∈ Uǫ i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' then (as E is ǫ/2-dense and as U contains an ǫ/2-ball) there is x ∈ U with x ∈ E so that ˜x = x and d(δγ(x), V ) < 2ǫ, and so � δγ(x) = αγ(x) ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This will guarantee condition (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Also notice that d(γ · ˜x, αγ(˜x)) < 7ǫ for any ˜x ∈ ˜E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 12 Then the partial dynamical system ( ˜E, Γ, {αγ}γ∈Γ, {Cγ}γ∈Γ) is (F r, 7ǫ)- approximating for the restricted partial action Γ ↷ � U∈Uǫ i U and every U ∈ Uǫ i contains at least one element of ˜E (since the set E was ǫ/2-dense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Moreover, if x ∈ U with x ∈ ˜E and αγ(x) ∈ V ∈ Uǫ i , and y ∈ U with y ∈ ˜E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' then if αγ(y) ∈ � U∈Uǫ i U, d(αγ(x), αγ(y)) < 15ǫ, and so αγ(y) ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' For γ ∈ Γ, restrict αγ to the points x ∈ ˜E such that there exists a word fm · · · f1 in F r which is equal to γ in Γ and such that the composition αfm ◦· · ·◦αf1(x) is defined (recall that since we are restricting a partial action of Γ, such compositions must agree with αγ whenever they are defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Having been restricted this way, each partial bijection αγ for γ ∈ Γ de- scends to a partial bijection βγ of Uǫ i , and form the partial dynamical system (Uǫ i , Γ, {Dγ}γ∈Γ, {βγ}γ∈Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We have that βγ ◦ βλ ⊆ βγλ since βγ is induced from αγ on a quotient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Explicitly, this definition amounts to saying βγ(U) = V iff there exists ˜x ∈ ˜E∩U and a sequence f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' , fn ∈ F r such that αfm◦· · ·◦αf1(x) is defined and in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This does not depend on ˜x by the preced- ing paragraph, and does not depend on f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' , fm since αfm ◦ · · · ◦ αf1 = αγ whenever the left hand side is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The descended self maps are still partial bijections since non-injectivity would imply d(γ · x, γ · y) < 15ǫ for x ∈ U ∈ Uǫ i , y ∈ V ∈ Uǫ i , and U ̸= V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' and this would contradict that the original action is isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' As noted in the third paragraph, these systems satisfy (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Now we make a further restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' For each γ ∈ Γ, replace Dγ with the subset of itself consisting of those U ∈ Uǫ i such that there exists a word fm · · · f1 in F r which is equal to γ in Γ and such that for all m0 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' , m} there exists V ∈ Uǫ i such that γm0 · · · γ1·U∩V ̸= ∅ and let θγ be the restriction of βγ to this new Dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Notice that this preserves (i) while now ensuring (ii) by the choice of ǫ we made at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Condition (ii) is needed since the partial dynamical systems constructed here are not free, so it is important that they be asymptotically free in an appropriate sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This condition is also used with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='4 to show these partial dynamical systems locally resemble Γ ↷ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Condition (i) ensures a (F r, M)- cover for the system on Uǫ i gives rise to a (F r, M)-cover for the restricted action Γ ↷ � U∈Uǫ i U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The sequences of actions constructed above are therefore similar to box spaces of residually finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Indeed, we are motivated by past results relating dynamic asymptotic dimension to the asymptotic dimension of box spaces [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 13 6 Application to DAD Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let Γ ↷ X and suppose there is a (d, F k, D)-cover for Γ ↷ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then there is δ > 0 such that for Uǫ as in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6 with ǫ < δ, r = k, and p = D +1, there is a (d, F r, D)-cover for (Uǫ i , Γ, {Di,ǫ γ }γ∈Γ, {θi,ǫ γ }γ∈Γ) for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Fix r > 0 and let V = {Vm}d m=0 be a (d, F r, D)-cover of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This cover has some Lebesgue number λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' For ǫ < λ/4 each element of Uǫ i is entirely contained in at least one Vm, so we can form a cover of Ui by d + 1 sets (by making some choices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Moreover, for each x ∈ U ∈ Ui with U ⊂ Vm for some m, any F r-chain x = x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' , xn with cardinality more than D must have some xj /∈ Vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Replace each Vm by its λ/3-interior, which can be done for all i while maintaining that the Vm cover X and that each U ∈ Ui is entirely contained in some Vm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Now any F r-chain as before with cardinality more than D must have some xj /∈ Nλ/3(Vm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then by taking ǫ < λ 3(D+1), any F r-chain in Ui with more than D elements beginning at U ⊂ Vm will contain some U′ entirely outside of Vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose Γ is finitely generated, that Γ ↷ X is free and iso- metric, and that X has finite doubling dimension and no isolated points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose further that DAD(Γ ↷ X) ≤ d for some d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then DAD(Γ ↷ X) ≤ asdimΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' More precisely, if F ⊂ Γ, then for every r > 0 there is a (asdimΓ, F r, D)-cover for Γ ↷ X and D depends only on K, r, CF(Γ), and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' As always, assume Γ has finite generating set F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If Γ is not amenable, then DAD(Γ ↷ X) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This follows from [9, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='27], which shows that a groupoid with finite dynamic asymptotic dimension must be amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Since amenable measure-preserving actions must be by amenable groups (and isometric actions fix measures), this means the dimension is infinite when Γ is non-amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If Γ is amenable, Γ ↷ X is residually finite by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2, and we can therefore apply 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' For each 0 ≤ i ≤ K, let n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=', and form a sequence (Uǫn i , Γ, {Di,ǫn γ }γ∈Γ, {θi,ǫn γ }γ∈Γ)n of partial dynamical system with the properties from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6 with r = n, P = n and ǫn < 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Since Γ ↷ X is finite dimensional, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='1 shows we have, for any r > 0, a (d, F r, Mr)-cover for these systems for n sufficiently large and all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then by 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6(ii), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6, we have a (asdimΓ, F r, M′ r)-cover for each of these systems and M′ r depends only on r, CF(Γ), and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then for n ≥ r, by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6(i), this gives a (asdimΓ, F r, M′ r)-cover for the restricted action Γ ↷ � U∈Ui U for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Define fi(r) := M′ r for all i and let ri and Ri be as in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then, letting n ≥ RK, we can apply 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2 to get a (asdimΓ, F r, RK)-cover for Γ ↷ X, while also ensuring n is sufficiently large that 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6(ii) holds with P = RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Since RK depends only on K, r, CF r(Γ), and d, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose Γ ↷ X is free and isometric and that X has finite doubling dimension and no isolated points (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' is a Riemannian manifld).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then either DAD(Γ ↷ X) = ∞ or DAD(Γ ↷ X) = asdimΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This follows from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 7 Further discussion It is now possible to give a complete description of the asymptotic dimension of translation actions on compact Lie groups in terms of the amenability and asymptotic dimension of the acting group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Let Γ ↷ G be a translation action by a finitely generated subgroup of a compact Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then DAD(Γ ↷ G) = asdimΓ if Γ is amenable and DAD(Γ ↷ G) = ∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' If Γ is not amenable, then DAD(Γ ↷ G) = ∞ by the same argument as in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2, as this action is Haar measure-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Assume therefore that Γ is amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Representation theory shows G ≤ Un is a subgroup of the n × n unitaries for some n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Since DAD does not increase when passing to a closed, invariant subset, we can assume G = Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Fix a finite subset F ⊂ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' The group Γ is a finitely generated, amenable subgroup of a compact Lie group, and is therefore virtually abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This follows from a combination of the Tits alternative, Lie’s theorem, and Engel’s theorem (see, for instance, [18] for a proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' In particular, this means Γ is virtually nilpotent, hence polynomial growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Moreover, Un ⊂ Mn(C) ∼= Cn2 has finite doubling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 15 The sequence of partial dynamical systems constructed in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='6 have uni- form polynomial growth by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='10 and so have asymptotic dimension uniformly ≤ d for some d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We can therefore apply 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We can similarly compute the DAD of many actions by Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Suppose Zd ↷ X is a free, isometric action and X has finite doubling dimension (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' is a Riemannian manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Then DAD(Zd ↷ X) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Apply 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='10 as in the proof above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Although previously suspected, it remains somewhat surprising that the dynamic asymptotic dimension does not depend on the topological complex- ity of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' Said another way, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='3 shows that dynamic asymptotic dimension can be equivalently defined using covers by Borel sets or even arbitrary sets (at least for actions satisfying the hypotheses of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This is spiritually sim- ilar to the results about Borel asymptotic dimension found in [5, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' We can also obtain more precise bounds for other dynamical dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' As mentioned earlier, the dynamic asymptotic dimension is related to other dimension theories for group actions by [10, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' This and corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content='3 show together that if dim(Γ ↷ X) is the amenability dimension (or the tower dimension), then either dim(Γ ↷ X) = ∞ or dim(Γ ↷ X) ≤ (asdimΓ + 1)(dim X + 1) − 1 where dim X is the covering dimension of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 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Response to stack exchange question, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} +page_content=' 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFOT4oBgHgl3EQf7jS_/content/2301.12963v1.pdf'} diff --git a/WtFJT4oBgHgl3EQf4y1Z/vector_store/index.faiss b/WtFJT4oBgHgl3EQf4y1Z/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8e2862ecd935e0f268406eac8fdee17429c468f8 --- /dev/null +++ b/WtFJT4oBgHgl3EQf4y1Z/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b7e704b1daa630f1fcd7dfc03ac25cbe77930693dad9262b84342d460bf9c822 +size 3735597 diff --git a/WtFJT4oBgHgl3EQf4y1Z/vector_store/index.pkl b/WtFJT4oBgHgl3EQf4y1Z/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..9b68a6cec16aa01d7e791f18af975660d235c2b3 --- /dev/null +++ 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Berghmans1,⋆, P. Antolin2, F. Auchère3, R. Aznar Cuadrado4, K. Barczynski5, 6, L. P. Chitta4, S. Gissot1, L. +Harra5, 6, Z. Huang4, M. Janvier7, 3, E. Kraaikamp1, D. M. Long8, S. Mandal4, M. Mierla1, S. Parenti3, 1, H. Peter4, L. +Rodriguez1, U. Schühle4, P. J. Smith8, S. K. Solanki4, K. Stegen1, L. Teriaca4, C. Verbeeck1, M. J. West9, +A. N. Zhukov1, 10, T. Appourchaux3, G. Aulanier11, 12, E. Buchlin3, F. Delmotte13, J. M. Gilles14, M. Haberreiter5, J.-P. +Halain14, 7, K. Heerlein4, J.-F. Hochedez15, 16, M. Gyo5, S. Poedts17, 18, and P. Rochus14 +1 Solar-Terrestrial Centre of Excellence – SIDC, Royal Observatory of Belgium, Ringlaan -3- Av. Circulaire, 1180 Brussels, Bel- +gium +2 Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, UK +3 Institut d’Astrophysique Spatiale, CNRS, Univ. Paris-Sud, Université Paris-Saclay, Bât. 121, 91405 Orsay, France +4 Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany +5 Physikalisch-Meteorologisches Observatorium Davos, World Radiation Center, 7260, Davos Dorf, Switzerland +6 ETH Zürich, Institute for Particle Physics and Astrophysics , Wolfgang-Pauli-Strasse 27, 8093 Zürich +7 European Space Agency, ESTEC, Keplerlaan 1, PO Box 299, NL-2200 AG Noordwijk, The Netherlands +8 UCL-Mullard Space Science Laboratory, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK +9 Southwest Research Institute, 1050 Walnut Street, Suite 300, Boulder, CO 80302, USA +10 Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia +11 Sorbonne Université, Observatoire de Paris - PSL, École Polytechnique, Institut Polytechnique de Paris, CNRS, Laboratoire de +physique des plasmas (LPP), 4 place Jussieu, F-75005 Paris, France +12 Rosseland Centre for Solar Physics, University of Oslo, P.O. Box 1029, Blindern, NO-0315 Oslo, Norway +13 Laboratoire Charles Fabry, Institut d’Optique Graduate School, Université Paris-Saclay, 91127 Palaiseau Cedex, France +14 Centre Spatial de Liège, Université de Liège, Av. du Pré-Aily B29, 4031 Angleur, Belgium +15 AESTER INCOGNITO, 75008 Paris, France +16 LATMOS, CNRS - UVSQ - Sorbonne Université, 78280, Guyancourt, France +17 Centre for mathematical Plasma Astrophysics, KU Leuven, 3001 Leuven, Belgium +18 Institute of Physics, University of Maria Curie-Skłodowska, Pl. M. Curie-Skłodowskiej 5, 20-031 Lublin, Poland +Received ; accepted +ABSTRACT +Context. The Extreme Ultraviolet Imager (EUI), onboard Solar Orbiter consists of three telescopes: the two High Resolution Imagers +in EUV (HRIEUV) and in Lyman-α (HRILya), and the Full Sun Imager (FSI). Solar Orbiter/EUI started its Nominal Mission Phase on +2021 November 27. +Aims. EUI images from the largest scales in the extended corona off limb, down to the smallest features at the base of the corona and +chromosphere. EUI is therefore a key instrument for the connection science that is at the heart of the Solar Orbiter mission science +goals. +Methods. The highest resolution on the Sun is achieved when Solar Orbiter passes through the perihelion part of its orbit. On 2022 +March 26, Solar Orbiter reached for the first time a distance to the Sun close to 0.3 au. No other coronal EUV imager has been this +close to the Sun. +Results. We review the EUI data sets obtained during the period 2022 March-April, when Solar Orbiter quickly moved from alignment +with the Earth (2022 March 6), to perihelion (2022 March 26), to quadrature with the Earth (2022 March 29). We highlight the first +observational results in these unique data sets and we report on the in-flight instrument performance. +Conclusions. EUI has obtained the highest resolution images ever of the solar corona in the quiet Sun and polar coronal holes. Several +active regions were imaged at unprecedented cadences and sequence durations. We identify in this paper a broad range of features +that require deeper studies. Both FSI and HRIEUV operate at design specifications but HRILya suffered from performance issues near +perihelion. We conclude emphasising the EUI open data policy and encouraging further detailed analysis of the events highlighted in +this paper. +Key words. Sun: UV radiation – Sun: transition region – Sun: corona – Instrumentation: high angular resolution +1. Introduction +The launch of the Atmospheric Imaging Assembly (AIA) on- +board the Solar Dynamics Observatory (SDO, Pesnell et al. +(2012)) in 2010 heralded the era of continuous full disc coro- +⋆ Corresponding +author: +David +Berghmans +e-mail: +david.berghmans@oma.be +nal imaging at high spatial resolution. In normal mode, AIA +images are produced with a cadence of 12 s at a spatial reso- +lution of 1.5′′, over a field of view (FOV) of (41′)2 (Lemen et al. +2012). Recent developments in coronal imagers have included +increased fields of view and higher spatial resolution. SWAP on +PROBA2 (Seaton et al. 2013a; Halain et al. 2013) and SUVI on +GOES (Darnel et al. 2022) have boosted observations with FOVs +Article number, page 1 of 19 +arXiv:2301.05616v1 [astro-ph.SR] 13 Jan 2023 + +A&A proofs: manuscript no. main +of (54′)2 and (53′)2 respectively. Thanks to these larger FOVs, +both SWAP and SUVI image the EUV structures and dynam- +ics well beyond the AIA FOV, into what has become known as +the Middle Corona (Seaton et al. 2021; West et al. 2022; Chitta +et al. 2022b). Meanwhile, the sounding rocket Hi-C (Kobayashi +et al. 2014) pushed the limits in terms of spatial resolution. In +its second successful flight (Rachmeler et al. 2019), Hi-C took +(subfield) images of an active region at a cadence of 4 s and a +spatial resolution better than 0.46′′ (330 km on the Sun). +Solar Orbiter (Müller et al. 2020) is in a highly elliptical or- +bit with perihelia below 0.3 au and, in later years of the nominal +mission, well out of the ecliptic, beyond 30◦ solar latitude. The +EUI instrument (Rochus et al. 2020) onboard Solar Orbiter will +use this unique orbit to observe the Sun from different vantage +points through three separate telescopes, imaging the outer solar +atmosphere at an even higher spatial resolution than Hi-C, and +over wider fields of view than SUVI and SWAP, further extend- +ing the middle corona discovery space. +The first EUI telescope, the Full Sun Imager (FSI) is a one- +mirror telescope taking alternating images in the 17.4 nm and +30.4 nm passbands. For a coronal EUV imager, FSI has an un- +precedented large FOV: (228′)2, which has a significant overlap +(Auchère et al. 2020a) with the Solar Orbiter coronagraph Metis +(Antonucci et al. 2020). At perihelion, this FOV corresponds to +(4 R⊙)2 such that the full solar disc is always seen, even at max- +imal offpoint (1 R⊙). This FOV is significantly wider than the +(3.34 R⊙)2 of EUVI (Howard et al. 2008) or the (3.38 R⊙)2 of +SWAP (Seaton et al. 2013b). When at 1 au (near aphelion), this +FOV corresponds to (14.3 R⊙)2 providing unique opportunities +to image the Middle Corona and eruptions that transit through +this region. +The other EUI telescopes are the two High Resolution Im- +agers (HRIs), HRIEUV and HRILya, which are two-mirror opti- +cal systems imaging through EUV and hydrogen Lyman-α pass- +bands respectively. HRIEUV images the corona at 17.4 nm, which +corresponds to the 17.4 nm channel of FSI. HRILya, which im- +ages in the Lyman-α line, shares its resonance formation process +for hydrogen with the 30.4 nm channel of FSI for helium. The +HRIEUV plate scale is 0.492′′, the HRILya plate scale is 0.514′′. +At the 2022 March 26 perihelion, Solar Orbiter reached a dis- +tance of 0.323 au from the Sun, giving (single) pixel values on +the Sun of (115 km)2 for HRIEUV, and (120 km)2 for HRILya. The +actual spatial resolution of the telescopes is discussed in Section +4. Both HRI cameras are capable of operating at cadences in the +1 s range, over 2048×2048 pixel arrays. The HRIs image through +a 17′ × 17′ FOV, corresponding to (1 R⊙)2, when observing at +1 au, and (0.28 R⊙)2 at perihelion. +Following its launch on 2020 February 10, Solar Orbiter +spent 4 months in the Near Earth Commissioning Phase, fol- +lowed by 17 months of Cruise Phase. During the Cruise Phase +only the in-situ instruments were collecting science grade data, +while the remote sensing instruments were undergoing extended +testing in preparation for the science phase of the mission. The +Nominal Mission Phase of Solar Orbiter started on 2021 Novem- +ber 27. During the Nominal Mission Phase, the remote sensing +instruments run a non-stop synoptic observation program inter- +leaved three times per orbit with 10 days periods of enhanced +observational activity. These periods are called “Remote Sensing +Windows” (RSWs) and are typically scheduled at the perihelion +of the orbit of Solar Orbiter and at the times of minimum and +maximum latitude. +In this paper we present an overview of the unique data sets +collected by EUI during the very first close perihelion RSWs, +covering the period from 2022 March 2 to 2022 April 6 (Fig. 1, +Fig. 1. Trajectory of Solar Orbiter in Geocentric Solar Ecliptic (GSE) +coordinates in black, starting at 2021 December 27 (square symbol). +The Remote Sensing Windows (RSW) correspond to the orange, red, +and blue parts of the trajectory. In this paper we cover the period +from 2022 March 02 (beginning of RSW1, orange) till 2022 April 06 +(end of RSW3, blue). The perihelion occurred at the end of the RSW2 +(red), around 2022 March 26. The trajectories of the ESA mission Bepi +Colombo and the NASA missions STEREO-A, and Parker Solar Probe +are indicated brown, green, and purple respectively. +reproduced from the ESA Solar Orbiter website1). As compared +to Earth, Solar Orbiter observed from a perspective of increasing +solar longitude (i.e. East to West), and transitioned from near- +alignment with Earth on 2022 March 6, to perihelion on 2022 +March 26, and then in a quadrature formation with the Earth, +observing the Sun above the West limb, on 2022 March 29. Dur- +ing that period, the distance to the Sun from Solar Orbiter varied +from 0.32 au to 0.55 au, closer to the Sun than any coronal EUV +imager before. Given the variable angle with the Earth, the vari- +able distance to the Sun, and the constraints of the low teleme- +try bandwidth, the instrument operations of the Remote Sensing +Payload on Solar Orbiter was highly non-synoptic. The aim of +the paper is to guide the EUI data user through the unique but +very variable EUI observations that were collected in the period +2022 March 2 to April 6. In Section 2 we present the various EUI +observation campaigns that were taken during the perihelion. In +Section 3 we give an overview of the observational highlights +that were completed in these campaigns. After that, in Section 4, +we describe for each EUI telescope the instrument performance. +Finally, in Section 5, we give an outlook for upcoming orbits. +2. From Science Goals to EUI Data sets +As Solar Orbiter is a deep space, non-synoptic probe, the activ- +ity scheduling of its instruments is coordinated well in advance. +This is done through Solar Orbiter Observing Plans (SOOPs), +which bring a group of Solar Orbiter instruments in a specific +mode to target a certain science goal at appropriate times during +1 https://issues.cosmos.esa.int/solarorbiterwiki/display/SOSP +Article number, page 2 of 19 + +SolarOrbiter-ParkerSolarProbe +STEREO-Ahead +BepiColombo +1.00 +0.75 +0.50 +0.25 +Y (AU) +0.00 +GSE +0.25 +0.50 +0.75 +1.00 +0.0 +0.5 +1.0 +1.5 +2.0 +GSE X (AU)D. Berghmans et al.: First Perihelion of EUI on the Solar Orbiter mission +the orbit. These science goals have been generically described in +Zouganelis et al. (2020) and updates are maintained on the ESA +Solar Orbiter website. +In order to document the intended purpose and context of +the collected observations, we here summarise the SOOPs that +the EUI instrument was involved in during the 2020 March 2 to +April 6 period. Grouping the SOOPs by science target and by +increasing operational complexity, we arrive at three themes: (1) +observing the Middle Corona off limb, (2) observing the build- +ing blocks of the solar atmosphere and, (3) making the connec- +tion from high resolution on-disk observations to in-situ mea- +surements of the solar wind. The full technical details of the re- +sulting EUI data sets can be found in Table A.1 in Appendix 5, +where entries are linked to corresponding SOOP names. +2.1. Observing the off limb Middle Corona +The first set of SOOPs focused on the off limb corona. Most of +these SOOPs are led by the Solar Orbiter coronagraph Metis, +which requires Sun center pointing. This makes the SOOPs op- +erationally simpler as no last-minute pointing corrections are +needed. For these SOOPs, EUI FSI images are prioritized due to +the large overlap with Metis (Auchère et al. 2020b) with Metis. +Optional HRI images are necessarily pointed near disc center. +The +aim +of +the +L_FULL_HRES_MCAD_Coronal-He- +Abundance SOOP on 2022 March 7 was to support observations +during the second launch of the Herschel sounding rocket, whose +first flight provided the first helium abundance maps of the so- +lar corona (Moses et al. 2020). The abundance is deduced from +the ratio of the resonantly scattered intensities of neutral hydro- +gen and singly ionized helium, as imaged by two coronagraphs: +SCORE (Romoli et al. 2003) and HeCOR (Auchère et al. 2007). +The method is model-dependent, and requires an independent +knowledge of the temperature of the scattering ions. This can be +constrained by simultaneous EUV observations, which was the +purpose of the FSI observations at 17.4 nm. In order to minimize +stray-light at large distances from the solar limb, the instrument +was used in coronagraph mode, with a movable disk masking +direct sunlight (see Auchère et al. 2005; Rochus et al. 2020). +One of the returned images is shown in Fig. 2, composited with +the closest in time disk image taken before the campaign. The +sounding rocket payload failed, but the FSI data are still very +useful to study the 17.4 nm emission of the extended corona. The +composite EUV image allows to link the magnetic structures on +the disk (coronal holes, plumes, active regions) to magnetic field +expansions in the extended corona, which appear always open +but far from being simply radially aligned. +The L_FULL_HRES_HCAD_CoronalDynamics SOOP is +designed to observe structures in the outer corona with the aim +of linking them to the solar wind observed in-situ. The SOOP +was run twice, on 2022 March 22 (at 0.33 au) and March 27 (at +0.32 au) just after perihelion. Given the outer corona focus, the +main contribution of EUI was through FSI observations in both +wavelengths which provide a large overlap with Metis. Never- +theless, HRI images of the quiet Sun at disc center were also +taken at relative low cadence (30 s to 60 s). While these HRI im- +ages are perhaps not directly useful for the SOOP goal, they are +unique observations as, given the close solar approach, they were +the sharpest quiet Sun EUV images ever taken during the first +perihelion passage. +The goal of the L_FULL_HRES_HCAD_Eruption-Watch +SOOP is to observe eruptive events and to contribute to the un- +derstanding of Coronal Mass Ejections. The SOOP was carried +out in two campaigns; the first one on 2022 March 22-23, and the +Fig. 2. FSI image taken in coronagraph mode (outer FOV) on 2022 +March 7 at 16:00:05 UT, composited with a regular disk image (cen- +ter) taken at 11:29:45 UT. The images were enhanced using the WOW +algorithm (Auchère et al. 2022) to reveal faint features. See Sect. 2.1. +second one between 2022 March 29-30 (close to Solar Orbiter in +quadrature at the west side of Earth). There was long term mon- +itoring with the FSI (every 6 min) while the HRIs operated in +30 min bursts. +Fig. 3. Observed active regions on 2022 March 2 (left), March 17 (mid- +dle) and April 2 (right). See Sects. 2.2 and 2.3. +2.2. Observing the building blocks of the on-disc solar +atmosphere +A second set of SOOPs targeted various features in the on- +disc solar atmosphere. These SOOPs are typically led by SPICE +(SPICE Consortium et al. 2020) and/or EUI and require point- +ing the spacecraft to targeted features. Pointing corrections were +made a few days in advance, commanded through the so-called +"Pointing-Very Short Term Planning" (pVSTP, Zouganelis et al. +(2020)) and based on Low Latency FSI images that are brought +to the ground as a high priority. These are typically less than a +day old. +The observation of small-scale EUV brightenings in the +quiet Sun was an early success of EUI (Berghmans et al. +2021). Therefore, particular attention was paid to scheduling the +R_BOTH_HRES_HCAD_Nanoflares SOOP that is focused on +surveying small impulsive events. There are many open ques- +tions on their origin and properties. Are they also located in ac- +tive regions and coronal holes? Similar events were observed +in active regions by the Hi-C, but it is not clear yet if they +Article number, page 3 of 19 + +5000" +Helioprojective Latitude (Solar-Y) +O" +-5000" +5000" +0" +5000" +Helioprojective Longitude (Solar-x)12967 +12975 +12956 +12958 +12957 +12976A&A proofs: manuscript no. main +are the same phenomenon. If this is the case, do they have the +same properties everywhere? Recent work suggests that a sub- +population of the small-scale EUV brightenings does not reach +the 1 MK (Dolliou et al. 2022, (this volume, in revision); Huang +et al. 2022, (this volume, in prep)). Answering these questions +will help us understand, for instance, their origin and relationship +with the small scale magnetic field structuring and evolution. +The R_BOTH_HRES_HCAD_Nanoflares SOOP was sched- +uled several times near the Sun-Earth line (2022 March 7), at an +angle with the Earth of roughly 30◦ and near quadrature (2022 +March 30). The spacecraft pointed alternatively to an active re- +gion and to the quiet Sun. This SOOP resulted in thousands of +HRIEUV and HRILya images obtained with a cadence of usually +3 s (HRIEUV) and 5 s (HRILya) for a duration of approximately +30 min (co-temporal in both channels). The HRIEUV initial plan +was to run at 2s cadence, however the first run on 2022 March +6 failed and the cadence was decreased to 3 s. This very high +cadence period was, in general, followed by longer periods at +lower cadence. This choice came from the allocated telemetry +and the need for long high cadence temporal sequences. These +campaigns were widely supported by other instruments on Solar +Orbiter but also by IRIS (De Pontieu et al. 2014), Hinode (Ko- +sugi et al. 2007), PROBA2/LYRA (Dominique et al. 2013) and +SDO/AIA. The latter instrument ran in a restricted configuration +(subfield, few wavelengths) to image at an enhanced 6s cadence +in 4 wavelengths (13.1 nm, 19.1 nm, 17.1 nm and 30.4 nm chan- +nels). +The R_SMALL_HRES_MCAD_Polar-Observations SOOP, +as described in Zouganelis et al. (2020), aims at observing the +polar magnetic fields by PHI (Solanki et al. 2020). In this early +phase of the mission however such observations are not ideal +yet, as the spacecraft is still at low solar latitudes. Instead, the +focus of the SOOP was to observe the polar coronal holes with +the high resolution imagers of EUI. Such observations are timely +as polar coronal holes will soon disappear with the rising solar +cycle. This SOOP was carried out three times, once when Solar +Orbiter was close to the Sun-Earth line (2022 March 6), once +when Solar Orbiter was close to quadrature with Earth (2022 +March 30) and once more when Solar Orbiter was at roughly +120◦ with the Earth (2022 April 4). In the first two campaigns, +the solar south pole coronal hole was the target but in the last +campaign the solar north pole coronal hole was more visible. +In particular the 2022 March 30 observation of the south pole +returned the ‘best ever’ EUV images of a polar coronal hole as +they were taken from 0.33 au with an imaging cadence of 3 s. +The +R_BOTH_HRES_MCAD_Bright-Points +SOOP +is +SPICE-focused and aims at observing coronal bright points. +This SOOP was carried out between 2022 March 8 08:10 and +14:10. During that time, FSI operated with a relatively high time +cadence (5 min). HRIEUV and HRILya observed during 2 hours +at 1 minute cadence. The EUI telescopes pointed at disc centre +(quiet Sun) and bright points were observed. +The R_SMALL_MRES_MCAD_AR-Long-Term was car- +ried out between 2022 March 31 and 2022 April 4. The goal was +to track the decay phase of an active region. Two good candidates +appeared on the solar disk a few days before the SOOP started: +NOAA AR2 12975 and NOAA AR 12976 (see Fig. 3). Com- +paring both regions, the leading polarity of NOAA AR 12976 +provided a good target and the EUI FOV was chosen centered +on it. There was long term monitoring with FSI (every 30 min) +and burst image sequences with both HRIs, typically with a time +cadence of 10 s and lasting 47 min. The regions observed dur- +2 Active region numbering by https://www.swpc.noaa.gov +ing these few days produced several flares, including an M-class +flare on 2022 April 2 that is discussed further below. +On 2022 March 7, Solar Orbiter crossed the Sun-Earth line, +at a distance of 0.49 au, allowing for cross-calibration with sim- +ilar Earth-bound instruments. For a complete inter-comparison, +a full range of scenes (quiet Sun area, an active region or a coro- +nal hole) was to be targeted within the small FOV of PHI/HRT, +HRIEUV, HRILya and SPICE. However, by having Solar Orbiter +point in a 5x5 pattern, these high resolution telescopes could +however make a Full Disc Mosaic of the whole solar disc, +thereby avoiding upfront guessing the position of the various +scenes. Solar Orbiter followed the 5x5 pointing pattern from +north-east (top left) to south-west (bottom right) in columns. Im- +ages in subsequent pointing positions are 10 min apart in the ver- +tical direction and 50 min apart in the horizontal direction. To en- +sure a maximum overlap between image panels, the HRIEUV im- +ages were commanded to be 2368x2368 in size. On average, the +images between dwells overlapped by 600 pixels. The HRIEUV +telescope took high-gain and low-gain image pairs every 30 s +within each dwell period, resulting in 9 such image pairs per +pointing position. The high- and low-gain images were taken +5 s apart and calibrated and combined on-ground into high dy- +namic range images (15-bit). To create a high resolution mo- +saic of the full Sun, these high dynamic range images from each +dwell position were aligned and stitched together using affine +image transformation making use of the spacecraft attitude in- +formation available in the source FITS files. The resulting panels +were then blended together manually in photo editing software, +minimizing image artifacts in the mosaic caused by changing +views in neighboring panels due to dynamic events and solar ro- +tation. The panels were blended together preferentially in quiet +Sun areas, avoiding the faster changing active regions where pos- +sible. The final mosaic has more than 83 million pixels, making +it the highest resolution image of the Sun’s full disc and corona +ever taken. An interactive version of the image can be found in +Kraaikamp (2022). +2.3. Making the connection from high resolution on disk to +in-situ +A third set of SOOPs aimed at finding the connection between +the smallest features imaged on disc to the corresponding in-situ +measurements of the out flowing solar wind. +The goal of the L_SMALL_MRES_MCAD_Connection- +Mosaic SOOP is to identify, with SPICE and the high resolution +imagers of EUI and PHI, the connection point on the solar disc +for the solar wind observed in situ. In order to increase the proba- +bility of successfully catching the connection point, this SOOP is +implemented in combination with a mosaic of spacecraft point- +ings. The SOOP was run twice, once when Solar Orbiter was +near the Sun-Earth line (2022 March 2-3, at 0.56 au) and once +when Solar Orbiter was in quadrature with Earth (2022 March +30, at 0.33 au). In the first instance, the spacecraft made a mo- +saic of 3 vertically aligned pointings, in the second instance, a +mosaic of 3x2 pointings was made. Each of these pointings was +maintained for several hours. It was later discovered that solar +rotation was not correctly compensated, so the EUI FOV slightly +shifts over this duration. The location of the mosaics was decided +a few days in advance with the help of various models and tools +(Rouillard et al. 2020). During the first instance of this SOOP, +EUI observed an M2 flare in the high resolution FOV, see sec- +tion 3.1.9. +The +L_SMALL_HRES_HCAD_Slow-Wind-Connection +SOOP (Yardley, S. et al, 2022, this issue, in preparation) was +Article number, page 4 of 19 + +D. Berghmans et al.: First Perihelion of EUI on the Solar Orbiter mission +HRIEUV | 2022-03-04 |10:48:50 +0 +100 +200 +300 +Mm +0 +100 +200 +300 +Mm +0 +100 +200 +300 +0 +100 +200 +300 +aaaa + +NOAA 12957 + + + +S1 +S1 +S1 +S1 +S2 +S2 +S2 +S2 +S3 +S3 +S3 +S3 +20 Mm +20 Mm +20 Mm +20 Mm +bbbb +Slit-S1 +0 +10 +20 +30 +40 +50 +Time (min) +0 +5 +10 +15 +Mm +cccc +Slit-S2 +0 +10 +20 +30 +40 +50 +Time (min) +0 +2 +4 +6 +8 +10 +12 +14 + +dddd +Slit-S3 +0 +10 +20 +30 +40 +50 +Time (min) +0 +5 +10 +15 + +eeee +Fig. 4. Example of decayless kink waves observed in AR12957 on 2022 March 04. Panel b shows a zoom into the loops in panel a. The boxes S1 +to S3 mark slits along which the temporal evolution is shown in the form of space-time diagrams in panels c to e. To enhance the appearance of +the oscillating threads in these space-time diagrams, a smooth version of the map (boxcar-smoothed in the vertical direction) is subtracted from +each original map. The black lines represent fits for the oscillation of the loops. See Sect. 3.1.1 for details. +designed to combine the remote-sensing and in-situ capabilities +of Solar Orbiter, observing the source of solar wind connected +to the spacecraft and then detecting the plasma released from the +Sun as it passed over the spacecraft several days later. Although +the primary instruments used in the SOOP are SPICE and SWA +(Owen et al. 2020), the EUI HRI observations provided high +resolution context images. Due to the need to identify where +material passing over the spacecraft was originally ejected from +the Sun, the SOOP coordinator relied heavily on the connectivity +tool developed by IRAP (Rouillard et al. 2020) to identify the +origin of the magnetic field predicted to be connected to Solar +Orbiter during the observing window. For the first observing +window (2022 March 3-6), this was the boundary between +NOAA AR 12957 and a nearby equatorial coronal hole (Fig. 3). +For the second window (2022 March 17-22), two different +targets were selected due to a change in the connectivity of the +spacecraft with respect to the solar magnetic field. The boundary +of the southern polar coronal hole was selected as the target for +the first part of the observing window, with the positive polarity +of NOAA AR 12967 in the northern hemisphere selected as the +target for the second part of the window. +The +L_BOTH_HRES_LCAD_CH-Boundary-Expansion +SOOP is similar to L_SMALL_HRES_HCAD_Slow-Wind- +Connection discussed above, but it specifically aims to study +coronal holes boundaries as possible sources of the slow solar +wind. The SOOP was active between 2022 March 25 19:40 +and 2022 March 27 00:00. FSI acquired 17.4 nm and 30.4 nm +images at 10 min cadence. The quiet Sun at disc centre was +observed in both campaigns. +3. Observational highlights +In the previous section, the EUI observations of the 2022 March +2–April 6 period were presented from a science planning per- +spective. Solar activity however seldom follows the science plan +so we additionally review the actual observational highlights +here that were identified so far in the collected data. While this +35-day period is longer than a solar rotation period, its sub-solar +Article number, page 5 of 19 + +A&A proofs: manuscript no. main +point ranged from 64◦ Carrington longitude in the beginning of +the period to 95◦ Carrington longitude at the end of the period, +due to the intrinsic motion of the spacecraft over the same pe- +riod. The longitudinal angle Earth-Sun-spacecraft ranged from +-8◦ to 122◦. In what follows we use EUI Data Release 5 (Mam- +paey et al. 2022). +3.1. Active region dynamics +Several active regions (see Fig. 3) were observed in the HRIEUV +and HRILya FOVs with imaging cadences sometimes as fast as +3 s. Below we highlight a sample of some particular dynamics.. +3.1.1. Decayless kink oscillations +NOAA AR 12957 was observed first as part of the mosaic pat- +tern of L_SMALL_MRES_MCAD_ConnectionMosaic (2022 +March 2, 3) and then as part of the daily high resolution bursts +of L_SMALL_HRES_HCAD_Slow-Wind-Connection on 2022 +March 3, 4 and 5. At this time, Solar Orbiter was at 0.544 au from +the Sun, resulting in an HRIEUV pixel footprint of 194 km on the +Sun. The core of the active region showed a myriad of counter- +streaming loops, of which some exhibited decayless kink oscil- +lations (Fig. 4). This event is of particular interest due to the fact +that these oscillating loops are rooted inside sunspots which are +generally devoid of supergranular flows, the commonly assumed +driver of such decayless kink oscillations. Moreover, these de- +cayless oscillations were only observed during specific time in- +tervals although the loop environment remained more-or-less +similar throughout. Further details on the magnetic configura- +tion of those loop footpoints, as well as the existence of other +possible wave drivers are presented in Mandal et al. (2022). +3.1.2. Braiding loops +As a part of the R_BOTH_HRES_HCAD_Nanoflares SOOP on +2022 March 17, HRIEUV observed active region AR12965 at a +cadence of 3 s. These are among the first highest cadence EUV +images of an active region ever observed. During this period, +Solar Orbiter was at a distance of 0.38 au. Thus the 2-pixel foot- +print of HRIEUV was about 270 km on the Sun. An overview of +the observed active region is displayed in Fig. 5a. These high- +resolution, high-cadence observations of this active region re- +vealed a number of impulsive EUV brightenings on timescales +of a few minutes or less. A closer look at some of the brighten- +ings revealed that they are associated with untangling of braided +coronal strands or loops. Most of these events are observed in +shorter, low-lying loop features. HRIEUV also observed untan- +gling of coronal braids in a more conventional loop system. A +sequence of this untangling of braided loops is shown in Fig. 5b– +j. More details on examples of braided structures observed by +HRIEUV and the implications for coronal heating are discussed +in Chitta et al. (2022a). +3.1.3. Highly dynamic cooler loops +The region of interest of R_SMALL_MRES_MCAD_AR-Long- +Term on 2022 April 1 was on the east limb of the Sun from the +vantage point of Solar Orbiter, which means that the same re- +gion appeared in the western hemisphere from the viewpoint of +Earth. Fig. 6a shows the full FOV of HRIEUV, with NOAA AR +12975 on the western side and NOAA AR 12976 on the east- +ern side. Besides imaging instances of coronal braids in low- + + + + +2022-03-17 UT 03:32:03 +(a) +20 Mm + + + + + + + + +UT 03:31:33 +(b) +2 Mm + + + + + + + + +UT 03:32:03 +(c) + + + + + + + + +UT 03:32:33 +(d) + + + + + + + + +UT 03:33:03 +(e) + + + + + + + + +UT 03:33:33 +(f) + + + + + + + + +UT 03:34:03 +(g) + + + + +UT 03:34:33 +(h) + + + + +UT 03:35:03 +(i) + + + + +UT 03:35:33 +(j) +Fig. 5. Example of a relaxation of braided coronal loops observed on +2022 March 17. Panels (b) to (j) display a zoom into the loop in panel +(a) (white box) and show the evolution of the untangling within the loop. +See Sect. 3.1.2 for details. +lying loop systems in both these active regions (see discussion +in Chitta et al. 2022a), HRIEUV captured a highly dynamical sys- +tem of cooler loops that appear darker in EUV due to absorption. +These dynamic loops were observed in the core of AR12975 +(see white box in Fig. 6). These are likely related to chromo- +spheric arch-filament systems associated with emerging flux re- +gions (van Driel-Gesztelyi & Green 2015). Individual strands or +loops within this system exhibited intermittent brightenings in +EUV. In addition, there are also repeated compact brightenings +on one end of this feature. The morphology of these compact +EUV brightenings appear akin to the transition region ultraviolet +bursts that are often observed in emerging flux regions (Young +et al. 2018). The evolution of this region over a period of 1 hour +is displayed in a sequence of images in Fig. 6b–j. +3.1.4. Brightening on border of dark material +During the R_SMALL_MRES_MCAD_AR-Long-Term SOOP +on 2022 April 1, HRIEUV observed brightening events at the top +of dark jet-like structures The angular separation between the +Earth and Solar Orbiter was 104◦ which allowed for simultane- +ous observation from SDO/AIA. Solar Orbiter was 0.35 au from +the Sun, resulting in a spatial resolution of HRIEUV images (two +Article number, page 6 of 19 + +D. Berghmans et al.: First Perihelion of EUI on the Solar Orbiter mission + + + + + + + + +(a) +2022-04-01 UT 10:28:51 +20 Mm + + + + + + + + +UT 09:24:42 +(b) +2 Mm + + + + + + + + +UT 09:33:02 +(c) + + + + + + + + +UT 09:41:21 +(d) + + + + + + + + +UT 09:49:41 +(e) + + + + + + + + +UT 09:58:02 +(f) + + + + + + + + +UT 10:06:21 +(g) + + + + +UT 10:14:41 +(h) + + + + +UT 10:23:01 +(i) + + + + +UT 10:31:21 +(j) +Fig. 6. Example of a highly dynamic cool loop system in the core of +active region AR12975 observed by HRIEUV on 2022 April 1. See Sect. +3.1.3 +pixel size) of 248 km. The dark jet-like structures appear to be +the so-called light walls (Hou et al. 2016) or fan-shaped jets +(Robustini et al. 2016), which are field-aligned long chromo- +spheric jets thought to be produced by magnetic reconnection +in the photosphere (Bharti 2015; Bai et al. 2019). Two kinds +of brightening events can be distinguished (see time-distance +map in Fig. 7). The first kind is continuously present at the +chromosphere-corona interface of the jets and is seen to oscillate +up and down with a ballistic motion, strongly suggesting that +this corresponds to the upward motion of the transition region +observed by HRIEUV (17.4 nm). Its narrow thickness (as small +as 200 km or less) and strong brightness is probably due in part +to the passage from high to low density and cool to hot plasma, +to which HRIEUV is particularly sensitive. The second kind of +brightening is far more impulsive, with life times on the order +of 10 s or less, and appears on top of the first kind as pertur- +bances propagating upwards at speeds of ≈ 100 km s−1. Both +brightening events are therefore likely due to slow mode shocks +generated from the reconnection events lower down, first propa- +gating in the chromosphere and then into the corona at the tube +speed. The second kind is seen to originate when the dark struc- +ture is at the lower end of the oscillation range, as expected from +chromospheric shocks leading to spicule-like events (Heggland +et al. 2007, 2011). +!"!!!"#!"$%"&'!('#!)*%)+,-./01 +" +2 +$" +$2 +!" +!2 +3" +4)5678 +" +2 +$" +$2 +!" +!2 +3" +9)5678 +!:# +!:; +!:( +3:" +3:! +3:# +3:; +<=>+?@/A@BC/9)5DEFB81 +Fig. 7. HRIEUV observations of the brightening on a border of dark ma- +terial in an active region on 2022 April 1. The white arrows in the top +panel mark the brightening location. The bottom panel shows a time- +distance map along one of the jets observed propagating from the bright +structure. Note the existence of 2 kinds of brightening events, the bright +edge of the dark structure oscillating up and down, and jets propagating +upwards (both indicated by the arrows). See Sect. 3.1.4. +3.1.5. Coronal rain +Coronal rain appears ubiquitous on-disk in AR NOAA 2975 and +2976 observed on 2022 March 30, April 1 and 2 (see Fig.8). +In HRIEUV, coronal rain can be clearly distinguished in EUV +absorption by its dynamics (with velocities close to 100 km s−1 +in the plane-of-the-sky) and its clumpy and multi-stranded mor- +phology (Antolin et al. 2015; Antolin 2020). At the spatial res- +olution of ≈250 km, individual coronal rain clumps only a few +pixels wide can be observed. Their morphology is strongly rem- +iniscent of Hα high-resolution observations (Antolin & Rouppe +van der Voort 2012), a similarity that has been predicted but so +far only observed at larger loop scales (Anzer & Heinzel 2005; +Yang et al. 2021). Coronal rain showers (composed of clumps) +can be observed in loop bundles rooted to moss, but both clumps +and showers (despite the large shower widths above 10 Mm) ap- +pear mostly unresolved in AIA passbands. This picture therefore +constitutes a major difference to previous EUV observations of +active regions. The on-disk observation at high resolution pro- +vides a connection to the chromosphere (and photosphere with +PHI), thus providing a unique insight of the heating events at the +footpoints that lead to thermal non-equilibrium and instability +associated with this phenomenon (Antolin & Froment 2022). A +Article number, page 7 of 19 + +A&A proofs: manuscript no. main +Fig. 8. Top: Field-of-view of the 2022 March 30 dataset observed by +HRIEUV showing an active region at the South-East limb. The solid +white curves follow several trajectories of coronal rain clumps seen in +EUV absorption. Bottom panels: Snapshots at various instances of a +coronal rain shower, corresponding to the small white rectangle shown +in the top panel. The black arrows indicate the head of the coronal rain +shower. See Sect. 3.1.5. +full paper reporting coronal rain observed with HRIEUV is avail- +able in Antolin et al. 2023 (this volume, in prep.). +3.1.6. Hints of torsional Alfvén waves in twisted coronal loops +In AR NOAA 2975 observed on 2022 April 2nd twisted, in- +tertwined coronal strands can be observed in the plane-of-the- +sky, appearing and disappearing on a timescale of 5 − 10 min. +The strands disappear by the end of the sequence with hints +of untwisting, ending in coronal rain falling onto bright foot- +points rooted in moss. The entire event is strongly reminiscent +of the coronal loop model of Díaz-Suárez & Soler (2021) in +which, torsional Alfvén waves propagate along a twisted flux +tube. The radially varying magnetic field due to the twist pro- +vides an Alfvén continuum that allows phase mixing to happen. +The Kelvin-Helmholtz instability is then generated due to the ve- +locity shear at the phase mixing layers, which leads to compres- +sion of the plasma and the generation of coronal strands that fol- +low the twisted flux tube (a process also observed for kink waves, +Antolin et al. 2014). The time-scale of appearance/disappearance +of strands, their morphology and change in orientation during the +oscillation is seen to match the one observed in this event with +HRIEUV (see Fig. 9). +Fig. 9. Top: Field-of-view of the 2022 April 2 dataset observed by +HRIEUV showing an active region at the South-East limb. The solid +white rectangle shows a twisted coronal loop where hints of torsional +Alfvén waves may be observed. Bottom panels: snapshots over 30 min +evolution of the loop observed in the field-of-view corresponding to +the white rectangle. Note the change in orientation of the twisted EUV +strands. See Sect. 3.1.6 +3.1.7. Large-scale reconfiguration of coronal loop +sub-structure +In AR NOAA 2975 observed on 2022 April 1, a large scale loop +bundle rooted in moss is composed of various strands. Without +the presence of any flare in the vicinity, the strands undergo a co- +herent reconfiguration akin to contraction following a flare (see +Fig. 10). The strands also exhibit continuous kink motions dur- +ing the global contracting motion. The overall event is accompa- +nied by coronal rain. +3.1.8. Coronal strands associated with coronal rain +In AR NOAA 2975 observed on 2022 April 1, coronal strands +rooted in moss are observed to appear and disappear on a short +time-scale of tens of minutes. Contrary to usual coronal strands, +these appear first near the loop apex, and are seen to extend dy- +namically towards the footpoints in a flow-like manner. This is +followed by localised dark or bright features at the loop apex +with the appearance and dynamics of coronal rain (see Fig. 11). +The entire event strongly resembles the 2.5D MHD numerical +modelling of coronal rain by Antolin et al. (2022). +Article number, page 8 of 19 + +20 Mm +2022-03-30UT00:28:00 +00:28:39 +00:28:54 +00:29:09 +00:29:24 +00:29:39 +00:29:54 +00:30:09 +00:30:2420 Mm +2022-04-02 UT 09:19:15 +09:35:55 +09:40:05 +09:44:15 +09:48:25D. Berghmans et al.: First Perihelion of EUI on the Solar Orbiter mission +Fig. 10. Top: A sub-field of the 2022 April 1 dataset observed by +HRIEUV (located on the bottom-right of Fig. 6). A loop bundle is seen +(inverted in this figure in order to have the apex on top), where a large +scale reconfiguration is observed. Bottom: Time-distance plot across the +apex of the loop (dashed white curve on top panel), revealing a down- +ward (inward) motion of many loops (akin to contraction), indicated by +the black arrows, accompanied by transverse oscillations. The time of +the snapshot on the top panel corresponds to the vertical white dashed +line. See Sect. 3.1.7. +3.1.9. M2 flare: 2022 March 2 +The chances to observe flares in the small FOVs of the HRIs, +which are only operated a small fraction of an orbit, are not +large. Nevertheless, on 2022 March 2, during the mosaic pat- +tern of L_SMALL_HRES_HCAD_Slow-Wind-Connection, an +M2 flare was observed in active region NOAA 12958 by HRIEUV +and HRILya (Fig. 12). This is the largest flare seen so far in the +HRI subfields. Since the HRIs FOV only covered the lower part +of this active region, we have in Fig. 12 also used AIA 17.1 nm +and 30.4 nm images to show the context of the evolution of the +entire active region. +Solar Orbiter was at 0.55 au from the Sun, meaning that +the spatial resolution of the HRIEUV images (two pixel size) +is 397 km. The imaging cadence was unfortunately only 2 min. +Fig. 12 shows the two ribbons shortly before the flare peaked at +17:34 and HRIEUV saturated with a front filter diffraction pat- +tern. The HRIEUV images show several small-scale brightenings +during the pre-flare time. Views from HRIEUV and HRILya are +shown within the white boxes in Fig. 12. Thanks to the high +resolution of HRIEUV (left hand side panels), we can observe +in detail the brightening structures in this active region, such as +the double J-shaped brightening in the core region. In the right +hand side panels, the lower resolution and the saturated signal of +the HRILya images only allow distinguishing the outline of the +brightening structures. After the peak time, some brightenings +can be found at the upper edge of the HRIs FOV appearing at +the source region and propagating from west to east, forming a +long thin bright band of several tens Mm, as shown in the right +hand side panels. Flare loops are clearly visible in the HRIEUV +Fig. 11. Top: A sub-field of the 2022 April 1 dataset observed by +HRIEUV (located on the right of Fig. 6). A loop bundle rooted on moss is +seen composed of various strands. The strands appear first near the apex +and exhibit bright and dark flows with dynamics characteristic of coro- +nal rain. Bottom: Time-distance plot along one of the observed strands +(dashed white curve on top panel). Note the fuzzy brightening events +along the middle of the strand followed by bright or dark flows toward +either footpoint of the strand (indicated by the white arrows). The time +of the snapshot on the top panel corresponds to the vertical white dashed +line. See Sect. 3.1.8. +image, while the flare ribbons (and flare-driven rain) are visible +as two parallel bright structures in the HRILya image. +Fig. 13 shows that the temporal variation of the intensities +observed by HRIEUV and HRILya are in qualitative agreement +with comparable observations by SDO/AIA and GOES/LYA. +The HRIEUV intensity corresponds well with the SDO/AIA +17.1 nm and 30.4 nm, only the emission observed in SDO/AIA +9.4 nm occurs few minutes after the rest of the presented lines. +The HRILya and GOES/LYA intensity curves corresponds well +too. +3.2. Quiet Sun features +In the current phase of the solar cycle, most active regions appear +beyond 15◦ solar latitude North or South, meaning that when- +ever Solar Orbiter was not off pointed away from disk center, the +HRIs had the quiet Sun in the FOV. Below we present a small +sample of quiet Sun features and events that illustrate the quality +of the obtained data. +Article number, page 9 of 19 + +HRIEUV - 20220401 - UT09:55:55 +60 +50 +40 +[Mm] +30 +20 +10 +0 +0 +20 +40 +60 +80 +100 +120 +[Mm] +40 +[Mm] +30 +20 +Distance +10 +0 +0 +20 +40 +60 +Time [min]HRIEUV - 20220401 - U109:44:15 +25 +20 +15 +[Mm] +10 +5 +0 +0 +10 +20 +30 +40 +[Mm] +40 +「Mm] +along path +30 +20 +Distance +10 +0 +0 +10 +20 +30 +40 +50 +Time [min]A&A proofs: manuscript no. main +AIA 171 & EUI-HRIEUV 174 +0 +100 +200 +300 +400 Solar Orbiter: 2022-03-02 17:34:01 UT +AIA 304 & EUI-HRILYA +Solar Orbiter: 2022-03-02 17:34:01 UT +AIA 171 & EUI-HRIEUV 174 +0 +100 +200 +300 +400 +X [Mm] +0 +100 +200 +300 +400 +Y [Mm] +Solar Orbiter: 2022-03-02 17:44:01 UT +AIA 304 & EUI-HRILYA +0 +100 +200 +300 +400 +Solar Orbiter: 2022-03-02 17:44:01 UT +Fig. 12. M2 flare on 2022 March 2. Left hand side panels: combination +of AIA 17.1 nm images and HRIEUV 17.4 nm, the latter being shown +within the white box (only few seconds apart). Right hand side panels: +Same, but for the combination of AIA 30.4 nm and HRILya images. The +bottom row is taken 10 min later than the top row images. The white +boxes delineate the boundaries of the areas showing data from EUI- +HRIs. See Sect. 3.1.9. +17:28 +17:30 +17:32 +17:34 +17:36 +17:38 +Start Time (02−Mar−22 17:28:00) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +normalized count rate (AIA, HRI) +7.50 +7.55 +7.60 +7.65 +7.70 +GOES/LYA at 1 AU [W/m 2]x10 −3 +AIA 304 +AIA 171 +AIA 94 +HRI/LYA +HRI/EUV +GOES/LYA +Fig. 13. Temporal evolution of intensity of the M2 flare on 2022 March +2. The colour-coded lines present average intensities at the flare region +observed with HRIEUV, HRILya, SDO/AIA and GOES. See Sect. 3.1.9. +3.2.1. Small-scale EUV brightenings +On 2022 March 26, Solar Orbiter reached its first perihelion dur- +ing the Nominal Mission Phase at a distance of 0.323 au from the +Sun. The HRIEUV observations closest to perihelion were taken +on March 27, starting at 19:40 (distance 0.324 au) and were part +of the L_FULL_HRES_HCAD_Coronal-Dynamics SOOP. On +this day, HRIEUV had a pixel footprint on the sun of (115 km)2. +Small EUV brightenings observed by HRIEUV, a.k.a. camp- +fires, were first identified by (Berghmans et al. 2021) in data +taken at 0.556 au. The observed campfires were typically elon- +gated structures from 0.2 Mm to 4 Mm with aspect ratios be- +tween 1 and 5. In Fig. 14 we show a subfield, particularly rich in +Fig. 14. Examples of a group of campfires observed near perihelion on +2022 March 27. See Sect. 3.2.1. +Fig. 15. A long and slender campfire observed on 2022 March 27. The +left image shows the calibrated images (L2) with the original sensor +pixelisation, each pixel corresponds to (115 km)2) on the sun. On the +right, we show various cross-cuts through the campfire demonstrating +that the FWHM of the feature is 1-2 pixels. See Sect. 3.2.1. +campfires, taken on 2022 March 27 at a distance of 0.324 au from +the Sun. The 2022 March 27 L_FULL_HRES_HCAD_Coronal- +Dynamics dataset has a cadence of 1 min but several of the +R_BOTH_HRES_HCAD_Nanoflares data sets have cadences +down to 3 s. The visually identified sample of campfires in +Fig. 14 appear somewhat smaller (none is larger than 2 Mm) but +this needs to be confirmed by objective algorithmic detection. +Fig. 15 shows an example of particular long and slender +campfire demonstrating that loop-like features are present in the +quiet Sun with a width of the order of 200 km. It also demon- +strates that the spatial resolution of HRIEUV is pixel-limited. +3.2.2. EUV network flares +Also at larger scales than campfires (say ≳ 10 Mm), flare-like +brightenings are frequently seen in the quiet Sun. Fig. 16 shows +(top-right) the location of 4 such flare-like brightenings in a typi- +cal quiet Sun scenery observed by HRIEUV on March 27/28. The +time evolution of the 4th event is shown in Fig. 17 with corre- +sponding SDO/AIA 17.1 nm imagery in quadrature showing the +off-limb evolution. In X-rays, such events have been called Net- +work Flares (Krucker et al. 1997; Attie et al. 2016). The HRIEUV +extreme high resolution EUV images of the quiet Sun confirm +that these brightenings do indeed show many of the usual flare at- +tributes such as pre-flare sigmoids, dimmings, ribbons and post- +flare loops. At least for some of these (like in Fig. 17) we can +also confirm jets and filament eruptions, making them candidate +sites for mini-CMEs (Innes et al. 2009; Sterling et al. 2015). +Article number, page 10 of 19 + +20:51 +20:52 +20:53 +20:54 +20:55 +20:56 +20:57 +20:58 +1 Mmprofile across feature +1.01 +0.8 +1-2 pixels FWHM +0.4 +0.2 +00 +0 +10 +15 +20 +25 +30 +pixelsD. Berghmans et al.: First Perihelion of EUI on the Solar Orbiter mission +Fig. 16. EUV network flares observed in the quiet Sun. The top-left +panel shows an FSI 17.4 nm image with the white rectangle indicat- +ing the HRIEUV FOV that is zoomed in on the top-right panel. On this +HRIEUV image, taken at a time without network flares, the location of +four network flares is indicated. Three of these network flares are shown +in the bottom row of the figure. The 4th event is shown in Fig. 17. The +HRIEUV pixels correspond to (115 km)2 on the Sun. See Sect. 3.2.2. +Fig. 17. Time evolution of an EUV network flare seen by HRIEUV on +2022 March 28 at 17.4 nm (left) near disc center and by SDO/AIA +17.1 nm near the limb. This event corresponds to location 4 in the top- +right panel of Fig. 16. The off limb data confirm the eruptive character +of this EUV network flare. See Sect. 3.2.2. +3.2.3. Polar coronal holes +Fine-scale structure and dynamics of coronal holes is of partic- +ular interest for studies of the fast solar wind origin (e.g. Cir- +tain et al. 2007; Poletto 2015). Due to the progression of the +ascending phase of the solar cycle, polar coronal holes were +shrinking in 2022, so it was important to observe these struc- +tures at high spatio-temporal resolution early in the Solar Orbiter +mission. During the first perihelion passage in 2022, this was +done on three occasions: on March 6, March 30, and April 4–5. +The R_SMALL_HRES_MCAD_Polar-Observations SOOP was +used (see Table A.1). +On March 30 Solar Orbiter was situated at the distance of +0.33 au from the Sun, and HRIEUV reached the two-pixel spa- +tial resolution of around 240 km. This was the highest ever spa- +tial resolution reached in coronal hole observations. An excellent +Fig. 18. Examples of filaments and prominences observed in HRIEUV in +2022 March. See Sect. 3.2.4. +Fig. 19. South polar coronal hole imaged by HRIEUV on 2022 March +30. Solar north is up, west is to the right. See Sect. 3.2.3. +cadence of 3 s allowed observing numerous dynamic fine-scale +structures in the south polar coronal hole (see Figure 19), includ- +ing “bright points”, plumes, plumelets, jetlets, and jets (Chitta et +al. 2022, submitted). Numerous campfires were visible in the ad- +jacent quiet Sun area. +Similar datasets were taken for the north polar coronal hole +on March 6 and April 4–5, when the distance from the space- +craft to the Sun was 0.5 au and 0.37 au respectively (Table A.1). +The HRIEUV spatial resolution of around 270 km was reached +on April 4. Even if 7 HRIEUV images were taken at the very high +cadence of 2 s on March 6, the typical cadence was 30 s in both +datasets. +Article number, page 11 of 19 + +2 +4 +3. +10 Mm +2022-03-28T07:59 +2022-03-28T07:59 +3 +10 Mm +2022-03-28T07:35 +2022-03-27T23:17 +2022-03-27T21:4407:05 +07:27 +10 Mm +07:09 +07:31 +07:15 +07:35 +07:19 +07:39 +07:23 +07:43 +10 Mm2022-03-30T04:39:58.046A&A proofs: manuscript no. main +3.2.4. Filaments and prominences observations +The first perihelion passage of Solar Orbiter has also permitted +EUI to take close up observations of filaments and prominences +at high cadence and spatial resolution. On the disk, the width of +their core fine structure has been resolved by HRIEUV down to +the limit of the Hα instrument resolution (≈ 0.2′′). In Hα the +threads have a width distribution centered at 0.3′′, which means +≈ 225 km on the Sun (for a review of the observations see Parenti +2014). Fig. 18 shows some example of filaments seen mostly in +absorption by HRIEUV. Similar absorption features are also ob- +served by HRIEUV in coronal rain events (see section 3.1.5). The +top left panel shows a filament which was followed for about half +of an hour on 2022 March 18 at 10:30UT at a cadence of 5 s, +allowing to detect fast intensity variation at small scales along +and across the structure. Full Sun images show local activity on +2022 March 17, which led to the formation of the filament and +the opening of a small coronal hole. The merging of the polar +coronal hole and the dimming region formed by the eruption of +the filament is discussed in Ngampoopun et al. (2022; in prepa- +ration, this issue). +The second panel on the top row of Fig. 18 shows a filament +close to the limb, that was observed during the Full Disc Mo- +saic campaign on 2022 March 7 (see Section 2.2). Both the on +disk and off disk part of the filament shows a complex fine struc- +ture, highlighted by dark and bright alternating wavy shaped fea- +tures. During the same campaign, we also observed the promi- +nence shown on the bottom left. This has a tornado-like shape, +with thin bright and dark threads and a bright extension at the +base of the coronal cavity. These observations are very promis- +ing for future quiescent prominence and filament studies, as they +provide elements to derive the fine scale morphology, and char- +acterize the dynamics from possible injected plasma and waves +activity. The bottom-right panel shows part of an AR filament +observed in 2022 March 30. During the 30 min HRIEUV high ca- +dence sequence, the filament was quite active, with brightenings +in threads and fine dark features levitating on top of the main +dark body. The cadence of 3 – 5 s that was chosen for all the se- +quences shown in Fig. 18 appear to be adequate for studying the +dynamics at such a small scales. +3.3. Eruptions +Due to varying distance to the Sun, the FSI FOV (from disc cen- +ter to edge) changed from 4 R⊙ on 2022 March 2, to 2.3 R⊙ at per- +ihelion and back to 2.8 R⊙ on 2022 April 6. This large FOV, un- +precedented for an EUV imaging telescope, allowed us to mon- +itor the early evolution of eruptions. Table 1 lists the approxi- +mate starting time, the shape and the greatest height reached by +the eruptions observed by FSI during the period. Some exam- +ple prominence eruptions are shown in Fig. 20. They appear in a +multitude of shapes: surge-like, loop-like, curled-like eruptions +and have different kinematic behavior, from slow rising to fast +eruptions. In the following subsections, we highlight a number of +eruptions with particularly good coverage by other instruments +or with space weather relevance. +3.3.1. C2.8 flare: 2022 March 10 +The C2.8 flare from active region NOAA 12962 on 2022 March +10 (GOES X-ray peak at 20:33) was observed by FSI as a clas- +sical two-ribbon flare, and later a post-flare arcade, with a ca- +dence of 10 min in the 304 channel and 30 min in the 174 chan- +nel (see Fig. 21). It was also observed by STIX (Krucker et al. +2022-03-11T01:00 +2022-03-04T19:30 +2022-03-10T01:00 +2022-03-16T15:30 +2022-03-19T12:00 +2022-03-19T07:00 +2022-03-21T07:30 +2022-03-25T05:20 +2022-03-30T17:30 +Text +2022-03-31T03:30 +2022-03-28T12:20 +2022-03-26T21:18 +Fig. 20. Mosaic of prominence eruptions observed by FSI in the 30.4 nm +passband in 2022 March. The ’enhance off limb’ functionality of JHe- +lioviewer (Müller et al. 2017) was used when creating these graphics. +See Sect. 3.3 and 3.3.3. +2020) and EPD (Rodríguez-Pacheco et al. 2020) on Solar Or- +biter. The Earth was separated by 7.8◦ from Solar Orbiter. From +the Earth’s perspective the flare was near the central meridian +and associated with a partial halo coronal mass ejection (CME) +that eventually led to a moderate geomagnetic storm (Kp = 6) +on March 13 and 14. The evolution of the CME shock and its ef- +fect on ion acceleration was studied by Walker et al., 2022 (this +volume, in prep). +Article number, page 12 of 19 + +20 +30 ++15 +15 +0 +30 +45 +60 +0 +-20 +-40 +-60 +-60 +-80 +-80 +2022-03-04T19:30:20.36700 +CF- +-20 +-40 +-60 +-80 +2022-03-10T01:00:20.27760 +40 +20 +-60-45 +-B0 +2022-03-11T01:00:20.17860 +40 +20 +-601-45 +-30 +75.11 +15 +0 +0 +-20 +2022-03-16T15:30:20.76530 +45 60 1.175 +15 +0 +-20 +-40 +-60 +-80 +2022-03-19T07:00:20.184-45 +-B0 +±15 +0 +-20 +-40 +-60 +-80 +2022-03-19T12:00:20.215-30 +15 +30 +60 +0 +-20 +-40 +-60 +-60 +-80 +-80 +2022-03-21T07:30:20.234-60-45 +-30 +-15 +0 +-20 +-40 +-60 +2022-03-25T05:20:20.212 +0880 +80 +60 +40 +20 +20215-26T21:18:20.Q8 +15 +30 +45 +607540 +20 +-60 +-45 +-30 +15 +0 +0 +-20 +-40 +2022-03-28T12:10:20.21640 +20 +-60 +-45 +-30 ++15 +0 +-20 +2022-03-30T17:30:20.30230 +0 +-20 +-40 +-60 +-80 +2022-03-31T03:30:20.242D. Berghmans et al.: First Perihelion of EUI on the Solar Orbiter mission +Table 1. Eruptions observed by FSI between 2022 March 2 and 2022 April 6. +Start Date +Position +FSI channel +Comments +2022 March 04 +SW and E +304 +two prominences at SW (18:00, loop-like opening, up to 1.65 R⊙) +and E (21:00, jet-like, up to 2.25 R⊙) +2022 March 05 +SE +304, 174 +small prominence (12:30, along a loop, up to 1.24 R⊙) +2022 March 06 +NE +304 +small prominence (03:00, loop-like, up to 1.31 R⊙) +2022 March 08 +NE +174 +NE (08:10, jet-like?, up to 2.2 R⊙) +2022 March 08 +SE +174 +to the outer FOV +(21:00, fan-like, extended concave-out, up to 3.12 R⊙) +2022 March 09 +SE +304 +far in the FOV (19:30, twisted, 2.45 R⊙) +2022 March 10 +NW quadrant +174 +on-disk (18:30 dimming; 21:30 post-eruption arcade) +2022 March 10 +E +304 +end FOV (19:00, jet-like, 3.24 R⊙) +2022 March 10 +E +304 +end FOV (23:30, fan-like, 3.15 R⊙) +2022 March 12 +W +174 +to the end FOV (06:00, elongated sinusoidal?, up to 2.85 R⊙) +2022 March 13 +E +304, 174 +two small (00:30, 05:00, loop opening, 1.50 R⊙) +2022 March 14 +SW +174 +big (17:20, loop-like opening, 2.24 R⊙) +2022 March 16 +E +304 +2 curled prominences (13:00, 14:30, 2.65 R⊙) +2022 March16 +SE, NE, SW +174 +elongated curved SE (08:00, 2.60 R⊙) +2 eruptions, 14:10 NE and SW +2022 March17 +W +174 +small (06:30, 2.10 R⊙) +2022 March 18 +W +174 +small (11:00, concave-out, 2.30 R⊙) +2022 March 19 +W, SE +304 +prominence W (06:00, fan-like, 2.6 R⊙) and SE (10:30, curled, 1.81 R⊙) +2022 March 19 +SE +174 +(10:00, curled, 1.8 R⊙) +2022 March 20 +NE +304, 174 +prominence (08:00, curled, 2.32 R⊙) +2022 March 21 +SW +304, 174 +(05:30 UT, fan-like, 3.25 R⊙) +2022 March 24 +SE +174 +small (11:30, loop-like, 1.50 R⊙) +2022 March 25 +SE +304, 174 +(05:00, loop-like, 1.77 R⊙) +2022 March 26 +NW +304, 174 +(19:30, loop-like, 2.27 R⊙) +2022 March 27 +E +304, 174 +2 small eruptions (13:00, 19:00, loop-like, 1.5 R⊙) +2022 March 28 +E +304, 174 +prominence (11:20, loop-like + fan, 2.3 R⊙) +M4 flare and halo CME arriving at Earth on 2022 March 31 +2022 March 30 +NW, E +304, 174 +at NW (05:30, loop-like, 2.07 R⊙), and E (14:00, ragged, 1.9 R⊙) +2022 March31 +SW +304, 174 +(02:30, loop-like, 2.30 R⊙) +2022 April 02 +NE, SE +304, 174 +prominences (13:00, ragged, 2.7 R⊙) +2022 April 03 +SE +304 +prominence (15:00, untwisting to loop-like, 2.7 R⊙) +2022 April 04 +SE +304 +big filament (10:30, faint elongated off-limb, 2.11 R⊙) +2022 April 05 +SW +174 +(13:00, fan-like, 2 R⊙) +2022 April 06 +SW +304, 174 +prominence (22:00, ragged, 1.7 R⊙) +3.3.2. Limb CME: 2022 March 21 +Starting around 05:30 UT on 2022 March 21, an eruption was +observed by FSI at the SW limb (see Fig. 22) that led to a partial +halo CME observed from the Earth. The CME was associated +with a Type-II radio burst (measured by RPW, Maksimovic et al. +2020), with X-ray emission (observed by STIX) and with a wide +SEP event measured by EPD, SOHO and STEREO-A (34◦ to the +east of the Earth). Solar Orbiter was at 0.34 au from the Sun, and +44◦ west of the Sun-Earth line. The source location of the CME +was located close to the west limb as seen from Solar Orbiter, +at least partially occulted. The CME was fast, with speeds above +1000 km s−1. At that time EUI was executing the Slow-Wind- +Connection SOOP and observed in the 30.4 nm passband of FSI +with a cadence of 30 min and in the 17.4 nm passband of FSI +with 10 min cadence. A time sequence of the eruption can be +seen in Fig. 22. +3.3.3. East limb eruption: 2022 March 30 +On 2022 March 30, at around 14:00 UT, FSI observed in the +30.4 nm channel a prominence erupting at the East limb (see +lower left panel of Fig. 20) which was further observed by +SolOHI (Howard et al. 2020). Prominence material is still vis- +ible in the FSI FOV at around 20:30 UT. At around 17:30 a flare +was observed on the disk (N15W30 Stonyhurst coordinates) and +a bright loop is visible off-limb overlapping with the prominence +but not disturbing its evolution. This indicates that the promi- +nence was situated far away from the flare. By inspecting the +Article number, page 13 of 19 + +A&A proofs: manuscript no. main +2022−03−10T22:05:20 UT (Earth) +EUI/FSI174 +0 +100 +200 +300 +400 +500 +x [Mm] +0 +100 +200 +300 +400 +500 +y [Mm] +0.5 +1.0 +1.5 +2.0 +log(Intensity [DN/s]) +Fig. 21. FSI observations in the 17.4 nm passband of the C2.8 flare on +2022 March 10. The images present the arcades of the post-flare loops. +See Sect.3.3.1. +. +Fig. 22. Time sequence of the eruption on 2022 March 21, as seen by +FSI in the 30.4 nm passband. The ’enhance off limb’ functionality of +JHelioviewer (Müller et al. 2017) was used when creating these graph- +ics. See Sect. 3.3.2. +SDO/AIA304 and STEREO-A/EUVI304 (Howard et al. 2008) +movies one could see an extended filament erupting at the East +of the flare. +FSI observed in the 17.4 nm passband faint material erupting +at around 14:00 at the East limb followed by a big dimming off- +limb at around 17:30. Material moving out is still observed at +20:50. A large EUV wave is observed on-disk. +The +eruption +was +associated +with +a +flux-rope +like +coronal mass ejection observed by STEREO-A/COR2 and +SOHO/LASCO-C2 (Brueckner et al. 1995) coronagraphs at +West limb at 18:23 and 18:12 respectively. +3.3.4. North-east limb eruption: 2022 April 2 +On 2022 April 2, FSI observed in the 30.4 nm channel a filament +erupting at the NE limb (as seen from Solar Orbiter) between +13:00 and 13:30 UT. It was associated with an M3.9-class flare. +The event was also captured by several other remote-sensing in- +struments on Solar Orbiter such as SPICE, STIX, and SoloHI. +Interestingly, the erupting filament was also monitored a few +days prior and during its eruption by Earth-based assets such +as Solar Dynamics Observatory, IRIS and Hinode. Its position +from the Sun-Earth line was N12W68. The flare recorded by +GOES soft-X ray observations indicates a start at 12:56:00, with +a peak at 13:55:00 and followed by a long duration event. +This event is particularly interesting for several reasons: first, +the large coverage available with different instruments allows us +to follow the pre-flare phase, during which the filament slowly +rises and pushes overlying coronal arcades away, as modelled +in 3D numerical simulations of eruptive flares. This may be +linked to the observed large-scale reconfiguration reported in +section 3.1.7. Second, Doppler velocity and intensity changes in +several lines are reported between the upper chromosphere and +transition regions (SPICE diagnostics) as well as coronal lines +(Hinode/EIS diagnostics) with different view points. This is the +first time that such an event is seen stereoscopically with dif- +ferent spectrometers. Finally, the extended coverage, from spec- +troscopy to EUV and X-Ray imaging allows us to understand +the evolution of the magnetic field changes during the different +phases of the flare. A dedicated study of this event is available in +Janvier et al. 2022 (this volume, in prep) and in Ho et al. 2022 +(this volume, in prep). +4. Instrument Performance at perihelion +The pre-flight instrument characterisation is discussed in tele- +scope specific papers in this issue (Auchère and EUI consor- +tium partners 2022 (this volume, in prep); Aznar Cuadrado and +EUI consortium partners 2022 (this volume, in prep); Gissot and +EUI consortium partners 2022 (this volume, in prep)). The pe- +riod around the 2022 March 27 perihelion was the first time the +instrument was operated in the environment for which it was pri- +marily designed. In this section we review how the instrument +was operated technically and the resulting performance. Over- +all, FSI and HRIEUV performed largely nominally while HRILya +suffered from a temporary degradation in throughput and resolu- +tion (see below). +4.1. Sensors +The three EUI telescopes share the same CMOS sensor design +(Rochus et al. 2020). The sensors consist of two parts of each +1536 x 3072 pixels, stitched together as a 3072 x 3072 array. +The HRI sensors are used sub-fielded to 2048 x 2048 pixels. +Careful inspection reveals that the stitching line remains visible +in all three telescopes but most noticeably in HRILya (see Fig. 25 +c) at x=180 Mm). +Each pixel has a high-gain and low-gain read-out which can +be brought to the ground independently or selected per pixel per +intensity threshold. Onboard electronics then re-scales the high- +gain and low-gain signals from different pixels into one coher- +ent intensity range over all pixels in a ’recombined’ image. In +Article number, page 14 of 19 + +2022-03-21T05:30 +2022-03-21T06:00 +2022-03-21T06:30 +2022-03-21T07:00D. Berghmans et al.: First Perihelion of EUI on the Solar Orbiter mission +the 2022 March/April period, FSI and HRIEUV have been nom- +inally operated in the combined gain mode, resulting in 15-bit +images (an intensity range in Level 1 files of 0–32767 DN for +FSI and 0–25600 DN for HRIEUV). The selection threshold be- +tween low-gain and high-gain read-out happens near a Level 1 +intensity level of 1097 DN for FSI and 1118 DN for HRIEUV. +This transition is weakly visible in the FSI and HRIEUV images +as a band of enhanced noise, which is to be expected given the +different photon statistics in the low-gain and high-gain read-out +channels. In contrast, HRILya has been operated exclusively in +low-gain read-out resulting in 12-bit images (an intensity range +of 0–4095 DN in the Level 1 image files) and which do not show +such transition. +Other sensor artefacts, affecting FSI, are dark vertical bands +in very faint areas, aligned with the brightest on-disk features. +This effect is assumed to be caused by saturation of the high-gain +read-out of pixels in the same column and is still under investi- +gation. A post-processing semi-empirical fix is being developed. +4.2. Onboard Processing +EUI is equipped with software controlled onboard calibration +electronics to correct the images pixel-wise for offset and flat +field before compression. For FSI, pre-flight offset and flat field +maps are available on board and have been applied until 2022 +March 16 when it was discovered that the flat field map was not +applied correctly. Correction for the flat field was turned off at +this time and subsequent FSI images have only the offset map +applied. +For HRIEUV, only a synthetic 4-column pattern is subtracted +that mimics the observed offset. For HRILya no onboard correc- +tion is applied. +Despite the close solar proximity, radiation hits on the sen- +sors have been very limited and the onboard cosmic ray corrector +has therefore not been employed. Enhanced radiation hits were +observed, e.g. following the 2022 March 21 event (see subsec- +tion 3.3.2). +4.3. Image resolution +During commissioning, the FWHM of the FSI Point Spread +Function (PSF) was estimated to be 1.5 pixels, or 6.66". As +shown by Fig. 23, there is no sign of changes so far. At closest +approach, i.e. at 0.3 au, this corresponds to a resolution of 2.5" +(two pixels) as seen from 1 au, similar to that of STEREO/EUVI +(2.4", Wülser et al. 2007). +The excellent resolving quality of HRIEUV was confirmed +during perihelion through identification of point-like features +(Fig. 24), and slender features (Fig. 15), with a FWHM width of +about 1.5 pixels. This is consistent with the spatial resolution of +the HRIEUV telescope being equal the Nyquist sampling limit of +2 pixels, 2 × 0.492′′. At perihelion just inside 0.3 au this corre- +sponds to about 2 × 100 km on the Sun. +From the beginning of the mission, the HRILya spatial reso- +lution was found to be lower than expected, with a first estimate +placing it at around 3′′ (see Berghmans et al. 2021). However, +during the perihelion approach of Solar Orbiter the telescope +has shown a further substantial degradation of spatial resolution, +contrast and throughput. Fig. 25 shows three images of quiet sun +regions taken on 2022 March 8 (where Solar Orbiter was at a dis- +tance to the Sun of 0.49 au), March 22 (at 0.33 au), and March +30 (at 0.34 au). All targets were selected to be near disk centre. +The loss in performance can be clearly seen in panels (b) and +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +Fig. 23. Enlargements of selected compact features observed by FSI at +17.4 nm on 2022 March 7 at 06:20:30 UT. Assuming that the sources +are unresolved, the green and red (horizontal and vertical respectively) +profiles indicate a FWHM width of the effective Point Spread Function +of 1.5 pixels. See Sect. 4.3. +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +0 +5 +10 +15 +Fig. 24. Enlargements of selected compact features observed by HRIEUV +on 2022 March 7 at 00:41:55 UT. Assuming that the sources are unre- +solved, the green and red (horizontal and vertical respectively) profiles +indicate a FWHM width of the effective Point Spread Function of 1.5 +pixels. See Sect. 4.3. +(c), immediately before and after the closest approach to the Sun +on 2022 March 26 (at 0.32 au). Most obvious is the resolution +degradation which may be a result of a heat effect on the en- +trance filter of the HRILya telescope. In addition, as Solar Orbiter +approaches the Sun, both the contrast and throughput degrade by +Article number, page 15 of 19 + +A&A proofs: manuscript no. main +approximately 37 % with respect to data taken before the perihe- +lion (around mid-February 2022), and these recover slowly after +perihelion passage. +4.4. Filters and light leaks +The reasons for the observed overall loss of performance of +HRILya during perihelion passage are currently under investiga- +tion. Experience from ground testing of the entrance filter with +heating to 200 ◦C has revealed a non-linear decline of its trans- +mission of up to 40 % as a function of temperature. Part of +the loss of the channel’s throughput may be associated with the +temperature dependency of the filter. Consistent with this, some +throughput and resolution recovery was observed further from +the Sun, on 2022 June 12, which was the first HRILya obser- +vations after 2022 April. A full assessment of the evolution of +throughput since launch and of its comparison with the expec- +tations from ground calibration will be the subject of a separate +publication. +In contrast to HRILya, the EUV channels may be affected by +light leaks. Two issues are known to affect the FSI filters: +– A faint light leak, likely caused by a pinhole in the front filter, +affects the images from both channels. Due to the specific +design of FSI, its visibility depends on the distance to the +Sun and pointing of the spacecraft. There is no quantitative +correction for this yet. This has no impact for morphological +studies, but care must be taken for photo-metric analysis off- +disk. +– A very faint light-leak, invisible in regular images, affects the +30.4 nm data taken in coronagraph mode. Its origin is still +unknown, but only a small number of images are affected. +4.5. Pointing error and jitter +The pointing information in the World Coordinate System +(WCS) keywords of the EUI Level 1 FITS files are based on the +as-flown Solar Orbiter spacecraft kernels. As such, these key- +words capture most of Solar Orbiter pointing instabilities, but +unfortunately not all. Even after correcting for the known point- +ing variation, occasional jitter remains visible from image to +image (as well as slower trends) in high cadence HRIEUV se- +quences. But in general, HRIEUV images do not seem to be af- +fected much by jitter blurring. +For FSI images, in which the solar limb is always visible, the +WCS pointing keywords are updated in the EUI Level 2 FITS +files with much more precise information from a procedure that +fits a circle to the solar limb. This is unfortunately not possible +for HRIEUV and HRILya image sequences and the data user is +advised to use alignment methods to remove the remaining jitter. +5. Conclusions +During the Solar Orbiter perihelion passage of 2022 March 26, +and the weeks before and after, EUI collected more than 35000 +images. Solar Orbiter reached a distance to the Sun as low as +0.32 au, closer to the Sun than any other coronal imager. Both +FSI and HRIEUV operated at design specifications during the per- +ihelion passage but HRILya suffered from an unexpected (but re- +versible) performance degradation near perihelion that needs to +be studied further. +EUI has achieved the highest resolution images ever of the +solar corona in the quiet Sun and polar coronal holes. Ubiquitous +EUV brightenings (a.k.a. campfires) and small scale jets were re- +covered down to the resolution limit of HRIEUV of about 200 km +on the solar surface. These smallest features require further in- +vestigations for their relevance to the heating of the corona and +the powering of the solar wind. +Whereas the Hi-C sounding rocket (Kobayashi et al. 2014; +Rachmeler et al. 2019) achieved comparable resolution in active +regions, HRIEUV imaged active regions at much longer sequence +durations (hours) at high cadence (3 s). Known phenomena such +as coronal braiding, decayless oscillations, coronal rain and flar- +ing activity were observed in unprecedented details. +The highest resolution full disc image ever was constructed +as a mosaic of 25 high resolution images. Together with the PHI +and SPICE instruments onboard Solar Orbiter, this full disc mo- +saic will be repeated twice per year when Solar Orbiter crosses +a distance of 0.5 au from the Earth. +Meanwhile, the big novelty of FSI, namely its very extended +FOV, allowed the imaging of eruptions off limb further than ever +before, with in particular the prominence eruption showing a be- +wildering variety in structural appearances. +Future perihelia will go another 10 % closer to the Sun, to a +distance of 0.29 au from the Sun, and as the mission progresses, +Solar Orbiter/EUI will also observe from increasing solar lati- +tudes. Many of the SOOPs and EUI observations presented in +this paper will be repeated from these upcoming vantage points. +Special attention will be paid to deepening joint observations +with other instruments on Solar Orbiter but also with Earth- +bound observatories in space and on the ground. +This paper presented how the EUI observations contributed +to the various Solar Orbiter Observations Programs (SOOPs) +that implement cross-instrument science goals. By highlighting +particular features and events, many of which require further +study, this paper intended to demonstrate the potential of the +EUI data and to inspire external users to take part in the EUI +data analysis. The EUI dataset presented in this paper has been +distributed as part of the EUI Data Release 5.0 (Mampaey et al. +2022) and is freely accessible. We encourage EUI data users to +read the release notes and get in contact with the EUI team for +specific support. +Acknowledgements. The building of EUI was the work of more than 150 indi- +viduals during more than 10 years. We gratefully acknowledge all the efforts +that have led to a successfully operating instrument. The authors thank the Bel- +gian Federal Science Policy Office (BELSPO) for the provision of financial +support in the framework of the PRODEX Programme of the European Space +Agency (ESA) under contract numbers 4000112292, 4000134088, 4000134474, +and 4000136424. The French contribution to the EUI instrument was funded by +the French Centre National d’Études Spatiales (CNES); the UK Space Agency +(UKSA); the Deutsche Zentrum für Luft- und Raumfahrt e.V. (DLR); and the +Swiss Space Office (SSO). PA and DML acknowledge funding from STFC +Ernest Rutherford Fellowships No. ST/R004285/2 and ST/R003246/1, respec- +tively. SP acknowledges the funding by CNES through the MEDOC data and op- +erations center. 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See Sect. 4.3. +Article number, page 17 of 19 + +(a) +(0.49 AU) +300 +100 +2022-03-08T00:43:24 UT +100 +200 +300 +X [Mm] +250 +(b) +(0.33 AU) +200 +150 +Y [Mm] +100 +50 +2022-03-22T16;26:10 UT +0 +50 +100 +150 +200 +250 +X [Mm] +250 +(c) +(0.34 AU) +200 +150 +[Mm] +Y +100 +50 + 2022-03-30T22:00;10 UT +0 +50 +100 +150 +200 +250 +X [Mm]A&A proofs: manuscript no. main +Table A.1. Summary of SOOPs and corresponding EUI datasets. In between the SOOPs, additional FSI synoptic images have been taken that are +not listed in this table. Some specific calibration datasets have also been omitted. +# images +cadence +start +end +comment +L_SMALL_MRES_MCAD_Connection-Mosaic +2022-03-01 18:00 +2022-03-03 03:21 +3 pointings +HRIEUV +810 +2 min +2022-03-02 00:00 +2022-03-03 03:00 +HRILya +809 +2 min +2022-03-02 00:00 +2022-03-03 03:00 +FSI174 +108 +15 min +2022-03-02 00:01 +2022-03-03 02:45 +FSI304 +108 +15 min +2022-03-02 00:01 +2022-03-03 02:46 +L_SMALL_MRES_MCAD_Connection-Mosaic +2022-03-30 07:55 +2022-03-31 17:40 +6 pointings +HRIEUV & +576 +30 s +2022-03-30 11:00 +2022-03-31 15:47 +6 bursts of 48 min at 11:00, 18:00, +HRILya +288 +60 s +22:00, 03:30, 09:00, 15:00 +FSI174 +188 +10 min +2022-03-30 08:00 +2022-03-31 17:30 +FSI304 +64 +30 min +2022-03-30 08:00 +2022-03-31 17:30 +L_SMALL_HRES_HCAD_Slow-Wind-Connection +2022-03-03 06:00 +2022-03-06 16:45 +various pointings +HRIEUV & +3x720 +5 s +2022-03-03 09:40 +2022-03-05 16:20 +1h bursts starting 3th 09:40, +HRILya +3x720 +5 s +4th 10:45, 5th 15:20 +FSI174 +479 +10 min +2022-03-03 06:00 +2022-03-06 15:11 +FSI304 +160 +30 min +2022-03-03 06:00 +2022-03-06 15:01 +L_SMALL_HRES_HCAD_Slow-Wind-Connection +2022-03-17 06:00 +2022-03-22 00:00 +various pointings +HRIEUV & +5x720 +5 s +2022-03-17 09:47 +2022-03-21 12:36 +1h bursts: 17th 09:47, 18th 10:10, +HRILya +5x720 +5 s +19th 10:36, 20th 11:27, 21th 11:36 +FSI174 +630 +10 min +2022-03-17 06:00 +2022-03-21 23:51 +FSI304 +210 +30 min +2022-03-17 06:00 +2022-03-21 23:30 +R_SMALL_HRES_MCAD_Polar-Observations +2022-03-06 16:45 +2022-03-06 21:50 +pointing: North pole +HRIEUV +7, 149 +2 s, 30 s +2022-03-06 17:34 +2022-03-06 18:51 +HRILya +105 +60 s +2022-03-06 17:05 +2022-03-06 18:50 +FSI174 +75 +90 s +2022-03-06 17:05 +2022-03-06 21:21 +FSI304 +5 +30 min +2022-03-06 19:20 +2022-03-06 21:20 +R_SMALL_HRES_MCAD_Polar-Observations +2022-03-30 03:30 +2022-03-30 07:00 +pointing: South Pole +HRIEUV +600 +3 s +2022-03-30 04:30 +2022-03-30 05:00 +HRILya +360 +5 s +2022-03-30 04:30 +2022-03-30 05:00 +FSI174 +18 +10 min +2022-03-30 03:30 +2022-03-30 07:01 +FSI304 +7 +30 min +2022-03-30 03:50 +2022-03-30 07:01 +regular spacing +R_SMALL_HRES_MCAD_Polar-Observations +2022-04-04 16:25 +2022-04-05 23:53 +pointing: North Pole +HRIEUV +90,755 +30 s & 60 s +2022-04-04 16:25 +2022-04-05 05:44 +variable cadence +HRILya +786 +60 s +2022-04-04 16:25 +2022-04-05 05:44 +FSI174 +190 +10 min +2022-04-04 16:30 +2022-04-05 23:50 +5 min cadence in last 3h +FSI304 +51 +30 min +2022-04-04 16:30 +2022-04-05 20:30 +R_BOTH_HRES_HCAD_Nanoflares +2022-03-06 21:50 +2022-03-07 03:00 +pointing: active region +HRIEUV +7, 1188, 149, 60 +2, 5, 12, 20 s +2022-03-07 00:29 +2022-03-07 03:00 +variable cadence +HRILya +783, 60 +12, 20 s +2022-03-07 00:00 +2022-03-07 03:00 +variable cadence +FSI174 +354 +30 s +2022-03-07 00:00 +2022-03-07 03:00 +R_BOTH_HRES_HCAD_Nanoflares +2022-03-08 00:00 +2022-03-08 03:00 +pointing: disc center, quiet Sun +HRIEUV +588, 1188, 149, 60 +3, 5, 12, 20 s +2022-03-08 00:00 +2022-03-08 03:00 +variable cadence +HRILya +783, 60 +12, 20 s +2022-03-08 00:00 +2022-03-08 03:00 +variable cadence +FSI174 +355 +30 s +2022-03-08 00:00 +2022-03-08 03:00 +R_BOTH_HRES_HCAD_Nanoflares +2022-03-17 00:00 +2022-03-17 02:55 +quiet Sun +HRIEUV +600 +3 s +2022-03-17 00:18 +2022-03-17 00:48 +HRILya +150 +12 s +2022-03-17 00:18 +2022-03-17 00:48 +FSI304 +28 +60 s +2022-03-17 00:03 +2022-03-17 01:04 +gap between 00:16 and 00:50 +R_BOTH_HRES_HCAD_Nanoflares +2022-03-17 03:00 +2022-03-17 05:55 +pointing: active region +HRIEUV +900 +3 s +2022-03-17 03:18 +2022-03-17 04:03 +HRILya +348 +5 s +2022-03-17 03:18 +2022-03-17 03:47 +some images missing +FSI304 +28 +60 s +2022-03-17 03:03 +2022-03-17 04:03 +gap between 03:16 and 03:50 +R_BOTH_HRES_HCAD_Nanoflares +2022-03-30 00:00 +2022-03-30 03:24 +pointing: active region +HRIEUV +900 +3 s +2022-03-30 00:03 +2022-03-30 00:48 +HRILya +360 +5 s +2022-03-30 00:18 +2022-03-30 00:48 +FSI174 +14 +10 min +2022-03-30 01:00 +2022-03-30 03:20 +03:10 missing +FSI304 +5 +30 min +2022-03-30 01:00 +2022-03-30 03:20 +Article number, page 18 of 19 + +D. Berghmans et al.: First Perihelion of EUI on the Solar Orbiter mission +Table A.2. Summary of SOOP instances and corresponding EUI datasets. In between the SOOPs, additional FSI synoptic images have been taken +that are not listed in this table. Some specific calibration datasets have also been omitted. +# images +cadence +start +end +comment +Full Disc Mosaic +2022-03-07 07:00 +2022-03-07 11:30 +25 pointings +HRIEUV +450 +- +2022-03-07 07:01 +2022-03-07 11:30 +18 images/pointing, HG/LG +HRILya +200 +- +2022-03-07 07:04 +2022-03-w07 11:29 +8 images/pointing +FSI174 +25 +11 min +2022-03-07 07:05 +2022-03-07 11:30 +FSI304 +25 +11 min +2022-03-07 07:05 +2022-03-07 11:30 +L_FULL_HRES_MCAD_Coronal-He-Abundance +2022-03-07 16:00 +2022-03-07 20:00 +FSI174 +8 +30 min +2022-03-07 16:00 +2022-03-07 19:31 +HG, occulted, exptime 1000s +R_BOTH_HRES_MCAD_Bright-Points +2022-03-08 08:10 +2022-03-08 16:45 +pointing: disc center +HRIEUV +120 +1 min +2022-03-08 08:10 +2022-03-08 10:10 +HRILya +120 +1 min +2022-03-08 08:10 +2022-03-08 10:10 +FSI174 +77 +5 min +2022-03-08 08:10 +2022-03-08 16:41 +5 min cadence till 14:05 +FSI304 +74 +5 min +2022-03-08 08:10 +2022-03-08 16:30 +5 min cadence till 14:05 +L_FULL_HRES_HCAD_Coronal-Dynamics +2022-03-22 03:10 +2022-03-22 16:30 +pointing: disc center +HRIEUV +1600 +30 s +2022-03-22 03:10 +2022-03-22 16:30 +HRILya +800 +60 s +2022-03-22 03:10 +2022-03-22 16:30 +FSI174 +68 +10 min +2022-03-22 03:18 +2022-03-22 16:21 +gap 15:00-16:00 +FSI304 +68 +10 min +2022-03-22 03:18 +2022-03-22 16:21 +gap 15:00-16:00 +L_FULL_HRES_HCAD_Coronal-Dynamics +2022-03-27 19:40 +2022-03-28 16:30 +pointing: disc center +HRIEUV +1249 +60 s +2022-03-27 19:40 +2022-03-28 16:30 +HRILya +795 +60 s +2022-03-27 19:40 +2022-03-28 09:00 +FSI174 +114 +10 min +2022-03-27 19:48 +2022-03-28 16:26 +gap 15:00-16:00 +FSI304 +114 +10 min +2022-03-27 19:48 +2022-03-28 16:26 +gap 15:00-16:00 +L_BOTH_HRES_LCAD_CH-Bounday-Expansion +2022-03-25 19:40 +2022-03-27 00:00 +pointing: disc center +FSI174 & +140 +10 min +2022-03-25 19:40 +2022-03-27 00:00 +gap 04:00-05:00, ... +FSI304 +140 +10 min +..., 15:00-16:00, 16:30 -19:30 +L_FULL_HRES_HCAD_Eruption-Watch +2022-03-22 19:40 +2022-03-23 16:30 +pointing: disc center +HRIEUV & +60 +30 s +2022-03-22 03:30 +2022-03-23 16:30 +30 min burst ... +HRILya +30 +60 s +... at 03:30, 16:00 +FSI174 +188 +6 min +2022-03-22 19:40 +2022-03-23 16:25 +gap 15:00-16:00 +FSI304 +188 +6 min +2022-03-22 19:40 +2022-03-23 16:25 +gap 15:00-16:00 +L_FULL_HRES_HCAD_Eruption-Watch +2022-03-29 03:10 +2022-03-30 00:00 +pointing: disc center +HRIEUV & +80 +30 s +2022-03-29 12:00 +2022-03-29 12:40 +30 min burst ... +HRILya +40 +60 s +... at 03:30, 16:00 +FSI174 +208 +6 min +2022-03-29 03:10 +2022-03-29 23:53 +gap 15:00-16:00 +FSI304 +208 +6 min +2022-03-29 03:10 +2022-03-29 23:53 +gap 15:00-16:00 +R_SMALL_MRES_MCAD_AR-Long-Term +2022-03-31 17:45 +2022-04-04 20:26 +pointing: Active Region +HRIEUV & +4x450 +10 s +2022-04-01 09:19 +2022-04-04 10:34 +75 min burst at 09:19 on ... +HRILya +4x150 +30 s +...April 1, 2, 3, 4 +FSI174 +538 +10 min +2022-03-31 17:50 +2022-04-04 20:26 +gap April 4 18:30-20:00 +FSI304 +179 +30 min +2022-03-31 18:00 +2022-04-04 20:01 +30 min cadence till April 4 18:30 +Article number, page 19 of 19 + diff --git a/_NFRT4oBgHgl3EQfsjc9/content/tmp_files/2301.13624v1.pdf.txt b/_NFRT4oBgHgl3EQfsjc9/content/tmp_files/2301.13624v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e9d42bf72c659ce403a13843a4b94f4bac2f55be --- /dev/null +++ b/_NFRT4oBgHgl3EQfsjc9/content/tmp_files/2301.13624v1.pdf.txt @@ -0,0 +1,801 @@ +A Kubernetes-Based Edge Architecture for Controlling the Trajectory of a +Resource-Constrained Aerial Robot by Enabling Model Predictive Control +Achilleas Santi Seisa, Sumeet Gajanan Satpute and George Nikolakopoulos +Abstract— In recent years, cloud and edge architectures have +gained tremendous focus for offloading computationally heavy +applications. From machine learning and Internet of Thing +(IOT) to industrial procedures and robotics, cloud computing +have been used extensively for data processing and storage +purposes, thanks to its “infinite” resources. On the other +hand, cloud computing is characterized by long time delays +due to the long distance between the cloud servers and the +machine requesting the resources. In contrast, edge computing +provides almost real-time services since edge servers are located +significantly closer to the source of data. This capability sets +edge computing as an ideal option for real-time applications, +like high level control, for resource-constrained platforms. In +order to utilize the edge resources, several technologies, with +basic ones as containers and orchestrators like Kubernetes, have +been developed to provide an environment with many features, +based on each application’s requirements. In this context, +this works presents the implementation and evaluation of a +novel edge architecture based on Kubernetes orchestration for +controlling the trajectory of a resource-constrained Unmanned +Aerial Vehicle (UAV) by enabling Model Predictive Control +(MPC). +Index Terms— Robotics; Edge Computing; Kubernetes; UAV; +MPC. +I. INTRODUCTION +Nowadays, as technology is progressing and the need for +computational resources is continuously increasing, different +computation layers have been evolved. We can differ these +layers into four distinct categories, cloud, fog, edge and +devices, while each one of them has its own different +characteristics and utilization. At the same time, all of them +can be combined with each other to create an ecosystem +for the utilization of external computational resources. As +these technologies mature, researchers and engineers use +them more and more to offload their applications, due to the +capabilities and features they provide [1]. Additionally, since +the mentioned computation layers have attracted tremendous +focus, several state-of-the-art technologies have been devel- +oped and are promising to revolutionize many technolog- +ical fields, like containerized applications and containers’ +orchestrators. In this framework, robotics can take a huge +advantage of the external resources and many resource- +constrained platforms can make the most out of them, since +they will be able to run algorithms that they can not run on +their onboard processors. In this context, edge is emerging +This work has been partially funded by the European Union’s Horizon +2020 Research and Innovation Programme AERO-TRAIN under the Grant +Agreement No. 953454. +The authors are with the Robotics and AI Team, Department of Computer, +Electrical and Space Engineering, Lule˚a University of Technology, Lule˚a +Corresponding Author’s email: achsei@ltu.se +since it can provide tremendous resources for enhancing +the performance, and the overall efficiency of autonomous +operations, and at the same time minimize the travel time +delays when transmitting data from UAVs, like in [2] and [3]. +Thus, edge can be established as a promising solution +for time critical operations, like offloading computationally +costly controllers, of resource-constrained platforms. In this +article, we propose an architecture where we offload the +Model Predictive Control method, which is a relatively heavy +controller, to the edge, as a containerized application, and we +use Kubernetes for managing the containers. +Researchers seek to utilize the advantages of edge com- +puting for the benefit of robots. However, researchers have +to overcome some limitations and challenges, in order to +use these technologies universally. In [4] an architecture +consisting of all four computation layers is used to offload +the localization and mapping problem from robots. in this +case, edge is operating as a layer between sensor devices, +gateways, and cloud servers for enhancing the quality of +services, while in [5] edge is used to design a search planner +algorithm using deep learning for UAVs. In [6] edge and +cloud were utilized in terms of storage and computational +resources for deep robot learning including object recogni- +tion, grasp planning and localization of the computational. +Some works that utilized edge for robotic applications by +implementing Kubernetes or container based architectures +can be summarized as it follows. In [7] researcher tried to +automate, by using Kubernetes orchestration, the process of +making decision in terms of placement of the expected work- +load to edge, fog and cloud, for robotic applications. In [8], a +methodology based on docker and Kubernetes for ROS-based +robotic application is presented. In that case, the architecture +was evaluated by experimental results obtained by a mobile +robot interacting with an industrial agile production chain. +In the works mentioned above, the approach regarding +edge computing is mainly towards non-time critical tasks. +High level controllers must operate almost real-time. In [9] +an architecture was proposed where the control method +consists of the combination of a LQR running on the device +and an MPC running the optimization both on edge and +cloud, while in [10] two complimentary MPCs are running, +one on a local edge and one on the cloud. In comparison to +our proposed work, these articles are partly offloading the +MPC method on the edge and are focused on evaluating the +system in terms of related latency, and the related uncertainty +for several cases. +The motivation behind this work is to fill the gap re- +garding edge enabled robotics. Even though edge comput- +arXiv:2301.13624v1 [cs.RO] 31 Jan 2023 + +ing has proven to be a promising technology to expand +the autonomous capabilities of resource-constrained robotic +platforms, especially when combined with 5G networks, +the research that has been done around this area is rel- +atively limited. Despite the fact that the great advantage +of edge computing is the ability of enabling almost real- +time operation by offloading the computing process on the +edge, most researchers have focused on utilizing edge for +offline procedures. Thus, the contribution of this article is +to present a novel edge architecture for enabling the time +sensitive operation of controlling the trajectory of a resource- +constrained UAV in real-time through MPC. Control is one +of the basic components of autonomy, thus the performance +and efficiency is the main criteria when choosing a controller. +Model predictive controllers are widely used on UAVs due +to their characteristics and optimal behavior, but they are +computationally costly, thus some UAVs, deployed with light +processors, like Raspberry Pi, can not handle them. By +utilizing the proposed architecture, we will be able to use +edge resources in order to offload the MPC and control +resource-constrained platforms by closing the loop over the +edge. Additionally, we are using Kubernetes orchestration +that provides best practices for cloud and edge applications +but inserts some challenges that we have to overcome. +The rest of the article unfolds in the following Sections. +In Section II, we describe the Kubernetes-based edge archi- +tecture, while in Section III, we give a brief overview of +the UAV and MPC model. In Section IV, we present the +simulation results of the edge architecture in terms of time +delays and performance. Finally, in Section V, we conclude +the article by highlighting the main points of this work, and +we propose future directions. +II. KUBERNETES EDGE ARCHITECTURE +The proposed architecture is based on Kubernetes. Ku- +bernetes is a container orchestrator, developed by Google. +Before we start analyzing the Kubernetes-based architecture, +we have to describe the containers developed for this work. +Afterward, we are going to present the system’s architecture +and the Robotic Operating System (ROS) framework that +was utilized for the UAV-MPC system. Finally, we will +describe the communication layer and network. +Containers are based on software that creates an operating +environment and are deployed only with the necessary and +chosen packages, libraries and dependencies, in order to +run a specific application. The application running in this +form is called a containerized application. Containers are +based on images that are the nominal state of containers +before they get deployed. An image can be used to deploy +many containers. For our system, we deployed two docker +containers. One container is responsible for running the +controller and all the necessary libraries and dependencies +for its smooth and reliable operation, and the other is +responsible for running the ROS master, which takes care of +the communication between the ROS nodes. To deploy the +two docker containers, we had to developed two different +docker images. For both images, we used ROS Noetic on +Ubuntu 20.04 entrypoint, and we built on top of them. For the +first image, we included several ROS packages and libraries, +as well as an optimization engine for the MPC containerized +application, while for the second image we just needed to +run the ROS master. For a more complex application, we +could split it into more containers, each one of them would +be assigned a specific task. +Fig. 1. +Diagram of the Kubernetes-based edge architecture for the UAV- +MPC system +Once we had developed the docker images, we were able +to deploy the docker containers inside the Kubernetes cluster. +We decided to use Kubernetes due to the features it provides +for our containers. Kubernetes gives us the capability to +manage our containers and automates the whole process +of deploying the containers, assign them resources, check +their health. The services and features that Kubernetes is +providing can be extremely helpful for our application, since +they give us the chance to manage and monitor our system +in an optimal way, and it can even more handful when we +have to deploy more containers and the system get more +and more complex. The Kubernetes architecture is depicted +in Fig. 1. The top part of the Kubernetes cluster consists +of four components that make the master node. These are +the kube-apiserver that exposes the Kubernetes Application +Programming Interface (API), the etcd that is used as the +backing store for all cluster data, the kubescheduler that +watches for newly created pods with no assigned node, and +selects a node for them to run, and finally the kubecontroller +that runs the control processes. Besides the master node, +we have the worker nodes. In our case, we have only one +worker node, inside which we have deployed our containers + +etcd +kubescheduler +kubecontroller +kube-apiserver +Master +Node +Worker +kube-proxy +kubelet +Node +roscore pod +ROS master node +registration +registration : +r(k) +MPC pod +ROS MPC node +u(k-d2) +- +Kubernetes Cluster +Robot +u(k-d3) +UAV dynamics node +y(k)in the form of pods. A pod is the basic operational unit +of Kubernetes and consist of a set of containers that share +storage, network resources, and specifications on how to run +the containers. The two pods we have deployed are related +to the ROS master and the MPC respectively. Apart from the +pods, the worker node consists of the kubelet which makes +sure that containers are running in a pod, and kube-proxy +which makes sure that network rules are met. +From Fig. 1 we can describe the block diagram of the +close loop system. Let’s assume that in the time step, k the +UAV dynamics node generates a signal x(k) that describes +the states of the UAV. These states are the position, velocity, +and orientation of the UAV. This signal will arrive at the +MPC ROS node, running on the edge, delayed, due to the +travel time the signal needs to travel from the UAV to the +edge. Thus, the signal carrying the information of x(k) will +arrive at the MPC ROS node as x(k − d1), while at the +same time, another signal regarding the desired states for the +UAV will arrive at the MPC ROS node as a reference signal +r(k). The controller will have to process this information and +generate the command signal u(k − d2). Given that u(k − +d2) is corresponding to the signals x(k − d1) and r(k), the +variable d2 is related to d1, as well as the execution time +of the MPC. This command signal has to travel from the +edge to the UAV in order to close the loop of the system. +Thus, the signal arriving to the UAV is denoted as u(k−d3), +where d3 is related to d1, d2, as well as to the travel time the +command signal needs to travel from the edge to the UAV. +Finally, the output of the system is denoted as y(k). +The communication between the UAV model simulation +and the controller is taken care by ROS. There should be +only one ROS master, and every ROS node has to register +to that ROS master to be able to run and communicate with +other ROS nodes. When two ROS nodes want to exchange +data by subscribing and publishing to the same ROS topic, +ROS master opens a random port and allows the two ROS +nodes to communicate through that port. Once ROS assigns +a random ports, different every time, the nodes running +on the edge and the nodes running on the robot try to +communicate with each other through these ports. Since the +containers are deployed on the Kubernetes cluster of the edge +machine (host), we have to specify which ports the containers +should be exposed to for communication purposes. The +challenge occurs because ROS master do not assign specific +ports for communication, but it assigns them randomly. To +overcome this issue, we used the host network option when +we deployed the containers on the Kubernetes cluster, in +order to expose all the host ports to the containers and vice +versa. That way, the containers can access all the traffic at +the host machine’s ports and the host machine can access the +traffic at the containers’ ports. Now, the data coming from +the UAV to the edge machine can be forwarded inside the +containers and the data from the containerized applications +can be exposed to the edge machine and then sent to the +UAV. +In this paper, both the edge machine and the UAV are on +the same network, thus we were able to use Wi-Fi. Wi-Fi +can be an efficient network option for the communication +between the UAV and the edge machine and has been used +widely, but it is not the optimal solution. 5G is a promising +technology that will provide essential features for secure, +robust and reliable networking, and can be the field of study +for future works. +III. MODEL PREDICTIVE CONTROL +Model predictive control is a standard method used for +high level control for UAVs, thus there are many works +describing in detail the behavior of the controller and the +kinematics of the UAV, like in [11], where authors suggested +a UAV model that could afford disturbances by stabilizing +its location in space. The preference on MPC in comparison +to other common controllers, like PID or LQR, is explained +by its predictive behavior and performance. Based on these +characteristics, we were prompted to use this controller for +controlling the trajectory of an UAV, and we were motivated +to offload it to the edge so resource-constrained UAVs and +robots in general, that can not afford to run this controller +onboard, would be able to take advantage of the benefits of +MPC. The UAV model and the implementation of the MPC +for this work are based on [12]. +A. UAV Model +In order to develop the MPC methodology, the first step is +to describe the UAV kinematics model, which is presented +through the Eq. 1. +˙p(t) = vz(t) +˙v(t) = Rx,y(θ, φ) +� +� +0 +0 +T +� +� + +� +� +0 +0 +−g +� +� − +� +� +Ax +0 +0 +0 +Ay +0 +0 +0 +Az +� +� u(t) +(1) +˙φ(t) = 1 +τφ +(Kφφd(t) − φ(t)) +˙θ(t) = 1 +τθ +(Kθθd(t) − θ(t)), +Fig. 2. +Coordinate frames, where W and B represent the world and body +coordinate frames respectively on gazebo simulation environment +where p = [px, py, pz]T and v = [vx, vy, vz]T are the +position and the linear velocity respectively based on the + +ZB +YB +X +ZWworld frame (W), as depicted in Fig. 2. We donate as +R(φ(t), θ(t)) ∈ SO(3) the rotation matrix that represents +the attitude. φ and θ ∈ [−π, π] are the roll and pitch angles, +while T ≥ 0 describes the total thrust. The acceleration +depends on the magnitude and angle of the thrust vector, +the gravity, and the linear damping terms Ax, Ay, Az ∈ R +g. φd and θd ∈ R are the desired roll and pitch inputs with +gains Kφ and Kθ ∈ R, and time constants τφ and τθ ∈ R. +B. Cost Function +Next step for the MPC methodology is to present the cost +function. x = [p, v, φ, θ]T and u = [T, φd, θd]T represent the +UAV’s state vector and the control input, respectively. The +sampling time of the system is δt ∈ Z+, while the forward +Euler method is used for each time instance (k + 1|k). The +predictive behavior of the MPC is based on the prediction +horizon, which considers a specified number of steps into +the future, and is represented as N. +In order to minimize the cost of the cost function, an +optimizer has been assigned to find the optimal set of +control actions. The cost function associates the cost of the +configuration of states and inputs at the current time and in +the prediction. xk+j|k represents the predicted states at the +time step k + j, produced at the time step k, while uk+j|k +represents the corresponding control actions. Furthermore, +xk represents the predicted states and uk represents the +corresponding control inputs along the prediction horizon. +The equation describing the cost function is presented in +Eq. 2. +J = +N +� +j=1 +(xd − xk+j|k)T Qx(xd − xk+j|k) +� +�� +� +state +cost ++ (ud − uk+j|k)T Qu(ud − uk+j|k) +� +�� +� +input +cost +(2) ++ uk+j|k − uk+j−1|k)T Qδu(uk+j|k − uk+j−1|k), +� +�� +� +control +actions +smoothness +cost +where the first term denotes the cost related to the de- +viation between the predicted states and the certain desired +states xd, while Qx ∈ R8x8 is a matrix describing the state +weights. The second term denotes the input cost, describing +hovering, and that penalizes a deviation from the steady-state +input ud = [g, 0, 0], while Qu is a matrix describing the input +weights. The third term is added to guarantee that the control +actions are smooth. This is achieved by comparing the input +at (k + j − 1|k) with the input at (k + j|k) and penalizing +the changing of the input from one time step to the next +one, with N ∈ N + to denote the control Horizon of the +MPC, while Qδu ∈ R3x3 is a matrix describing the input +rate weights. +IV. SIMULATION RESULTS +In this section, we are presenting the simulation results of +the proposed architecture. For the simulation, we used the +gazebo environment and the UAV simulation model hum- +mingbird of the rotor simulation ROS package, as depicted +in Fig. 2. For the edge, we utilized a powerful machine and +microk8s was running on the edge, which is a lightweight +Kubernetes software that was used as the Kubernetes or- +chestrator. The specifications of the edge are: 1) Processor: +Intel Core i5-8400 CPU@2.80GHz×6, 2) Memory: 32 GB 3) +Operating System: Ubuntu 20.04 LTS and 4) Disk Capacity: +2.5 TB. +For the following simulations, the MPC horizon was set +at 100 steps and the MPC rate was set at 100Hz. We were +able to select this values, because the MPC is running on +the edge and we are using its capabilities. UAVs’ onboard +processors would not be able to handle an MPC with +these high values since they increase the complexity of the +controller (solution of the optimization problem), thus the +computational demands. +In Fig. 3, the responses for the three different tested +trajectories of the UAV are depicted. The first line of figures +depicts the 3D response of the circular, spiral and helical +trajectories while the second, third and fourth lines depict +the responses of the X, Y and Z axis respectively, for each +Fig. 3. +UAV responses based on the Kubernetes-based architecture for circular, spiral and helical trajectories. A1)Depicts the 3D response of the circular +trajectory while A2, A3, A4 depict the responses of the X, Y and Z axis respectively. B1) Depicts the 3D response of the spiral trajectory. C1) Depicts +the 3D response of the helical trajectory. The blue line represents the real trajectory of the UAV, while the blue line represents the reference points for the +desired trajectory of the UAV. + +A1. 3D Response +B1. 3D Response +C1. 3D Response +10 +L 0 +5 +() z +y (m) +x (m) +y (m) +x (m) +y (m) +x (m) +C2 +≤. 0 +X-5 +20 +30 +40 +50 +60 +70 +80 +10 +20 +30 +40 +60 +80 +0 +10 +20 +30 +40 +50 +0 +70 +0 +60 +A3. 三 0 +B3. +0 +C3. +Y-5 +20 +20 +30 +40 +60 +70 +80 +10 +40 +60 +10 +30 +40 +10 +50 +60 +0 +20 +70 +70 +So +30 +8o +0 +80 +10F +10 +A4. 2.5 +B4. +5 +70 +80 +60 +10 +30 +50 +60 +20 +30 +50 +60 +10 +20 +30 +40 +50 +80 +20 +10 +70 +80 +Time (s) +Time (s)different trajectory. The blue line represents the real trajec- +tory of the UAV, while the blue line represents the reference +points for the desired trajectory of the UAV. From these +figures, we can notice that the UAV simulation model can +successfully follow the desired trajectory. The time delays +seem to not have a significant effect on the performance of +the controller. On the next figures, we are investigating in +more detail these time delays. +Fig. 4. +Euclidean error between UAV position and reference point for +each time step of the Kubernetes-based architectures, for A) the circular, B) +spiral, and C) helical trajectory. The blue line represents the error and the +red line represents the error tolerance +Fig. 4 depicts the Euclidean error between the UAV +position and the reference point for each time step of the +Kubernetes-based architectures, for the circular, spiral and +helical trajectories. The blue line represents the error and +the red line represents the error tolerance. The controller is +responsible to keep the error below the tolerance value. If the +error goes above the tolerance, the controller will correct it +and the UAV will continue following the desired trajectory. +The tolerance was set at 0.4 meters for each axis, thus in +total of +√ +0.68 meters. +In Fig. 5, the deviation of the different types of time delays +for the spiral trajectory are presented. In the left figure, the +deviation for the travel time of a signal from the UAV to the +edge, in the middle figure the deviation for the execution time +of the MPC, and in the right figure the deviation for the travel +time of a signal from the edge to the UAV, are depicted. The +average measured travel time from the UAV to the edge is +0.0089 seconds, and the maximum 0.1700 seconds. For the +execution time, the average measured time is 0.0141 seconds +and the maximum is 0.2200 seconds. Finally, for the travel +time, from the edge to the UAV, the measured travel time is +0.0161 seconds and the maximum is 0.2600 seconds. +Fig. 6. +Edge resources usage during the spiral trajectory. The red bar +represents the user space and the blue bar represents the system kernel- +space. +To end the evaluation of the system, we measured the +resource usage for the execution of the MPC on the edge +and the data are depicted in Fig. 6. The red bars represent +the time the CPU spends executing processes in user-space +(us). Similarly, the blue bar represents the time spent on +running system kernel-space (sy) processes. From the figure +we can observe that by utilizing the edge machine, the edge +does not get overloaded, and the maximum reached value is +84.50% which occurs when the values us and sy are 46.70% +and 37.80% respectively. The maximum values that us and +sy reach independently are 54.40% and 37.80% respectively, +and their average values are 20.225% for the us and 4.582 +Fig. 5. +Deviation of the different types of time delays for the spiral trajectory: A) Deviation for the travel time of a signal from the UAV to the edge. +B) Deviation for the execution time of the MPC. C) Deviation for the travel time of a signal from the edge to the UAV. + +A. Error of Euclidean distance +0.8 +0.7 +0.6 +0.3 +0.2 +0.1 +0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Time (s) +B. Error of Euclidean distance +0.8 +0.7 +0.6 +0.3 +0.2 +0.1 +0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Time (s) +C. Error of Euclidean distance +0.8 +0.7 +0.6 +0.3 +0.2 +0.1 +0 +0 +10 +20 +30 +40 +50 +60 +70 +80 +Time (s)A. Deviation of travel time +B.Deviation of MPC execution +C. Deviation of travel time +0.21 +0.16 +0.24 +0.14 +0.18 +0.12 +0.15 + 0.16 +0.1 +0.12 + 0.08 +0.09 +900 +0.08 + 0.06 + 0.04 +0 0.04 + 0.02 +0.03 +50 +0 +10 +20 +30 +60 +70 +10 +30 +40 +50 +60 +87 +10 +20 +30 +50 +60 +70 +40 +70 +80Resources usage of the edge +100% +system kernel-space +user space +80% +Percentage +60% +40% +20% +0 +10 +20 +30 +40 +50 +60 +70 +Time (s)for the sy. From these measurements and figure, we can +notice that the relatively immense assigned edge resources +are adequate in order to run the computationally demanding +controller, but even in this case, during the 35th second of the +trajectory, the usage of resources were almost at 90%. This +means that computational light units, like UAVs’ onboard +processors, might not be able to execute that controller +smoothly. +V. CONCLUSIONS AND FUTURE WORK +In this work, we presented a novel edge architecture +to control the trajectory of an UAV through the edge by +enabling an MPC methodology. This architecture can be +beneficial for expanding the computational capabilities of +resource-constrained platforms like aerial robots, that in +many cases are deployed with light microprocessors onboard, +like Raspberry Pi, and can not afford to run computationally +expensive processes onboard. By utilizing edge, we were +able to offload the controller there, and control the trajectory +of the UAV in real-time by closing the loop of the system +through the edge. Furthermore, we evaluated the proposed +architecture, through a series of experiments, through which +we examined the performance of the system, as well as the +overall time delays. +Edge computing is a promising technology for the field +of robotics. In the current article, we offloaded the computa- +tionally costly MPC, while future works can move towards +offloading other time sensitive robotic application, like sensor +fusion for online perception, or offload applications that +require many resources in order to operate in real-time, like +map merging from multiple agents. The end goal would be +to create an ecosystem through which multiple agents will be +able not only to use edge resources to expand their autonomy +capacity, but also communicate and collaborate through the +edge. +REFERENCES +[1] A. Panousopoulou, G. Nikolapoulos, and A. Tzes, “Experimental +controller tuning and qos optimization of a wireless transmission +scheme for real-time remote control applications,” in 2004 IEEE +International Conference on Industrial Technology, 2004. IEEE ICIT +’04., vol. 2, 2004, pp. 801–806 Vol. 2. +[2] R. B. Ernesto Bernardo, Rocco Palamara, “Uav and soft computing +methodology for monitoring landslide areas (susceptibility to land- +slides and early warning),” in WSEAS Transactions on Environment +and Development, vol. 17, 2021, pp. 490–501. +[3] S. C. Ernesto Bernardo, Stefano Bonfa, “Techniques of geomatics +and soft computing for the monitoring of infrastructures and the +management of big data,” in WSEAS Transactions on Environment +and Development, vol. 17, 2021, pp. 371–385. +[4] V. K. Sarker, J. Pe˜na Queralta, T. N. Gia, H. Tenhunen, and T. Wester- +lund, “Offloading slam for indoor mobile robots with edge-fog-cloud +computing,” in 2019 1st International Conference on Advances in +Science, Engineering and Robotics Technology (ICASERT), 2019, pp. +1–6. +[5] A. Barnawi, M. Alharbi, and M. Chen, “Intelligent search and find +system for robotic platform based on smart edge computing service,” +IEEE Access, vol. 8, pp. 108 821–108 834, 2020. +[6] A. K. Tanwani, N. Mor, J. Kubiatowicz, J. E. Gonzalez, and +K. Goldberg, “A fog robotics approach to deep robot learning: +Application to object recognition and grasp planning in surface +decluttering,” 2019. [Online]. Available: https://arxiv.org/abs/1903. +09589 +[7] P. +Kochovski, +R. +Sakellariou, +M. +Bajec, +P. +Drobintsev, +and +V. Stankovski, “An architecture and stochastic method for database +container placement in the edge-fog-cloud continuum,” in 2019 +IEEE International Parallel and Distributed Processing Symposium +(IPDPS). +IEEE, 2019, pp. 396–405. +[8] F. Lumpp, M. Panato, F. Fummi, and N. Bombieri, “A container-based +design methodology for robotic applications on kubernetes edge-cloud +architectures,” in 2021 Forum on specification & Design Languages +(FDL). +IEEE, 2021, pp. 01–08. +[9] P. Skarin, J. Eker, M. Kihl, and K.-E. ˚Arz´en, “Cloud-assisted model +predictive control,” in 2019 IEEE International Conference on Edge +Computing (EDGE), 2019, pp. 110–112. +[10] P. Skarin, J. Eker, and K.-E. ˚Arz´en, “A cloud-enabled rate-switching +mpc architecture,” in 2020 59th IEEE Conference on Decision and +Control (CDC), 2020, pp. 3151–3158. +[11] R. K. Lucjan Setlak, “Research and analysis of the issue of flight +stability of an unmanned aerial vehicle,” in WSEAS Transactions on +Applied and Theoretical Mechanics, vol. 15, 2020, pp. 11–18. +[12] B. Lindqvist, S. S. Mansouri, and G. Nikolakopoulos, “Non-linear +mpc based navigation for micro aerial vehicles in constrained envi- +ronments,” in 2020 European Control Conference (ECC), 2020, pp. +837–842. + diff --git a/_NFRT4oBgHgl3EQfsjc9/content/tmp_files/load_file.txt b/_NFRT4oBgHgl3EQfsjc9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2e874ad14f24f121275354d7de4d8b1c303e9b2 --- /dev/null +++ b/_NFRT4oBgHgl3EQfsjc9/content/tmp_files/load_file.txt @@ -0,0 +1,356 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf,len=355 +page_content='A Kubernetes-Based Edge Architecture for Controlling the Trajectory of a Resource-Constrained Aerial Robot by Enabling Model Predictive Control Achilleas Santi Seisa, Sumeet Gajanan Satpute and George Nikolakopoulos Abstract— In recent years, cloud and edge architectures have gained tremendous focus for offloading computationally heavy applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' From machine learning and Internet of Thing (IOT) to industrial procedures and robotics, cloud computing have been used extensively for data processing and storage purposes, thanks to its “infinite” resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' On the other hand, cloud computing is characterized by long time delays due to the long distance between the cloud servers and the machine requesting the resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In contrast, edge computing provides almost real-time services since edge servers are located significantly closer to the source of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' This capability sets edge computing as an ideal option for real-time applications, like high level control, for resource-constrained platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In order to utilize the edge resources, several technologies, with basic ones as containers and orchestrators like Kubernetes, have been developed to provide an environment with many features, based on each application’s requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In this context, this works presents the implementation and evaluation of a novel edge architecture based on Kubernetes orchestration for controlling the trajectory of a resource-constrained Unmanned Aerial Vehicle (UAV) by enabling Model Predictive Control (MPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Index Terms— Robotics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Edge Computing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Kubernetes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' UAV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' INTRODUCTION Nowadays, as technology is progressing and the need for computational resources is continuously increasing, different computation layers have been evolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' We can differ these layers into four distinct categories, cloud, fog, edge and devices, while each one of them has its own different characteristics and utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' At the same time, all of them can be combined with each other to create an ecosystem for the utilization of external computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' As these technologies mature, researchers and engineers use them more and more to offload their applications, due to the capabilities and features they provide [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Additionally, since the mentioned computation layers have attracted tremendous focus, several state-of-the-art technologies have been devel- oped and are promising to revolutionize many technolog- ical fields, like containerized applications and containers’ orchestrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In this framework, robotics can take a huge advantage of the external resources and many resource- constrained platforms can make the most out of them, since they will be able to run algorithms that they can not run on their onboard processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In this context, edge is emerging This work has been partially funded by the European Union’s Horizon 2020 Research and Innovation Programme AERO-TRAIN under the Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 953454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The authors are with the Robotics and AI Team, Department of Computer, Electrical and Space Engineering, Lule˚a University of Technology, Lule˚a Corresponding Author’s email: achsei@ltu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='se since it can provide tremendous resources for enhancing the performance, and the overall efficiency of autonomous operations, and at the same time minimize the travel time delays when transmitting data from UAVs, like in [2] and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Thus, edge can be established as a promising solution for time critical operations, like offloading computationally costly controllers, of resource-constrained platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In this article, we propose an architecture where we offload the Model Predictive Control method, which is a relatively heavy controller, to the edge, as a containerized application, and we use Kubernetes for managing the containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Researchers seek to utilize the advantages of edge com- puting for the benefit of robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' However, researchers have to overcome some limitations and challenges, in order to use these technologies universally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In [4] an architecture consisting of all four computation layers is used to offload the localization and mapping problem from robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' in this case, edge is operating as a layer between sensor devices, gateways, and cloud servers for enhancing the quality of services, while in [5] edge is used to design a search planner algorithm using deep learning for UAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In [6] edge and cloud were utilized in terms of storage and computational resources for deep robot learning including object recogni- tion, grasp planning and localization of the computational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Some works that utilized edge for robotic applications by implementing Kubernetes or container based architectures can be summarized as it follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In [7] researcher tried to automate, by using Kubernetes orchestration, the process of making decision in terms of placement of the expected work- load to edge, fog and cloud, for robotic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In [8], a methodology based on docker and Kubernetes for ROS-based robotic application is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In that case, the architecture was evaluated by experimental results obtained by a mobile robot interacting with an industrial agile production chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In the works mentioned above, the approach regarding edge computing is mainly towards non-time critical tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' High level controllers must operate almost real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In [9] an architecture was proposed where the control method consists of the combination of a LQR running on the device and an MPC running the optimization both on edge and cloud, while in [10] two complimentary MPCs are running, one on a local edge and one on the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In comparison to our proposed work, these articles are partly offloading the MPC method on the edge and are focused on evaluating the system in terms of related latency, and the related uncertainty for several cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The motivation behind this work is to fill the gap re- garding edge enabled robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Even though edge comput- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='13624v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='RO] 31 Jan 2023 ing has proven to be a promising technology to expand the autonomous capabilities of resource-constrained robotic platforms, especially when combined with 5G networks, the research that has been done around this area is rel- atively limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Despite the fact that the great advantage of edge computing is the ability of enabling almost real- time operation by offloading the computing process on the edge, most researchers have focused on utilizing edge for offline procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Thus, the contribution of this article is to present a novel edge architecture for enabling the time sensitive operation of controlling the trajectory of a resource- constrained UAV in real-time through MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Control is one of the basic components of autonomy, thus the performance and efficiency is the main criteria when choosing a controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Model predictive controllers are widely used on UAVs due to their characteristics and optimal behavior, but they are computationally costly, thus some UAVs, deployed with light processors, like Raspberry Pi, can not handle them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' By utilizing the proposed architecture, we will be able to use edge resources in order to offload the MPC and control resource-constrained platforms by closing the loop over the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Additionally, we are using Kubernetes orchestration that provides best practices for cloud and edge applications but inserts some challenges that we have to overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The rest of the article unfolds in the following Sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In Section II, we describe the Kubernetes-based edge archi- tecture, while in Section III, we give a brief overview of the UAV and MPC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In Section IV, we present the simulation results of the edge architecture in terms of time delays and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Finally, in Section V, we conclude the article by highlighting the main points of this work, and we propose future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' KUBERNETES EDGE ARCHITECTURE The proposed architecture is based on Kubernetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Ku- bernetes is a container orchestrator, developed by Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Before we start analyzing the Kubernetes-based architecture, we have to describe the containers developed for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Afterward, we are going to present the system’s architecture and the Robotic Operating System (ROS) framework that was utilized for the UAV-MPC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Finally, we will describe the communication layer and network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Containers are based on software that creates an operating environment and are deployed only with the necessary and chosen packages, libraries and dependencies, in order to run a specific application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The application running in this form is called a containerized application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Containers are based on images that are the nominal state of containers before they get deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' An image can be used to deploy many containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' For our system, we deployed two docker containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' One container is responsible for running the controller and all the necessary libraries and dependencies for its smooth and reliable operation, and the other is responsible for running the ROS master, which takes care of the communication between the ROS nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' To deploy the two docker containers, we had to developed two different docker images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' For both images, we used ROS Noetic on Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='04 entrypoint, and we built on top of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' For the first image, we included several ROS packages and libraries, as well as an optimization engine for the MPC containerized application, while for the second image we just needed to run the ROS master.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' For a more complex application, we could split it into more containers, each one of them would be assigned a specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Diagram of the Kubernetes-based edge architecture for the UAV- MPC system Once we had developed the docker images, we were able to deploy the docker containers inside the Kubernetes cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' We decided to use Kubernetes due to the features it provides for our containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Kubernetes gives us the capability to manage our containers and automates the whole process of deploying the containers, assign them resources, check their health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The services and features that Kubernetes is providing can be extremely helpful for our application, since they give us the chance to manage and monitor our system in an optimal way, and it can even more handful when we have to deploy more containers and the system get more and more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The Kubernetes architecture is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The top part of the Kubernetes cluster consists of four components that make the master node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' These are the kube-apiserver that exposes the Kubernetes Application Programming Interface (API), the etcd that is used as the backing store for all cluster data, the kubescheduler that watches for newly created pods with no assigned node, and selects a node for them to run, and finally the kubecontroller that runs the control processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Besides the master node, we have the worker nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In our case, we have only one worker node, inside which we have deployed our containers etcd kubescheduler kubecontroller kube-apiserver Master Node Worker kube-proxy kubelet Node roscore pod ROS master node registration registration : r(k) MPC pod ROS MPC node u(k-d2) Kubernetes Cluster Robot u(k-d3) UAV dynamics node y(k)in the form of pods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' A pod is the basic operational unit of Kubernetes and consist of a set of containers that share storage, network resources, and specifications on how to run the containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The two pods we have deployed are related to the ROS master and the MPC respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Apart from the pods, the worker node consists of the kubelet which makes sure that containers are running in a pod, and kube-proxy which makes sure that network rules are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 1 we can describe the block diagram of the close loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Let’s assume that in the time step, k the UAV dynamics node generates a signal x(k) that describes the states of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' These states are the position, velocity, and orientation of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' This signal will arrive at the MPC ROS node, running on the edge, delayed, due to the travel time the signal needs to travel from the UAV to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Thus, the signal carrying the information of x(k) will arrive at the MPC ROS node as x(k − d1), while at the same time, another signal regarding the desired states for the UAV will arrive at the MPC ROS node as a reference signal r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The controller will have to process this information and generate the command signal u(k − d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Given that u(k − d2) is corresponding to the signals x(k − d1) and r(k), the variable d2 is related to d1, as well as the execution time of the MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' This command signal has to travel from the edge to the UAV in order to close the loop of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Thus, the signal arriving to the UAV is denoted as u(k−d3), where d3 is related to d1, d2, as well as to the travel time the command signal needs to travel from the edge to the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Finally, the output of the system is denoted as y(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The communication between the UAV model simulation and the controller is taken care by ROS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' There should be only one ROS master, and every ROS node has to register to that ROS master to be able to run and communicate with other ROS nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' When two ROS nodes want to exchange data by subscribing and publishing to the same ROS topic, ROS master opens a random port and allows the two ROS nodes to communicate through that port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Once ROS assigns a random ports, different every time, the nodes running on the edge and the nodes running on the robot try to communicate with each other through these ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Since the containers are deployed on the Kubernetes cluster of the edge machine (host), we have to specify which ports the containers should be exposed to for communication purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The challenge occurs because ROS master do not assign specific ports for communication, but it assigns them randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' To overcome this issue, we used the host network option when we deployed the containers on the Kubernetes cluster, in order to expose all the host ports to the containers and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' That way, the containers can access all the traffic at the host machine’s ports and the host machine can access the traffic at the containers’ ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Now, the data coming from the UAV to the edge machine can be forwarded inside the containers and the data from the containerized applications can be exposed to the edge machine and then sent to the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In this paper, both the edge machine and the UAV are on the same network, thus we were able to use Wi-Fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Wi-Fi can be an efficient network option for the communication between the UAV and the edge machine and has been used widely, but it is not the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 5G is a promising technology that will provide essential features for secure, robust and reliable networking, and can be the field of study for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' MODEL PREDICTIVE CONTROL Model predictive control is a standard method used for high level control for UAVs, thus there are many works describing in detail the behavior of the controller and the kinematics of the UAV, like in [11], where authors suggested a UAV model that could afford disturbances by stabilizing its location in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The preference on MPC in comparison to other common controllers, like PID or LQR, is explained by its predictive behavior and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Based on these characteristics, we were prompted to use this controller for controlling the trajectory of an UAV, and we were motivated to offload it to the edge so resource-constrained UAVs and robots in general, that can not afford to run this controller onboard, would be able to take advantage of the benefits of MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The UAV model and the implementation of the MPC for this work are based on [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' UAV Model In order to develop the MPC methodology, the first step is to describe the UAV kinematics model, which is presented through the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' ˙p(t) = vz(t) ˙v(t) = Rx,y(θ, φ) � � 0 0 T � � + � � 0 0 −g � � − � � Ax 0 0 0 Ay 0 0 0 Az � � u(t) (1) ˙φ(t) = 1 τφ (Kφφd(t) − φ(t)) ˙θ(t) = 1 τθ (Kθθd(t) − θ(t)), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Coordinate frames, where W and B represent the world and body coordinate frames respectively on gazebo simulation environment where p = [px, py, pz]T and v = [vx, vy, vz]T are the position and the linear velocity respectively based on the ZB YB X ZWworld frame (W), as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' We donate as R(φ(t), θ(t)) ∈ SO(3) the rotation matrix that represents the attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' φ and θ ∈ [−π, π] are the roll and pitch angles, while T ≥ 0 describes the total thrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The acceleration depends on the magnitude and angle of the thrust vector, the gravity, and the linear damping terms Ax, Ay, Az ∈ R g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' φd and θd ∈ R are the desired roll and pitch inputs with gains Kφ and Kθ ∈ R, and time constants τφ and τθ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Cost Function Next step for the MPC methodology is to present the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' x = [p, v, φ, θ]T and u = [T, φd, θd]T represent the UAV’s state vector and the control input, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The sampling time of the system is δt ∈ Z+, while the forward Euler method is used for each time instance (k + 1|k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The predictive behavior of the MPC is based on the prediction horizon, which considers a specified number of steps into the future, and is represented as N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In order to minimize the cost of the cost function, an optimizer has been assigned to find the optimal set of control actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The cost function associates the cost of the configuration of states and inputs at the current time and in the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' xk+j|k represents the predicted states at the time step k + j, produced at the time step k, while uk+j|k represents the corresponding control actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Furthermore, xk represents the predicted states and uk represents the corresponding control inputs along the prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The equation describing the cost function is presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' J = N � j=1 (xd − xk+j|k)T Qx(xd − xk+j|k) � �� � state cost + (ud − uk+j|k)T Qu(ud − uk+j|k) � �� � input cost (2) + uk+j|k − uk+j−1|k)T Qδu(uk+j|k − uk+j−1|k), � �� � control actions smoothness cost where the first term denotes the cost related to the de- viation between the predicted states and the certain desired states xd, while Qx ∈ R8x8 is a matrix describing the state weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The second term denotes the input cost, describing hovering, and that penalizes a deviation from the steady-state input ud = [g, 0, 0], while Qu is a matrix describing the input weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The third term is added to guarantee that the control actions are smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' This is achieved by comparing the input at (k + j − 1|k) with the input at (k + j|k) and penalizing the changing of the input from one time step to the next one, with N ∈ N + to denote the control Horizon of the MPC, while Qδu ∈ R3x3 is a matrix describing the input rate weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' SIMULATION RESULTS In this section, we are presenting the simulation results of the proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' For the simulation, we used the gazebo environment and the UAV simulation model hum- mingbird of the rotor simulation ROS package, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' For the edge, we utilized a powerful machine and microk8s was running on the edge, which is a lightweight Kubernetes software that was used as the Kubernetes or- chestrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The specifications of the edge are: 1) Processor: Intel Core i5-8400 CPU@2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='80GHz×6, 2) Memory: 32 GB 3) Operating System: Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='04 LTS and 4) Disk Capacity: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='5 TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' For the following simulations, the MPC horizon was set at 100 steps and the MPC rate was set at 100Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' We were able to select this values, because the MPC is running on the edge and we are using its capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' UAVs’ onboard processors would not be able to handle an MPC with these high values since they increase the complexity of the controller (solution of the optimization problem), thus the computational demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 3, the responses for the three different tested trajectories of the UAV are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The first line of figures depicts the 3D response of the circular, spiral and helical trajectories while the second, third and fourth lines depict the responses of the X, Y and Z axis respectively, for each Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' UAV responses based on the Kubernetes-based architecture for circular, spiral and helical trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' A1)Depicts the 3D response of the circular trajectory while A2, A3, A4 depict the responses of the X, Y and Z axis respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' B1) Depicts the 3D response of the spiral trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' C1) Depicts the 3D response of the helical trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The blue line represents the real trajectory of the UAV, while the blue line represents the reference points for the desired trajectory of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 3D Response B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 3D Response C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 3D Response 10 L 0 5 () z y (m) x (m) y (m) x (m) y (m) x (m) C2 ≤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 0 X-5 20 30 40 50 60 70 80 10 20 30 40 60 80 0 10 20 30 40 50 0 70 0 60 A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 三 0 B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 0 C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Y-5 20 20 30 40 60 70 80 10 40 60 10 30 40 10 50 60 0 20 70 70 So 30 8o 0 80 10F 10 A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='5 B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 5 70 80 60 10 30 50 60 20 30 50 60 10 20 30 40 50 80 20 10 70 80 Time (s) Time (s)different trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The blue line represents the real trajec- tory of the UAV, while the blue line represents the reference points for the desired trajectory of the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' From these figures, we can notice that the UAV simulation model can successfully follow the desired trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The time delays seem to not have a significant effect on the performance of the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' On the next figures, we are investigating in more detail these time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Euclidean error between UAV position and reference point for each time step of the Kubernetes-based architectures, for A) the circular, B) spiral, and C) helical trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The blue line represents the error and the red line represents the error tolerance Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 4 depicts the Euclidean error between the UAV position and the reference point for each time step of the Kubernetes-based architectures, for the circular, spiral and helical trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The blue line represents the error and the red line represents the error tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The controller is responsible to keep the error below the tolerance value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' If the error goes above the tolerance, the controller will correct it and the UAV will continue following the desired trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The tolerance was set at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='4 meters for each axis, thus in total of √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='68 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 5, the deviation of the different types of time delays for the spiral trajectory are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In the left figure, the deviation for the travel time of a signal from the UAV to the edge, in the middle figure the deviation for the execution time of the MPC, and in the right figure the deviation for the travel time of a signal from the edge to the UAV, are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The average measured travel time from the UAV to the edge is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='0089 seconds, and the maximum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='1700 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' For the execution time, the average measured time is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='0141 seconds and the maximum is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='2200 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Finally, for the travel time, from the edge to the UAV, the measured travel time is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='0161 seconds and the maximum is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='2600 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Edge resources usage during the spiral trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The red bar represents the user space and the blue bar represents the system kernel- space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' To end the evaluation of the system, we measured the resource usage for the execution of the MPC on the edge and the data are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The red bars represent the time the CPU spends executing processes in user-space (us).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Similarly, the blue bar represents the time spent on running system kernel-space (sy) processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' From the figure we can observe that by utilizing the edge machine, the edge does not get overloaded, and the maximum reached value is 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='50% which occurs when the values us and sy are 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='70% and 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='80% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The maximum values that us and sy reach independently are 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='40% and 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='80% respectively, and their average values are 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='225% for the us and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='582 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Deviation of the different types of time delays for the spiral trajectory: A) Deviation for the travel time of a signal from the UAV to the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' B) Deviation for the execution time of the MPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' C) Deviation for the travel time of a signal from the edge to the UAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Error of Euclidean distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='1 0 0 10 20 30 40 50 60 70 80 Time (s) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Error of Euclidean distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='1 0 0 10 20 30 40 50 60 70 80 Time (s) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Error of Euclidean distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='1 0 0 10 20 30 40 50 60 70 80 Time (s)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Deviation of travel time B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='Deviation of MPC execution C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Deviation of travel time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='09 900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='04 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content='03 50 0 10 20 30 60 70 10 30 40 50 60 87 10 20 30 50 60 70 40 70 80Resources usage of the edge 100% system kernel-space user space 80% Percentage 60% 40% 20% 0 10 20 30 40 50 60 70 Time (s)for the sy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' From these measurements and figure, we can notice that the relatively immense assigned edge resources are adequate in order to run the computationally demanding controller, but even in this case, during the 35th second of the trajectory, the usage of resources were almost at 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' This means that computational light units, like UAVs’ onboard processors, might not be able to execute that controller smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK In this work, we presented a novel edge architecture to control the trajectory of an UAV through the edge by enabling an MPC methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' This architecture can be beneficial for expanding the computational capabilities of resource-constrained platforms like aerial robots, that in many cases are deployed with light microprocessors onboard, like Raspberry Pi, and can not afford to run computationally expensive processes onboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' By utilizing edge, we were able to offload the controller there, and control the trajectory of the UAV in real-time by closing the loop of the system through the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Furthermore, we evaluated the proposed architecture, through a series of experiments, through which we examined the performance of the system, as well as the overall time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' Edge computing is a promising technology for the field of robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' In the current article, we offloaded the computa- tionally costly MPC, while future works can move towards offloading other time sensitive robotic application, like sensor fusion for online perception, or offload applications that require many resources in order to operate in real-time, like map merging from multiple agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' The end goal would be to create an ecosystem through which multiple agents will be able not only to use edge resources to expand their autonomy capacity, but also communicate and collaborate through the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFRT4oBgHgl3EQfsjc9/content/2301.13624v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': 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Bauwelinckx1, Jan Dhaene1, Milan van den Heuvel 2, and Tim +Verdonck1,3 +1Department of Economics, Katholieke Universiteit Leuven, Belgium +2Department of Economics, Universiteit Gent, Belgium +3IMEC, Universiteit Antwerpen, Belgium +Contact: Milan.vandenHeuvel@UGent.be (Address: Tweekerkenstraat 2, 9000, Ghent, Belgium). +1 +arXiv:2301.01109v1 [cs.LG] 3 Jan 2023 + +1 +Introduction +To make sense of the complexities of reality, and make optimal decisions accordingly, organ- +isations and researchers have always striven to come up with models that can accurately +represent observed phenomena (e.g. consumption behaviour, loan defaults). In the past, +these models were defined by the analyst and calibrated to (small) data. Recently, however, +during the so-called machine learning revolution, the focus shifted to a more data-driven, +algorithmic approach. Machine learning algorithms now search for the optimal model by +finding support for it in the data instead of being chosen by the analyst. This approach has +increased the collection of, investment in, demand for, reliance on, and value of data for +organisations and research significantly (McKinsey & Company, 2021). It has also brought +the tension between utility of data and privacy of its subjects to the forefront of public +discussion (European Commission, 2021). Recently developed generative modelling meth- +ods, which generate data with a distribution similar to the original but without containing +any of the real data, have been proposed as a potential solution (Gartner Research, 2021; +Castellanos, 2021). Decision-making is, however, almost always a causal question and little +is known about the replication capabilities of these methods beyond correlations. For this +reason, this paper seeks to fill the gap by performing an investigation of the causal replication +capabilities of data replication methods as well as defining a path forward to making them a +viable option for decision-making. +There are a lot of advantages to the algorithmic approach to modelling, the most impor- +tant being increased performance and the opportunity for analysts to be systematic and +transparent about the process by which the model was selected (Athey, 2018). The power +of this approach has been apparent in several fields that have had incredible advances in +replicating reality due to the availability of large amounts of data. One of the most famous +examples is ImageNet, a database with millions of hand-labelled pictures, enabling revo- +lutionary progress in image recognition (Krizhevsky et al., 2012). GPT-3, a multi-purpose +natural language model, similarly achieved impressive results after learning from a data set +containing 45 TB of plain text (Brown et al., 2020). Besides these topics focused around +machine learning, examples can also be found in other fields such as physics and astron- +omy which have collected ever-growing volumes of data to learn from. Projects like the +Large Hadron Collider (Evans, 2009) and the imaging of the black hole at the centre of +galaxy M87 (Castelvecchi, 2019) handle data in the order of petabytes. However, such large +amounts of data are not always readily available. In many fields centring around individuals, +such as the social and health sciences (e.g. finance, insurance, medical fields), the collecting +or sharing of such datasets is far from trivial due to ethical and privacy concerns (Koenecke +and Varian, 2020). One recently emerging option to alleviating such concerns is generative +modelling. +Generative models are models that try to learn a representation of the (high-dimensional) +distribution of a dataset. Once this representation is learned, it can then be used to generate +new samples that maintain the original dataset’s distribution of features but that did not +appear in the original dataset 1. Generative methods are thus capable of simulating non- +existent but realistic-looking data, also referred to as synthetic data, that can be shared more +1Note that to have such privacy guarantees, one needs to explicitly include an optimisation for it in the +model fitting step such as in Yoon et al. (2019). Else there are cases when replication could occur (Feng et al., +2022). +2 + +freely. A well-known use-case are pictures of human faces for computer vision applications. +Even in the possession of a large dataset of pictures of human faces, sharing this freely could +present issues concerning privacy. However, generative models are capable of constructing +fake but human-looking faces that can, due to their non-existence, be shared more freely to +further the quality of applications. +While generative modelling has been around for decades, a major breakthrough in the +ability to efficiently training such models was achieved in 2014 with Generative Adversarial +Networks (GANs) (Goodfellow et al., 2014). This method increased our capacity to fit high- +dimensional distributions of data, like images and video data. The GAN framework has found +widespread applications throughout computer vision, like image generation (Karras et al., +2017; Radford et al., 2015), text to image translation (Zhang et al., 2016), the blending of +images (Wu et al., 2017), enhancing quality of pictures (Ledig et al., 2016a), filling in blanks +in pictures (Pathak et al., 2016), and a more infamous example of deepfakes (Mirsky and +Lee, 2022). While these are noteworthy variations and applications of the GAN framework, +the common factor here is the focus on computer vision. In contrast, GANs have found +limited adoption within the human sciences, like economics. +The main reason for this is that in these fields, most questions are inherently about identifi- +cation of causal effects. Neural networks, which are at the centre of the GAN framework, in +contrast, still focus mostly on high-dimensional correlations. An example of this is shown in +the paper by Beery et al. (2018), where they analyse a neural network trained to classify +images. The neural network appears to be able to accurately identify whether or not there is +a cow in a picture, until you ask the network to classify a picture of a cow in an uncommon +environment. The model is, for instance, not able to recognise a cow on a beach, because of +the spurious correlation between cows and grasslands. Learning to label images with grass +in it are shortcuts that expose the lack of generalisation of the neural network. Recently, +a field has emerged called Causal Machine Learning where researchers try to make steps +towards making machine learning models more causal (Scholkopf et al., 2021). While this +field is promising, due to the inverse problem nature of finding causality in observational +data, it is currently still in its infancy in regards to applicability. As we will show below. +The most prevalent used loss-functions for GANs are some form of binary cross-entropy (Good- +fellow et al., 2014; Yoon et al., 2019; Radford et al., 2015; Wiese et al., 2020) or Wasserstein +distance (Arjovsky et al., 2017; Xu et al., 2019; Athey et al., 2019). These losses indicate in +some form or another the difference between two joint probability functions. Replicating +the joint probability distribution, however, does not guarantee replication of the underlying +causal process. Finding the causal structure from observational data is an inverse problem, +finding the cause from the effect. Consider this example in The Black Swan from Taleb (Taleb, +2010): +"Operation 1 (the melting ice cube): Imagine an ice cube and consider how it may melt over +the next two hours while you play a few rounds of poker with your friends. Try to envision the +shape of the resulting puddle. +Operation 2 (where did the water come from?): Consider a puddle of water on the floor. +Now try to reconstruct in your mind’s eye the shape of the ice cube it may once have been. Note +that the puddle may not have necessarily originated from an ice cube." +3 + +Operation 1 is an example of the forward way of thinking, where the effect (the water) is to +be predicted from the cause (ice cube). With the right models it is possible to accurately +come up with the resulting pool of water. In contrast, operation 2 asks the inverse, finding +the shape of the cube (cause) from the pool of water (effect). There are however an almost +infinite amount of possible ice cubes that could have led to that pool of water. This example +also translates to joint probability distributions and underlying causal models. For a given +joint distribution there are a multitude of possible underlying causal models. +In this paper, we survey the literature on generative adversarial networks, being the dominant +model among generative models for synthetic data, and evaluate their capacity to preserve +certain causal structures (i.e. cross-sectional, time series, and full structural) in the synthetic +datasets they generate. We do so by first generating a dataset where the data-generating +function, and thus the structural causal model, is know. Secondly, we make a synthetic copy +of this with a specific GAN method and perform different causal analyses with an increasingly +lenient set of assumptions, from cross-sectional to time-series to structural. Lastly, we check +if the results in the real data align with those in the synthetic data to evaluate the causality +preserving capabilities. +We find that for relationships in data where the assumptions hold such that correlation +equals causation, inference on the real and synthetic data yield the same results only in +the case where the actual causal structure aligns with the most simple model that can +replicate the correlations in the data. In more complex cases, for instance when a variable +has time-dependence and both influences cross-sectional features as well as itself, we find +that the generative model converges on a model with the same general distribution, but that +it does so with a simpler underlying causal structure. Our results point at the reason being +the often-used regularisation in machine learning that builds in a preference for smaller +models (as posited in occam’s razor) which is not necessarily a valid principle in causality. +Finally, when the whole causal structure is considered, it becomes apparent that currently +the applicability is still limited due to the stringent assumptions that need to be met in order +to overcome the challenges of the inverse problem. +The remainder of this paper is structured as follows. In Chapter 2, we lay-out the problem +setup and discuss the structural approach we take to evaluate the causal replication capacity +of GAN-based models. In Chapter 3, we give a general introduction to the inner workings +of GAN-models and detail three different GAN variations that we take as representative +for the different streams in the GAN literature that aim to capture increasingly complex +correlations (i.e. cross-sectional correlations, time-series correlations, full causal structure). +In Chapter 4, we present the results of our evaluation. In Chapter 5, we discuss some of +the additional real-world challenges that we abstracted away from but that need to be +considered where these methods to be used in real-world cases. Lastly, in Chapter 6, we +summarise and conclude our findings. +4 + +2 +Problem setup +The goal of the evaluation in this paper is to see if current data replication methods are +useful when causal analyses are to be performed on the resulting synthetic data. To simulate +realistic use-cases of synthetic data for sectors/fields where decisions require causal inference, +we take popular causal inference methods from the field of econometrics. These models +have the benefit of having well-known sets of assumptions under which the estimated +parameters may be interpreted as causal effects. First, we consider a cross-sectional model, +namely ordinary least squares. This is the simplest case in which the researcher assumes that +observations at different timestamps are independent and the functional form of the model +is linear in the parameters. While being the simplest model to estimate in econometrics, it +also comes with the most stringent assumptions to be interpreted as causal, discussed in +Section 2.1. Next, we look at the class of time-series models that allow dependence between +observations at different timestamps in Section 2.2. Lastly, in Section 2.3, we discuss the +case where a full structural equation model is the goal of the estimation. Here, an approach +from the field of Causal Discovery is needed since none such data-driven method exists in +econometrics. +The evaluation setup is shown in Figure 1. First, a dataset is generated with known causal +structure. The design of this dataset is discussed in Section 2.4. The generated data is then +used to train a chosen type of GAN, designed to capture the distribution of the generated +data, further discussed in Chapter 3. The trained GAN is then sampled to construct a +synthetic dataset. Note that, in the remainder of this paper, artificial data we generate from +the known model are referred to as generated data while data sampled from the GAN models +are referred to as synthetic data. Next, a causal inference method, with its accompanying +assumptions, is selected to apply to both the generated data and the synthetic data. We first +validate that the chosen causal inference method is appropriate to estimate (a subset of) +causal connections in our defined causal structure. This is done by comparing the estimated +parameters of the model on the generated data to those we defined in the structural model +to generate it. Alignment of the two indicates that the causal inference method is indeed +appropriate to estimate causality for (a subset of) the underlying causal relationships. If +the GAN method therefore generates a synthetic dataset where the same causal inference +method does not estimate the same parameters as in the generated data, this difference +can be entirely attributed to the GAN method. This approach allows us to examine the +causality-preservation capabilities of the GAN methods. +5 + +Figure 1: Experiment setup for each choice of assumptions and GAN method +2.1 +Cross-sectional +The first type of causal relationships are those on a cross-sectional level. Ordinary least +squares (OLS) is a popular regression model to find causal effects in cross-sectional data. In +this case we assume that a variable can be represented by an OLS model. The OLS model +produces valid causal inference under the following assumptions: +Assumption 1: Linear in Parameters +The model can be written in the form: +y = β0 + β1x1 + β2x2 + ... + βkxk + ε +(1) +Assumption 2: Random Sampling +The sample of n observations (xi1, xi2,..., xik, yi) : i = 1,2,..., n is drawn randomly from the +model. +Assumption 3: No Perfect Collinearity +An independent variable in (1) cannot be an exact linear combination of the other indepen- +dent variables. +Assumption 4: Zero Conditional Mean +The expected value of the error ε should be zero, given any values of the independent +variables. +E(ε|x1, x2,..., xk) = 0 +Assumption 5: Homoskedasticity +6 + +ChooseGANand +Choose modeland +Generatedata +Generated data +train on generated +Syntheticdata +assumptions +according to model +data +Fit model to data +Fit model to data +Model parameters +Model parameters +CompareThe error ε has the same variance, given any values of the independent variables. +Var(ε|x1, x2,..., xk) = σ2 +Assumption 6: Normality +The error ε is normally distributed a zero mean and variance σ2 and independent of the +explanatory variables x1, x2,..., xk. +ε ∼ Normal(0,σ2) +2.2 +Time-series +In cross-sectional modelling observations have no time aspect, this changes when considering +time-series models. Here we consider the popular class of linear autoregressive models. The +assumptions to perform valid causal inference with these models are as follows: +Assumption 1: Linear in Parameters +The stochastic process (xt1, xt2,..., xtk, yt) : t = 1,2,..., n can be written in the form: +yt = β0 + β1xt1 + β2xt2 + ... + βkxtk + εt +(2) +Assumption 2: No Perfect Collinearity +An independent variable in (2) cannot be an exact linear combination of the other indepen- +dent variables. +Assumption 3: Zero Conditional Mean +For each t, the expected value of the error εt should be zero, given any values of the +independent variables. +E(εt|xt1, xt2,..., xtk) = 0, t = 1,2,..., n +Assumption 4: Homoskedasticity +The error εt has the same variance, given any values of the independent variables. +Var(ε|xt1, xt2,..., xtk) = σ2, t = 1,2,..., n +Assumption 5: No Serial Correlation +Given the independent variables xt1, xt2,..., xtk, errors in two different time steps are not +correlated. +7 + +Corr(εs,εt|xt1, xt2,..., xtk) = 0,∀t ̸= s +Assumption 6: Normality +The error εt is normally distributed a zero mean and variance σ2 and independent of the +explanatory variables xt1, xt2,..., xtk. +u ∼ Normal(0,σ2) +Most assumptions are very similar to the previous OLS assumptions. There are two main +differences. First is the absence of OLS Assumption 2 specifying observations to be randomly +sampled. Under time-series assumptions observations have an order determined by the time +step t. Second, time-series Assumption 5 is added, requiring the error term to have no serial +correlation. +In the time-series we will consider, autoregressive terms are included as well. We make an +additional assumption for this autoregressive time-series to be weakly dependent, meaning +the correlation between yt and yt+s is almost 0 for s large enough. In other words, as the +variables get farther away from each other in time, the correlation decreases. +yt = αyt−1 + ε +In the case above of an autoregressive model lagged for one period, this assumption is +satisfied if |α| < 1. +2.3 +Structural model +Lastly, the case remains where the whole causal structure is considered. Here, the goal is to +attempt to reconstruct the full structural causal model from the data. As far as we know, no +such methods exist in econometrics 2. For this reason, we adopt a method from the recently +developing field of causal discovery, situated mostly in the computer science literature, that +tries to accomplish this task. +Recovering the causal model from observational data is far from trivial. Recall the example +above of trying to figure out the shape of the ice cube from a pool of water. As many forms +of ice cubes can result in the same pool of water, many structural causal models can result +in the same observational data. Therefore, picking one of all possible models is dependent +on further assumptions made by each causal discovery algorithm. The general approach is +to embed known features of causality, such as environment independence (Arjovsky et al., +2020) or acyclicity (Zheng et al., 2018), into the loss function that a machine learning +algorithm optimises for. Even then, it is sometimes only possible to provide a set of possible +structural causal models that are all equally able to generate the observational data, also +called Markov equivalent. A recent trend is to extend the data to also include interventions +2In economics, and many other fields that model complex phenomena, a structural model is defined from +theory and then calibrated to data instead of trying to infer the complete model itself from the data. +8 + +and their outcomes (Vowels et al., 2022). This extra information can be used to exclude +certain Markov equivalent models and decrease the set of potential underlying causal models. +One of the more frequently used causal discovery algorithms is LiNGAM (Shimizu et al., +2006), which assumes that the causal effects are linear, the generating causal graph is acyclic, +that the distribution of the noise is non-gaussian and no unobserved confounders. The +LiNGAM model can be expressed in matrix form as follows: +x = Bx + e +with the observed variables x, the connection strength matrix B and exogenous variables e. +The condition of acyclicity allows the matrix B to be permuted to become lower triangular +with a zero-diagonal. With the additional assumption of the exogenous variables e, or in +other words the noise, being non-Gaussian, the matrix B can be uniquely identified using only +the data x. This identifiability thus means that the algorithms results in a single causal graph. +Different variations on this method exist like models with hidden common causes (Hoyer +et al., 2008), time-series (Hyvärinen et al., 2010) or non-linearity (Zhang and Hyvärinen, +2009). +In the case of Gaussian noise, only a set of Markov equivalent causal models can be estimated, +while under the assumption of non-Gaussian noise this set can be reduced to one full +causal model. This assumption is, however, in contrast with the assumption off Gaussian +noise needed in many inference methods for valid causal inference, including the OLS and +autoregressive models we discussed above. +2.4 +Generated dataset +We define the following model: +yt = αyt−1 + β1x1,t + β2x2,t + ε1 +x1,t = β3z1,t + β4z2,t + ε2 +x2,t = β5z2,t + ε3 +z1,t = ε4 +z2,t = ε5 +(3) +A graphic representation of this structural model, also called the causal graph, is shown in +Figure 2. For the estimation of this causal structure with the different inference methods, +we will always assume full observability. +The variables x1 and x2 are a linear combinations of the contemporaneous values of z1 and +z2. The underlying models for these two variables therefore meet the assumptions of the +cross-sectional ordinary least squared (OLS) model. OLS should therefore be an appropriate +method to estimate the causal effects of z1 and z2 on x1 and x2. We confirm this in the +Results section. +For the variable yt, extending the assumption on the data to allow for autocorrelation, a +first order autoregressive model can infer α on β1 and β2. +9 + +Finally a variant of LiNGAM for time-series can be used to infer the causal structural model. +While the model was specifically chosen to contain both cross-sectional and time-series +causality, it is easy to think of real-world model that follow this functional form. One example +is a simple income process, where the monthly income now depends on the income last +month and some contemporaneous features (e.g. employment sector, location) which in +turn are distributed according to (conditionally) random distributed preferences. +Figure 2: Full causal model of the generated dataset. +10 + +Z1 +X1 +y +Z2 +X23 +Generative adversarial networks +Generative adversarial networks, or GANs, is a framework for generative machine learning +first introduced by Goodfellow et al. in 2014 (Goodfellow et al., 2014). A generative model +takes a training dataset drawn from a real world distribution as input and tries to replicate +this data distribution. The framework has shown great success in generating synthetic +images indistinguishable from real images (Ledig et al., 2016b; Brock et al., 2019; Karras +et al., 2020). While the focus has been on the improvement of the framework for image +generation and manipulation, the GAN framework has recently also gathered attention for +its possibilities with numerical and categorical data, like tabular and time series data. +3.1 +Framework +A generative adversarial network consists of two competing neural networks: a generator +G, which generates fake data, and a discriminator D, that is trained to discern which data +is fake (made by the generator) and which data is real. The process can be described as +a zero-sum game between the generator and discriminator. During the training process +the generator adapts to better fool the discriminator and the discriminator in turn adapts +to better detect the fake data. The resulting trained generator can then be extracted to +replicate the distribution of the original data. +3.1.1 +Architecture +Figure 3 shows the basic structure of a GAN. The generator G learns to map a latent space +pz to a more complex distribution pg, which is the distribution meant to mimic the real data +distribution pdata. Typically, this latent space is a high-dimensional space with each variable +drawn from a Gaussian distribution with a mean of zero and a standard deviation of one. +The concept is thus that one can insert sample of noise (z) into the generator, which it will +learn to map onto a sample of the distribution of the real data. The generating function can +then be described by +Figure 3: Generative Adversarial Networks diagram. +11 + +Real +Sample +Real Data +Pdata +Discriminator +RealorFake +Loss +Fake +Generator +Sample +~Pz +TrainingG(z) = X g +(4) +where X g are samples created by the generator. The discriminator D has the task of dis- +tinguishing the fake data X g from the real data Xdata. The generator and discriminator +are trained by playing a non-cooperative game against each other. The main aim of the +generator is to produce samples which are similar to the real data. On the other hand, the +main aim of the discriminator is to distinguish between fake samples from the generator +and samples from the real data. The discriminator D receives both samples and tries to +determine which comes from the real data distribution by assigning a probability D(x), +which signifies the certainty the discriminator has in its decision. If D(x) = 1, the sample x +is thought to come from pdata. On the other hand, if D(x) = 0, the discriminator judges the +sample to be from pg. This prediction from the discriminator and the known ground truth is +then used to improve both the generator and the discriminator. During the joint training of +the generator and discriminator, G will start to generate increasingly realistic samples to +fool the discriminator, while the discriminator learns to better differentiate the real and fake +samples. The end goal of the GAN as a whole is that the discriminator can no longer tell the +difference between the generated samples X g (D(x) = 1/2) and the real data samples Xdata +with the discriminator no longer able to improve itself. +3.1.2 +Loss function +The objective function of the GAN tries to match the real data distribution pdata with pg. The +original GAN (Goodfellow et al., 2014) uses two objective functions. The objective for D is +to maximize the probability of assigning the correct label to both real and fake samples. This +done by minimizing the negative log-likelihood for binary classification. Simultaneously +G is trained to minimize log(1-D(G(z))), thus maximizing the probability of the generated +samples being classified as real by the discriminator. This results in a mini-max game with +objective function V(G,D): +min +G max +D +V(D, G) = �x∼pdata[logD(x)] + �z∼pz[log(1 − D(G(z)))] +(5) +The value function V(G,D) is known as the binary cross entropy function, commonly used in +binary classification tasks. +3.2 +GAN extentions +Many different variations of GANs have been proposed since its inception. In this section +different relevant adaptions are presented, ordered by which level of causality they are +aiming to improve. +3.2.1 +TimeGAN +TimeGAN by Yoon et al. (Yoon et al., 2019) is an adaptation of the original GAN framework +that aims to improve the preservation of temporal dynamics for time-series data. This means +that newly generated sequences should respect the original relationships between variables +across time. Two main ideas are combined in the TimeGAN framework, the flexibility of the +12 + +Figure 4: TimeGAN diagram +unsupervised GAN framework and a more controllable supervised autoregressive model. +Figure 4 shows the structure of TimeGAN. +The TimeGAN framework contains the components of a generative adversarial network, +as well as an auto-encoder. The latter takes as input a vector of static features, s, and a +vector of temporal features, x1:T. The encoder is then trained to map the feature space, +which s and x1:T belong to, to a latent space. This allows the adversarial network to learn +the underlying temporal dynamics of the data via lower-dimensional representations. The +output of the encoder are the latent vectors hs and ht, being lower-dimensional latent codes +of the input s and x1:T. In the opposite direction, the decoder takes the static and temporal +latent vectors back to their feature representations. The reconstructed static and temporal +features are respectively denoted as ˜s and ˜xt. +The other main component in the framework, the generative adversarial network, has a +generator that takes as input random noise vectors and outputs latent vectors ˆhs and ˆht. +The generator in this framework is autoregressive, meaning it also uses its previous outputs +ˆh1:t−1 for the construction of ˆht. A key difference with a regular GAN architecture is that the +generator maps to this latent space instead of the usual feature space. Both the real latent +codes hs and ht and the synthetic latent codes ˆhs and ˆht are received by the discriminator, +which has the task to classify these codes as either real or fake. +The resulting framework has three loss functions. First, the reconstruction loss. This loss is +linked to the auto-encoder component of the framework, quantifying the difference between +original features s, xt and the reconstructed features ˜s and ˜xt. +�R = �s,x1:T ∼p[||s − ˜s||2 + +� +t +||xt − ˜xt||2] +(6) +Second, the unsupervised loss is the same type of loss used in the original GAN framework, +13 + +Encoder +Decoder +Reconstruction +S, 1:T +loss +Supervised +loss +Generator +Discriminator +Unsupervised +Zs, Z1:T +lossmaximising (discriminator) or minimising (generator) the likelihood of providing correct +classifications. Notations y and ˆy denote classifications by the discriminator as respectively +real or synthetic data. +�U = �s,x1:T ∼p[log ys + +� +t +log yt] + �s,x1:T ∼ˆp[log(1 − ˆys) + +� +t +log(1 − ˆyt)] +(7) +Lastly, the supervised loss is introduced. The addition of this loss is motivated by the idea +that the regular feedback from the discriminator, the unsupervised loss, may be insufficient +incentive for the generator to capture the step-wise conditional distributions in the data. +To calculate this loss, the autoregressive generator g uses the real latent codes hs and ht−1 +instead of the synthetic ˆhs and ˆht−1 to generate ˆht, or g(hs,ht−1,zt), as shown in (8). +�S = �s,x1:T ∼p[ +� +t +||ht − g(hs,ht−1,zt)||2] +(8) +A linear combination of �U and �S is used to train the generator and the discriminator. +�U guides the generator to create realistic sequence, while �S uses ground-truth targets +to ensure that the stepwise transitions are similar. To train the autoencoder components, +the encoder and the decoder, a linear combination of �R and �S is used. By combining the +different objectives, TimeGAN is trained to simultaneously encode feature vectors, generate +latent codes for these feature vectors, and iterate across time. +3.2.2 +CausalGAN +CausalGAN is a generative adversarial framework proposed by Kocaoglu et al. (Kocaoglu +et al., 2017). CausalGAN is an implicit causal generative model that replicates data constraint +to a given causal graph. Implicit generative models, which the original GAN model is part of, +can sample from a probability distribution, without the ability to provide likelihoods for the +samples (Mohamed and Lakshminarayanan, 2016). Causal implicit generative models can +not only sample from a probability distribution but also from conditional and interventional +distributions, which causal graphs embeds. +Consider a simple causal graph, A −→ C ←− B, as depicted in Figure 5a. The parent nodes, +A and B are assumed to have no other variables influencing their distribution and can be +written as A = GA(ZA) and B = GB(ZB), where Z∗ is some chosen noise distribution (e.g. +Gaussian), and G∗ is a function mapping this distribution to the distribution of the variable. +The variable C has two parent nodes and can be written as C = GC(A, B, ZC), being a function +of both A and B, as well as a chosen distribution. This representation is similar to how the +generator of the original GAN framework is structured. Figure 5b shows how a generator +can be constructed to represent a given causal graph. For each variable a feedforward neural +network is used represent functions G∗, resulting in a larger generator network consisting +of linked individual generators. By building in the causal graph into the generator, it will +constrain the data generation data to the actual causal model and not only reproduce joint +probabilities but also the causal relationships. For the implementation of CausalGAN in +this paper Causal-TGAN (Wen et al., 2021) is used. This version uses the same core idea as +CausalGAN, with some added adjustments for tabular data. +14 + +(a) Causal graph A −→ C ←− B +(b) Generator architecture for A −→ C ←− B +Figure 5 +The downside is of course that both data and the relevant causal graph needs to be known +to train and use the generator. To this end, we use a causal discovery method, here the +standard- and time-variant of LiNGAM to provide us with the causal graph of the data. +15 + +BZA +GA +G +B +GB +ZB4 +Results +Consider the model described in Section 2.4 with the following parameters: +Parameter +Value +α +0.5 +β1,β2,β3,β4,β5 +1 +σ1,σ2 +1 +where ε∗ ∼ N(0,0.5). From this model we sample 10,000 observations to use for further +experiments. These observations will further be referred to as generated data and will +be used to both train the different GAN models and give baseline values for estimated +parameters. The experiments assume a perfect scenario where the model is known. Each +experiment is done, in its entirety, 10 times and reported results show averages and standard +deviations over these 10 runs. +4.1 +GAN +First, we train a standard GAN with the generated data. From this GAN, we generate 10,000 +samples to preserve the statistical power of our inference results. These latter samples will +be referred to as the synthetic data. The first model we fit on both datasets is OLS for the +following variables: +x1 = β3z1 + β4z2 + ε2 +x2 = β5z2 + ε3 +The resulting parameters can be seen in Table 1. The results show that on a cross-sectional +level, with the underlying model meeting the assumptions in 2.1, the GAN methodology can +replicate data with similar causal relationships. While on average the causal relationships +detected in the synthetic data are less accurate than the causal relationships in the generated +data, the results are not significantly different from the true parameters. +While the data the GAN is trained on is time-ordered, the synthetic data produced by the +GAN is sampled randomly, without any notion of time. So, as expected, when running an +autoregressive model on the y variable in our model, it does not find any time-correlation +(α coefficient for y) in the synthetic data. Interestingly, it does capture the cross-sectional +relationships for y (β1 and β2). +4.2 +TimeGAN +Next, TimeGAN is trained on the generated dataset, after which we again sample 10,000 +datapoints for a new synthetic dataset. Note that the sampled datapoints are now ordered +in time instead of randomly sampled as in section 4.1. +As can be seen in Table 1, the synthetic data produced by TimeGAN does not properly +maintain causal relationships, neither on a cross-sectional level nor over time. The results +16 + +Model +Par. +Real +GAN +TimeGAN +CausalGAN +OLS +β3 +0.9990 ± 0.0051 +1.0209 ± 0.0715 +0.3762 ± 0.4320 +0.9869 ± 0.1087 +β4 +1.0017 ± 0.0052 +1.0797 ± 0.1272 +1.2249 ± 0.3362 +0.9666 ± 0.1029 +β5 +0.9996 ± 0.0057 +1.0157 ± 0.1266 +1.1066 ± 0.0179 +1.0006 ± 0.1625 +TS +α +0.5004 ± 0.0011 +0.0030 ± 0.0020 +0.0233 ± 0.1331 +0.0011 ± 0.0064 +β1 +0.9993 ± 0.0040 +1.0007 ± 0.1773 +1.0597 ± 1.5236 +0.9635 ± 0.1896 +β2 +0.9982 ± 0.0045 +1.1439 ± 0.1682 +0.8796 ± 2.0436 +0.9927 ± 0.2035 +Table 1: Results for all GANs +are far from what would be expected and also vary significantly from run to run, resulting in a +higher standard deviations in the results. This is likely due to there being no auto-correlation +in the variables outside of y, and TimeGAN attempting to find time dependent structure +where none exists. To confirm this, we also consider the following alternate causal structure, +where all variables have some sort of time-dependence (direct or indirect). +yt = αyt−1 + β1x1,t + β2x2,t + ε1 +x1,t = β3z1,t + β4z2,t + ε2 +x2,t = β5z2,t + ε3 +z1,t = z1,t−1 + ε4 +z2,t = z2,t−1 + ε5 +(9) +Table 2 shows the results for TimeGAN in the case of the alternative structure. In this case +TimeGAN is able to accurately capture the causal relationships on a cross-sectional level (β3, +β4, β5) but still fails to capture the structure in y (α, β1 and β2). However, it does not seem +like the model completely missed the mark. When we look at the original formulation for y, +with the chosen parameters for the experiment, it can be rewritten as follows: +yt = 0.5yt−1 + x1,t + x2,t + εt +yt = 0.25yt−2 + (x1,t + 0.5x1,t−1) + (x2,t + 0.5x2,t−1) + (εt + 0.5εt−1) +yt = 0.125yt−3 + (x1,t + 0.5x1,t−1 + 0.25x1,t−2) + +(x2,t + 0.5x2,t−1 + 0.25x2,t−2) + (εt + 0.5εt−1 + 0.25εt−2) +This decomposition of y can be continued further until the autoregressive part for y is +negligible. Now, if the change in x1 and x2 in each time step is limited and thus x1,t ≈ x1,t−1 +and x2,t ≈ x2,t−1, as is the case here due to the stationarity of y, and using +� +n=0( 1 +2)n = 2, +we can write: +yt ≈ 2x1,t + 2x2,t + ε +with ε ∼ N(0, 4 +3). The results shown in Table 2 thus suggest that TimeGAN has learned this +smaller representation of y, using only x1 and x2, that results in the same expected values +17 + +Model +Parameter +Real +TimeGAN +OLS +β3 +0.9999 ± 0.0002 +0.9967 ± 0.0247 +β4 +1.0000 ± 0.0001 +1.0049 ± 0.0219 +β5 +0.9999 ± 0.0001 +1.0005 ± 0.0024 +TS +α +0.4999 ± 0.0008 +-0.0128 ± 0.0207 +β1 +1.0000 ± 0.0016 +2.0719 ± 0.0680 +β2 +1.0000 ± 0.0016 +2.0002 ± 0.1550 +Table 2: Result for TimeGAN on the second model +of y over time. This representation, however, does not represent the actual causal model +underlying y. +4.3 +CausalGAN +Lastly, the full structural causal model is considered. Here, a model can not be directly +trained to the data since no such method exists as far as the authors are aware. A two-step +approach is taken where first the causal structure is identified with LiNGAM. This extracted +structure is then compared to our data generating model (Eq. 3) to check if LiNGAM is an +appropriate and efficient causal discovery method for our case. Then CausalGAN is used to +generate data that follows this structure. Lastly, LiNGAM is applied to the synthetic data +and its output is compared to the causal structure retrieved from the generated data. +As noted before, LiNGAM uses the assumption of non-Gaussian noise, which is incorrect for +model (3) used previously in this section. To start from a correct causal structure for this +experiment, we adjust the distribution of the noise our data structure (3) to be uniformly +distributed, ε∗ ∼ U(−1,1). Under these conditions the time-variant of LiNGAM is able to +find the underlying causal model correctly. However, CausalGAN is not equipped to deal +with time-series, so we are forced to only consider the cross-sectional causal relations here. +Table 3 shows all causal relationships detected by LiNGAM in both the generated dataset and +the synthetic dataset produced by CausalGAN. Additionally, we show the causal relationships +detected in synthetic data from a basic GAN trained on the real data. For this one repre- +sentative example is chosen since the use of means and standard deviations give warped +representations of the results. The synthetic data sampled from CausalGAN consistently +maintains causal relationships relatively well. Some deterioration can be seen, as well +as introducing small additional causal effects. The basic GAN framework is however not +capable of retaining the causal relationships when the whole causal structure is considered. +Causal discovery on the synthetic data of the basic GAN gives varying results even when +performed multiple times on one synthetic dataset. None of the resulting graphs are close to +the original causal graph. This shows that adding the additional information of the (correct) +underlying causal graph through the CausalGAN model does help maintaining the causal +structure. +18 + +Causal effect +Real +CausalGAN +GAN +z1 −→ x1 +1.00 +0.93 +1.03 +z2 −→ x1 +1.01 +0.80 +1.07 +z2 −→ x2 +0.99 +0.83 +0.16 +x1 −→ y +1.02 +1.04 +0.14 +x2 −→ y +1.01 +1.00 +0.39 +z1 −→ z2 +0 +0 +-1.11 +z1 −→ x2 +0 +0 +-0.47 +z1 −→ y +0 +0 +0.86 +z2 −→ y +0 +0.14 +-0.10 +x2 −→ x1 +0 +0.14 +0.65 +Table 3: Causal effects detected by LiNGAM on both the generated dataset and the synthetic +dataset generated by CausalGAN. The table contains all significant causal effects (> 0.1). +Causal effects of less significance (< 0.1) are simplified to 0. Bold number indicate that the +causal effect is reversed +19 + +5 +Real world challenges +In our tests of the causality replicating capabilities of GANs, we have purposely abstracted +away from many of the additional challenge that come with working with real-world data. +In this section we address three of the most important challenges and give an overview of +the variations on the GAN framework that have been proposed to tackle them. +5.1 +Privacy +Privacy concerns are one of the main drivers for the recent rise in interest in synthetic data. +While in general synthetic data is sampled from a reconstruction of the distribution of the +original data, fear of replicating real samples due to overfitting remain (Webster et al., 2019; +Feng et al., 2022). Membership inference attacks also form a common concern in the field of +privacy (Hayes et al., 2017; Chen et al., 2020). These attacks leverage the fact that machine +learning models generally perform better on the data it was trained on to reconstruct the +training data. +These concerns have sparked the search for GAN variants that give certain privacy guarantees. +One such guarantee is differential privacy. An algorithm is differentially private if an observer +seeing the output can not tell if a particular datapoint was used in the computation. In the +case where the observer has access to the generated samples but not the generator, recent +work has shown that the base form of GAN has some privacy guarantees in terms of both +differential privacy and robustness to membership inference attacks (Lin et al., 2022). These +guarantees get stronger for larger training datasets. If additionally the generator is available, +several differential privacy GANs have been proposed, such as DPGAN (Xie et al., 2018), +PPGAN (Liu et al., 2019) and PATE-GAN (Yoon et al., 2019). +Privacy guarantees, however, come at the price of replication quality since you in some form +or another adding noise to the data by limiting the impact a training sample can have on the +model, even though it might be highly informative (Huang et al., 2017; Lin et al., 2020). +5.2 +Fairness +Machine learning has an increasingly large impact on current day decision making, scaling +decisions made on a micro-scale to a macro-scale in an often opaque manner. This trend +has raised concerns about building in, or scaling up biases in decisions. Fairness in machine +learning is a recently growing area of research that studies how to ensure that such biases and +model inaccuracies do not lead to discriminatory models on the basis of sensitive attributes +such as gender or ethnicity. Using synthetic data can help by debiasing the data before it +even gets used for further analysis. In such a framework a generative model is trained on +unfair data to generate synthetic fair data. +A first challenge to fairness is defining what it actually is, which is often highly dependent +on the context of the business decisions that is being made with the model. One often used +interpretation is that certain features, also called protected or sensitive features (e.g. gender, +ethnicity), should not have any impact on the outcome of the model. This orthogonalisation +of the model outcome and the protected features comes with two major challenges. First, +it requires outside definition of what the protected features are. Second, if you want to +20 + +rid observational data of such biases, it is not enough to just delete the features, you need +to know the relevant causal structures to exclude both the direct and indirect impact the +protected attribute has on the outcome (Kusner et al., 2017; Zhang and Bareinboim, 2018). +Otherwise the model can just learn the protected features by using different proxies which +are correlated to them (van Breugel et al., 2021). CFGAN (Xu et al., 2019) and DECAF (van +Breugel et al., 2021) are two methods to generate fair data that are rooted in this approach +to fairness. Both methods therefore require a causal graph as additional input, something +we saw in our results is not generally feasible with current causal discovery methods. +FairGAN (Xu et al., 2018) and Fairness GAN (Sattigeri et al., 2018) have also been suggested +for the purpose of generating fair data. FairGAN uses an additional discriminator on top +of the classical GAN architecture to determine whether samples are from the protected or +unprotected group. Fairness GAN uses an added loss function that encourages demographic +parity. Demographic parity is satisfied if the decisions made from the data are not dependent +on a given sensitive attribute. This requires a specification of the explanatory variables x, +the target variables y and the sensitive variables s, where y does not need specification in +other methods. FairGAN is applied to low-dimensional structured data, while Fairness GAN +is applied to high dimensional image data. +5.3 +Tabular data +Tabular data is data that contains both discrete and continuous columns and is one of the most +commonly encountered data formats in both business and research (Xu and Veeramachaneni, +2018). Tabular data, and especially the discrete features within them are challenging for +GAN methods since the continuous functions used in neural nets are ill-equipped to fit the +non-continuous distributions of discrete variables. +The generator of a regular GAN cannot generate discrete samples because the generator is +trained by the loss from the discriminator via backpropagation (Goodfellow et al., 2014). +To tackle this problem, MedGAN (Choi et al., 2018) adds an autoencoder model to the +regular GAN framework to generate high-dimensional discrete variables. Both TGAN (Xu and +Veeramachaneni, 2018) and TableGAN (Park et al., 2018) look to improve the performance +on the continuous distributions as well. TGAN clusters numerical variables to deal with the +multi-modal distribution for continuous features and adjusts the loss function to effectively +generate discrete features. TableGAN uses a classifier neural network to predict synthetic +records’ labels to improve consistency in generated records. An additional loss, information +loss, is introduced as well. This loss is the difference in key statistical values of both the real +and synthetic data. In the paper the mean and standard deviation are used as key statistical +properties. +Besides the mix of continuous and discrete columns, the distributions of data often differs +from the standard Gaussian-like distributions found in typical generative applications like +image generation. To this end CTGAN (Xu et al., 2019) addresses additional concerns +about non-Gaussian and multi-modal distributions, and imbalanced categorical columns. +CTAB-GAN (Zhao et al., 2021) looks further into these issues and tackles data imbalance and +long-tail distributions. The previously mentioned Causal-TGAN (Wen et al., 2021) combines +ideas of CTGAN and CausalGAN (Kocaoglu et al., 2017) to leverage knowledge about the +causal structure for a better performance. +21 + +6 +Conclusion +Data has become a driving force in both business, research, and policy. And rightfully so if +we see how increased access to data has furthered our ability to understand and support +decision making in complex environments. While some fields are just collecting more and +more data in labs or in nature, most of the decision making that occurs in business is in +regards to actual human beings. This rightfully raises concerns about privacy and ethics. +Should companies just be allowed to collect, buy, sell, and share more and more data on the +behaviour and features of actual human beings just for the sake of making better business- +decisions? The answer is obviously no, and regulatory bodies are acting accordingly by +setting in place boundaries on what is and is not allowed in regards to data on individuals. +How do we balance the benefits of increased accuracy and understanding with the privacy +and ethics concerns that both come with having more data? One solution that has gained +a lot of traction is synthetic data, which are data sampled from generative methods that +are meant to replicate high-dimensional distributions of data. After all, the improvements +in modelling complex phenomena come from sufficient coverage of the high-dimensional +distribution of relevant features, and not from knowing someone’s exact name or address. +So if we could generate data with the same distribution as the original, but not containing +any identifiable features as well as different enough in exact values such that no individual +could be uniquely linked to one sample from the data, we could have all the benefits without +introducing risks to privacy or ethics. +While this is true for predictive models, that solely map correlations to an outcome, many +decisions intend an intervention to influence the outcome. The difference lies in that the +former asks an observational question: “If I observe X, what will Y be?” and the latter +asks an interventional question: “If I do X, how will the outcome Y change”. Apart from a +group of so-called policy prediction problems (Kleinberg et al., 2015), which only require a +prediction to make a decision, the latter requires causal inference. Once we enter into this +territory, it no longer just matters that the synthetic data has the right distribution, but also +that it was generated with the correct underlying causal relationships. and because there +can exist multiple underlying structures that generate the same distribution, there are no +intrinsic guarantees that current generative modelling methods converge on the correct one. +We evaluate the causal replication capabilities of the generative modelling techniques that +are typically used for synthetic data. As far as we know, we are the first to do so with a focus +on causality. We find that in the case where the assumptions are met that make correlation +equal causation, causal inference on the real and synthetic data yield the same results only +if the simplest model that can generate the distribution of the features equals the real one. +This points at the principle of occam’s razor, that is the foundation for regularisation in +machine learning to counter overfitting, is actually working against us in the case where we +want to replicate causal relationships. +When nothing is known about the causal structure, and the analyst can thus not easily +construct a functional form to test with classic causal inference methods like OLS, causal +discovery can be used. Causal discovery tries to find the complete causal structure in +observational data, which can then be used as input for a generative model that can generate +synthetic data explicitly according to the causal structure. We find that, while this works in +simple cases (e.g. in the case of cross-sectional correlation with non-gaussian noise), the +22 + +necessary assumptions on both the causal discovery and generation side seem too restrictive +to be widely applicable in real-world contexts. +A path forward seems to be to augment the observational data fed to the GAN models with +additional information such as knowledge on different environment in which the data was +collected or interventional data from experiments (Scholkopf et al., 2021). While this can +present a way forward for many fields, it is often not applicable in the context of businesses +related to people’s finances or health. +Organisations that want to improve their decision making by leveraging synthetic data +should thus be careful about what the current state-of-the-art is actually capable of. +23 + +References +Arjovsky, M., L. Bottou, I. Gulrajani, and D. Lopez-Paz (2020). Invariant risk minimization. +Arjovsky, M., S. 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Dags with no tears: Continuous +optimization for structure learning. +27 + diff --git a/adAzT4oBgHgl3EQfLPsZ/content/tmp_files/load_file.txt b/adAzT4oBgHgl3EQfLPsZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd96a60caefd4986ae4a978133a1ee42de239d2a --- /dev/null +++ b/adAzT4oBgHgl3EQfLPsZ/content/tmp_files/load_file.txt @@ -0,0 +1,1074 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf,len=1073 +page_content='On the causality-preservation capabilities of generative modelling Yves-Cédric Bauwelinckx1, Jan Dhaene1, Milan van den Heuvel 2, and Tim Verdonck1,3 1Department of Economics, Katholieke Universiteit Leuven, Belgium 2Department of Economics, Universiteit Gent, Belgium 3IMEC, Universiteit Antwerpen, Belgium Contact: Milan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='vandenHeuvel@UGent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='be (Address: Tweekerkenstraat 2, 9000, Ghent, Belgium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='01109v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='LG] 3 Jan 2023 1 Introduction To make sense of the complexities of reality, and make optimal decisions accordingly, organ- isations and researchers have always striven to come up with models that can accurately represent observed phenomena (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' consumption behaviour, loan defaults).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In the past, these models were defined by the analyst and calibrated to (small) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Recently, however, during the so-called machine learning revolution, the focus shifted to a more data-driven, algorithmic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Machine learning algorithms now search for the optimal model by finding support for it in the data instead of being chosen by the analyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This approach has increased the collection of, investment in, demand for, reliance on, and value of data for organisations and research significantly (McKinsey & Company, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' It has also brought the tension between utility of data and privacy of its subjects to the forefront of public discussion (European Commission, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Recently developed generative modelling meth- ods, which generate data with a distribution similar to the original but without containing any of the real data, have been proposed as a potential solution (Gartner Research, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Castellanos, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Decision-making is, however, almost always a causal question and little is known about the replication capabilities of these methods beyond correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' For this reason, this paper seeks to fill the gap by performing an investigation of the causal replication capabilities of data replication methods as well as defining a path forward to making them a viable option for decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' There are a lot of advantages to the algorithmic approach to modelling, the most impor- tant being increased performance and the opportunity for analysts to be systematic and transparent about the process by which the model was selected (Athey, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The power of this approach has been apparent in several fields that have had incredible advances in replicating reality due to the availability of large amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' One of the most famous examples is ImageNet, a database with millions of hand-labelled pictures, enabling revo- lutionary progress in image recognition (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' GPT-3, a multi-purpose natural language model, similarly achieved impressive results after learning from a data set containing 45 TB of plain text (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Besides these topics focused around machine learning, examples can also be found in other fields such as physics and astron- omy which have collected ever-growing volumes of data to learn from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Projects like the Large Hadron Collider (Evans, 2009) and the imaging of the black hole at the centre of galaxy M87 (Castelvecchi, 2019) handle data in the order of petabytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' However, such large amounts of data are not always readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In many fields centring around individuals, such as the social and health sciences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' finance, insurance, medical fields), the collecting or sharing of such datasets is far from trivial due to ethical and privacy concerns (Koenecke and Varian, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' One recently emerging option to alleviating such concerns is generative modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Generative models are models that try to learn a representation of the (high-dimensional) distribution of a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Once this representation is learned, it can then be used to generate new samples that maintain the original dataset’s distribution of features but that did not appear in the original dataset 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Generative methods are thus capable of simulating non- existent but realistic-looking data, also referred to as synthetic data, that can be shared more 1Note that to have such privacy guarantees, one needs to explicitly include an optimisation for it in the model fitting step such as in Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Else there are cases when replication could occur (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 2 freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' A well-known use-case are pictures of human faces for computer vision applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Even in the possession of a large dataset of pictures of human faces, sharing this freely could present issues concerning privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' However, generative models are capable of constructing fake but human-looking faces that can, due to their non-existence, be shared more freely to further the quality of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' While generative modelling has been around for decades, a major breakthrough in the ability to efficiently training such models was achieved in 2014 with Generative Adversarial Networks (GANs) (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This method increased our capacity to fit high- dimensional distributions of data, like images and video data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The GAN framework has found widespread applications throughout computer vision, like image generation (Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2015), text to image translation (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2016), the blending of images (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2017), enhancing quality of pictures (Ledig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2016a), filling in blanks in pictures (Pathak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2016), and a more infamous example of deepfakes (Mirsky and Lee, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' While these are noteworthy variations and applications of the GAN framework, the common factor here is the focus on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In contrast, GANs have found limited adoption within the human sciences, like economics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The main reason for this is that in these fields, most questions are inherently about identifi- cation of causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Neural networks, which are at the centre of the GAN framework, in contrast, still focus mostly on high-dimensional correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' An example of this is shown in the paper by Beery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' (2018), where they analyse a neural network trained to classify images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The neural network appears to be able to accurately identify whether or not there is a cow in a picture, until you ask the network to classify a picture of a cow in an uncommon environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The model is, for instance, not able to recognise a cow on a beach, because of the spurious correlation between cows and grasslands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Learning to label images with grass in it are shortcuts that expose the lack of generalisation of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Recently, a field has emerged called Causal Machine Learning where researchers try to make steps towards making machine learning models more causal (Scholkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' While this field is promising, due to the inverse problem nature of finding causality in observational data, it is currently still in its infancy in regards to applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' As we will show below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The most prevalent used loss-functions for GANs are some form of binary cross-entropy (Good- fellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Wiese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2020) or Wasserstein distance (Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Athey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' These losses indicate in some form or another the difference between two joint probability functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Replicating the joint probability distribution, however, does not guarantee replication of the underlying causal process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Finding the causal structure from observational data is an inverse problem, finding the cause from the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Consider this example in The Black Swan from Taleb (Taleb, 2010): "Operation 1 (the melting ice cube): Imagine an ice cube and consider how it may melt over the next two hours while you play a few rounds of poker with your friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Try to envision the shape of the resulting puddle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Operation 2 (where did the water come from?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' ): Consider a puddle of water on the floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Now try to reconstruct in your mind’s eye the shape of the ice cube it may once have been.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Note that the puddle may not have necessarily originated from an ice cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='" 3 Operation 1 is an example of the forward way of thinking, where the effect (the water) is to be predicted from the cause (ice cube).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' With the right models it is possible to accurately come up with the resulting pool of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In contrast, operation 2 asks the inverse, finding the shape of the cube (cause) from the pool of water (effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' There are however an almost infinite amount of possible ice cubes that could have led to that pool of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This example also translates to joint probability distributions and underlying causal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' For a given joint distribution there are a multitude of possible underlying causal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In this paper, we survey the literature on generative adversarial networks, being the dominant model among generative models for synthetic data, and evaluate their capacity to preserve certain causal structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' cross-sectional, time series, and full structural) in the synthetic datasets they generate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' We do so by first generating a dataset where the data-generating function, and thus the structural causal model, is know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Secondly, we make a synthetic copy of this with a specific GAN method and perform different causal analyses with an increasingly lenient set of assumptions, from cross-sectional to time-series to structural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Lastly, we check if the results in the real data align with those in the synthetic data to evaluate the causality preserving capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' We find that for relationships in data where the assumptions hold such that correlation equals causation, inference on the real and synthetic data yield the same results only in the case where the actual causal structure aligns with the most simple model that can replicate the correlations in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In more complex cases, for instance when a variable has time-dependence and both influences cross-sectional features as well as itself, we find that the generative model converges on a model with the same general distribution, but that it does so with a simpler underlying causal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Our results point at the reason being the often-used regularisation in machine learning that builds in a preference for smaller models (as posited in occam’s razor) which is not necessarily a valid principle in causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Finally, when the whole causal structure is considered, it becomes apparent that currently the applicability is still limited due to the stringent assumptions that need to be met in order to overcome the challenges of the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In Chapter 2, we lay-out the problem setup and discuss the structural approach we take to evaluate the causal replication capacity of GAN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In Chapter 3, we give a general introduction to the inner workings of GAN-models and detail three different GAN variations that we take as representative for the different streams in the GAN literature that aim to capture increasingly complex correlations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' cross-sectional correlations, time-series correlations, full causal structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In Chapter 4, we present the results of our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In Chapter 5, we discuss some of the additional real-world challenges that we abstracted away from but that need to be considered where these methods to be used in real-world cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Lastly, in Chapter 6, we summarise and conclude our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 4 2 Problem setup The goal of the evaluation in this paper is to see if current data replication methods are useful when causal analyses are to be performed on the resulting synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' To simulate realistic use-cases of synthetic data for sectors/fields where decisions require causal inference, we take popular causal inference methods from the field of econometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' These models have the benefit of having well-known sets of assumptions under which the estimated parameters may be interpreted as causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' First, we consider a cross-sectional model, namely ordinary least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This is the simplest case in which the researcher assumes that observations at different timestamps are independent and the functional form of the model is linear in the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' While being the simplest model to estimate in econometrics, it also comes with the most stringent assumptions to be interpreted as causal, discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Next, we look at the class of time-series models that allow dependence between observations at different timestamps in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Lastly, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='3, we discuss the case where a full structural equation model is the goal of the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Here, an approach from the field of Causal Discovery is needed since none such data-driven method exists in econometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The evaluation setup is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' First, a dataset is generated with known causal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The design of this dataset is discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The generated data is then used to train a chosen type of GAN, designed to capture the distribution of the generated data, further discussed in Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The trained GAN is then sampled to construct a synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Note that, in the remainder of this paper, artificial data we generate from the known model are referred to as generated data while data sampled from the GAN models are referred to as synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Next, a causal inference method, with its accompanying assumptions, is selected to apply to both the generated data and the synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' We first validate that the chosen causal inference method is appropriate to estimate (a subset of) causal connections in our defined causal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This is done by comparing the estimated parameters of the model on the generated data to those we defined in the structural model to generate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Alignment of the two indicates that the causal inference method is indeed appropriate to estimate causality for (a subset of) the underlying causal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' If the GAN method therefore generates a synthetic dataset where the same causal inference method does not estimate the same parameters as in the generated data, this difference can be entirely attributed to the GAN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This approach allows us to examine the causality-preservation capabilities of the GAN methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 5 Figure 1: Experiment setup for each choice of assumptions and GAN method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1 Cross-sectional The first type of causal relationships are those on a cross-sectional level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Ordinary least squares (OLS) is a popular regression model to find causal effects in cross-sectional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In this case we assume that a variable can be represented by an OLS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The OLS model produces valid causal inference under the following assumptions: Assumption 1: Linear in Parameters The model can be written in the form: y = β0 + β1x1 + β2x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' + βkxk + ε (1) Assumption 2: Random Sampling The sample of n observations (xi1, xi2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', xik, yi) : i = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', n is drawn randomly from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Assumption 3: No Perfect Collinearity An independent variable in (1) cannot be an exact linear combination of the other indepen- dent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Assumption 4: Zero Conditional Mean The expected value of the error ε should be zero, given any values of the independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' E(ε|x1, x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', xk) = 0 Assumption 5: Homoskedasticity 6 ChooseGANand Choose modeland Generatedata Generated data train on generated Syntheticdata assumptions according to model data Fit model to data Fit model to data Model parameters Model parameters CompareThe error ε has the same variance, given any values of the independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Var(ε|x1, x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', xk) = σ2 Assumption 6: Normality The error ε is normally distributed a zero mean and variance σ2 and independent of the explanatory variables x1, x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' ε ∼ Normal(0,σ2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='2 Time-series In cross-sectional modelling observations have no time aspect, this changes when considering time-series models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Here we consider the popular class of linear autoregressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The assumptions to perform valid causal inference with these models are as follows: Assumption 1: Linear in Parameters The stochastic process (xt1, xt2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', xtk, yt) : t = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', n can be written in the form: yt = β0 + β1xt1 + β2xt2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' + βkxtk + εt (2) Assumption 2: No Perfect Collinearity An independent variable in (2) cannot be an exact linear combination of the other indepen- dent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Assumption 3: Zero Conditional Mean For each t, the expected value of the error εt should be zero, given any values of the independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' E(εt|xt1, xt2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', xtk) = 0, t = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', n Assumption 4: Homoskedasticity The error εt has the same variance, given any values of the independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Var(ε|xt1, xt2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', xtk) = σ2, t = 1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', n Assumption 5: No Serial Correlation Given the independent variables xt1, xt2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', xtk, errors in two different time steps are not correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 7 Corr(εs,εt|xt1, xt2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', xtk) = 0,∀t ̸= s Assumption 6: Normality The error εt is normally distributed a zero mean and variance σ2 and independent of the explanatory variables xt1, xt2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', xtk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' u ∼ Normal(0,σ2) Most assumptions are very similar to the previous OLS assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' There are two main differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' First is the absence of OLS Assumption 2 specifying observations to be randomly sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Under time-series assumptions observations have an order determined by the time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Second, time-series Assumption 5 is added, requiring the error term to have no serial correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In the time-series we will consider, autoregressive terms are included as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' We make an additional assumption for this autoregressive time-series to be weakly dependent, meaning the correlation between yt and yt+s is almost 0 for s large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In other words, as the variables get farther away from each other in time, the correlation decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' yt = αyt−1 + ε In the case above of an autoregressive model lagged for one period, this assumption is satisfied if |α| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='3 Structural model Lastly, the case remains where the whole causal structure is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Here, the goal is to attempt to reconstruct the full structural causal model from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' As far as we know, no such methods exist in econometrics 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' For this reason, we adopt a method from the recently developing field of causal discovery, situated mostly in the computer science literature, that tries to accomplish this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Recovering the causal model from observational data is far from trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Recall the example above of trying to figure out the shape of the ice cube from a pool of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' As many forms of ice cubes can result in the same pool of water, many structural causal models can result in the same observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Therefore, picking one of all possible models is dependent on further assumptions made by each causal discovery algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The general approach is to embed known features of causality, such as environment independence (Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2020) or acyclicity (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2018), into the loss function that a machine learning algorithm optimises for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Even then, it is sometimes only possible to provide a set of possible structural causal models that are all equally able to generate the observational data, also called Markov equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' A recent trend is to extend the data to also include interventions 2In economics, and many other fields that model complex phenomena, a structural model is defined from theory and then calibrated to data instead of trying to infer the complete model itself from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 8 and their outcomes (Vowels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This extra information can be used to exclude certain Markov equivalent models and decrease the set of potential underlying causal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' One of the more frequently used causal discovery algorithms is LiNGAM (Shimizu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2006), which assumes that the causal effects are linear, the generating causal graph is acyclic, that the distribution of the noise is non-gaussian and no unobserved confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The LiNGAM model can be expressed in matrix form as follows: x = Bx + e with the observed variables x, the connection strength matrix B and exogenous variables e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The condition of acyclicity allows the matrix B to be permuted to become lower triangular with a zero-diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' With the additional assumption of the exogenous variables e, or in other words the noise, being non-Gaussian, the matrix B can be uniquely identified using only the data x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This identifiability thus means that the algorithms results in a single causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Different variations on this method exist like models with hidden common causes (Hoyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2008), time-series (Hyvärinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2010) or non-linearity (Zhang and Hyvärinen, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In the case of Gaussian noise, only a set of Markov equivalent causal models can be estimated, while under the assumption of non-Gaussian noise this set can be reduced to one full causal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This assumption is, however, in contrast with the assumption off Gaussian noise needed in many inference methods for valid causal inference, including the OLS and autoregressive models we discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='4 Generated dataset We define the following model: yt = αyt−1 + β1x1,t + β2x2,t + ε1 x1,t = β3z1,t + β4z2,t + ε2 x2,t = β5z2,t + ε3 z1,t = ε4 z2,t = ε5 (3) A graphic representation of this structural model, also called the causal graph, is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' For the estimation of this causal structure with the different inference methods, we will always assume full observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The variables x1 and x2 are a linear combinations of the contemporaneous values of z1 and z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The underlying models for these two variables therefore meet the assumptions of the cross-sectional ordinary least squared (OLS) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' OLS should therefore be an appropriate method to estimate the causal effects of z1 and z2 on x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' We confirm this in the Results section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' For the variable yt, extending the assumption on the data to allow for autocorrelation, a first order autoregressive model can infer α on β1 and β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 9 Finally a variant of LiNGAM for time-series can be used to infer the causal structural model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' While the model was specifically chosen to contain both cross-sectional and time-series causality, it is easy to think of real-world model that follow this functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' One example is a simple income process, where the monthly income now depends on the income last month and some contemporaneous features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' employment sector, location) which in turn are distributed according to (conditionally) random distributed preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Figure 2: Full causal model of the generated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 10 Z1 X1 y Z2 X23 Generative adversarial networks Generative adversarial networks, or GANs, is a framework for generative machine learning first introduced by Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' in 2014 (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' A generative model takes a training dataset drawn from a real world distribution as input and tries to replicate this data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The framework has shown great success in generating synthetic images indistinguishable from real images (Ledig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Brock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' While the focus has been on the improvement of the framework for image generation and manipulation, the GAN framework has recently also gathered attention for its possibilities with numerical and categorical data, like tabular and time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1 Framework A generative adversarial network consists of two competing neural networks: a generator G, which generates fake data, and a discriminator D, that is trained to discern which data is fake (made by the generator) and which data is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The process can be described as a zero-sum game between the generator and discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' During the training process the generator adapts to better fool the discriminator and the discriminator in turn adapts to better detect the fake data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The resulting trained generator can then be extracted to replicate the distribution of the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1 Architecture Figure 3 shows the basic structure of a GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The generator G learns to map a latent space pz to a more complex distribution pg, which is the distribution meant to mimic the real data distribution pdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Typically, this latent space is a high-dimensional space with each variable drawn from a Gaussian distribution with a mean of zero and a standard deviation of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The concept is thus that one can insert sample of noise (z) into the generator, which it will learn to map onto a sample of the distribution of the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The generating function can then be described by Figure 3: Generative Adversarial Networks diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 11 Real Sample Real Data Pdata Discriminator RealorFake Loss Fake Generator Sample ~Pz TrainingG(z) = X g (4) where X g are samples created by the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The discriminator D has the task of dis- tinguishing the fake data X g from the real data Xdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The generator and discriminator are trained by playing a non-cooperative game against each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The main aim of the generator is to produce samples which are similar to the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' On the other hand, the main aim of the discriminator is to distinguish between fake samples from the generator and samples from the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The discriminator D receives both samples and tries to determine which comes from the real data distribution by assigning a probability D(x), which signifies the certainty the discriminator has in its decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' If D(x) = 1, the sample x is thought to come from pdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' On the other hand, if D(x) = 0, the discriminator judges the sample to be from pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This prediction from the discriminator and the known ground truth is then used to improve both the generator and the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' During the joint training of the generator and discriminator, G will start to generate increasingly realistic samples to fool the discriminator, while the discriminator learns to better differentiate the real and fake samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The end goal of the GAN as a whole is that the discriminator can no longer tell the difference between the generated samples X g (D(x) = 1/2) and the real data samples Xdata with the discriminator no longer able to improve itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='2 Loss function The objective function of the GAN tries to match the real data distribution pdata with pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The original GAN (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2014) uses two objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The objective for D is to maximize the probability of assigning the correct label to both real and fake samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This done by minimizing the negative log-likelihood for binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Simultaneously G is trained to minimize log(1-D(G(z))), thus maximizing the probability of the generated samples being classified as real by the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This results in a mini-max game with objective function V(G,D): min G max D V(D, G) = �x∼pdata[logD(x)] + �z∼pz[log(1 − D(G(z)))] (5) The value function V(G,D) is known as the binary cross entropy function, commonly used in binary classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='2 GAN extentions Many different variations of GANs have been proposed since its inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In this section different relevant adaptions are presented, ordered by which level of causality they are aiming to improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1 TimeGAN TimeGAN by Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' (Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2019) is an adaptation of the original GAN framework that aims to improve the preservation of temporal dynamics for time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This means that newly generated sequences should respect the original relationships between variables across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Two main ideas are combined in the TimeGAN framework, the flexibility of the 12 Figure 4: TimeGAN diagram unsupervised GAN framework and a more controllable supervised autoregressive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Figure 4 shows the structure of TimeGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The TimeGAN framework contains the components of a generative adversarial network, as well as an auto-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The latter takes as input a vector of static features, s, and a vector of temporal features, x1:T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The encoder is then trained to map the feature space, which s and x1:T belong to, to a latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This allows the adversarial network to learn the underlying temporal dynamics of the data via lower-dimensional representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The output of the encoder are the latent vectors hs and ht, being lower-dimensional latent codes of the input s and x1:T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In the opposite direction, the decoder takes the static and temporal latent vectors back to their feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The reconstructed static and temporal features are respectively denoted as ˜s and ˜xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The other main component in the framework, the generative adversarial network, has a generator that takes as input random noise vectors and outputs latent vectors ˆhs and ˆht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The generator in this framework is autoregressive, meaning it also uses its previous outputs ˆh1:t−1 for the construction of ˆht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' A key difference with a regular GAN architecture is that the generator maps to this latent space instead of the usual feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Both the real latent codes hs and ht and the synthetic latent codes ˆhs and ˆht are received by the discriminator, which has the task to classify these codes as either real or fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The resulting framework has three loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' First, the reconstruction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This loss is linked to the auto-encoder component of the framework, quantifying the difference between original features s, xt and the reconstructed features ˜s and ˜xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' �R = �s,x1:T ∼p[||s − ˜s||2 + � t ||xt − ˜xt||2] (6) Second, the unsupervised loss is the same type of loss used in the original GAN framework, 13 Encoder Decoder Reconstruction S, 1:T loss Supervised loss Generator Discriminator Unsupervised Zs, Z1:T lossmaximising (discriminator) or minimising (generator) the likelihood of providing correct classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Notations y and ˆy denote classifications by the discriminator as respectively real or synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' �U = �s,x1:T ∼p[log ys + � t log yt] + �s,x1:T ∼ˆp[log(1 − ˆys) + � t log(1 − ˆyt)] (7) Lastly, the supervised loss is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The addition of this loss is motivated by the idea that the regular feedback from the discriminator, the unsupervised loss, may be insufficient incentive for the generator to capture the step-wise conditional distributions in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' To calculate this loss, the autoregressive generator g uses the real latent codes hs and ht−1 instead of the synthetic ˆhs and ˆht−1 to generate ˆht, or g(hs,ht−1,zt), as shown in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' �S = �s,x1:T ∼p[ � t ||ht − g(hs,ht−1,zt)||2] (8) A linear combination of �U and �S is used to train the generator and the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' �U guides the generator to create realistic sequence, while �S uses ground-truth targets to ensure that the stepwise transitions are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' To train the autoencoder components, the encoder and the decoder, a linear combination of �R and �S is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' By combining the different objectives, TimeGAN is trained to simultaneously encode feature vectors, generate latent codes for these feature vectors, and iterate across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='2 CausalGAN CausalGAN is a generative adversarial framework proposed by Kocaoglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' (Kocaoglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' CausalGAN is an implicit causal generative model that replicates data constraint to a given causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Implicit generative models, which the original GAN model is part of, can sample from a probability distribution, without the ability to provide likelihoods for the samples (Mohamed and Lakshminarayanan, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Causal implicit generative models can not only sample from a probability distribution but also from conditional and interventional distributions, which causal graphs embeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Consider a simple causal graph, A −→ C ←− B, as depicted in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The parent nodes, A and B are assumed to have no other variables influencing their distribution and can be written as A = GA(ZA) and B = GB(ZB), where Z∗ is some chosen noise distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Gaussian), and G∗ is a function mapping this distribution to the distribution of the variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The variable C has two parent nodes and can be written as C = GC(A, B, ZC), being a function of both A and B, as well as a chosen distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This representation is similar to how the generator of the original GAN framework is structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Figure 5b shows how a generator can be constructed to represent a given causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' For each variable a feedforward neural network is used represent functions G∗, resulting in a larger generator network consisting of linked individual generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' By building in the causal graph into the generator, it will constrain the data generation data to the actual causal model and not only reproduce joint probabilities but also the causal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' For the implementation of CausalGAN in this paper Causal-TGAN (Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2021) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This version uses the same core idea as CausalGAN, with some added adjustments for tabular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 14 (a) Causal graph A −→ C ←− B (b) Generator architecture for A −→ C ←− B Figure 5 The downside is of course that both data and the relevant causal graph needs to be known to train and use the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' To this end, we use a causal discovery method, here the standard- and time-variant of LiNGAM to provide us with the causal graph of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 15 BZA GA G B GB ZB4 Results Consider the model described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='4 with the following parameters: Parameter Value α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='5 β1,β2,β3,β4,β5 1 σ1,σ2 1 where ε∗ ∼ N(0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' From this model we sample 10,000 observations to use for further experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' These observations will further be referred to as generated data and will be used to both train the different GAN models and give baseline values for estimated parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The experiments assume a perfect scenario where the model is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Each experiment is done, in its entirety, 10 times and reported results show averages and standard deviations over these 10 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1 GAN First, we train a standard GAN with the generated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' From this GAN, we generate 10,000 samples to preserve the statistical power of our inference results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' These latter samples will be referred to as the synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The first model we fit on both datasets is OLS for the following variables: x1 = β3z1 + β4z2 + ε2 x2 = β5z2 + ε3 The resulting parameters can be seen in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The results show that on a cross-sectional level, with the underlying model meeting the assumptions in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1, the GAN methodology can replicate data with similar causal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' While on average the causal relationships detected in the synthetic data are less accurate than the causal relationships in the generated data, the results are not significantly different from the true parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' While the data the GAN is trained on is time-ordered, the synthetic data produced by the GAN is sampled randomly, without any notion of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' So, as expected, when running an autoregressive model on the y variable in our model, it does not find any time-correlation (α coefficient for y) in the synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Interestingly, it does capture the cross-sectional relationships for y (β1 and β2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='2 TimeGAN Next, TimeGAN is trained on the generated dataset, after which we again sample 10,000 datapoints for a new synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Note that the sampled datapoints are now ordered in time instead of randomly sampled as in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' As can be seen in Table 1, the synthetic data produced by TimeGAN does not properly maintain causal relationships, neither on a cross-sectional level nor over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The results 16 Model Par.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Real GAN TimeGAN CausalGAN OLS β3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='9990 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0051 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0209 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='3762 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='4320 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='9869 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1087 β4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0017 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0052 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0797 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1272 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='2249 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='3362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='9666 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1029 β5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='9996 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0057 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0157 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1266 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1066 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0179 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0006 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1625 TS α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='5004 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0030 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0233 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1331 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0011 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0064 β1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='9993 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0040 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0007 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1773 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0597 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='5236 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='9635 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1896 β2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='9982 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0045 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1439 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1682 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='8796 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='9927 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='2035 Table 1: Results for all GANs are far from what would be expected and also vary significantly from run to run, resulting in a higher standard deviations in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This is likely due to there being no auto-correlation in the variables outside of y, and TimeGAN attempting to find time dependent structure where none exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' To confirm this, we also consider the following alternate causal structure, where all variables have some sort of time-dependence (direct or indirect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' yt = αyt−1 + β1x1,t + β2x2,t + ε1 x1,t = β3z1,t + β4z2,t + ε2 x2,t = β5z2,t + ε3 z1,t = z1,t−1 + ε4 z2,t = z2,t−1 + ε5 (9) Table 2 shows the results for TimeGAN in the case of the alternative structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In this case TimeGAN is able to accurately capture the causal relationships on a cross-sectional level (β3, β4, β5) but still fails to capture the structure in y (α, β1 and β2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' However, it does not seem like the model completely missed the mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' When we look at the original formulation for y, with the chosen parameters for the experiment, it can be rewritten as follows: yt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='5yt−1 + x1,t + x2,t + εt yt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='25yt−2 + (x1,t + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='5x1,t−1) + (x2,t + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='5x2,t−1) + (εt + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='5εt−1) yt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='125yt−3 + (x1,t + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='5x1,t−1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='25x1,t−2) + (x2,t + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='5x2,t−1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='25x2,t−2) + (εt + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='5εt−1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='25εt−2) This decomposition of y can be continued further until the autoregressive part for y is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Now, if the change in x1 and x2 in each time step is limited and thus x1,t ≈ x1,t−1 and x2,t ≈ x2,t−1, as is the case here due to the stationarity of y, and using � n=0( 1 2)n = 2, we can write: yt ≈ 2x1,t + 2x2,t + ε with ε ∼ N(0, 4 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The results shown in Table 2 thus suggest that TimeGAN has learned this smaller representation of y, using only x1 and x2, that results in the same expected values 17 Model Parameter Real TimeGAN OLS β3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='9999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='9967 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0247 β4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0049 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0219 β5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='9999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0005 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0024 TS α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='4999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0128 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0207 β1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0016 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0719 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0680 β2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0016 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='0002 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1550 Table 2: Result for TimeGAN on the second model of y over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This representation, however, does not represent the actual causal model underlying y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='3 CausalGAN Lastly, the full structural causal model is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Here, a model can not be directly trained to the data since no such method exists as far as the authors are aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' A two-step approach is taken where first the causal structure is identified with LiNGAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This extracted structure is then compared to our data generating model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 3) to check if LiNGAM is an appropriate and efficient causal discovery method for our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Then CausalGAN is used to generate data that follows this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Lastly, LiNGAM is applied to the synthetic data and its output is compared to the causal structure retrieved from the generated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' As noted before, LiNGAM uses the assumption of non-Gaussian noise, which is incorrect for model (3) used previously in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' To start from a correct causal structure for this experiment, we adjust the distribution of the noise our data structure (3) to be uniformly distributed, ε∗ ∼ U(−1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Under these conditions the time-variant of LiNGAM is able to find the underlying causal model correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' However, CausalGAN is not equipped to deal with time-series, so we are forced to only consider the cross-sectional causal relations here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Table 3 shows all causal relationships detected by LiNGAM in both the generated dataset and the synthetic dataset produced by CausalGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Additionally, we show the causal relationships detected in synthetic data from a basic GAN trained on the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' For this one repre- sentative example is chosen since the use of means and standard deviations give warped representations of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The synthetic data sampled from CausalGAN consistently maintains causal relationships relatively well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Some deterioration can be seen, as well as introducing small additional causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The basic GAN framework is however not capable of retaining the causal relationships when the whole causal structure is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Causal discovery on the synthetic data of the basic GAN gives varying results even when performed multiple times on one synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' None of the resulting graphs are close to the original causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This shows that adding the additional information of the (correct) underlying causal graph through the CausalGAN model does help maintaining the causal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 18 Causal effect Real CausalGAN GAN z1 −→ x1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='93 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='03 z2 −→ x1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='07 z2 −→ x2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='16 x1 −→ y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='14 x2 −→ y 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='39 z1 −→ z2 0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='11 z1 −→ x2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='47 z1 −→ y 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='86 z2 −→ y 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='10 x2 −→ x1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='65 Table 3: Causal effects detected by LiNGAM on both the generated dataset and the synthetic dataset generated by CausalGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The table contains all significant causal effects (> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Causal effects of less significance (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1) are simplified to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Bold number indicate that the causal effect is reversed 19 5 Real world challenges In our tests of the causality replicating capabilities of GANs, we have purposely abstracted away from many of the additional challenge that come with working with real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In this section we address three of the most important challenges and give an overview of the variations on the GAN framework that have been proposed to tackle them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='1 Privacy Privacy concerns are one of the main drivers for the recent rise in interest in synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' While in general synthetic data is sampled from a reconstruction of the distribution of the original data, fear of replicating real samples due to overfitting remain (Webster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Membership inference attacks also form a common concern in the field of privacy (Hayes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' These attacks leverage the fact that machine learning models generally perform better on the data it was trained on to reconstruct the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' These concerns have sparked the search for GAN variants that give certain privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' One such guarantee is differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' An algorithm is differentially private if an observer seeing the output can not tell if a particular datapoint was used in the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In the case where the observer has access to the generated samples but not the generator, recent work has shown that the base form of GAN has some privacy guarantees in terms of both differential privacy and robustness to membership inference attacks (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' These guarantees get stronger for larger training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' If additionally the generator is available, several differential privacy GANs have been proposed, such as DPGAN (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2018), PPGAN (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2019) and PATE-GAN (Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Privacy guarantees, however, come at the price of replication quality since you in some form or another adding noise to the data by limiting the impact a training sample can have on the model, even though it might be highly informative (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='2 Fairness Machine learning has an increasingly large impact on current day decision making, scaling decisions made on a micro-scale to a macro-scale in an often opaque manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This trend has raised concerns about building in, or scaling up biases in decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Fairness in machine learning is a recently growing area of research that studies how to ensure that such biases and model inaccuracies do not lead to discriminatory models on the basis of sensitive attributes such as gender or ethnicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Using synthetic data can help by debiasing the data before it even gets used for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In such a framework a generative model is trained on unfair data to generate synthetic fair data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' A first challenge to fairness is defining what it actually is, which is often highly dependent on the context of the business decisions that is being made with the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' One often used interpretation is that certain features, also called protected or sensitive features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' gender, ethnicity), should not have any impact on the outcome of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This orthogonalisation of the model outcome and the protected features comes with two major challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' First, it requires outside definition of what the protected features are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Second, if you want to 20 rid observational data of such biases, it is not enough to just delete the features, you need to know the relevant causal structures to exclude both the direct and indirect impact the protected attribute has on the outcome (Kusner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Zhang and Bareinboim, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Otherwise the model can just learn the protected features by using different proxies which are correlated to them (van Breugel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' CFGAN (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2019) and DECAF (van Breugel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2021) are two methods to generate fair data that are rooted in this approach to fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Both methods therefore require a causal graph as additional input, something we saw in our results is not generally feasible with current causal discovery methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' FairGAN (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2018) and Fairness GAN (Sattigeri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2018) have also been suggested for the purpose of generating fair data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' FairGAN uses an additional discriminator on top of the classical GAN architecture to determine whether samples are from the protected or unprotected group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Fairness GAN uses an added loss function that encourages demographic parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Demographic parity is satisfied if the decisions made from the data are not dependent on a given sensitive attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This requires a specification of the explanatory variables x, the target variables y and the sensitive variables s, where y does not need specification in other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' FairGAN is applied to low-dimensional structured data, while Fairness GAN is applied to high dimensional image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='3 Tabular data Tabular data is data that contains both discrete and continuous columns and is one of the most commonly encountered data formats in both business and research (Xu and Veeramachaneni, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Tabular data, and especially the discrete features within them are challenging for GAN methods since the continuous functions used in neural nets are ill-equipped to fit the non-continuous distributions of discrete variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The generator of a regular GAN cannot generate discrete samples because the generator is trained by the loss from the discriminator via backpropagation (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' To tackle this problem, MedGAN (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2018) adds an autoencoder model to the regular GAN framework to generate high-dimensional discrete variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Both TGAN (Xu and Veeramachaneni, 2018) and TableGAN (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2018) look to improve the performance on the continuous distributions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' TGAN clusters numerical variables to deal with the multi-modal distribution for continuous features and adjusts the loss function to effectively generate discrete features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' TableGAN uses a classifier neural network to predict synthetic records’ labels to improve consistency in generated records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' An additional loss, information loss, is introduced as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This loss is the difference in key statistical values of both the real and synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' In the paper the mean and standard deviation are used as key statistical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Besides the mix of continuous and discrete columns, the distributions of data often differs from the standard Gaussian-like distributions found in typical generative applications like image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' To this end CTGAN (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2019) addresses additional concerns about non-Gaussian and multi-modal distributions, and imbalanced categorical columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' CTAB-GAN (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2021) looks further into these issues and tackles data imbalance and long-tail distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The previously mentioned Causal-TGAN (Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2021) combines ideas of CTGAN and CausalGAN (Kocaoglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2017) to leverage knowledge about the causal structure for a better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 21 6 Conclusion Data has become a driving force in both business, research, and policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' And rightfully so if we see how increased access to data has furthered our ability to understand and support decision making in complex environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' While some fields are just collecting more and more data in labs or in nature, most of the decision making that occurs in business is in regards to actual human beings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This rightfully raises concerns about privacy and ethics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Should companies just be allowed to collect, buy, sell, and share more and more data on the behaviour and features of actual human beings just for the sake of making better business- decisions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The answer is obviously no, and regulatory bodies are acting accordingly by setting in place boundaries on what is and is not allowed in regards to data on individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' How do we balance the benefits of increased accuracy and understanding with the privacy and ethics concerns that both come with having more data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' One solution that has gained a lot of traction is synthetic data, which are data sampled from generative methods that are meant to replicate high-dimensional distributions of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' After all, the improvements in modelling complex phenomena come from sufficient coverage of the high-dimensional distribution of relevant features, and not from knowing someone’s exact name or address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' So if we could generate data with the same distribution as the original, but not containing any identifiable features as well as different enough in exact values such that no individual could be uniquely linked to one sample from the data, we could have all the benefits without introducing risks to privacy or ethics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' While this is true for predictive models, that solely map correlations to an outcome, many decisions intend an intervention to influence the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' The difference lies in that the former asks an observational question: “If I observe X, what will Y be?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' and the latter asks an interventional question: “If I do X, how will the outcome Y change”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Apart from a group of so-called policy prediction problems (Kleinberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2015), which only require a prediction to make a decision, the latter requires causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Once we enter into this territory, it no longer just matters that the synthetic data has the right distribution, but also that it was generated with the correct underlying causal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' and because there can exist multiple underlying structures that generate the same distribution, there are no intrinsic guarantees that current generative modelling methods converge on the correct one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' We evaluate the causal replication capabilities of the generative modelling techniques that are typically used for synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' As far as we know, we are the first to do so with a focus on causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' We find that in the case where the assumptions are met that make correlation equal causation, causal inference on the real and synthetic data yield the same results only if the simplest model that can generate the distribution of the features equals the real one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' This points at the principle of occam’s razor, that is the foundation for regularisation in machine learning to counter overfitting, is actually working against us in the case where we want to replicate causal relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' When nothing is known about the causal structure, and the analyst can thus not easily construct a functional form to test with classic causal inference methods like OLS, causal discovery can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Causal discovery tries to find the complete causal structure in observational data, which can then be used as input for a generative model that can generate synthetic data explicitly according to the causal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' We find that, while this works in simple cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' in the case of cross-sectional correlation with non-gaussian noise), the 22 necessary assumptions on both the causal discovery and generation side seem too restrictive to be widely applicable in real-world contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' A path forward seems to be to augment the observational data fed to the GAN models with additional information such as knowledge on different environment in which the data was collected or interventional data from experiments (Scholkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' While this can present a way forward for many fields, it is often not applicable in the context of businesses related to people’s finances or health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Organisations that want to improve their decision making by leveraging synthetic data should thus be careful about what the current state-of-the-art is actually capable of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 23 References Arjovsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Bottou, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Gulrajani, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Lopez-Paz (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} 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effects using linear non-gaussian causal models with hidden variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' International Journal of Approximate Reasoning 49(2), 362–378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Special Section on Probabilistic Rough Sets and Special Section on PGM’06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Kairouz, X.' metadata={'source': 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+page_content=' AUAI Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Zhao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Kunar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Van der Scheer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Birke, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Chen (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Ctab-gan: Effective table data synthesizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Zheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Aragam, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Ravikumar, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Xing (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' Dags with no tears: Continuous optimization for structure learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} +page_content=' 27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/adAzT4oBgHgl3EQfLPsZ/content/2301.01109v1.pdf'} diff --git a/b9AyT4oBgHgl3EQfwfkG/content/tmp_files/2301.00647v1.pdf.txt b/b9AyT4oBgHgl3EQfwfkG/content/tmp_files/2301.00647v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4cd93bea008f74e7ee91621de3af6223cca3b59 --- /dev/null +++ b/b9AyT4oBgHgl3EQfwfkG/content/tmp_files/2301.00647v1.pdf.txt @@ -0,0 +1,987 @@ +A tau-leaping method for computing joint probability +distributions of the first-passage time and position of a Brownian +particle. +Jaroslav Albert +Abstract +First passage time (FPT), also known as first hitting time, is the time a particle, subject to some +stochastic process, hits or crosses a closed surface for the very first time. τ-leaping methods are a +class of stochastic algorithms in which, instead of simulating every single reaction, many reactions +are “leaped” over in order to shorten the computing time. In this paper we developed a τ-leaping +method for computing the FPT and position in arbitrary volumes for a Brownian particle governed +by the Langevin equation. The τ-leaping method proposed here works as follows. A sphere is +inscribed within the volume of interest (VOI) centered at the initial particle’s location. On this +sphere, the FPT is sampled, as well as the position, which becomes the new initial position. Then, +another sphere, centered at this new location, is inscribed. This process continues until the sphere +becomes smaller than some minimal radius Rmin. When this occurs, the τ-leaping switches to +the conventional Monte Carlo, which runs until the particle either crosses the surface of the VOI +or finds its way to a position where a sphere of radius > Rmin can be inscribed. The switching +between τ-leaping and MC continues until the particle crosses the surface of the VOI. The purpose +of a minimal radius is to avoid having to sample the velocities, which become irrelevant when the +particle diffuses beyond a certain distance, i. e. Rmin The size of this radius depends on the system +parameters and on one’s notion of accuracy: the larger this radius the more accurate the τ-leaping +method, but also less efficient. This trade off between accuracy and efficiency is discussed. For +two VOI, the τ-leaping method is shown to be accurate and more efficient than MC by at least +a factor of 10 and up to a factor of about 110. However, while MC becomes exponentially slower +with increasing VOI, the efficiency of the τ-leaping method remains relatively unchanged. Thus, +the τ-leaping method can potentially be many orders of magnitude more efficient than MC. +1 +arXiv:2301.00647v1 [q-bio.MN] 2 Jan 2023 + +INTRODUCTION +First passage time (FPT) is the time that a certain event occurs for the first time during +an evolution of a system. +In molecular biology it is often desirable to know the FPT +distributions for a molecule, such as protein, for crossing a surface, e. +g. +that of the +cell nucleus or the cell membrane, or for finding its target site on the DNA [1]. Although +mean first passage times for these types of events have been worked out to various degrees of +approximations [2, 3], the non-trivial shapes of volumes and obstacle-riddled environments +in which biological molecules have to navigate makes computations of FPT distributions +difficult. The usual strategy in such efforts is to simulate the molecular dynamics using +Monte Carlo methos, which do get the job done but are notoriously inefficient. +In this paper we draw inspiration from computational analysis of stochastic gene expres- +sion – an area of research that has produced many alternative methods to brute Monte +Carlo simulations. In particular, we focus on two such methods: τ-leaping [4–16] and hy- +brid stochastic simulation algorithms (HSSA) [17–30]. A τ-leaping method approximates +the evolution of a system over many small steps in a MC simulation by taking larger steps +or leaps, thereby reducing the overall number of steps that need to be taken. The HSSAs +on the other hand, work by employing a form of τ-leaping method on a part of the system +(a subset of molecular species and chemical reactions), while using good old MC on the +rest of the system. In this paper we apply these concepts to Brownian motion described +by the Langevin equation in volumes of arbitrary shapes with the goal to compute joint +distributions of the FPT and position. More specifically, we take advantage of the fact that +LEs can be solved approximately for a spherical volume of certain minimal size, which can +be used to fill parts of the larger volume of interest. Sampling the FPT and position for this +spherical volume, we generate another sphere centered at the sampled position. When this +process brings the particle within a certain distance from the boundary, we switch to the +MC. Thus, with each sphere, we effectively τ-leap over τ/dt number of steps, where dt is the +temporal size of each step in the MC simulation. With full details about what happens near +the boundary, we show the accuracy and efficiency of our method on two examples volumes. +2 + +BROWNIAN MOTION AND THE LANGEVIN EQUATION +When a large particle is immersed in a medium (gas or liquid) of many smaller particles +at equilibrium, it moves in a jittery fashion due to density fluctuations in that medium. One +model of such motion is called Brownian, and is described by the Langevin equation, +mdv(t) +dt +− v(t) +τB += f(t) +(1) +where v(t) is the particle’s velocity at time t, m is its mass, τB is the relaxation time, and +f(t) is a random force. This random force changes magnitude and direction at time intervals +separated by dt and follows a Gaussian distribution +P(f) = +1 +(2πσ2 +f)3/2e−f·f/(2σ2 +f), +(2) +where σ2 +f = 2kBTm/(dtτB), and kB and T are the Boltzman constant and temperature, +respectively. +The relaxation time τB is related to the mass m, viscosity of the medium +ν, and the particle’s size rB via this expression: τB = m/(6πνrB). Hence, coupled with +the definition of velocity, v = dr/dt, Eq. (1) can be used to simulate the evolution of a +Brownian particle’s velocity and position r by iteration. The time step, dt, must be chosen +to satisfy τs ≪ dt, where τs is the average collision time between the Brownian particle and +the molecules of the medium. +Another approach to studying Brownian motion is via a Master Equation for the joint +probability distribution, P(r, v, t), which is given by the Klein-Kramers equation (also re- +ferred to as Fokker-Planck equation) [31]: +∂P +∂t + v · ∇rP − 1 +τB +v · ∇vP − kBT +τBm∇2 +vP = 0. +(3) +The solution to Eq. +(3) with infinite boundaries and the initial conditions P(r, v, 0) = +δ(3)(r − r′)δ(3)(v − v′) is given by [32, 33]: +P(r, v, t) = +1 +(2πσXσV +� +1 − β2)3 × +exp +� +− +1 +2(1 − β2) +�|r − µµµX|2 +σ2 +X ++ |v − µµµV |2 +σ2 +V +− 2β(r − µµµX) · (v − µµµV ) +σXσV +�� +, (4) +3 + +where +σ2 +X = kBTτ 2 +B +m +� +1 + 2t/τB − +� +2 − e−t/τB�2� +(5) +σ2 +V = kBT +m +� +1 − e−2t/τB� +(6) +β = kBTτB +σXσV +� +1 − e−t/τB�2 +µµµX = r′ + (1 − e−t/τB)τBv′ +µµµV = v′e−t/τB. +We can obtain the probability for the particle’s position by integrating Eq. (4) over v: +P(r, t) = +� ∞ +−∞ +P(r, v, t)dv = +1 +(2πσX(t)2)3exp +� +−|r − µµµX(t)|2 +2σX(t)2 +� +. +(7) +For t ≫ τB, σX(t)2 → 2(kBT/m)t and µµµX(t) → r′ + τBv′, which allows us to replace the +Brownian model with a diffusion model: +∂P(r, t) +∂t += 1 +D∇2P(r, t), +(8) +where D = kBT/m, subject to the initial conditions P(r, 0) = δ(3)(r − r′ − τBv′). +We +can quantify the discrepancy between the Langevin model and the diffusion model via this +expression: +w(t) = 1 − σX(t)2 +2Dt . +(9) +If we set w(t) to some small value εw, we can solve Eq. (9) for the minimal time the system +must evolve before we can treated as diffusive: tmin = 3τB/(2εw). For example, if εw = 0.03, +we get tmin = 50τB. Thus, if we are only interested in times > tmin, we are free to use +Eq. (8) as our model. Although one can chose v′ in the initial conditions to be any value, +it is useful to consider the magnitude of the term τBv′ for a realistic scenario, e. g. v′ +being the result of a Brownian particle having arrived at position r′ at time t = 0, after +traveling for a time > tmin. According to Eq. (6), the distribution of velocities for such +a particle would have the standard deviation σ2 +V = kBT/m. Thus, the maximum speed +of the arriving Brownian particle would be ∼ 3 +� +kBT/m. For a large enough volume, we +can assume the term 3τB +� +kBT/m to be negligible, i. e. if 3τB +� +kBT/m/R ≪ 1, where +R is the radius of our sphere. By choosing the smallness of εR = 3τB +� +kBT/m/R, e. g. +εR = 0.03, we can determine the minimum radius R for which the initial velocity can be +neglected: Rmin = 3εR +� +m/(kBT)τB. Thus, provided the particle takes significantly longer +4 + +on average than tmin to reach a distance Rmin, we can replace the Brownian model with a +diffusion model for t > tmin. In a moment we will see that the minimal time to reach a +distance Rmin is indeed much longer than tmin. With these criteria we can compute the FPT +for a Brownian particle using Eq. (8) and the initial condition P(r, 0) = δ(3)(r). In spherical +coordinates, Eq. (8) reads: +∂P(ξ, t) +∂T += 1 +ξ2 +∂ +∂ξ +� +ξ2∂P(ξ, t) +∂ξ +� +, +(10) +where ξ = r/R and T = Dt/R2. Thanks to spherical symmetry, P(r, t) is independent of the +longitudinal and azimuthal angles, φ and θ. The initial condition becomes P(r, 0) = δ(r)/r2, +or P(ξ, 0) = δ(ξ)/ξ2. To compute the FPT, we also need to add the absorbing boundary +condition P(ξ = 1, T) = 0, for which the solution is: +P(ξ, T) = lim +M→∞ PM(ξ, T), +(11) +where +PM(ξ, T) = +M +� +n=1 +An(M)sin(πnξ) +ξ +e−(πn)2T +(12) +and +An(M) = 2πne−(πn/M)2. +(13) +The survivor’s probability S∞(T), which is the probability that the particle remains inside +R for a period of time T, is given by +S∞(T) = +� 1 +0 +PM(ξ, T)ξ2dξ = lim +M→∞ +M +� +n=1 +2(−1)n+1e−(πn)2(T+1/M). +(14) +The subscript ∞ serves as a reminder that M → ∞. +The FPT distribution is simply +F∞(T) = 1−S∞(T). In practice, however, the summation limit can be cut off at some finite +value of M: FM(T) = 1 − SM(T). Since the exponential term e−(πn)2(T+1/M) decays very +rapidly for large n, we can take the limit (T + 1/M) → T, while cutting the summation off +at some finite M to obtain: +˜FM(T) = 1 − +M +� +n=1 +2(−1)n+1e−(πn)2T. +(15) +Figures 1 a) and b) show the behaviors of PM(ξ, 0), FM(T) and ˜FM(T) for different values of +M. Evidently, to make PM(ξ, 0) sharply peaked near r = 0 and FM(T) converge to F∞(T) +5 + +FIG. 1. a) Initial distributions PM(ξ, 0) for different M; b) Distributions FM(T) (solid lines) and +˜FM(T) (dashed lines) for different M. +requires large values of M, while the function ˜FM(T) does not: for M = 10 it already +behaves correctly for T larger than ∼ 0.004. +To sample ˜FM(T), one needs only to generate a random real number η = (0, 1] and solve +˜FM(T) − η = 0 for T. For nontrivial ˜FM(T), this could be done by minimizing |FM(T) − η|. +However, if we compute the average T, +⟨T⟩ = +� ∞ +0 +∞ +� +n=1 +2(−1)n+1e−(πn)2TdT = +∞ +� +n=1 +2(−1)n+1 +(πn)2 += 1 +6, +(16) +we notice, by examining Figure 1 b) again, that the function ˜F1(T) = 1 − 2e−π2T behaves +correctly for T > 1/6 - the average. Hence, we could speed up the minimization procedure +by first checking whether 1 − 2e−π2/6 is greater or smaller than η. If it is the latter, we can +set 1 − 2e−π2T to η and solve for T analytically: +T = 1 +π2 ln +� +2 +1 − η +� +. +(17) +If it is the former, we only need to search for T in the range [0, 1/6]. +Before we continue we must circle back and check that the time to reach a distance Rmin +is much greater than tmin. We can do this by requiring that ˜F10(T) be less than some chosen +value, e. g. 0.001, which corresponds to T = 0.04. If we recall that T = tMRmin/(kBTτB), +Rmin = 3τB +� +kBT/m/εR and tmin = 3τB/(2εw), we obtain t/tmin = 6 × 0.04(εw/ε2 +R). For +εd = εR = 0.03, we get t/tmin = 8. +6 + +a) +b) +40 +1.0 +M=10 +0.8 F +FM(T) +FM(T) +30 +M=20 +0.6F +M=50 +()W/L)W +M=1 +M=1 +PM(5.0) +0.4 +1 +M=2 +M=2 +20 +M=100 +M=3 +M=3 +0.2 + M=4 +M=4 +10F +0.0 +M=10 +M=10 + M=100 +0.2 F +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +5 + = 1/6 +TFIG. 2. A 2-dimensional illustration of the τ-leaping method. The particle starts out at position +x0. The smallest possible circle centered at x0, Circle 1, is generated and a point on its surface, +x1, is sampled. The same process is repeated for Circle 2, Circle 3 and Circle 4. However, the +radius of Circle 4 is smaller than Rmin, so we must switch to Monte Carlo. In Scenario 1, the +particle diffuses a distance ≥ Rmin (blue dashed circle) to the point y1; a sphere centered at y1 is +generated but its radius is smaller than Rmin, so Monte Carlo continues until the particle crosses +the boundary (black thick line). In Scenario 2, the particle diffuses a distance ≥ Rmin to the point +y1; a sphere centered at y1 is generated with radius < Rmin, so we continue with Monte Carlo; +the particle diffuses to a point y2, where a sphere of radius > Rmin is generated, and we switch to +τ-leaping to continue the process. +τ-LEAPING +Now that we have an analytical expression for the FPT for a sphere, we can use it to speed +up simulation of Brownian motion. The scheme is shown in Figure 2 on a two-dimensional +example. First, we give the surface of the volume of interest a skin of thickness Rmin on +the inside (the purpose of which will be explained shortly). Next, starting from some initial +point x0, we generate a sphere centered at x0 such that its surface and the skin share a +7 + +Circle 1 +Circle 2 +Circle 3 +Circle 4 +Switch to +MonteCarlo +Scenario 1 +Scenario 1continued +Scenario2 +Scenario 2continuedunique point. This is equivalent to finding the smallest sphere whose surface touches the +skin. Then, we sample the FPT and the particle’s position on the surface of the sphere, +(t1, x1). Centered at x1, we generate another sphere whose surface touches the skin. We +sample the FPT and surface position, (t2, x2), and continue this process in this manner until +we generate a sphere with a radius > Rmin. When this happens, we switch to Monte Carlo, +with the initial conditions ˜x0 = xi and v = 0. Of course, in reality there is no reason to +expect v to be 0, unless we get very lucky. However, if we let the particle evolve past a +radius Rmin, we do not need to worry about its initial velocity and may set to zero. This +is where the skin guarantees accuracy: in the (unlikely) event of sampling a position that +falls on the skin, we are guaranteed that, should the particle evolve past the outer surface, it +will have traveled at least the distance Rmin. With this quality check in place, we can write +down the steps of this procedure in more detail. +0 : +Choose a volume whose enclosing surface is given by a vector g(λ1, λ2), +parametrized by λ1 and λ2. Also choose (T, m, τB, Rmin) and the step size dt. +1 : +Set (p, n) = 0, where p and n are counters, and choose initial time tp (e. g. zero) and +an initial position xp. +2 : +Generate a sphere of radius R by minimizing |g(λ1, λ2) − xp|. If R ≥ Rmin, set +p = p + 1 and go to step 3; otherwise go to step 6. +3 : +Sample T by generating a random real number η = [0, 1). If η > 1 − 2eπ2/6, +set T = 1/π2 ln[2/(1 − η)]; otherwise set T =min| ˜F10(T ′) − η|. Set tp = R2T/D and +record it. +4 : +Sample a point on a sphere, r, from a uniform distribution by generating two +random numbers q1 = [0, 2π] and q2 = [0, 1] and set +r = [R sin θ cos φ, R sin θ sin φ, R cos θ], where θ = q1 and φ = arccos(1 − 2q2) [34]. +5 : +Set xp = xp−1 + r and go to step 2. +6 : +Simulate Eq. (1) using Monte Carlo with the initial conditions Xn = xp and Vn = 0, +until a) Xn reaches the outside of the volume; or b) |Xn − xp| ≥ Rmin. +If a) is satisfied, go to step 8; otherwise, set xp = Xn and go to step 2. +8 : +Record Xn and the FPT t = +p +� +i=0 +ti + ndt. +8 + +FIG. 3. Volume of interest (right) generated by rotating a curve (left) around the z-axes. The +parameters λ1 and λ2 play the role of the polar and azimuthal angle. +VALIDATION +In this section we apply our method to two example volumes and compare the results to +Monte Carlo simulations. +Example 1 +We chose a volume by revolving the curve +h(λ1) = 1 − e−4(λ1−1)2 +2 +. +(18) +around the z-axes, where λ1 has a range [0, π]. The corresponding volume is given by the +vector +g(λ1, λ2) = (h(λ1) sin λ1 cos λ2, h(λ1) sin λ1 sin λ2, h(λ1) cos λ1), +(19) +shown in Fig. 3. +To give this volume a skin, we need to subtract Rminu(λ1, λ2) from g, where u(λ1, λ2) +is the unit vector perpendicular to the surface at the point (λ1, λ2). Since the horizontal +cross section of the volume is a circle, we can write u(λ1, λ2) and g(λ1, λ2) in cylindrical +coordinates (ρ, z, φ), where ρ = +� +x2 + y2, as u(λ1) = (uρ(λ1), uz(λ1), 0) and g(λ1, λ2) = +9 + +Z +z +个 +1.0 +0.5 +g(11/ 12) +M +d +0.2 +0.4 +0.6 +0.8 +-0.5 +y +V +x +1.0(f(λ1) sin λ1, f(λ1) cos λ1, gφ(λ1, λ2)), respectively. +Finding (uρ(λ1) and uz(λ1)) is then a +matter of solving the equation +dg(λ1, λ2) +dλ1 +· u(λ1) = 0, +(20) +or +uρ[h(λ1) cos λ1 + h′(λ1) sin λ1] + uz(λ1)[−h(λ1) sin λ1 + h′(λ1) cos λ1] = 0. +(21) +Coupled with the condition that u(λ1) has a unit length, i. e. u2 +ρ + u2 +z = 1, we obtain +uρ(λ1) = +1 +� +1 + H(λ1)2 +uz(λ1) = − +H(λ1) +� +1 + H(λ1)2, +where +H(λ1) = h(λ1) cos λ1 + h′(λ1) sin λ1 +−h(λ1) sin λ1 + h′(λ1) cos λ1 +. +To generate a sphere centered at x0 that touches the skin at a single point, we only need to +minimize its radius, or, equivalently, its square radius: +R(λ1)2 = [h(λ1) sin λ1 − Rminuρ(λ1)]2 + [h(λ1) cos λ1 − Rminuz(λ1)]2. +(22) +To sample T, we set it to (1/π2) ln(2/1 − η) if η > 1 − 2e−π2/6, otherwise we used +the minimizer “fminbnd” for the function [ ˜F10(T ′) − η]2 in the range [0.01, 1/6]. We used +“fminbnd” to minimize R(λ1)2 as well, but in two steps: first we searched λ1 in the range +[0, π/2] and then in the range [π/2, π]. +For both, MC and τ-leaping, the condition that determines whether the particle is inside +or outside of the VOI is as follows: +If |g(θ′)| − |x| > 0, +particle inside +If |g(θ′)| − |x| ≤ 0, +particle outside, +where x is the particle’s position vector and θ′ is its polar angle, which can be computed +from its components (x, y, z): +θ′ = +� +� +� +� +� +� +� +� +� +arctan +√ +x2+y2 +z +, +if z > 1 +π + arctan +√ +x2+y2 +z +, if z < 0 +π +2, +if z = 0 and x ̸= y ̸= 0. +(23) +10 + +FIG. 4. Volume of interest (left) and a horizontal cross-section at λ1 = π/2 (right). +Example 2 +Let us now generate a more complicated volume by allowing the length of the vector g +to depend on λ2 as well: +g(λ1, λ2) = (h(λ1)f(λ2) sin λ1 cos λ2, h(λ1)f(λ2) sin λ1 sin λ2, h(λ1) cos λ1) +(24) +where +f(λ2) = 1 − cos(4λ2) +4 +. +(25) +The corresponding volume is shown in Fig. 4. To find the unit vector u(λ1, λ2) perpendicular +to the surface, we can vary g(λ1, λ2) in an arbitrary direction and demand that +δg(λ1, λ2) · u(λ1, λ2) = ∂g(λ1, λ2) +∂λ1 +· u(λ1, λ2)δλ1 + ∂g(λ1, λ2) +∂λ2 +· u(λ1, λ2)δλ2 = 0. +(26) +Since δλ1 and δλ2 are arbitrary, albeit infinitesimal, each of the two terms on the right in +Eq. (26) must be zero. Hence, +∂g(λ1, λ2) +∂λ1 +· u(λ1, λ2) = 0 +∂g(λ1, λ2) +∂λ2 +· u(λ1, λ2) = 0, +11 + +y +g1(11, 12) +0.5 +0.5 +-x +0.5 +0.5 +=π/2which, when coupled with the condition that u2 +x + u2 +y + u2 +z = 1, yields a unique solution to +ux, uy and uz: +ux(λ1, λ2) = [h(λ1) sin λ1 − cos λ1h′(λ1)][cos λ1f(λ2) + sin λ1f ′(λ2)]/K(λ1, λ2) +uy(λ1, λ2) = [h(λ1) sin λ1 − cos λ1h′(λ1)][f(λ2) sin λ2 − cos λ2f ′(λ2)]/K(λ1, λ2) +uz(λ1, λ2) = [f(λ2)2(cos λ1h(λ1) + sin λ1h′(λ1)]/K(λ1, λ2), +where +K(λ1, λ2) = +� +f(λ1)4[cos λ1h(λ1) + sin λ1h′(λ1)]2 + [h(λ1) sin λ1 − cos λ1h′(λ1)]2[f(λ2)2 + f ′(λ2)2] +�1/2 . +(27) +The square radius to be minimized is now +R(λ1, λ2)2 = [h(λ1)f(λ2) sin λ1 cos λ2 − Rminux(λ1, λ2)]2 ++ [h(λ1)f(λ2) sin λ1 sin λ2 − Rminuy(λ1, λ2)]2 ++ [h(λ1) cos λ1 − Rminuy(λ1, λ2)]2. +(28) +To minimize R(λ1, λ2)2, we formed a grid by dividing λ1 into two sections - [0, π/2] and +[π/2, π] - and λ2 into five sections -[0, 2π/5], [2π/5, 4π/5], [4π/5, 6π/5], [6π/5, 8π/5] and +[8π/5, 2π] - and used “fmincon”, with the initial search point being in the middle of each +pixel. +The condition that determines whether the particle is inside or outside of the VOI is now +a function of two variables: +If |g(θ′, φ′)| − |x| > 0, +particle inside +If |g(θ′, φ′)| − |x| ≤ 0, +particle outside, +where x is the particle’s position vector and θ′ and φ′ are its polar and azimuthal angles, +12 + +FIG. 5. Example volume 1: Monte Carlo (black) and τ-leaping for three values of εR - 0.03 +(purple), 0.05 (green) and 0.1 (orange) - for a) probability for FPT; b) cumulative probability for +the FPT; c) probability for the distance between the initial position and the point of crossing; and +d) probability for the speed, i. e. distance between the initial position and the point of crossing +divided by the FPT. The bin sizes are: a) 1, b) 1, c) 0.005, and d) 0.01. +and can be computed from its components (x, y, z): +φ′ = +� +� +� +� +� +� +� +� +� +� +� +arcsin +y +√ +x2+y2, +if x > 0 and y > 0 +π − arcsin +y +√ +x2+y2, +if x < 0 and y ̸= 0 +2π + arcsin +y +√ +x2+y2, if x > 0 and y < 0 +(29) +θ′ = +� +� +� +� +� +� +� +� +� +arctan +√ +x2+y2 +zf(φ′) , +if z > 0 +π + arctan +√ +x2+y2 +zf(φ′) , if z < 0 +π +2, +if z = 0 and x ̸= y ̸= 0. +(30) +13 + +a) +c) +0.07 ++ Monte Carlo +0.035 +0.06 +. r-leaping, g = 0.03 +0.030 +0.05 +r-leaping, g = 0.05 +0.025E +P +0.04 +r-leaping, &g = 0.1 +0.020 +P +0.03 +P0.015 +0.02 +0.010 +0.01 +0.005 +0.00E +0.000 +0 +50 +100 +150 +200 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +FPT +p +b) +d) +1.0 +0.14 +0.12 +0.8 +0.10E +(Id) +0.6 +/FPT) +0.08F +0.06 +0.4 +P +0.04 +0.2 +0.02 +0.0E +0.00E +0 +50 +100 +150 +200 +0.00 +0.05 +0.10 +0.15 +0.20 +FPT +d/FPTFIG. 6. Example volume 2: Monte Carlo (black) and τ-leaping for three values of εR - 0.03 +(purple), 0.05 (green) and 0.1 (orange) - for a) probability for FPT; b) cumulative probability for +the FPT; c) probability for the distance between the initial position and the point of crossing; and +d) probability for the speed, i. e. distance between the initial position and the point of crossing +divided by the FPT. The bin sizes are: a) 1, b) 1, c) 0.005, and d) 0.01. +Results +The parameter values for all simulations were chosen to be: kBT = 4.14×10−9kg·µm2·s−2, +m = 10−10kg, viscosity ν = 1.7 × 10−9kg·µm−1·s−1, and particle’s size rB = 58.6µm. These +values render the relaxation time τB = 5.31 × 10−5s. In all simulations, dt was chosen to be +5 × 10−6s. In the two examples above, the initial positions were chosen to be (0, 0.4, 0) and +(0.5, 0.5, 0) respectively. Figures 5 and 6 shows the comparisons between Monte Carlo and +the τ-leaping method for example volumes 1 and 2, respectively. +14 + +a) +c) +0.08 ++ Monte Carlo +0.05 +0.06 +r-leaping, g = 0.05 +0.04F +PT) +r-leaping, g = 0.1 +0.03 +0.04 +P +0.02 +0.02 +0.01 +0.00 +0.00E +0 +20 +40 +60 +80 +100 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +FPT +p +b) +d) +1.0F +0.08 +0.8 +（/FPT） +0.06 +E +0.6 +0.04 +Pc +0.4 +P +0.2 +0.02 +0.00E +0 +20 +40 +60 +80 +100 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +FPT +d/FPTDISCUSSION +We have presented a τ-leaping method to compute the first passage time (FPT) and +position of a Brownian particle. The “leaping” was done by sampling the FPT and position +for a sphere inscribed in the volume of interest (VOI) and centered at the last sampled +position of the particle. By setting a lower limit on the size of such a sphere, Rmin, and +repeating the “leaping” procedure, we eventually arrive at a position (near the surface of the +VOI) where the size of the sphere is less than Rmin; at such a point, the method switches to +regular Monte Carlo simulation until the particle either leaves the VOI, or reaches a position +where a sphere of radius greater than Rmin can be generated. The purpose of setting a lower +limit on the size of the spheres was to avoid having to sample the velocity of the particle: +the larger the sphere, the less important the initial velocity for the sampling of FPT and +position. Hence, Rmin is chosen based on one’s notion of accuracy. Another important step +in this method is to give the VOI an inner skin of thickness Rmin. This, again, is to avoid +having to sample velocities: by generating spheres that are inscribed by the volume bounded +by the inner surface of the skin, we are guaranteed (within an accuracy we have chosen by +setting Rmin) that the particle’s velocity at the last sampling will not be important in the +Monte Carlo simulation when the particle evolves to a distance greater than or equal to +Rmin. We have demonstrated this method, on two example volumes and three thicknesses of +skin to be as accurate and much more efficient than Monte Carlo, as shown in Table 1. The +last column gives the percentage values of the average distance between the probabilities for +the FPT of Monte Carlo and τ-leaping: +Accuracy = 100 +� +1 − +Nt +� +n=1 +|Pτ(Tn) − PMC(Tn)|/Nt +� +, +(31) +where Nt is the number of bins in the histograms in Figures 5a and 6a. +Although the +accuracy for the three choices of εR is essentially the same, the efficiency varies significantly. +According to the condition t/tmin = 6 × 0.04(εd/ε2 +R) ≫ 1 (see the last paragraph of section +“Brownian motion and the Langevin equation”), the three values of εR, 0.03, 0.05 and 0.1, +give t/tmin =8, 2.88 and 0.72, respectively, only the first of which can be said to satisfy the +condition t/tmin ≫ 1. What this tells us is that the condition itself might be too strict and +further analysis is needed to refine it. +We should point out that the size of the particle we have chosen as our test subject was +15 + +TABLE I. Values for efficiency of Monte Carlo simulations and the τ-leaping method (column 4) as +a function of volume of interest and Rmin. Column 5 shows the accuracy of the τ-leaping method +relative to Monte Carlo. +Method +Volume # +εr, Rmin (µm) +Average efficiency +Accuracy +(seconds/run) +Monte Carlo +1 +NA +155.52 +NA +2 +NA +156.25 +NA +τ-leaping +0.03, 0.034 +12.10 +99.917% +1 +0.05, 0.02 +1.84 +99.914% +0.1, 0.01 +1.38 +99.9% +0.03, 0.034 +16.9 +99.929% +2 +0.05, 0.02 +13.69 +99.911% +0.1, +0.01 +12.0 +99.904% +∼ 60µm, while the enclosing volumes were ∼ 1µm large. This may seem like a geometric +impossibility; however, it is not, since the volumes are imaginary and only serve to facilitate +a comparison between two methods. A more realistic scenario would have been to chose a +volume much larger than the particle’s size, in which case the volume could be treated as a +real physical enclosure. However, this would make Monte Carlo simulations infeasible: for +a volume 10 times larger than the particle’s radius (∼ 600µm) 1000 simulations would take +about 4.5×1034 hours. On the other hand, because the efficiency of our method is hindered +only by the thickness of the skin, which does not change with scaling of the volume, it would +be effected hardly at all. Another realistic scenario would have been to make the Brownian +particle much smaller, while keeping the volumes fixed. For example, mass and viscosity +typical of biological cells, m = 10−20kg, and ν = 1.7 × 10−8kg·µm−1·s−1, and an average +protein size ∼ 5.86 × 10−4µm, would give τB = 5.31 × 10−11s and the values for Rmin ten +times smaller than used in this paper, which would make the τ-leaping method faster still +by a factor of ∼10. +The relatively simple structure of our method makes it ideal for simulations that combine +interactions of a particle with not only boundaries, but also objects within the boundaries. +For example, a protein, seeking a binding site on DNA, would typically bounce or slide along +16 + +the chromatin, thus effectively reducing the search space from three to two (or even one, +for unwound chromatin) dimensions. Our method can be easily applied in this scenario by +simply generating a skin around the chromatin. +[1] Chou T, D’Orsogna MR (2008) First-Passage Phenomena and Their Applications World Sci- +entific, pp. 306-345 +[2] Singer A, Schuss Z, Holcman D (2008) Narrow escape and leakage of brownian particles Phys. +Rev. E. 78, 051111 +[3] Ward M, Peirce A, Kolokolnikov T (2010) An Asymptotic Analysis of the Mean First Passage +Time for Narrow Escape Problems: Part I: Two-Dimensional Domains SIAM Journal on +Multiscale Modeling and Simulation, DOI: 10.1137/090752511 +[4] Gillespie DT (2001) Approximate accelerated stochastic simulation of chemically reacting +systems J. Chem. Phys. 115, 1716–1733 +[5] Rathinam M, Petzold LR, Cao Y, Gillespie DT (2003) Stiffness in stochastic chemically re- +acting systems: The implicit tau-leaping method J. Chem. Phys. 119(24), 12784 +[6] Tian T, Burrage K (2004) Binomial leap methods for simulating stochastic chemical kinetics +J. Chem. 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(2015) http://corysimon.github.io/articles/uniformdistn-on-sphere/ +19 + diff --git a/b9AyT4oBgHgl3EQfwfkG/content/tmp_files/load_file.txt b/b9AyT4oBgHgl3EQfwfkG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1eeda98440d0c42fd4623aab3aca1ed2e911064 --- /dev/null +++ b/b9AyT4oBgHgl3EQfwfkG/content/tmp_files/load_file.txt @@ -0,0 +1,488 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf,len=487 +page_content='A tau-leaping method for computing joint probability distributions of the first-passage time and position of a Brownian particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Jaroslav Albert Abstract First passage time (FPT), also known as first hitting time, is the time a particle, subject to some stochastic process, hits or crosses a closed surface for the very first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' τ-leaping methods are a class of stochastic algorithms in which, instead of simulating every single reaction, many reactions are “leaped” over in order to shorten the computing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' In this paper we developed a τ-leaping method for computing the FPT and position in arbitrary volumes for a Brownian particle governed by the Langevin equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The τ-leaping method proposed here works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' A sphere is inscribed within the volume of interest (VOI) centered at the initial particle’s location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' On this sphere, the FPT is sampled, as well as the position, which becomes the new initial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Then, another sphere, centered at this new location, is inscribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' This process continues until the sphere becomes smaller than some minimal radius Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' When this occurs, the τ-leaping switches to the conventional Monte Carlo, which runs until the particle either crosses the surface of the VOI or finds its way to a position where a sphere of radius > Rmin can be inscribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The switching between τ-leaping and MC continues until the particle crosses the surface of the VOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The purpose of a minimal radius is to avoid having to sample the velocities, which become irrelevant when the particle diffuses beyond a certain distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Rmin The size of this radius depends on the system parameters and on one’s notion of accuracy: the larger this radius the more accurate the τ-leaping method, but also less efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' This trade off between accuracy and efficiency is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' For two VOI, the τ-leaping method is shown to be accurate and more efficient than MC by at least a factor of 10 and up to a factor of about 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' However, while MC becomes exponentially slower with increasing VOI, the efficiency of the τ-leaping method remains relatively unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Thus, the τ-leaping method can potentially be many orders of magnitude more efficient than MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='00647v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='MN] 2 Jan 2023 INTRODUCTION First passage time (FPT) is the time that a certain event occurs for the first time during an evolution of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' In molecular biology it is often desirable to know the FPT distributions for a molecule, such as protein, for crossing a surface, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' that of the cell nucleus or the cell membrane, or for finding its target site on the DNA [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Although mean first passage times for these types of events have been worked out to various degrees of approximations [2, 3], the non-trivial shapes of volumes and obstacle-riddled environments in which biological molecules have to navigate makes computations of FPT distributions difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The usual strategy in such efforts is to simulate the molecular dynamics using Monte Carlo methos, which do get the job done but are notoriously inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' In this paper we draw inspiration from computational analysis of stochastic gene expres- sion – an area of research that has produced many alternative methods to brute Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' In particular, we focus on two such methods: τ-leaping [4–16] and hy- brid stochastic simulation algorithms (HSSA) [17–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' A τ-leaping method approximates the evolution of a system over many small steps in a MC simulation by taking larger steps or leaps, thereby reducing the overall number of steps that need to be taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The HSSAs on the other hand, work by employing a form of τ-leaping method on a part of the system (a subset of molecular species and chemical reactions), while using good old MC on the rest of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' In this paper we apply these concepts to Brownian motion described by the Langevin equation in volumes of arbitrary shapes with the goal to compute joint distributions of the FPT and position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' More specifically, we take advantage of the fact that LEs can be solved approximately for a spherical volume of certain minimal size, which can be used to fill parts of the larger volume of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Sampling the FPT and position for this spherical volume, we generate another sphere centered at the sampled position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' When this process brings the particle within a certain distance from the boundary, we switch to the MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Thus, with each sphere, we effectively τ-leap over τ/dt number of steps, where dt is the temporal size of each step in the MC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' With full details about what happens near the boundary, we show the accuracy and efficiency of our method on two examples volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 2 BROWNIAN MOTION AND THE LANGEVIN EQUATION When a large particle is immersed in a medium (gas or liquid) of many smaller particles at equilibrium, it moves in a jittery fashion due to density fluctuations in that medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' One model of such motion is called Brownian, and is described by the Langevin equation, mdv(t) dt − v(t) τB = f(t) (1) where v(t) is the particle’s velocity at time t, m is its mass, τB is the relaxation time, and f(t) is a random force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' This random force changes magnitude and direction at time intervals separated by dt and follows a Gaussian distribution P(f) = 1 (2πσ2 f)3/2e−f·f/(2σ2 f), (2) where σ2 f = 2kBTm/(dtτB), and kB and T are the Boltzman constant and temperature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The relaxation time τB is related to the mass m, viscosity of the medium ν, and the particle’s size rB via this expression: τB = m/(6πνrB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Hence, coupled with the definition of velocity, v = dr/dt, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (1) can be used to simulate the evolution of a Brownian particle’s velocity and position r by iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The time step, dt, must be chosen to satisfy τs ≪ dt, where τs is the average collision time between the Brownian particle and the molecules of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Another approach to studying Brownian motion is via a Master Equation for the joint probability distribution, P(r, v, t), which is given by the Klein-Kramers equation (also re- ferred to as Fokker-Planck equation) [31]: ∂P ∂t + v · ∇rP − 1 τB v · ∇vP − kBT τBm∇2 vP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (3) The solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (3) with infinite boundaries and the initial conditions P(r, v, 0) = δ(3)(r − r′)δ(3)(v − v′) is given by [32, 33]: P(r, v, t) = 1 (2πσXσV � 1 − β2)3 × exp � − 1 2(1 − β2) �|r − µµµX|2 σ2 X + |v − µµµV |2 σ2 V − 2β(r − µµµX) · (v − µµµV ) σXσV �� , (4) 3 where σ2 X = kBTτ 2 B m � 1 + 2t/τB − � 2 − e−t/τB�2� (5) σ2 V = kBT m � 1 − e−2t/τB� (6) β = kBTτB σXσV � 1 − e−t/τB�2 µµµX = r′ + (1 − e−t/τB)τBv′ µµµV = v′e−t/τB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' We can obtain the probability for the particle’s position by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (4) over v: P(r, t) = � ∞ −∞ P(r, v, t)dv = 1 (2πσX(t)2)3exp � −|r − µµµX(t)|2 2σX(t)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (7) For t ≫ τB, σX(t)2 → 2(kBT/m)t and µµµX(t) → r′ + τBv′, which allows us to replace the Brownian model with a diffusion model: ∂P(r, t) ∂t = 1 D∇2P(r, t), (8) where D = kBT/m, subject to the initial conditions P(r, 0) = δ(3)(r − r′ − τBv′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' We can quantify the discrepancy between the Langevin model and the diffusion model via this expression: w(t) = 1 − σX(t)2 2Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (9) If we set w(t) to some small value εw, we can solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (9) for the minimal time the system must evolve before we can treated as diffusive: tmin = 3τB/(2εw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' For example, if εw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='03, we get tmin = 50τB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Thus, if we are only interested in times > tmin, we are free to use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (8) as our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Although one can chose v′ in the initial conditions to be any value, it is useful to consider the magnitude of the term τBv′ for a realistic scenario, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' v′ being the result of a Brownian particle having arrived at position r′ at time t = 0, after traveling for a time > tmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (6), the distribution of velocities for such a particle would have the standard deviation σ2 V = kBT/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Thus, the maximum speed of the arriving Brownian particle would be ∼ 3 � kBT/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' For a large enough volume, we can assume the term 3τB � kBT/m to be negligible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' if 3τB � kBT/m/R ≪ 1, where R is the radius of our sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' By choosing the smallness of εR = 3τB � kBT/m/R, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' εR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='03, we can determine the minimum radius R for which the initial velocity can be neglected: Rmin = 3εR � m/(kBT)τB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Thus, provided the particle takes significantly longer 4 on average than tmin to reach a distance Rmin, we can replace the Brownian model with a diffusion model for t > tmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' In a moment we will see that the minimal time to reach a distance Rmin is indeed much longer than tmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' With these criteria we can compute the FPT for a Brownian particle using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (8) and the initial condition P(r, 0) = δ(3)(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' In spherical coordinates, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (8) reads: ∂P(ξ, t) ∂T = 1 ξ2 ∂ ∂ξ � ξ2∂P(ξ, t) ∂ξ � , (10) where ξ = r/R and T = Dt/R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Thanks to spherical symmetry, P(r, t) is independent of the longitudinal and azimuthal angles, φ and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The initial condition becomes P(r, 0) = δ(r)/r2, or P(ξ, 0) = δ(ξ)/ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' To compute the FPT, we also need to add the absorbing boundary condition P(ξ = 1, T) = 0, for which the solution is: P(ξ, T) = lim M→∞ PM(ξ, T), (11) where PM(ξ, T) = M � n=1 An(M)sin(πnξ) ξ e−(πn)2T (12) and An(M) = 2πne−(πn/M)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (13) The survivor’s probability S∞(T), which is the probability that the particle remains inside R for a period of time T, is given by S∞(T) = � 1 0 PM(ξ, T)ξ2dξ = lim M→∞ M � n=1 2(−1)n+1e−(πn)2(T+1/M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (14) The subscript ∞ serves as a reminder that M → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The FPT distribution is simply F∞(T) = 1−S∞(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' In practice, however, the summation limit can be cut off at some finite value of M: FM(T) = 1 − SM(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Since the exponential term e−(πn)2(T+1/M) decays very rapidly for large n, we can take the limit (T + 1/M) → T, while cutting the summation off at some finite M to obtain: ˜FM(T) = 1 − M � n=1 2(−1)n+1e−(πn)2T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (15) Figures 1 a) and b) show the behaviors of PM(ξ, 0), FM(T) and ˜FM(T) for different values of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Evidently, to make PM(ξ, 0) sharply peaked near r = 0 and FM(T) converge to F∞(T) 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' a) Initial distributions PM(ξ, 0) for different M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' b) Distributions FM(T) (solid lines) and ˜FM(T) (dashed lines) for different M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' requires large values of M, while the function ˜FM(T) does not: for M = 10 it already behaves correctly for T larger than ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' To sample ˜FM(T), one needs only to generate a random real number η = (0, 1] and solve ˜FM(T) − η = 0 for T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' For nontrivial ˜FM(T), this could be done by minimizing |FM(T) − η|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' However, if we compute the average T, ⟨T⟩ = � ∞ 0 ∞ � n=1 2(−1)n+1e−(πn)2TdT = ∞ � n=1 2(−1)n+1 (πn)2 = 1 6, (16) we notice, by examining Figure 1 b) again, that the function ˜F1(T) = 1 − 2e−π2T behaves correctly for T > 1/6 - the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Hence, we could speed up the minimization procedure by first checking whether 1 − 2e−π2/6 is greater or smaller than η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' If it is the latter, we can set 1 − 2e−π2T to η and solve for T analytically: T = 1 π2 ln � 2 1 − η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (17) If it is the former, we only need to search for T in the range [0, 1/6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Before we continue we must circle back and check that the time to reach a distance Rmin is much greater than tmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' We can do this by requiring that ˜F10(T) be less than some chosen value, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='001, which corresponds to T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' If we recall that T = tMRmin/(kBTτB), Rmin = 3τB � kBT/m/εR and tmin = 3τB/(2εw), we obtain t/tmin = 6 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='04(εw/ε2 R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' For εd = εR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='03, we get t/tmin = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 6 a) b) 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0 M=10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='8 F FM(T) FM(T) 30 M=20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='6F M=50 ()W/L)W M=1 M=1 PM(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='4 1 M=2 M=2 20 M=100 M=3 M=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='2 M=4 M=4 10F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0 M=10 M=10 M=100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='2 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='5 5 = 1/6 TFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' A 2-dimensional illustration of the τ-leaping method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The particle starts out at position x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The smallest possible circle centered at x0, Circle 1, is generated and a point on its surface, x1, is sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The same process is repeated for Circle 2, Circle 3 and Circle 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' However, the radius of Circle 4 is smaller than Rmin, so we must switch to Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' In Scenario 1, the particle diffuses a distance ≥ Rmin (blue dashed circle) to the point y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' a sphere centered at y1 is generated but its radius is smaller than Rmin, so Monte Carlo continues until the particle crosses the boundary (black thick line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' In Scenario 2, the particle diffuses a distance ≥ Rmin to the point y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' a sphere centered at y1 is generated with radius < Rmin, so we continue with Monte Carlo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' the particle diffuses to a point y2, where a sphere of radius > Rmin is generated, and we switch to τ-leaping to continue the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' τ-LEAPING Now that we have an analytical expression for the FPT for a sphere, we can use it to speed up simulation of Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The scheme is shown in Figure 2 on a two-dimensional example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' First, we give the surface of the volume of interest a skin of thickness Rmin on the inside (the purpose of which will be explained shortly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Next, starting from some initial point x0, we generate a sphere centered at x0 such that its surface and the skin share a 7 Circle 1 Circle 2 Circle 3 Circle 4 Switch to MonteCarlo Scenario 1 Scenario 1continued Scenario2 Scenario 2continuedunique point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' This is equivalent to finding the smallest sphere whose surface touches the skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Then, we sample the FPT and the particle’s position on the surface of the sphere, (t1, x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Centered at x1, we generate another sphere whose surface touches the skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' We sample the FPT and surface position, (t2, x2), and continue this process in this manner until we generate a sphere with a radius > Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' When this happens, we switch to Monte Carlo, with the initial conditions ˜x0 = xi and v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Of course, in reality there is no reason to expect v to be 0, unless we get very lucky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' However, if we let the particle evolve past a radius Rmin, we do not need to worry about its initial velocity and may set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' This is where the skin guarantees accuracy: in the (unlikely) event of sampling a position that falls on the skin, we are guaranteed that, should the particle evolve past the outer surface, it will have traveled at least the distance Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' With this quality check in place, we can write down the steps of this procedure in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 0 : Choose a volume whose enclosing surface is given by a vector g(λ1, λ2), parametrized by λ1 and λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Also choose (T, m, τB, Rmin) and the step size dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 1 : Set (p, n) = 0, where p and n are counters, and choose initial time tp (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' zero) and an initial position xp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 2 : Generate a sphere of radius R by minimizing |g(λ1, λ2) − xp|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' If R ≥ Rmin, set p = p + 1 and go to step 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' otherwise go to step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 3 : Sample T by generating a random real number η = [0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' If η > 1 − 2eπ2/6, set T = 1/π2 ln[2/(1 − η)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' otherwise set T =min| ˜F10(T ′) − η|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Set tp = R2T/D and record it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 4 : Sample a point on a sphere, r, from a uniform distribution by generating two random numbers q1 = [0, 2π] and q2 = [0, 1] and set r = [R sin θ cos φ, R sin θ sin φ, R cos θ], where θ = q1 and φ = arccos(1 − 2q2) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 5 : Set xp = xp−1 + r and go to step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 6 : Simulate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (1) using Monte Carlo with the initial conditions Xn = xp and Vn = 0, until a) Xn reaches the outside of the volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' or b) |Xn − xp| ≥ Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' If a) is satisfied, go to step 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' otherwise, set xp = Xn and go to step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 8 : Record Xn and the FPT t = p � i=0 ti + ndt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Volume of interest (right) generated by rotating a curve (left) around the z-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The parameters λ1 and λ2 play the role of the polar and azimuthal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' VALIDATION In this section we apply our method to two example volumes and compare the results to Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Example 1 We chose a volume by revolving the curve h(λ1) = 1 − e−4(λ1−1)2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (18) around the z-axes, where λ1 has a range [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The corresponding volume is given by the vector g(λ1, λ2) = (h(λ1) sin λ1 cos λ2, h(λ1) sin λ1 sin λ2, h(λ1) cos λ1), (19) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' To give this volume a skin, we need to subtract Rminu(λ1, λ2) from g, where u(λ1, λ2) is the unit vector perpendicular to the surface at the point (λ1, λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Since the horizontal cross section of the volume is a circle, we can write u(λ1, λ2) and g(λ1, λ2) in cylindrical coordinates (ρ, z, φ), where ρ = � x2 + y2, as u(λ1) = (uρ(λ1), uz(λ1), 0) and g(λ1, λ2) = 9 Z z 个 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='5 g(11/ 12) M d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='5 y V x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0(f(λ1) sin λ1, f(λ1) cos λ1, gφ(λ1, λ2)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Finding (uρ(λ1) and uz(λ1)) is then a matter of solving the equation dg(λ1, λ2) dλ1 u(λ1) = 0, (20) or uρ[h(λ1) cos λ1 + h′(λ1) sin λ1] + uz(λ1)[−h(λ1) sin λ1 + h′(λ1) cos λ1] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (21) Coupled with the condition that u(λ1) has a unit length, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' u2 ρ + u2 z = 1, we obtain uρ(λ1) = 1 � 1 + H(λ1)2 uz(λ1) = − H(λ1) � 1 + H(λ1)2, where H(λ1) = h(λ1) cos λ1 + h′(λ1) sin λ1 −h(λ1) sin λ1 + h′(λ1) cos λ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' To generate a sphere centered at x0 that touches the skin at a single point, we only need to minimize its radius, or, equivalently, its square radius: R(λ1)2 = [h(λ1) sin λ1 − Rminuρ(λ1)]2 + [h(λ1) cos λ1 − Rminuz(λ1)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (22) To sample T, we set it to (1/π2) ln(2/1 − η) if η > 1 − 2e−π2/6, otherwise we used the minimizer “fminbnd” for the function [ ˜F10(T ′) − η]2 in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='01, 1/6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' We used “fminbnd” to minimize R(λ1)2 as well, but in two steps: first we searched λ1 in the range [0, π/2] and then in the range [π/2, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' For both, MC and τ-leaping, the condition that determines whether the particle is inside or outside of the VOI is as follows: If |g(θ′)| − |x| > 0, particle inside If |g(θ′)| − |x| ≤ 0, particle outside, where x is the particle’s position vector and θ′ is its polar angle, which can be computed from its components (x, y, z): θ′ = � � � � � � � � � arctan √ x2+y2 z , if z > 1 π + arctan √ x2+y2 z , if z < 0 π 2, if z = 0 and x ̸= y ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (23) 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Volume of interest (left) and a horizontal cross-section at λ1 = π/2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Example 2 Let us now generate a more complicated volume by allowing the length of the vector g to depend on λ2 as well: g(λ1, λ2) = (h(λ1)f(λ2) sin λ1 cos λ2, h(λ1)f(λ2) sin λ1 sin λ2, h(λ1) cos λ1) (24) where f(λ2) = 1 − cos(4λ2) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (25) The corresponding volume is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' To find the unit vector u(λ1, λ2) perpendicular to the surface, we can vary g(λ1, λ2) in an arbitrary direction and demand that δg(λ1, λ2) · u(λ1, λ2) = ∂g(λ1, λ2) ∂λ1 u(λ1, λ2)δλ1 + ∂g(λ1, λ2) ∂λ2 u(λ1, λ2)δλ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (26) Since δλ1 and δλ2 are arbitrary, albeit infinitesimal, each of the two terms on the right in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (26) must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Hence, ∂g(λ1, λ2) ∂λ1 u(λ1, λ2) = 0 ∂g(λ1, λ2) ∂λ2 u(λ1, λ2) = 0, 11 y g1(11, 12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='5 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='5 =π/2which, when coupled with the condition that u2 x + u2 y + u2 z = 1, yields a unique solution to ux, uy and uz: ux(λ1, λ2) = [h(λ1) sin λ1 − cos λ1h′(λ1)][cos λ1f(λ2) + sin λ1f ′(λ2)]/K(λ1, λ2) uy(λ1, λ2) = [h(λ1) sin λ1 − cos λ1h′(λ1)][f(λ2) sin λ2 − cos λ2f ′(λ2)]/K(λ1, λ2) uz(λ1, λ2) = [f(λ2)2(cos λ1h(λ1) + sin λ1h′(λ1)]/K(λ1, λ2), where K(λ1, λ2) = � f(λ1)4[cos λ1h(λ1) + sin λ1h′(λ1)]2 + [h(λ1) sin λ1 − cos λ1h′(λ1)]2[f(λ2)2 + f ′(λ2)2] �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (27) The square radius to be minimized is now R(λ1, λ2)2 = [h(λ1)f(λ2) sin λ1 cos λ2 − Rminux(λ1, λ2)]2 + [h(λ1)f(λ2) sin λ1 sin λ2 − Rminuy(λ1, λ2)]2 + [h(λ1) cos λ1 − Rminuy(λ1, λ2)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (28) To minimize R(λ1, λ2)2, we formed a grid by dividing λ1 into two sections - [0, π/2] and [π/2, π] - and λ2 into five sections -[0, 2π/5], [2π/5, 4π/5], [4π/5, 6π/5], [6π/5, 8π/5] and [8π/5, 2π] - and used “fmincon”, with the initial search point being in the middle of each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The condition that determines whether the particle is inside or outside of the VOI is now a function of two variables: If |g(θ′, φ′)| − |x| > 0, particle inside If |g(θ′, φ′)| − |x| ≤ 0, particle outside, where x is the particle’s position vector and θ′ and φ′ are its polar and azimuthal angles, 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Example volume 1: Monte Carlo (black) and τ-leaping for three values of εR - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='03 (purple), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='05 (green) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='1 (orange) - for a) probability for FPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' b) cumulative probability for the FPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' c) probability for the distance between the initial position and the point of crossing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' and d) probability for the speed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' distance between the initial position and the point of crossing divided by the FPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The bin sizes are: a) 1, b) 1, c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='005, and d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' and can be computed from its components (x, y, z): φ′ = � � � � � � � � � � � arcsin y √ x2+y2, if x > 0 and y > 0 π − arcsin y √ x2+y2, if x < 0 and y ̸= 0 2π + arcsin y √ x2+y2, if x > 0 and y < 0 (29) θ′ = � � � � � � � � � arctan √ x2+y2 zf(φ′) , if z > 0 π + arctan √ x2+y2 zf(φ′) , if z < 0 π 2, if z = 0 and x ̸= y ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' (30) 13 a) c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='07 + Monte Carlo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='06 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' r-leaping, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='05 r-leaping, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='025E P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='04 r-leaping, &g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='020 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='03 P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='00E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='000 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='4 FPT p b) d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='10E (Id) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='6 /FPT) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='08F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='4 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='00E 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='20 FPT d/FPTFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Example volume 2: Monte Carlo (black) and τ-leaping for three values of εR - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='03 (purple), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='05 (green) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='1 (orange) - for a) probability for FPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' b) cumulative probability for the FPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' c) probability for the distance between the initial position and the point of crossing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' and d) probability for the speed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' distance between the initial position and the point of crossing divided by the FPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The bin sizes are: a) 1, b) 1, c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='005, and d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Results The parameter values for all simulations were chosen to be: kBT = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='14×10−9kg·µm2·s−2, m = 10−10kg, viscosity ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='7 × 10−9kg·µm−1·s−1, and particle’s size rB = 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='6µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' These values render the relaxation time τB = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='31 × 10−5s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' In all simulations, dt was chosen to be 5 × 10−6s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' In the two examples above, the initial positions were chosen to be (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='4, 0) and (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='5, 0) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Figures 5 and 6 shows the comparisons between Monte Carlo and the τ-leaping method for example volumes 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' 14 a) c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='08 + Monte Carlo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='06 r-leaping, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='04F PT) r-leaping, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='04 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='00E 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='4 FPT p b) d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='8 （/FPT） 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='06 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='04 Pc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='4 P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='00E 0 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='30 FPT d/FPTDISCUSSION We have presented a τ-leaping method to compute the first passage time (FPT) and position of a Brownian particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The “leaping” was done by sampling the FPT and position for a sphere inscribed in the volume of interest (VOI) and centered at the last sampled position of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' By setting a lower limit on the size of such a sphere, Rmin, and repeating the “leaping” procedure, we eventually arrive at a position (near the surface of the VOI) where the size of the sphere is less than Rmin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' at such a point, the method switches to regular Monte Carlo simulation until the particle either leaves the VOI, or reaches a position where a sphere of radius greater than Rmin can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The purpose of setting a lower limit on the size of the spheres was to avoid having to sample the velocity of the particle: the larger the sphere, the less important the initial velocity for the sampling of FPT and position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Hence, Rmin is chosen based on one’s notion of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Another important step in this method is to give the VOI an inner skin of thickness Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' This, again, is to avoid having to sample velocities: by generating spheres that are inscribed by the volume bounded by the inner surface of the skin, we are guaranteed (within an accuracy we have chosen by setting Rmin) that the particle’s velocity at the last sampling will not be important in the Monte Carlo simulation when the particle evolves to a distance greater than or equal to Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' We have demonstrated this method, on two example volumes and three thicknesses of skin to be as accurate and much more efficient than Monte Carlo, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The last column gives the percentage values of the average distance between the probabilities for the FPT of Monte Carlo and τ-leaping: Accuracy = 100 � 1 − Nt � n=1 |Pτ(Tn) − PMC(Tn)|/Nt � , (31) where Nt is the number of bins in the histograms in Figures 5a and 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Although the accuracy for the three choices of εR is essentially the same, the efficiency varies significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' According to the condition t/tmin = 6 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='04(εd/ε2 R) ≫ 1 (see the last paragraph of section “Brownian motion and the Langevin equation”), the three values of εR, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='1, give t/tmin =8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='88 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='72, respectively, only the first of which can be said to satisfy the condition t/tmin ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' What this tells us is that the condition itself might be too strict and further analysis is needed to refine it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' We should point out that the size of the particle we have chosen as our test subject was 15 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Values for efficiency of Monte Carlo simulations and the τ-leaping method (column 4) as a function of volume of interest and Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Column 5 shows the accuracy of the τ-leaping method relative to Monte Carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Method Volume # εr, Rmin (µm) Average efficiency Accuracy (seconds/run) Monte Carlo 1 NA 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='52 NA 2 NA 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='25 NA τ-leaping 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='034 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='10 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='917% 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='84 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='914% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='38 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='9% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='034 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='929% 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='02 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='69 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='911% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='01 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='0 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='904% ∼ 60µm, while the enclosing volumes were ∼ 1µm large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' This may seem like a geometric impossibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' however, it is not, since the volumes are imaginary and only serve to facilitate a comparison between two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' A more realistic scenario would have been to chose a volume much larger than the particle’s size, in which case the volume could be treated as a real physical enclosure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' However, this would make Monte Carlo simulations infeasible: for a volume 10 times larger than the particle’s radius (∼ 600µm) 1000 simulations would take about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='5×1034 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' On the other hand, because the efficiency of our method is hindered only by the thickness of the skin, which does not change with scaling of the volume, it would be effected hardly at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' Another realistic scenario would have been to make the Brownian particle much smaller, while keeping the volumes fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' For example, mass and viscosity typical of biological cells, m = 10−20kg, and ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='7 × 10−8kg·µm−1·s−1, and an average protein size ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='86 × 10−4µm, would give τB = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content='31 × 10−11s and the values for Rmin ten times smaller than used in this paper, which would make the τ-leaping method faster still by a factor of ∼10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AyT4oBgHgl3EQfwfkG/content/2301.00647v1.pdf'} +page_content=' The 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b/b9E4T4oBgHgl3EQfow0v/content/tmp_files/2301.05186v1.pdf.txt @@ -0,0 +1,2473 @@ +Astronomy & Astrophysics manuscript no. ms +© ESO 2023 +January 13, 2023 +Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +R. Poitevineau1,⋆, G. Castignani2,3, and F. Combes1,4 +1 Observatoire de Paris, LERMA, PSL University, Sorbonne Université, CNRS, F-75014, Paris, France +2 Dipartimento di Fisica e Astronomia ”Augusto Righi”, Alma Mater Studiorum Università di Bologna, Via Gobetti 93/2, I-40129 +Bologna, Italy +3 INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3, I-40129, Bologna, Italy +4 Collège de France, 11 Place Marcelin Berthelot, 75231 Paris, France +Received 21 July 2022 +ABSTRACT +There exists a well known relation between the mass of the supermassive black hole (SMBH) in the center of galaxies and their +bulge mass or central velocity dispersion. This suggests a co-evolution between SMBH and their galaxy hosts. Our aim is to study this +relation specifically for radio loud galaxies, and as a function of redshift z. We selected a sample of 42 radio galaxies and active galactic +nuclei (AGN) with broad emission lines and spectroscopic redshifts between z = 0.3 − 4, by cross-matching the low radio frequency +sources from VLA FIRST with spectroscopically confirmed galaxies from wide field surveys including SDSS DR14 ugriz and DES +DR2 grzY in optical, WISE in infrared, and the Galaxy And Mass Assembly (GAMA) spectroscopic survey. We characterised the +stellar mass (M⋆), star formation, and black hole properties (mass of the central SMBH, the Eddington ratio η and the jet power, Qjet). +The relation between SMBH mass, M⋆, η and z are put into context by comparing them with scaling relations (MBH–M⋆, MBH/M⋆–z, +MBH–Qjet and Qjet–η) from the literature. On the basis of a multi-wavelength spectral energy distribution modeling, our radio sources +are broadly consistent with being on the star-forming main sequence. They have sub-Eddington accretion rates, η ≃ 1% on average, +as typically found in Type I AGN, while higher accretion rates favor more powerful jets to be launched by the central engine. We find +the presence of overmassive SMBHs in (17 ± 5)% of our radio sources, similarly to previous studies on nearby early-type galaxies. +Altogether, an evolutionary scenario where radio-mode AGN feedback regulates the accretion onto the SMBHs and the stellar mass +assembly of the radio sources is discussed, which may explain the observed phenomenology. This pilot study represents a benchmark +for future ones using wide field surveys such as Euclid and the Vera Rubin telescope. +Key words. galaxies: active – galaxies: bulges – galaxies: nuclei – (galaxies) quasars: supermassive black holes – infrared: galaxies +– radio continuum: galaxies +1. Introduction +Super Massive Black Holes (SMBHs), characterized by masses +in the range ∼ 106 M⊙ to ∼ 1010 M⊙, are observed to lie at the +center of most, if not all, massive galaxies (e.g., Graham 2016). +When the central regions of galaxies are sources of radiation, +because of accretion onto their SMBHs, they are called Active +Galactic Nuclei (AGN). AGN are among the strongest proofs +for the existence of SMBHs, together with the direct measure +of compact densities in our Galactic center (e.g. Genzel et al. +2010), and the direct observation of the SMBH shadow at the +center of M87 and of the Milky Way itself (The Event Horizon +Telescope Collaboration et al. 2019, 2022). +There exists an intrinsic co-evolution between AGN activ- +ity, SMBH growth, galaxy stellar content and star formation his- +tory (e.g. Kormendy & Ho 2013, for a review). In some cases +AGN are jetted, and thus called radio-loud AGN. They consti- +tute only 10% of the whole AGN population, but their fraction +varies with the stellar mass of the host, from 0 to 30% (Best et al. +2005). Large-scale radio jets are even able to impact the global +Mpc-scale environmental properties, via radio-mode AGN feed- +back, as for example at the center of galaxy (proto-)clusters (e.g., +Miley & De Breuck 2008; Fabian 2012; Magliocchetti 2022, for +a review). +⋆ e-mail: remi.poitevineau@obspm.fr +The mode of SMBH accretion ultimately regulates the exci- +tation properties of radio-loud AGN. It is indeed possible to dis- +tinguish two main classes of activity among the radio-loud AGN, +HE (High excitation) and LE (Low Excitation) Radio Galaxies +(RG) according to their accretion rate: HERGs typically have ac- +cretion rates between 1 and 10% of their Eddington rate, whereas +LERGs predominately accrete at a rate below 1% Eddington +(Best & Heckman 2012). In the former case of HERGs, the ma- +terial is thus losing progressively angular momentum in a geo- +metrically thin disk around the SMBH; this disk is usually op- +tically thick and radiates efficiently. When the accretion rate is +below 0.01 the Eddington rate, the AGN is instead characterized +by an advection-dominated accretion flow (ADAF, e.g. Narayan +& McClintock 2008), which radiates inefficiently. Radio-loud +AGN are in majority in the low-luminosity regime, and fre- +quently ADAFs. +A major observational breakthrough for what concerns the +co-evolution of galaxies and AGN with their SMBHs was the +discovery of a tight correlation, in the local universe, between +the SMBH mass and the mass of their host spheroids (Magorrian +et al. 1998; Ferrarese & Merritt 2000). The existence of this re- +lation implies a remarkable connection between the assembly of +galaxies and the formation and growth of SMBHs at their center +(e.g., Heckman & Best 2014, for a review). Models and simu- +lations (Menci et al. 2006; Marulli et al. 2008; Hopkins et al. +2006; Volonteri & Natarajan 2009) have attempted to explain +this correlation and its evolution with redshift, as found in sev- +1 +arXiv:2301.05186v1 [astro-ph.GA] 12 Jan 2023 + +R. Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +eral observational studies (e.g. Shields et al. 2006; Sarria et al. +2010; Wang et al. 2010; Merloni et al. 2010; Jahnke et al. 2009; +Cisternas et al. 2011; Schramm & Silverman 2013). +There are, however, a number of still related open issues. +Among them there is the existence of local ellipticals with over- +massive SMBHs (Kormendy et al. 1997; van den Bosch et al. +2012; Savorgnan & Graham 2016; Dullo et al. 2021). These +over-massive SMBH occur preferentially in galaxy clusters, and +in brightest cluster galaxies in particular (BCGs, e.g., McConnell +& Ma 2013), where environment effects strip galaxies from their +gas, stop star formation and the growth of bulges. Galaxies are +then called massive relics, with particularly old stellar popu- +lation (Trujillo et al. 2014; Martín-Navarro et al. 2015; Ferré- +Mateu et al. 2015, 2017). The very discovery of massive SMBHs +(MBH ≳ 109 M⊙) in bright quasars at the epoch of reionization +(e.g., Bañados et al. 2018; Farina et al. 2022) is a mystery, as it +shows that extreme SMBHs can form within 1 Gyr after the Big +Bang. The rapid formation of such high-z SMBHs might be ex- +plained invoking some extreme scenarios such as the growth of +a 102−5 M⊙ seed via super-Eddington accretion (Valiante et al. +2016b; Pezzulli et al. 2017), the direct collapse of an initial gas +condensation into a black hole of ∼105 M⊙ (Visbal et al. 2014; +Regan et al. 2017), or the merger of massive proto-galaxies (e.g., +Mayer et al. 2010, 2015; Ferrara et al. 2013; Bonoli et al. 2014). +Altogether, while existing studies show a tight co-evolution +between SMBHs, AGN, and their host galaxies with cosmic +time, this interplay is still substantially debated and uncon- +strained. This is at least partially due to the difficulty in building +large samples of distant AGN with well characterized stellar and +black hole properties. +In order to better understand the growth of SMBHs with cos- +mic time, and their co-evolution with their host galaxies, in this +work we have therefore built a sample of distant radio-loud AGN +spanning about 9 Gyr of cosmic time, between z ∼ 0.3 − 4, with +available radio-to-ultraviolet spectro-photometric data. Thanks +to this multi-wavelength dataset we assess the properties of the +AGN sample in terms for example of black hole and stellar +masses, jet power, and Eddington ratio. As radio-loud AGN are +associated with the most massive black holes and host galaxies +(e.g., Best et al. 2005; Chiaberge & Marconi 2011; Shen et al. +2011; Shaw et al. 2012), they are excellent sources to investi- +gate the galaxy, AGN, and SMBH co-evolution at the high-mass +regime. +The paper is organized as follows. In Sect. 2 we describe +our sample selection as well as its multi-wavelength dataset and +properties. In Sect. 3 we report estimates for the black hole, jet, +accretion, and stellar properties. In Sect. 4 we describe our com- +parison sample. The results, in terms of black hole - jet - host +galaxy scaling relations are reported in Sect. 5. In Sect. 6 we +summarize the results and draw our conclusions. +Throughout this work, we adopt a flat ΛCDM cosmology +with matter density Ωm = 0.30, dark energy density ΩΛ = 0.70 +and Hubble constant h = H0/100 km s−1 Mpc−1 = 0.70. +2. The Radio-Loud AGN sample +We selected a sample of radio-loud AGN by cross-matching the +Very Large Array Faint Images of the Radio Sky at Twenty- +centimeters (VLA FIRST) source catalog (Becker et al. 1995) +with infrared-to-optical spectro-photometric surveys. As further +described in the following, the use of infrared-to-ultraviolet pho- +tometry enables the modeling of the spectral energy distribution +(SED), which allows us to ultimately obtain a good characteri- +sation of the galaxy properties such as the stellar mass (M⋆) and +the star formation rate (SFR). +2.1. The Dark Energy Survey +We start by considering the Dark Energy Survey (DES +Collaboration 2005; Dark Energy Survey Collaboration et al. +2016), which is composed of two distinct multi-band imaging +surveys: a ∼5000 deg2 wide-area grizY survey and a deep super- +nova griz survey made by six distinct deep fields (Hartley et al. +2021). The coadded source catalog and images from the process- +ing of all six years of DES wide-area survey observations and all +five years of DES supernova survey observations have been re- +cently made public with the DES data release 2 (Abbott et al. +2021)1. +To build our sample of distant radio-loud AGN we limit our- +selves to equatorial DES supernova fields that overlap with the +VLA FIRST survey at low radio frequencies. The selection is +similar to that of our previous work (Castignani et al. 2019), +which we refer for further details. However, in that study we +focused only on two radio sources, and we investigated their +molecular gas content, their cluster environment, as well as the +stellar and star formation properties. In this work we consider +instead a more extended sample, as further outlined in the fol- +lowing. +2.2. Radio, optical, and spectroscopic selection +As we are interested in building a sample of extra-galactic radio +sources we consider the VLA FIRST survey (Becker et al. 1995), +which observed 10,000 deg2 of the North and South Galactic +Caps at low radio frequencies (1.4 GHz), down to a point source +detection limit of ∼1 mJy. +We therefore further limit ourselves to the Stripe 82 area, +that is a 300 deg2 equatorial field that was imaged multiple times +by the Sloan Digital Sky Survey (SDSS) and overlaps with the +VLA FIRST survey. Similarly, for our search we additionally +considered DES supernova deep fields numbered 2, 3, and 5, as +outlined in Sect. 2.1 +We have cross-matched the low radio frequency VLA FIRST +radio source catalog with both the SDSS DR14 ugriz and DES +DR1 grizY source catalogs within the considered fields with a +search radius of 3 arcsec, consistently with the positional ac- +curacy ∼ 1 arcsec of FIRST sources. As we are interested in +secure distant radio sources, we further restrict ourselves to +those sources with SDSS DR14 spectroscopic redshifts z > 0.3. +The search yields 158 spectroscopically confirmed radio sources +with unique optical counterparts from both SDSS and DES. +2.3. Infrared selection: WISE +We further look for infrared emission of the radio sources, +as found by the W4 filter of the Wide-field Infrared Survey +Explorer (WISE, Wright et al. 2010). To this aim we have cross- +correlated our radio sources with the allWISE source catalog2 by +adopting a search radius of 6.5 arcsec, consistently with previ- +ous work on extra-galactic radio sources (e.g., Castignani et al. +2013). The search yields 154 sources with unique WISE coun- +terparts and W4 magnitudes with signal-to-noise ratio S/N > 1. +1 https://des.ncsa.illinois.edu/releases/dr2 +2 http://wise2.ipac.caltech.edu/docs/release/allwise/ +2 + +R. Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +2.4. Broad Emission Lines from SDSS +As we are interested in assessing black hole masses of the con- +sidered radio-loud AGN we further restrict ourselves to those +sources with evidence of broad emission lines in the SDSS spec- +tra. To this aim we have further selected those sources that +have Hα, Hβ, Mg ii, or C iv +emission line fluxes at signal- +to-noise ratio S/N> 3, as well as full width at half maximum +FWHM>1000 km s−1, typical of the broad line region lines. We +use the spZline file3 that contains the results of the emission-line +fits for the BOSS spectra of SDSS sources (Bolton et al. 2012). +The Gaussian line width σ is reported and we converted it into +the FWHM = 2σ +� +2 log 2. This additional spectroscopic selec- +tion yields 21 sources at z ∼ 0.3 − 3.8. +Fig. 1: Top. 1.4 GHz luminosity density as a function of red- +shift. Points are color coded according the WISE color-based +classification, as in the bottom panel. The horizontal line is at +Lν = 2 × 1032 erg s−1 Hz−1, and separates low luminosity ra- +dio sources from high luminosity radio sources. Bottom. WISE +color-color plot, where sources are distinguished between AGN +(circles), starbursts (triangles) +, disks and spheroids (squares) according to the color-based +classification by (Stern et al. 2012; Jarrett et al. 2017). +2.5. Radio-Loud AGN in GAMA DR3 +In addition to the radio-loud AGN selected as described in the +previous sub-sections, we searched for distant radio sources +from the third data release (DR) of the Galaxy And Mass +Assembly (GAMA) spectroscopic survey. GAMA DR3 (Baldry +et al. 2018) provides in fact spectra obtained with the AAOmega +3 https://data.sdss.org/datamodel/files/BOSS_SPECTRO_REDUX/ +RUN2D/PLATE4/RUN1D/spZline.html +multi-object spectrograph on the Anglo-Australian Telescope +(AAT) as well as a wealth of ancillary information for more than +200 thousands sources. +Similarly to what we did concerning the SDSS spectra, we +first selected 632 sources at z > 0.3, with available GAMA DR3 +spectra, Hα or Hβ line fluxes at S/N> 3 as well as line widths +FWHM> 1000 km s−1, as inferred from single Gaussian model- +ing.4 +We selected these sources within the equatorial area of +180 deg2 covered by GAMA DR3 as well as by the 1.4 GHz +FIRST VLA survey. Among the 632 spectroscopic sources from +GAMA, we further limited ourselves to the subsample of 39 +galaxies with available 1.4 GHz fluxes from the FIRST VLA +survey. Multiple spectra are often available in the GAMA DR3 +database. We then inspected each of the available spectra and +discarded those galaxies where the evidence of emission lines +was dubious, and thus the associated fits. This analysis yields +21 GAMA DR3 radio-sources at z ∼ 0.3 − 0.8. For them, we +assigned unique WISE counterparts using a 6.5 arcsec search +radius, as in Sect. 2.3. +By combining these galaxies, denoted hereafter with the pre- +fix G, with the 21 radio sources with SDSS spectra, denoted in- +stead with the prefix RS, our final sample comprises 42 sources +that we consider hereafter for the present study. The main prop- +erties of this sample of galaxies are listed in Tables A.1. +2.6. Radio and infrared properties +We now investigate the low frequency radio luminosities and the +infrared colors of our sources. To this aim, similarly to previous +studies (Condon 1989; Chiaberge et al. 2009; Castignani et al. +2014), we first assumed a power-law for the radio spectrum, i.e., +S ν ∝ ν−α, where S ν is the radio flux density at the observer +frequency ν and the spectral index α is fixed to α = 0.8. We then +converted the 1.4 GHz VLA radio fluxes S 1.4 GHz into rest frame +1.4 GHz luminosity densities as follows: +L1.4 GHz = 4π S 1.4 GHz DL(z)2 (1 + z)α−1 , +(1) +where DL is the luminosity distance. +Figure 1 (top) displays our sources in the L1.4 GHz vs. redshift +plane. They all have L1.4 GHz ≳ 3 × 1030 erg s−1 Hz−1 typical +of radio-loud AGN, while purely starburst galaxies have lower +L1.4 GHz < 1030 erg s−1 Hz−1 (Chiaberge et al. 2009). +Furthermore, the majority (71%, i.e., 30/42) of our sources +have high radio powers, greater than L1.4 GHz += +2 × +1032 erg s−1 Hz−1, which we use to distinguish between Low +Luminosity Radio Sources (LLRS) from High Luminosity Radio +Sources (HLRS), similarly to what has been done in previous +studies (e.g., Chiaberge et al. 2009; Castignani et al. 2014). As +the radio galaxy population has a bimodal distribution in radio +power, it is worth mentioning that the adopted LLRS/HLRS lu- +minosity threshold corresponds to the fiducial radio power which +fairly separates FR I from FR II radio galaxies (Fanaroff & Riley +1974; Zirbel 1996). Furthermore, as a result of the Malmquist +bias associated with the VLA FIRST flux limit of ∼ 1 mJy, the +fraction of HLRS increases with redshift and reaches unity at +z > 1. +Figure 1 (bottom) shows instead the sources in our sample in +the WISE color-color diagram, where sources are distinguished +using the color-based classification by Jarrett et al. (2017), as +highlighted in the Figure. Interestingly, our sample populates +4 http://www.gama-survey.org/dr3/ +3 + +High Luminosity Radio Sources +34 +zH +33 +s +[erg +Lv +32 +Log +31 +Low Luminosity Radio Sources +0.2 +0.3 +0.4 +0.5 +0.8 +1 +2 +3 +4 +5 +Z2.0 +AGN +1.5 +V2-W3[mag] +ULIRGs +1.0 +0.5 +0.0 +-0.4 +Spheroids +Intermediate disks +Starbursts & LIRGs +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +W1-W2[mag]R. Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +only three regions in the diagram. The majority (28/42) of our +sources are classified as AGN, based on WISE colors. This is +not surprising as they have been selected as distant and powerful +radio sources at z > 0.3. Based on WISE colors the remaining +sources are fairly equally distributed between the intermediate +disk (9/42) and starburst (5/42) classes. +Furthermore, as shown in Fig. 1 (top) the vast majority of +z > 1 sources have WISE infrared colors consistent with AGN +contribution. They also show high 1.4 GHz radio luminosities +typical of radio-loud quasars (QSOs). The majority (22/42, i.e., +52%) of our sources are in fact classified as quasars in the NED +database, e.g., with counterparts in the 2dF–SDSS LRG And +QSO (2SLAQ, Croom et al. 2009) catalog, or with X-ray con- +terparts (XMM, Rosen et al. 2016), as outlined in Table A.1. +3. Black hole, jet, accretion, and stellar properties +3.1. Black Hole masses +One of the main goals of this work is to investigate the co- +evolution of central black-holes with the host galaxies of the +radio-loud AGN in our sample. +To this aim we estimated black hole masses using the widely +used Single Epoch (SE) method, that is particularly suited for +distant Type 1 AGN. According to this method, black hole +masses MBH can be estimated under the assumption that the +Broad Line Region (BLR) is in virial equilibrium, as follows: +MBH = f RBLR∆V2 +G +, +(2) +where RBLR is the BLR radius, ∆V is the velocity of the BLR +clouds that can be estimated from the broad emission line width, +f is the virial coefficient that depends on the geometry and kine- +matics of the BLR, and G is the gravitational constant. The SE +method then uses the relation that exists between the BLR size +and the AGN optical/ultraviolet continuum luminosity empiri- +cally found from reverberation mapping (Peterson et al. 2004; +Kaspi et al. 2007; Bentz et al. 2009), as well as the tight correla- +tion between the continuum luminosity and that of broad emis- +sion lines (e.g. Shen et al. 2011). With these considerations, the +black hole mass can be expressed as: +log +� MBH +M⊙ +� += a + b log +� +L +1044 erg s−1 +� ++ c log +�FWHM +km s−1 +� +, +(3) +where the coefficients a, b, and c are empirically calibrated +against local AGNs with reverberation mapping masses or using +different lines. L and FWHM are the line luminosity and width. +In this work we use the coefficients obtained for Hα, Hβ, Mg ii, +and C iv broad emission lines by Shen et al. (2011) and Shaw +et al. (2012). These are widely used lines which are redshifted +in the optical domain, depending on the redshift of the source. +These lines indeed enable estimates of black hole masses over a +wide range of redhifts. Similarly to previous studies (Shaw et al. +2012; Castignani et al. 2013), we used Hα, Hβ, and Mg ii for +sources at z < 1, and to the Mg ii and C iv lines for sources at +higher redshifts. In the case where multiple broad emission lines +are available for a given sources, we adopted the following order +of preference: Hα, Hβ, Mg ii, and C iv(see e.g.Shen & Liu (2012) +for more details). In Table 1 we report the coefficients used in +this work, while in Table A.2 we list the black hole masses. +a +b +c +Hα +1.24 +0.43 +2.1 +Hβ +1.63 +0.49 +2.0 +Mg ii +1.70 +0.63 +2.0 +C iv +1.52 +0.46 +2.0 +Table 1: Coefficients to estimate black hole masses using broad +emission lines and Eq. 3. Values for Hβ, MgII, and CIV lines +are from Shaw et al. (2012). The coefficients for Hα come from +Shen et al. (2011) +3.2. Jet Power +The sources in our sample are radio-loud AGN that are typi- +cally characterized by jetted outflows which strongly emit at ra- +dio wavelengths mainly via synchrotron emission. +By studying jet properties such as its total power we will +investigate the complex interplay between the jet, the black hole, +and the gas accretion on it, which is commonly referred to as +radio-mode AGN feedback (e.g., Fabian 2012, for a review). +Following previous work by Willott et al. (1999), we esti- +mate the jet power as: +Q jet = 3 × 1045 ξ3/2 +������ +L151 MHz +1035 erg s−1 Hz−1 sr−1 +������ +6/7 +erg s−1 , +(4) +where L151 MHz is the extended total radio luminosity density +at 151 MHz in the rest frame, and ξ is a factor ranging be- +tween 10 and 20. In this work, we used an intermediate value +ξ = 15. To estimate L151 MHz we extrapolated the L1.4 GHz lumi- +nosity densities assuming an isotropic emission and a power law +with α = 0.8, as further described in Sect. 2.6. The resulting jet +powers are reported in Table A.2. +3.3. Eddington ratio +As we want to link the gas accretion onto the black hole with the +AGN properties we estimated the Eddington ratio, that is defined +as +η = Ldisc +LEdd +, +(5) +where Ldisc and LEdd are the disc and Eddington luminosities. +The latter can be expressed as: +LEdd = 1.26 × 1038 +� MBH +M⊙ +� +erg s−1 . +(6) +To estimate the disc luminosity, we followed instead the pre- +scriptions described in Celotti et al. (1997). First, we assume that +BLR contributes to ∼ 10% of the total disc luminosity, that is +Ldisc ≃ 10 LBLR. To estimate the BLR luminosity, we then used +the line ratios reported in Francis et al. (1991), which are typical +line luminosities of bright QSOs, normalized to that of the Lyα +emission. The BLR luminosity can therefore be estimated as : +LBLR =< L∗ +BLR > +� +i Li,obs +� +i L∗ +i,est +, +(7) +where Li,obs is the luminosity of the observed i-th line in the BLR +and L∗ +i,est is the line ratio of the i-th line presented in Francis et al. +(1991) table. With these prescriptions, the total normalized BLR +luminosity is equal to < L∗ +BLR >= L∗ +Hα + L∗ +Hβ + L∗ +C iv + L∗ +Mg ii + +4 + +R. Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +Fig. 2: Examples of spectral Energy Distributions of the radio sources in our sample with black hole mass estimates. Data-points are +from GALEX (brown dots), SDSS (red pentagons), DES (blue squares), WISE (green triangles), and IRAS (yellow upper limits). +Dashed and solid lines are the best fit models for the stellar and dust components, respectively. +L∗ +Lyα + L∗ +Lyβ + L∗ +Hγ + L∗ +Al III + L∗ +Si IV + L∗ +C II + L∗ +O I = 390.3, where +L∗ +Lyα = 100 is fixed as a reference. +The resulting Eddington ratios are reported in Table A.2. +They are mostly in the range log η ∼ (−4; −1), with a median += -1.9, as typically found for Type 1 radio-loud AGN, but lower +than those of Type 2 quasars (e.g., Castignani et al. 2013; Kong +& Ho 2018). +3.4. SED modeling +The radio sources in our sample have a broad multi-wavelength +photometric coverage, from the ultraviolet (UV) to the infrared +(IR), which enables the determination of stellar masses and star +formation rate (SFR) estimates via Spectral Energy Distribution +(SED) modeling. +For the GAMA sources in our sample we considered the +SED fits by Driver et al. (2018) performed with MAGPHYS +(da Cunha et al. 2008). Photometric data include GALEX +(Martin et al. 2005; Morrissey et al. 2007) in the UV, SDSS +(York et al. 2000) in optical, as well as VISTA Kilo-degree +Infrared Galaxy Survey (VIKING, Edge et al. 2013), WISE +(Wright et al. 2010), and Herschel- ATLAS (Eales et al. 2010; +Valiante et al. 2016a) in the near- to far-IR. +For the sources in the DES SN deep fields available pho- +tometry includes GALEX in the UV, ugriz (SDSS) and grizY +(DES) magnitudes in the optical, WISE data in the near-IR, as +well as IRAS upper limits in the far-IR, that we gathered as in +Castignani et al. (2019), which we refer for further details. In +this previous work we followed-up in molecular gas two radio +sources in dense Mpc-scale environments at z = 0.4 and z = 0.6 +within the DES SN deep fields, while in the present study we +enlarge the sample to investigate the co-evolution of black holes +with the radio sources. +Analogously to Castignani et al. (2019), we then performed +fits to the SEDs using LePhare (Arnouts et al. 1999; Ilbert et al. +2006). Following the prescriptions provided for the LePhare +code, we fitted the far-IR data separately to account for possi- +ble dust emission, using the Chary & Elbaz (2001) library con- +sisting of 105 templates. The remaining photometric data-points +at shorter wavelengths were fitted using the CE_NEW_MOD li- +brary that consists of 66 galaxy templates based on linear inter- +polation of the four original SEDs of Coleman et al. (1980). We +then converted the rest frame (8.0-1000) µm infrared (dust) lu- +5 + +Log(Λe/A) +3 +4 +5 +6 +7 +-12 +RS 81 (z=0.779) +G- +-14 +XXL-N 062 013 +-16 +18 +-20 +-22 +3 +4 +5 +6 +7 +Log(Λobs/A)Log(Λe/A) +3 +4 +5 +6 +7 +-12 +RS 83 (z=0.682) +G- +-14 +PMN J0225-0536 +-16 +-18 +-20 +-22 +3 +4 +5 +6 +7 +Log(Λobs/A)Log(Λe/A) +3 +4 +5 +6 +-12 +RS 113 (z=2.082) +[G- +-14 +2SLAQJ024531.53-002612.2 +-16 +-18 +-20 +-22 +3 +4 +5 +6 +7 +Log(Λobs/A)Log(入e/A) +3 +4 +5 +6 +7 +-12 +RS 237 (z=0.953) +-14 +2SLAQJ024923.20-005437.7 +-16 +-18 +20 +-22 +3 +4 +5 +6 +7 +Log(Λobs/A)R. Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +Fig. 3: SFR versus stellar mass plot. Sources are color-coded ac- +cording to their redshift, while the different symbols correspond +to the different WISE classes (AGN as circles, starbursts as pen- +tagons, squares for the rest), as in Fig. 1. Upper limits to the +SFRs are indicated with arrows. The diagonal lines correspond +to the MS model prescriptions by Speagle et al. (2014) at z = 0.2, +1, and 2. The red cross at the bottom right shows the typical un- +certainties of ∼0.3 dex for both SFRs and stellar masses. +minosity into an SFR estimate by using the Kennicutt (1998) re- +lation, calibrated to a Chabrier (2003) initial mass function. The +SEDs of four of our radio sources are shown in Fig. 2. RS 81 +and 83 have prominent elliptical type emission in the optical do- +main, while their IR emission is consistent with dust emission +due to star formation. They are indeed classified as intermediate +disks based on WISE colors. RS 113 and 237 are WISE AGN +and show steep SEDs at near-IR-to-optical wavelengths, which +suggest that the emission is contaminated by non-thermal AGN +emission. +3.5. Star formation rate vs. stellar mass +Figure 3 displays the radio sources of our sample in the star for- +mation rate (SFR) versus stellar mass (M⋆) plane, resulting from +the SED fits. The sources are color-coded according to the red- +shift, while the different symbols correspond to the WISE clas- +sification. +Overall, the sources are massive, with log(M⋆/M⊙) ≃ 10.3− +12.0 (median=10.1), which is in agreement with being radio- +loud AGN, that are indeed typically hosted by massive ellipticals +(Best et al. 2005). Our galaxies also lie mostly along the star +forming main sequence (MS), although with a large scatter. The +mean specific SFR is sSFR=0.40±0.44 Gyr−1, where the root +mean square (rms) dispersion is reported as uncertainty. +Galaxies at higher redshifts tend to have higher SFRs, in +agreement with the MS model prescriptions (Speagle et al. +2014). However, as highlighted in Sect. 2.6, the fraction of +sources with AGN contamination also increases with redshift, +which may result in biased-high SFRs. The latter may be the +case where the optical-IR SED is steep, and thus the IR emission +is likely dominated by the AGN contribution, more than star for- +mation. To overcome this limitation, we conservatively reconsid- +ered the SFR estimates and assigned upper limits when the SFRs +largely exceed 100 M⋆/yr or in the cases of steep spectrum SEDs +(e.g., as for RS 113 and 237 mentioned above). Namely, we con- +sidered as steep spectra those AGN for which their optical-IR +SED has a characteristic power-law behavior Fλ ∝ λ−1. We ver- +ified a posteriori that these radio sources are indeed mostly lo- +cated in the upper part of the MS and are classified as WISE +AGN. +4. Comparison Sample +To put the AGN in our sample into a context, we additionally +considered a compilation of sources with available black hole +and stellar mass estimates: +• 30 nearby galaxies from Häring & Rix (2004). Galaxy +masses were derived by the authors through Jeans equation +modeling or adopted from dynamical models in the litera- +ture, while black hole masses are from Tremaine et al. (2002) +and references therein. +• 35 nearby galaxies from sample of McConnell & Ma (2013), +who expanded and revised available galaxy bulge masses and +dynamical measurements of black hole masses. +• 32 Type 1 AGN at z = 0.3 − 0.9 from Cisternas et al. (2011), +drawn from the XMM-COSMOS survey. Available stellar +masses are based on the modeling of HST images, taking +into account both AGN and host galaxy contributions, while +black hole masses are from Hβ (Trump et al. 2009). +• 18 broad-line X-ray AGN 0.5 +< +z +< +1.2 in the +Extended Chandra Deep Field-South Survey from Schramm +& Silverman (2013) who estimated Mg ii-based black hole +masses, as well as HST color-based stellar mass estimates. +• 78 radio-quite Type 1 AGN at z ≃ 1 − 2 from the COSMOS +survey Merloni et al. (2010). Stellar masses were determined +via SED fitting, while black hole masses are based on the +Mg ii emission lines of VIMOS/VLT spectra. +• 10 Type 1 AGN at 1 < z < 2 in COSMOS, from Jahnke +et al. (2009), who estimated HST color- based stellar masses, +while virial black hole masses come from the spectroscopic +COSMOS Magellan/IMACS and zCOSMOS surveys. +• 53 radio-quiet QSOs at z +< +3 from Decarli et al. +(2010a,b). Virial black hole masses come from Hβ, Mg ii, +and C iv emission lines, while stellar masses have been esti- +mated by the authors assuming a stellar R-band mass-to-light +ratio. +• Two SDSS luminous quasars at z +∼ +4 from Targett +et al. (2012). Virial black hole mass estimates come from +C iv emission, while stellar masses were estimated on the +basis of Bruzual & Charlot (2003) evolutionary synthesis +models. +• 9 distant z ∼ 1.4−6.4 QSOs from Shields et al. (2006). Black +hole masses were derived from broad emission lines, while +they used CO emission line widths to infer the dynamical +bulge masses. +• The 7 QSOs at z ≃ 6 from Wang et al. (2010). They cal- +culated the stellar mass as the difference between the bulge +dynamical mass and the CO molecular gas mass. For these +QSOs, we used the black hole masses adopted by the authors +and estimated using the AGN continuum luminosity (Jiang +et al. 2006; Wang et al. 2008). +In addition to the sources listed above, a second group of +galaxies that we use as a comparison is composed by powerful +AGN with available estimates of the black hole mass, the jet +power, and the Eddington ratio: +• 44 radio-loud AGN studied in Le et al. (2018), at redshifts +z < 0.2, thus lower than those of the radio sources in our +sample. These sources have estimates of the jet powers (Le +et al. 2018) and of their black hole masses (Allen et al. 2006; +Balmaverde et al. 2008). +6 + +Z +4 +3.5 +3.0 +2.5 +3 +2.0 +log[SFR/(M o /yr)] +z=1 +1.5 +z=0.2 +1.0 +0 +0.5 +10.4 +10.6 +10.8 +11.0 +11.2 +11.4 +11.6 +11.8 +12.0 +log(M* /M o)R. Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +• 208 γ-ray Fermi blazars at 0 < z < 3.1 from Xiong & Zhang +(2014). Virial black-hole mass estimates mostly come from +different broad emission lines, and the rest from scaling rela- +tions. Jet powers Qjet are mostly from Nemmen et al. (2012), +and were estimated using the correlation between the ex- +tended radio emission and the jet power. Alternatively, Xiong +& Zhang (2014) calculated Qjet using the scaling relation +provided by Nemmen et al. (2012) between the γ-ray lumi- +nosity and the kinetic power. +• 146 radio-loud QSOs at 0.1 < z < 2.5, from Liu et al. (2006), +classified as flat spectrum (54%) or steep spectrum radio +quasars (46%). The black hole virial masses come from Hβ, +Mg ii, or C iv emission lines. The jet power were calculated +by the authors using low-frequency radio emission, follow- +ing Punsly (2005). +These sources outlined above are radio-loud AGN. However +we verified that none of them is in our sample. While these stud- +ies investigated the black hole and jet properties of large sam- +ples of radio sources, they did not characterize their infrared- +to-optical SEDs, as done here for our smaller sample of radio +sources. +Fig. 4: Black hole vs. stellar mass scatter plot. Filled red dots +correspond to our sample of radio sources, while those in com- +parison sample are shown as open symbols. Scaling relations +are overlaid (Sani et al. 2011; DeGraf et al. 2015; Häring & +Rix 2004). The legend at the right displays the adopted color +code. The red cross at the top left shows the typical uncertain- +ties ∼0.4 dex and ∼0.3 dex for the black hole and stellar masses, +respectively. +5. Results +In this section we report different scaling relations for the radio +sources in our sample including black hole and stellar masses, jet +powers, Eddington ratios, and the redshifts. We also include as a +comparison the sources from the literature as outlined in Sect. 4 +as well as scaling relations derived in previous studies. +5.1. Black hole versus stellar mass relation +We start by considering black hole and stellar masses and their +relative evolution with redshift. Figure 4 displays the black hole +mass (MBH) versus the stellar mass (M⋆). Interestingly, our +radio-loud sources nicely follow the trend previously observed +for different samples of both local galaxies and distant AGN, +overplotted in the Figure, as well as those inferred by the scaling +relations, which also reported (Sani et al. 2011; Häring & Rix +2004; DeGraf et al. 2015). +In particular, our sources densely populate the high +log(MBH/M⊙) ≃ 7.1 − 10.0 and high log(M⋆/M⊙) ≃ 10.2 − 12.0 +region in Fig. 4, which is in agreement with the fact that radio- +loud AGN are almost invariably associated with the most mas- +sive galaxies and black holes. Interestingly, a substantial frac- +tion of our sources 9/42 (i.e., 21%) have black hole masses +log(MBH/M⊙) > 9 well above the scaling relations for both local +(Häring & Rix 2004; Sani et al. 2011) and distant sources at the +median resdhift z = 0.6 of our sample (DeGraf et al. 2015). This +behaviour suggests that the growth of black hole masses in radio +loud AGN largely occurs at early z > 1 epochs, while the early +stellar mass assembly may not be equally effective. Substantial +growth of the stellar mass may take place even at lower red- +shifts, in order to flatten the observed MBH-M⋆ scaling relation +by z = 0. Previous studies indeed suggested that massive ellip- +ticals may indeed double their stellar mass between z = 1 and +z = 0 (Ilbert et al. 2010; Lidman et al. 2012). +We further investigate this evolutionary scenario in Fig. 5, +which shows the MBH/M⋆ ratio as a function of redshift. The +large majority (36/42, i.e., 86%) of our radio sources have +MBH/M⋆ ratios which are similar to those of AGN in the com- +parison sample at similar redshifts, and are in agreement with +model prescriptions by McLure et al. (2006), which are over- +plotted as dashed lines in Fig. 5. +It is worth mentioning that our sample of radio sources is +flux limited. However, we expect the Malmquist bias to have a +marginal impact on the (MBH/M⋆) ratio. Indeed, both MBH and +M⋆ scale with the BLR line and the infrared-to-optical lumi- +nosities, respectively, and therefore have a similar dependence +on redshift, via the luminosity distance. Furthermore, as shown +in Fig. 5, at fixed redshift the MBH/M⋆ ratios of both our ra- +dio sources and those in the comparison sample span a broad +range. Similarly, when plotting M⋆ and MBH against redshift, +separately, we did not find any clear trend, as indeed the points +are scattered, at fixed redshift. These findings suggest that any +possible Malmquist bias likely has a sub dominant effect. +There are, however, five clear outliers among our radio +sources. RS 197 at the highest redshift z = 3.79 has a low +log(MBH/M⋆) = −3.2, well below the expected range of val- +ues, according to the McLure et al. (2006) model prescription. +Furthermore, a substantial fraction (5 sources, i.e., 12%) of our +radio loud AGN, namely RS 214, RG 237, G 537618, G 721940, +and G 746605, at redshifts between z ∼ 0.37 − 0.95, have high +log MBH/M⋆ ratios in the range ∼ (−1.69; −1.0), well above the +model predictions displayed in Fig. 5, as well as higher than +the ratios found in AGN at similar redshifts. Indeed, in the red- +shift range z ∼ 0.3 − 3.8 spanned by our radio sources, there +are only 3/186 (i.e. 1.6%) AGN with log MBH/M⋆ > −1.69 +in our comparison sample, while the proportion is significantly +higher (12%) for our radio-loud AGN. These results suggest that +7 + +11 +10 - +Sani et al.(2011) +口 +9 +Haring & Rix(2004) +DeGraf et al.(2015) +Haring & Rix(2004) +Shields et al.(2006) +Wang et al.(2010) +0 +Cisternas et al.(2010) +口 +Merloni et al.(2010) +8 +0 + Jahnke et al.(2009) +口 +Schramm & Silverman(2013) +Decarli et al. (2010ab) +McConnell et al.(2013) +Targett et al.(2012) +Nesvadba et al.(2010) +This work +10 +11 +12R. Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +Fig. 5: Black hole to stellar mass ratio as a function of redshift for our sample of radio sources (filled red dots) and for the galaxies +in our comparison sample (open symbols). The dashed black lines correspond to the evolutionary model described by McLure et al. +(2006), along with the associated 1σ uncertainty. The color coding is reported in the legend at the bottom right. The error bars to +the left of the legend show the typical ∼ 0.5 dex uncertainty in log(MBH/M⋆). +a non-negligible fraction of radio-loud AGN may experience a +different stellar mass assembly path than radio-quiet AGN. We +stress that these five radio sources are a subsample of the 9 out- +liers of Fig. 4, discussed above, and have high S/N line fluxes +in Hβ or Mg ii, which yielded robust MBH estimates. The only +exception is represented by G 721940, for which the Hβ emis- +sion and associated FWHM are at lower S/N∼ 2, as highlighted +in Table A.2. +The excess of overmassive SMBHs in radio-loud AGN sug- +gests that their stellar and SMBH mass built up is regulated by +their large scale radio jets. A possible scenario is that SMBHs of +the sub-population with high MBH/M⋆ are mature, that is, their +mass has been effectively assembled already by redshift z = 1, +via accretion (e.g. Delvecchio et al. 2018). On the other hand +their stellar mass growth may have not occurred as effectively +as in the overall AGN population, plausibly because of radio- +mode AGN feedback (Fabian 2012). While the accretion of hot +gas onto the SMBH sustains the AGN activity and the SMBH +growth, the large scale radio jets may prevent accretion and cool- +ing of the inter-galactic medium gas, which is ultimately respon- +sible for the stellar mass assembly. Altogether, we suggest that +radio-mode AGN feedback results in the observed high values +for MBH/M⋆ in radio-loud AGN. +In order to investigate further this scenario, in the next +Sections we link accretion and jet properties to the black hole +mass by considering both the jet power and Eddington ratio +of our radio sources. We stress that the usual MBH − Mbulge or +MBH − σ relations typically refer to the central spheroid, and not +to the total stellar mass (e.g., Kormendy & Ho 2013). However, +our radio-loud AGN sample is composed in a large majority of +early-type galaxies, where the spheroid constitutes most of the +stellar mass, and this approximation is justified. Furthermore, +because of the potential AGN contamination to the SED, the stel- +lar mass may be biased high. This implies that MBH/M⋆ ratios +can be even higher than reported. By considering MBH/M⋆ ra- +tios as lower limits we would have an even stronger discrepancy, +in particular for the subsample of high MBH/M⋆ radio sources +mentioned above, with respect to the model prescriptions and the +comparison sample of distant AGN, at fixed resdshift. All these +results seem to corroborate the scenario that SMBH growth is +more rapid than stellar mass assembly, and this is particularly +true for distant radio sources, in comparison to the overall AGN +population. +5.2. Jet power, black hole mass, and accretion +As mentioned in the previous sections, mechanical radio-mode +AGN feedback can regulate the cooling of hot gas in the inter- +galactic medium, and thus the stellar mass growth of the host +galaxy itself as well as the accretion onto the central SMBH. +To better understand the interplay between jet, black hole, and +accretion properties, in Fig. 6 we show the jet power Qjet (see +Sect. 3.2), plotted against the black hole mass MBH. The ra- +dio sources of our sample are highlighted, while we also over- +plot the comparison sources outlined in Sect. 4 (Liu et al. 2006; +Balmaverde et al. 2008; Xiong & Zhang 2014; Chen et al. 2015). +Our radio-loud AGN densely populate the upper right region +of the Qjet-MBH plane, which is occupied by sources with high +values of both the black hole mass (MBH ≳ 108M⊙) and the jet +power (Qjet ≳ 1043 erg s−1). Sources in the comparison sam- +ple similarly occupy this region, while they extend as well to +lower values of MBH (Liu et al. 2006; Xiong & Zhang 2014) and +jet power (Balmaverde et al. 2008). These results suggest that +the distant radio-loud AGN, quite independently of the redshift, +are almost invariably associated with massive black holes and +powerful radio jets. This is in agreement with the tight connec- +tion existing between black hole accretion and jet production in +powerful radio-loud AGN (e.g., Ghisellini et al. 2014; Sbarrato +et al. 2014; Inoue et al. 2017). +Furthermore, HLRSs are characterized by a jet power that +is typically higher with respect to LLRSs. As discussed in +Sect. 2.6, these two classes have indeed 1.4 GHz rest frame +luminosity densities typical of FR I and FR II radio galaxies, +respectively. As Qjet increases with the radio luminosity den- +sity (Eq. 4), high luminosity radio sources have higher Qjet val- +ues than low luminosity ones. Furthermore, the two classes of +LLRGs and HLRGs are also delimited in the Qjet-MBH plane by +the relation found in previous studies (Wu & Cao 2008; Chen +8 + +-0.5 +-1.0 +V +-1.5 +-2.0 +)60/ +中 +口 +3.0 +0 +VO +-3.5 - +★ +★ +McLure et al.(2006) +口 +Merloni et al. (2010) +☆ +McConnell et al.(2013) +-4.0 - +Haring & Rix(2004) +Jahnke et al. (2009) +Targett et al. (2012) +Shields et al. (2006) +Schramm & Silverman (2013) +Nesvadba et al. (2010) +Wang et al. (2010) +Decarli et al. (2010ab) +This work +Cisternas et al. (2010) +4.5 +0 +3 +4 +5 +6 +ZR. Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +et al. 2015), originally used to distinguish between FR I and +FR II radio galaxy populations. The clear separation of LLRGs +and HLRGs in the Qjet vs. MBH plane can be explained by com- +bining both the MBH vs. Mbulge relation for elliptical galaxies and +the relation between Qjet and the host galaxy optical luminosity +(Ledlow & Owen 1996) that separates the FR I and FR II radio +galaxies. The combination of these two relations also implies the +observed spread of our sources in Fig. 6. Indeed, we did not find +any significant correlation (as measured with the Spearman test) +between Qjet and MBH for our radio sources. +Figure 7 displays instead the jet power, plotted against the +Eddington ratio η (see Sect. 3.3) for both our radio sources and +the galaxies in the comparison sample (Xiong & Zhang 2014; +Liu et al. 2006). Higher accretion rates favor more powerful jets +to be launched by the central engine, as indeed the jet power +increases with increasing Eddington ratio. For our sample of ra- +dio sources we find that the two quantities are well correlated +at a level of 2.9-σ (p − value = 3.30 × 10−3), by means of the +Spearman test. No clear distinction in terms of η is found be- +tween the two classes of low and high luminosity radio sources, +which are distinguished in Fig. 7. However, as pointed out in +Sect. 3.3 our radio sources have, on average, an Eddington ra- +tio of log η = −1.9. This value is typical of radiatively effi- +cient accretion disks, such as the Shakura & Sunyaev (1973) +optically thick and geometrically thin accretion disk, which is +commonly invoked to explain the optical-ultraviolet emission in +Type I AGN (e.g., Ghisellini et al. 2010; Castignani et al. 2013). +We can then estimate the accretion rate ˙M = Ldisc/(ϵ c2), +where ϵ is the mass-to-light conversion efficiency for which we +adopt the standard value ϵ = 0.1, typical of radiatively efficient +disks. For our radio sources we obtain a median (mean) accre- +tion rate of 0.16 M⊙ yr−1 (0.6 M⊙ yr−1), which corresponds to +a substantial SMBH mass growth of ∆MBH = 1.6 × 106 M⊙, +(6.0 × 106 M⊙), over an AGN duty cycle with typical duration of +∼ 107 yr. +Altogether, the fact that the SMBHs of the radio sources in +our sample accrete at a sub-Edddington rate, irrespectively of +their redshift, suggests that most of their mass has been likely +built up at earlier epochs. Furthermore, while on one hand the +observed accretion state sustains both the nuclear activity and +the SMBH growth at sub-parsec scales, on the other hand it also +ultimately favors the persistence of large scale radio jets, which +may prevent the host galaxy to accrete gas at kilo-parsec scales +and thus form stars effectively. This radio-mode AGN feedback +may be responsible for the presence of overmassive SMBHs in +our sample of radio-loud AGN. Indeed, it is worth mentioning +that the five z ∼ 0.37 − 0.95 radio sources with high MBH/M⋆ +ratios, discussed in Sect. 5.1, accrete a sub-Eddington rate of +η ∼ 1%, while, on average, they have a normal jet power Qjet ∼ +1044 erg s−1. +5.3. Radio-loud AGN and their environments +A substantial fraction (17 ± 5)% of our radio sources have high +MBH/M⋆ ratios (Sect. 5.1), and may be early type galaxies at +the center of clusters. For these galaxies the stellar mass as- +sembly may have been halted, with reduced star formation ac- +tivity, as typically found in cluster core ellipticals, while their +black holes continue to grow via accretion. This interpretation +is consistent with earlier studies (e.g., Trujillo et al. 2014) as +well as with the substantial fraction (19%) of radio sources in +our sample with low SFR < 5 M⊙/yr, while many others have +SFR upper limits. It is indeed known that cluster core early type +Fig. 6: Jet power versus black hole mass. The diagonal dashed +region represents the model reported found in previous studies +(Wu & Cao 2008; Chen et al. 2015) that distinguishes between +FRI and FRII radio galaxies. Sources from our sample are dis- +tinguished between high luminosity and low-luminosity radio +sources. Sources from our comparison sample are also shown. +We refer to the legend for the color code adopted. +Fig. 7: Plot of the jet power versus the Eddington ratio (η) of the +associated black hole. In addition to our sample the data found +in Xiong & Zhang (2014) and Liu et al. (2006) are plotted. +galaxies tend to be outliers in the MBH vs Mbulge relation (e.g., +McConnell & Ma 2013). A famous example of a possibly over- +massive black hole is the case of NGC 1277, a lenticular galaxy +in the Perseus cluster (e.g., van den Bosch et al. 2012; Emsellem +2013; Scharwächter et al. 2016). +Motivated by these studies we looked in the literature +for clusters around the radio sources in our sample, search- +ing by coordinates in the NASA/IPAC Extragalactic Database +(NED). NED includes several catalogs of clusters identified +in wide field surveys (Goto et al. 2002; Koester et al. 2007; +McConnachie et al. 2009; Hao et al. 2010; Durret et al. 2011, +2015; Wen et al. 2012; Radovich et al. 2017; Rykoff et al. +2012; Oguri et al. 2018). Our search yielded three matches. +Radio sources RS 49, G 372455, and G 748815 are in the +cores (at cluster-centric distances ≲ 0.5 Mpc) of the clusters +[LIK2015] J034.16359-04.73395 (z = 0.89, Lee et al. 2015), +WHL J090325.6+011215 (z = 0.31, Wen et al. 2012; Wen +& Han 2015), and HSCS J142538+002320 (z = 0.33, Oguri +et al. 2018), respectively. These clusters have redshifts that are +9 + +47 +☆★ +☆☆☆ +☆ +☆ +☆ +口 +☆ +46 +☆ +☆ +口 +口 +口 +口 +口 +中 +中 +口 +口 +45 +☆ +M +☆ +电 +☆ +Qjet +erg/s +口 +Φ +中 +口 +口 +☆ +☆ +★ +)60/ +口 +44 +口 +43 +☆ +★ +8 +Chen et al. (2015) +口 +Xiong & Zhang (2014) +42 +Liu et al. (2006) +Balmaverde et al.(2008) +This work(High Radio Luminosity) +Thiswork(LowRadio Luminosity) +41 +7 +8 +MBH +9 +10 +11 +10 +MoXiong & Zhang (2014) +47 +☆ +Liu et al. (2006) +☆ +★ +☆☆ +This work(Low Radio Luminosity) +★ +This work(High Radio Luminosity) +★ +★ +★ +46 +★★ +口 +口 +口 +口 +'erg/s +口 +45 +口 +口 +)60/ +口 +☆ +口 +口 +口 +★ +口 +44 +8 +: +口 +☆ +口口 +43 +42 +-6 +-4 +0 +2 +log(n)R. Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +consistent with those of the radio sources as well as richness- +based masses M200 ∼ (0.9 − 3.0) × 1014 M⊙. These are there- +fore moderately massive clusters at intermediate-to-high red- +shifts. The three associated radio sources have instead moder- +ately overmassive black holes log(MBH/M⊙) ≃ 8.6 − 9.3, in +particular in comparison to the stellar masses of the systems +−2.65 ≲ log(MBH/M⋆) ≲ −2.24 (see Fig. 5). +These results support the above mentioned interpretation that +the cluster environments tend to prevent the stellar mass assem- +bly of cluster early-type galaxies, resulting in observed over- +massive black holes. Nevertheless, it is worth mentioning that +only three radio sources of our sample are found in clusters. +However, this is not surprising as clusters at higher redshifts +(z ≳ 1) or with lower masses M200 ≲ 1 × 1014 M⊙ typical of +rich groups are more difficult to find with current surveys and +observational facilities. It is thus possible that additional galax- +ies are hosted in clusters, as distant radio sources are often found +in dense mega-parsec scale environments (e.g. Galametz et al. +2012; Castignani et al. 2014; Malavasi et al. 2015; Golden-Marx +et al. 2019; Moravec et al. 2020). +6. Discussion and Conclusions +In this work we have investigated the evolution of distant radio- +loud active galactic nuclei (AGN), as well their co-evolution +with their host galaxies and their super massive black holes +(SMBHs) at their center. To this aim we have built a sam- +ple of 42 radio-loud AGN, with spectroscopic redshift between +z ∼ 0.3 − 3.8, by cross matching the 1.4 GHz VLA FIRST +point source catalog with available infrared-to-optical spectro- +photometric surveys including SDSS and DES in optical, WISE +in infrared, and the Galaxy And Mass Assembly (GAMA) spec- +troscopic survey. As we are interested in assessing the SMBH +masses, the 42 galaxies have been further selected by requiring +broad emission lines in Hα, Hβ, Mg ii, or C iv, with full width at +half maximum FWHM > 1000 km s−1. Thanks to the available +multi-wavelength photometry we modeled the spectral energy +distributions (SEDs) of the sources in the sample, and then de- +rived estimates to the stellar mass (M⋆) and the star formation +rate (SFR) for all sources, while for GAMA sources we took +them from the literature. We find that the 42 radio sources are +broadly consistent with the star forming main sequence. +For all sources we then estimated i) the black hole mass MBH, +based on single-epoch broad-line region spectra, ii) the black +hole to stellar mass ratio MBH/M⋆, iii) the jet power Qjet, on +the basis of the low frequency radio continuum emission, and +iv) the Eddington ratio η. Although samples of distant AGN +with SMBH mass estimates are rapidly growing (e.g., Shen et al. +2011; Shaw et al. 2012; Dabhade et al. 2020; Rakshit et al. 2020; +Gloudemans et al. 2021; Li et al. 2022), the present study still +represents one of the first where all the above quantities are de- +rived simultaneously for a single sample of distant radio-loud +AGN. +Our radio sources have log(MBH/M⊙) ≃ 7.1 − 10.0 and high +log(M⋆/M⊙) ≃ 10.2 − 12.0, which is in agreement with the fact +that radio-loud AGN are almost invariably associated with the +most massive galaxies and black holes (e.g., Best et al. 2005; +Chiaberge & Marconi 2011). While overall our sources follow +the expected trends previously found in the literature, a sub- +stantial fraction of our sources 9/42 (i.e., 21%) have black hole +masses log(MBH/M⊙) > 9 well above the values predicted by the +scaling relations (Häring & Rix 2004; Sani et al. 2011; DeGraf +et al. 2015). In particular, five sources out of the nine (12% of the +full radio source sample) are clearly overmassive outliers, hav- +ing MBH/M⋆ > 2%. This fraction is remarkably higher than that +of 1.6% found for AGN at similar redshifts from the literature. +These overmassive SMBHs are thus the high-z counterparts of +low-z overmassive SMBHs found in previous studies of nearby +early type galaxies (e.g., McConnell & Ma 2013; Trujillo et al. +2014). +Our results imply that the growth of black hole masses in +at least a substantial fraction of radio loud AGN largely occurs +at early epochs, while the early stellar mass assembly may not +be so efficient. This population of radio-loud AGN with high +MBH/M⋆ ratios have likely experienced a different stellar mass +growth than other types of AGN, and we further investigated this +scenario in terms of additional complementary probes. +Following early studies on nearby galaxies (e.g., McConnell +& Ma 2013; Trujillo et al. 2014), we found that three of +our radio-loud AGN with moderately overmassive SMBHs are +hosted in clusters from the literature, while clusters and groups +around the majority of the remaining radio-loud AGN will likely +be detected with forthcoming surveys such as Euclid (Euclid +Collaboration et al. 2019). These results suggest that the cluster +environments tend to prevent the stellar mass assembly of cluster +early-type galaxies, possibly via radio-mode AGN feedback. +Concerning instead the nuclear accretion and jet properties, +we found that the SMBHs of the radio sources in our sample +accrete, on average, at a sub-Eddington rate (η ∼ 1%), where +higher accretion rates favor more powerful jets to be launched +by the central engine. We also find that high jet powers (Qjet ≳ +1045 erg s−1) are invariably associated with high radio luminosity +sources (L1.4 GHz > 2 × 1032 erg s−1 Hz−1). Altogether, the ob- +served accretion state sustains both the nuclear activity and the +SMBH growth at sub-parsec scales, while it ultimately favors the +persistence of large scale radio jets, which may prevent the host +galaxy to accrete gas at kilo-parsec scales and thus form stars +effectively. Radio-mode AGN feedback may be responsible for +the presence of overmassive SMBHs in our sample of radio-loud +AGN. +Targeted observations of the ionized and the molecular gas +are nevertheless needed to further investigate the proposed radio- +mode AGN feedback scenario. Future studies on larger and +higher-redshift samples of radio-loud AGN will become pos- +sible with the advent of forthcoming radio-to-optical surveys +such as The Vera Rubin telescope and Euclid in infrared-optical, +SKA in radio, as well as its pathfinders and precursors (LOFAR, +ASKAP, and MeerKAT). +Acknowledgements. We thank the anonymous referee for helpful comments +which contributed to improve the paper. GC acknowledges the support +from the grants ASI n.2018-23-HH.0 and PRIN-MIUR 2017 WSCC32. We +thank Christophe Benoist for helpful discussion about the exploitation of +DES data. This publication makes use of data products from the Wide- +field Infrared Survey Explorer, which is a joint project of the University of +California, Los Angeles, and the Jet Propulsion Laboratory/California Institute +of Technology, and NEOWISE, which is a project of the Jet Propulsion +Laboratory/California Institute of Technology. WISE and NEOWISE are funded +by the National Aeronautics and Space Administration. This research has +made use of the NASA/IPAC Extragalactic Database (NED), which is op- +erated by the Jet Propulsion Laboratory, California Institute of Technology, +under contract with the National Aeronautics and Space Administration. +This project used public archival data from the Dark Energy Survey (DES). +Funding for the DES Projects has been provided by the U.S. Department +of Energy, the U.S. National Science Foundation, the Ministry of Science +and Education of Spain, the Science and Technology FacilitiesCouncil of +the United Kingdom, the Higher Education Funding Council for England, +the National Center for Supercomputing Applications at the University of +Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics +at the University of Chicago, the Center for Cosmology and Astro-Particle +Physics at the Ohio State University, the Mitchell Institute for Fundamental +Physics and Astronomy at Texas A&M University, Financiadora de Estudos +10 + +R. Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado +do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e +Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche +Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy +Survey. The Collaborating Institutions are Argonne National Laboratory, the +University of California at Santa Cruz, the University of Cambridge, Centro +de Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the +University of Chicago, University College London, the DES-Brazil Consortium, +the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) +Zürich, Fermi National Accelerator Laboratory, the University of Illinois at +Urbana-Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the +Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory, +the Ludwig-Maximilians Universität München and the associated Excellence +Cluster Universe, the University of Michigan, the National Optical Astronomy +Observatory, the University of Nottingham, The Ohio State University, the +OzDES Membership Consortium, the University of Pennsylvania, the University +of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the +University of Sussex, and Texas A&M University. 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Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +ID +RA +Dec. +z +log +L1.4 GHz +erg s−1 Hz−1 +log(Ldust/L⊙) +log(M⋆/M⊙) +SFR +WISE class +Type +Name +hh:mm:ss.ss +dd:mm:ss.ss +(M⊙/yr) +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +(10) +(11) +RS 16 +02:23:34.94 +-06:37:22.99 +1.217 +33.65 ± 0.01 +11.48 +10.40 +32 +AGN +Radio Source +SDSS J022334.90-063722.9 +RS 49 +02:16:40.74 +-04:44:04.97 +0.875 +33.51 ± 0.01 +13.65 +11.95 +<4801 +AGN +QSO +FBQS J0216-0444 +RS 52 +02:19:38.89 +-05:18:05.28 +2.207 +32.65 ± 0.04 +12.34 +11.73 +<235 +AGN +QSO +3XMM J021938.8-051805 +RS 60 +02:19:54.63 +-05:49:22.32 +0.322 +30.63 ± 0.05 +9.08 +10.15 +0.13 +AGN +QSO +SDSS J021954.62-054922.2 +RS 62 +02:22:47.91 +-04:33:31.02 +1.635 +32.60 ± 0.02 +12.43 +11.02 +<289 +AGN +QSO +XXL-N 027_020 +RS 79 +02:22:55.95 +-05:18:15.85 +1.756 +34.63 ± 0.01 +12.32 +11.48 +<225 +AGN +QSO +PKS 0220-055 +RS 81 +02:26:07.42 +-05:32:09.52 +0.779 +32.30 ± 0.01 +11.53 +10.76 +36 +Intermediate disk +X-ray source +XXL-N 062_013 +RS 82 +02:25:56.39 +-05:34:51.44 +2.879 +33.68 ± 0.01 +12.75 +11.25 +<605 +AGN +QSO +XXL-N 062_009 +RS 83 +02:25:05.12 +-05:36:47.94 +0.682 +33.60 ± 0.01 +9.58 +11.11 +0.41 +Intermediate disk +QSO +PMN J0225-0536 +RS 104 +02:27:12.99 +-04:46:36.36 +0.981 +32.32 ± 0.01 +12.62 +11.15 +<448 +AGN +X-ray source +SPIRE 13430 +RS 113 +02:45:31.51 +-00:26:12.32 +2.082 +32.83 ± 0.02 +12.52 +11.75 +<356 +AGN +QSO +2SLAQ J024531.53-002612.2 +RS 151 +02:27:40.56 +-04:02:51.16 +2.603 +32.42 ± 0.07 +13.69 +11.54 +<5265 +AGN +QSO +XXL-N 044_070 +RS 159 +02:29:15.79 +-04:42:15.95 +1.074 +34.12 ± 0.01 +12.29 +10.28 +<21 +AGN +QSO +3XMM J022915.7-044216 +RS 177 +02:51:56.32 ++00:57:06.53 +0.471 +31.80 ± 0.01 +11.74 +10.45 +<59 +AGN +QSO +LBQS 0249+0044 +RS 190 +02:51:15.50 ++00:31:35.45 +1.978 +33.93 ± 0.01 +12.02 +11.10 +<114 +AGN +QSO +WISEA J025115.50+003135.4 +RS 195 +02:47:06.66 ++00:23:18.10 +0.363 +30.85 ± 0.03 +11.67 +10.94 +<50 +AGN +QSO +FBQS J0247+0023 +RS 197 +02:46:16.61 ++00:19:53.11 +3.791 +33.46 ± 0.02 +11.61 +11.11 +44 +Starburst/LIRG +QSO +SDSS J024616.60+001953.6 +RS 205 +02:53:40.94 ++00:11:10.04 +1.683 +33.04 ± 0.01 +12.73 +11.23 +<577 +AGN +QSO +LBQS 0251-0001 +RS 206 +02:48:54.80 ++00:10:53.84 +1.145 +33.49 ± 0.01 +12.92 +11.05 +<904 +AGN +QSO +2SLAQ J024854.80+001053.9 +RS 214 +02:50:48.66 ++00:02:07.46 +0.766 +32.67 ± 0.01 +11.93 +10.43 +<91 +AGN +QSO +FBQS J0250+0002 +RS 237 +02:49:23.22 +-00:54:38.04 +0.953 +31.63 ± 0.06 +11.69 +10.27 +<53 +AGN +QSO +2SLAQ J024923.20-005437.7 +G 55673 +12:11:55.32 +-00:20:19.38 +0.436 +33.16 ± 0.04 +11.55 +11.19 +23 +Starburst/LIRG +G +SDSS J121155.31-002019.4 +G 71277 +12:16:12.27 +00:04:17.91 +0.316 +32.29 ± 0.03 +10.01 +11.23 +0.49 +Intermediate disk +G +SDSS J121612.26+000417.8 +G 165213 +12:01:13.77 +-02:42:41.34 +0.307 +31.90 ± 0.08 +11.64 +11.27 +34 +AGN +QSO +SDSS J120113.76-024241.3 +G 196970 +08:53:52.21 +-00:45:31.12 +0.323 +31.99 ± 0.06 +10.00 +11.08 +0.74 +Intermediate disk +G +SDSS J085352.21-004531.1 +G 208794 +08:40:44.47 +00:03:05.27 +0.449 +31.84 ± 0.10 +11,59 +10.92 +28 +AGN +G +SDSS J084044.10+000307.2 +G 249591 +14:08:22.42 +02:08:53.70 +0.432 +32.45 ± 0.02 +10.64 +11.31 +4.3 +Intermediate disk +G +SDSS J140822.76+020853.1 +G 251343 +14:32:57.38 +02:03:24.46 +0.761 +32.66 ± 0.01 +12.22 +11.54 +<1119 +Starburst/LIRG +G +SDSS J143257.64+020321.3 +G 298359 +14:37:31.86 +01:18:58.14 +0.342 +34.06 ± 0.01 +10.81 +10.92 +3.1 +Intermediate disk +Radio Source +GAMA J143257.41+020329.1 +G 372455 +09:03:25.57 +01:12:14.86 +0.311 +32.68 ± 0.01 +9.842 +11.02 +0.75 +Intermediate disk +Radio Source +GAMA J090325.48+011214.1 +G 537618 +12:23:48.39 +-00:52:50.43 +0.490 +32.57 ± 0.02 +12.06 +10.79 +43 +Starburst/LIRG +Radio Source +SDSS J122347.89-005249.2 +G 714133 +14:25:33.03 +01:07:37.73 +0.556 +32.48 ± 0.02 +12.35 +10.80 +<238 +AGN +Radio Source +NVSS J142533+010739 +G 714228 +14:31:20.49 +01:14:56.44 +0.343 +33.10 ± 0.01 +10.33 +10.67 +5.9 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+0.547 +32.28 ± 0.03 +10.47 +11.55 +2.4 +Intermediate disk +G +SDSS J143702.15-010357.2 +Table A.1: Properties of the radio sources in our sample. Column description. (1) galaxy ID; (2-3) RA and Dec. coordinates; +(4) spectroscopic redshift; (5) 1.4 GHz rest frame luminosity density; (6-7) SED-based dust luminosity and stellar mass; (8) star +formation rate; (9) WISE color-based class; (10-11) source type and name as found in the NED. +13 + +R. Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +ID +z +Line +FWHM +log(Lline/(erg s−1)) +log(MBH/M⊙) +log(Qjet/(erg s−1)) +log(LBLR/(erg s−1)) +log η +(103 km s−1) +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +RS 16 +1.22 +Mg ii +4.08 ± 0.42 +42.53 ± 0.07 +8.00 ± 0.12 +45.81 +43.50 +-1.60 +RS 49 +0.88 +Hβ +6.77 ± 0.11 +44.02 ± 0.01 +9.30± 0.04 +45.69 +45.06 +-1.34 +RS 52 +2.21 +Mg ii +6.00 ± 0.43 +43.41 ± 0.14 +8.88 ± 0.13 +44.94 +44.51 +-1.48 +RS 60 +0.32 +Hα +1.74 ± 0.07 +41.88 ± 0.02 +7.13 ± 0.04 +43.22 +42.44 +-1.80 +RS 62 +1.64 +Mg ii +3.20 ± 0.11 +43.26 ± 0.02 +8.25 ± 0.08 +44.90 +44.27 +-1.07 +RS 79 +1.76 +Mg ii +6.48 ± 0.16 +43.47 ± 0.03 +8.99 ± 0.08 +46.60 +44.73 +-1.30 +RS 81 +0.78 +Mg ii +12.47 ± 2.07 +42.14 ± 0.14 +8.72 ± 0.18 +44.60 +43.30 +-2.52 +RS 82 +2.88 +C iv +5.26 ± 0.37 +43.49 ± 0.12 +8.73 ± 0.24 +45.83 +44.28 +-1.55 +RS 83 +0.68 +Hβ +7.03 ± 1.67 +41.87 ± 0.14 +8.28 ± 0.23 +45.70 +40.57 +-4.80 +RS 104 +0.98 +Hβ +6.31 ± 0.30 +43.47 ± 0.03 +8.97 ± 0.06 +44.60 +44.56 +-1.52 +RS 113 +2.08 +Mg ii +5.89 ± 0.24 +43.62 ± 0.06 +9.00 ± 0.09 +45.10 +44.67 +-1.43 +RS 151 +2.60 +Mg ii +7.68 ± 0.30 +44.47 ± 0.08 +9.76 ± 0.09 +44.75 +45.13 +-1.73 +RS 159 +1.07 +Mg ii +3.95 ± 0.21 +42.94 ± 0.03 +8.22 ± 0.10 +46.21 +43.95 +-1.37 +RS 177 +0.47 +Hβ +5.31 ± 0.14 +42.60 ± 0.01 +8.39 ± 0.06 +44.22 +43.65 +-1.84 +RS 190 +1.98 +C iv +14.66 ± 1.29 +44.40 ± 0.10 +10.03 ± 0.24 +46.05 +45.05 +-2.08 +RS 195 +0.36 +Hβ +2.90 ± 0.03 +42.42 ± 0.01 +7.78 ± 0.06 +43.40 +43.89 +-0.99 +RS 197 +3.79 +C iv +1.93 ± 0.39 +43.64 ± 0.09 +7.92 ± 0.28 +45.61 +44.43 +-0.50 +RS 205 +1.68 +Mg ii +7.22 ± 0.18 +43.93 ± 0.02 +9.38 ± 0.07 +45.28 +45.19 +-1.29 +RS 206 +1.15 +Mg ii +4.12 ± 0.11 +43.54 ± 0.01 +8.64 ± 0.07 +45.60 +44.58 +-1.10 +RS 214 +0.77 +Hβ +11.05 ± 0.48 +43.42 ± 0.02 +9.43 ± 0.06 +44.96 +44.36 +-2.10 +RS 237 +0.95 +Mg ii +9.89 ± 0.99 +43.24 ± 0.07 +9.21 ± 0.12 +44.07 +44.30 +-2.01 +G 55673 +0.44 +Hβ +1.20 ± 0.19 +41.66 ± 0.09 +8.45 ± 0.16 +43.51 +33.91 +-2.60 +G 71277 +0.32 +Hα +1.05 ± 0.11 +41.99 ± 0.05 +8.36 ± 0.09 +43.35 +33.61 +-2.85 +G 165213 +0.31 +Hα +1.19 ±0.93 a +42.79 ± 0.01 +8.82 ± 0.71 +42.99 +34.45 +-2.47 +G 196970 +0.32 +Hβ +1.20 ±0.60a +40.89 ± 0.07 +8.08 ± 0.45 +43.11 +32.74 +-3.44 +G 208794 +0.45 +Hβ +1.20 ±0.11 a +41.35 ± 0.04 +8.30 ± 0.12 +43.26 +33.6 +-2.80 +G 249591 +0.43 +Hβ +1.20 ±0.62 a +41.66 ± 0.14 +8.45 ± 0.46 +43.75 +33.91 +-2.64 +G 251343 +0.76 +Hβ +1.19 ±0.15a +43.96 ± 0.02 +9.57 ± 0.12 +44.42 +36.21 +-1.46 +G 298359 +0.34 +Hα +1.19 ±0.18 +42.07 ± 0.06 +8.51 ± 0.14 +44.93 +33.76 +-2.85 +G 372455 +0.31 +Hα +1.19 ±0.60a +42.66 ± 0.08 +8.77 ± 0.08 +43.67 +34.29 +-2.58 +G 537618 +0.49 +Hβ +1.10 ± 0.04 +43.86 ± 0.02 +9.45 ± 0.05 +43.96 +36.11 +-1.44 +14 + +R. Poitevineau et al.: Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.3 − 4 +ID +z +Line +FWHM +log(Lline/(erg s−1)) +log(MBH/M⊙) +log(Qjet/(erg s−1)) +log(LBLR/(erg s−1)) +log η +(103 km s−1) +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +(9) +G 714133 +0.56 +Hβ +1.20 ±0.19 a +42.25 ± 0.01 +8.74 ± 0.10 +43.99 +34.50 +-2.34 +G 714228 +0.34 +Hβ +1.20 ±0.11 a +40.86±0.06 +8.06 ± 0.13 +44.11 +33.11 +-3.05 +G 720847 +0.42 +Hβ +1.19 ±0.24 a +41.92 ± 0.02 +8.57 ± 0.19 +44.07 +34.17 +-2.5 +G 721940 +0.64 +Hβ +1.20 ±0.60a +43.31 ± 0.04 +9.25 ± 0.44 +44.26 +35.56 +-1.79 +G 745066 +0.38 +Hβ +1.19 ±0.60a +41.73 ± 0.02 +8.47 ± 0.45 +43.30 +33.77 +-2.8 +G 746605 +0.37 +Hβ +1.24 ± 0.09 +43.37 ± 0.04 +9.31 ± 0.08 +43.03 +35.62 +-1.79 +G 748144 +0.59 +Hβ +1.22 ± 0.31a +43.08± 0.09 +9.16 ± 0.23 +43.95 +35.33 +-1.93 +G 748815 +0.33 +Hβ +1.19 ±0.07a +42.05±0.01 +8.63 ± 0.09 +44.68 +33.94 +-2.79 +G 804203 +0.51 +Hβ +1.20 ±0.60a +41.44±0.11 +8.34 ± 0.45 +43.82 +33.69 +-2.75 +G 835899 +0.76 +Hβ +1.19 ±0.16a +42.24±0.02 +8.72 ± 0.13 +43.90 +34.49 +-2.33 +G 887308 +0.55 +Hβ +1.20 ±0.19a +43.17±0.03 +9.18 ± 0.15 +43.80 +35.42 +-1.87 +Table A.2: Black hole, accretion, and jet properties of the radio sources in our sample. Column description. (1) source ID; (2) +spectroscopic redshift; (3-5) broad emission line, FWHM, and line luminosity; (6) single-epoch SMBH mass; (7) jet power; (8) +BLR luminosity; (9) Eddington ratio. Sources marked with the symbol a in column (4) have uncertain FWHM errors from GAMA, +when the lines were fitted with a single Gaussian component. To overcome this limitation we considered the GAMA fits that include +both narrow and broad components of the emission line. We then assumed a FWHM relative error equal to that resulting from the +fit to the broad component of the line. +15 + diff --git a/b9E4T4oBgHgl3EQfow0v/content/tmp_files/load_file.txt b/b9E4T4oBgHgl3EQfow0v/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..db47fb6cba5126ebdfbf294616481ae409376380 --- /dev/null +++ b/b9E4T4oBgHgl3EQfow0v/content/tmp_files/load_file.txt @@ -0,0 +1,2531 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf,len=2530 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' ms © ESO 2023 January 13, 2023 Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau1,⋆, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Castignani2,3, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Combes1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 1 Observatoire de Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' LERMA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' PSL University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Sorbonne Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' F-75014,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' France 2 Dipartimento di Fisica e Astronomia ”Augusto Righi”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Alma Mater Studiorum Università di Bologna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Via Gobetti 93/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' I-40129 Bologna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Italy 3 INAF - Osservatorio di Astrofisica e Scienza dello Spazio di Bologna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' via Gobetti 93/3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' I-40129,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Bologna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Italy 4 Collège de France,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 11 Place Marcelin Berthelot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 75231 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' France Received 21 July 2022 ABSTRACT There exists a well known relation between the mass of the supermassive black hole (SMBH) in the center of galaxies and their bulge mass or central velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This suggests a co-evolution between SMBH and their galaxy hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Our aim is to study this relation specifically for radio loud galaxies, and as a function of redshift z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We selected a sample of 42 radio galaxies and active galactic nuclei (AGN) with broad emission lines and spectroscopic redshifts between z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4, by cross-matching the low radio frequency sources from VLA FIRST with spectroscopically confirmed galaxies from wide field surveys including SDSS DR14 ugriz and DES DR2 grzY in optical, WISE in infrared, and the Galaxy And Mass Assembly (GAMA) spectroscopic survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We characterised the stellar mass (M⋆), star formation, and black hole properties (mass of the central SMBH, the Eddington ratio η and the jet power, Qjet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The relation between SMBH mass, M⋆, η and z are put into context by comparing them with scaling relations (MBH–M⋆, MBH/M⋆–z, MBH–Qjet and Qjet–η) from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' On the basis of a multi-wavelength spectral energy distribution modeling, our radio sources are broadly consistent with being on the star-forming main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' They have sub-Eddington accretion rates, η ≃ 1% on average, as typically found in Type I AGN, while higher accretion rates favor more powerful jets to be launched by the central engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We find the presence of overmassive SMBHs in (17 ± 5)% of our radio sources, similarly to previous studies on nearby early-type galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Altogether, an evolutionary scenario where radio-mode AGN feedback regulates the accretion onto the SMBHs and the stellar mass assembly of the radio sources is discussed, which may explain the observed phenomenology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This pilot study represents a benchmark for future ones using wide field surveys such as Euclid and the Vera Rubin telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' galaxies: active – galaxies: bulges – galaxies: nuclei – (galaxies) quasars: supermassive black holes – infrared: galaxies – radio continuum: galaxies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Introduction Super Massive Black Holes (SMBHs), characterized by masses in the range ∼ 106 M⊙ to ∼ 1010 M⊙, are observed to lie at the center of most, if not all, massive galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Graham 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' When the central regions of galaxies are sources of radiation, because of accretion onto their SMBHs, they are called Active Galactic Nuclei (AGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' AGN are among the strongest proofs for the existence of SMBHs, together with the direct measure of compact densities in our Galactic center (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2010), and the direct observation of the SMBH shadow at the center of M87 and of the Milky Way itself (The Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2019, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' There exists an intrinsic co-evolution between AGN activ- ity, SMBH growth, galaxy stellar content and star formation his- tory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Kormendy & Ho 2013, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In some cases AGN are jetted, and thus called radio-loud AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' They consti- tute only 10% of the whole AGN population, but their fraction varies with the stellar mass of the host, from 0 to 30% (Best et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Large-scale radio jets are even able to impact the global Mpc-scale environmental properties, via radio-mode AGN feed- back, as for example at the center of galaxy (proto-)clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Miley & De Breuck 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Fabian 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Magliocchetti 2022, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' ⋆ e-mail: remi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='poitevineau@obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='fr The mode of SMBH accretion ultimately regulates the exci- tation properties of radio-loud AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' It is indeed possible to dis- tinguish two main classes of activity among the radio-loud AGN, HE (High excitation) and LE (Low Excitation) Radio Galaxies (RG) according to their accretion rate: HERGs typically have ac- cretion rates between 1 and 10% of their Eddington rate, whereas LERGs predominately accrete at a rate below 1% Eddington (Best & Heckman 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In the former case of HERGs, the ma- terial is thus losing progressively angular momentum in a geo- metrically thin disk around the SMBH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' this disk is usually op- tically thick and radiates efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' When the accretion rate is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 the Eddington rate, the AGN is instead characterized by an advection-dominated accretion flow (ADAF, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Narayan & McClintock 2008), which radiates inefficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Radio-loud AGN are in majority in the low-luminosity regime, and fre- quently ADAFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' A major observational breakthrough for what concerns the co-evolution of galaxies and AGN with their SMBHs was the discovery of a tight correlation, in the local universe, between the SMBH mass and the mass of their host spheroids (Magorrian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Ferrarese & Merritt 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The existence of this re- lation implies a remarkable connection between the assembly of galaxies and the formation and growth of SMBHs at their center (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Heckman & Best 2014, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Models and simu- lations (Menci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Marulli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Volonteri & Natarajan 2009) have attempted to explain this correlation and its evolution with redshift, as found in sev- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='05186v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='GA] 12 Jan 2023 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' : Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 eral observational studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Shields et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Sarria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Merloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Jahnke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Cisternas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Schramm & Silverman 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' There are, however, a number of still related open issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Among them there is the existence of local ellipticals with over- massive SMBHs (Kormendy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Savorgnan & Graham 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Dullo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These over-massive SMBH occur preferentially in galaxy clusters, and in brightest cluster galaxies in particular (BCGs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', McConnell & Ma 2013), where environment effects strip galaxies from their gas, stop star formation and the growth of bulges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Galaxies are then called massive relics, with particularly old stellar popu- lation (Trujillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Martín-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Ferré- Mateu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2015, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The very discovery of massive SMBHs (MBH ≳ 109 M⊙) in bright quasars at the epoch of reionization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Bañados et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Farina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2022) is a mystery, as it shows that extreme SMBHs can form within 1 Gyr after the Big Bang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The rapid formation of such high-z SMBHs might be ex- plained invoking some extreme scenarios such as the growth of a 102−5 M⊙ seed via super-Eddington accretion (Valiante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Pezzulli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2017), the direct collapse of an initial gas condensation into a black hole of ∼105 M⊙ (Visbal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Regan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2017), or the merger of massive proto-galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2010, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Ferrara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Bonoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Altogether, while existing studies show a tight co-evolution between SMBHs, AGN, and their host galaxies with cosmic time, this interplay is still substantially debated and uncon- strained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This is at least partially due to the difficulty in building large samples of distant AGN with well characterized stellar and black hole properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In order to better understand the growth of SMBHs with cos- mic time, and their co-evolution with their host galaxies, in this work we have therefore built a sample of distant radio-loud AGN spanning about 9 Gyr of cosmic time, between z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4, with available radio-to-ultraviolet spectro-photometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Thanks to this multi-wavelength dataset we assess the properties of the AGN sample in terms for example of black hole and stellar masses, jet power, and Eddington ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' As radio-loud AGN are associated with the most massive black holes and host galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Best et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Chiaberge & Marconi 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2012), they are excellent sources to investi- gate the galaxy, AGN, and SMBH co-evolution at the high-mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2 we describe our sample selection as well as its multi-wavelength dataset and properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 3 we report estimates for the black hole, jet, accretion, and stellar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 4 we describe our com- parison sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The results, in terms of black hole - jet - host galaxy scaling relations are reported in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 6 we summarize the results and draw our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Throughout this work, we adopt a flat ΛCDM cosmology with matter density Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='30, dark energy density ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='70 and Hubble constant h = H0/100 km s−1 Mpc−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The Radio-Loud AGN sample We selected a sample of radio-loud AGN by cross-matching the Very Large Array Faint Images of the Radio Sky at Twenty- centimeters (VLA FIRST) source catalog (Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 1995) with infrared-to-optical spectro-photometric surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' As further described in the following, the use of infrared-to-ultraviolet pho- tometry enables the modeling of the spectral energy distribution (SED), which allows us to ultimately obtain a good characteri- sation of the galaxy properties such as the stellar mass (M⋆) and the star formation rate (SFR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The Dark Energy Survey We start by considering the Dark Energy Survey (DES Collaboration 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Dark Energy Survey Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2016), which is composed of two distinct multi-band imaging surveys: a ∼5000 deg2 wide-area grizY survey and a deep super- nova griz survey made by six distinct deep fields (Hartley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The coadded source catalog and images from the process- ing of all six years of DES wide-area survey observations and all five years of DES supernova survey observations have been re- cently made public with the DES data release 2 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2021)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' To build our sample of distant radio-loud AGN we limit our- selves to equatorial DES supernova fields that overlap with the VLA FIRST survey at low radio frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The selection is similar to that of our previous work (Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2019), which we refer for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' However, in that study we focused only on two radio sources, and we investigated their molecular gas content, their cluster environment, as well as the stellar and star formation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In this work we consider instead a more extended sample, as further outlined in the fol- lowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Radio, optical, and spectroscopic selection As we are interested in building a sample of extra-galactic radio sources we consider the VLA FIRST survey (Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 1995), which observed 10,000 deg2 of the North and South Galactic Caps at low radio frequencies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz), down to a point source detection limit of ∼1 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We therefore further limit ourselves to the Stripe 82 area, that is a 300 deg2 equatorial field that was imaged multiple times by the Sloan Digital Sky Survey (SDSS) and overlaps with the VLA FIRST survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Similarly, for our search we additionally considered DES supernova deep fields numbered 2, 3, and 5, as outlined in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1 We have cross-matched the low radio frequency VLA FIRST radio source catalog with both the SDSS DR14 ugriz and DES DR1 grizY source catalogs within the considered fields with a search radius of 3 arcsec, consistently with the positional ac- curacy ∼ 1 arcsec of FIRST sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' As we are interested in secure distant radio sources, we further restrict ourselves to those sources with SDSS DR14 spectroscopic redshifts z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The search yields 158 spectroscopically confirmed radio sources with unique optical counterparts from both SDSS and DES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Infrared selection: WISE We further look for infrared emission of the radio sources, as found by the W4 filter of the Wide-field Infrared Survey Explorer (WISE, Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' To this aim we have cross- correlated our radio sources with the allWISE source catalog2 by adopting a search radius of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 arcsec, consistently with previ- ous work on extra-galactic radio sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The search yields 154 sources with unique WISE coun- terparts and W4 magnitudes with signal-to-noise ratio S/N > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 1 https://des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='ncsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='edu/releases/dr2 2 http://wise2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='edu/docs/release/allwise/ 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' : Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Broad Emission Lines from SDSS As we are interested in assessing black hole masses of the con- sidered radio-loud AGN we further restrict ourselves to those sources with evidence of broad emission lines in the SDSS spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' To this aim we have further selected those sources that have Hα, Hβ, Mg ii, or C iv emission line fluxes at signal- to-noise ratio S/N> 3, as well as full width at half maximum FWHM>1000 km s−1, typical of the broad line region lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We use the spZline file3 that contains the results of the emission-line fits for the BOSS spectra of SDSS sources (Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The Gaussian line width σ is reported and we converted it into the FWHM = 2σ � 2 log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This additional spectroscopic selec- tion yields 21 sources at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 1: Top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz luminosity density as a function of red- shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Points are color coded according the WISE color-based classification, as in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The horizontal line is at Lν = 2 × 1032 erg s−1 Hz−1, and separates low luminosity ra- dio sources from high luminosity radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' WISE color-color plot, where sources are distinguished between AGN (circles), starbursts (triangles) , disks and spheroids (squares) according to the color-based classification by (Stern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Jarrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Radio-Loud AGN in GAMA DR3 In addition to the radio-loud AGN selected as described in the previous sub-sections, we searched for distant radio sources from the third data release (DR) of the Galaxy And Mass Assembly (GAMA) spectroscopic survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' GAMA DR3 (Baldry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2018) provides in fact spectra obtained with the AAOmega 3 https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='org/datamodel/files/BOSS_SPECTRO_REDUX/ RUN2D/PLATE4/RUN1D/spZline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='html multi-object spectrograph on the Anglo-Australian Telescope (AAT) as well as a wealth of ancillary information for more than 200 thousands sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Similarly to what we did concerning the SDSS spectra, we first selected 632 sources at z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3, with available GAMA DR3 spectra, Hα or Hβ line fluxes at S/N> 3 as well as line widths FWHM> 1000 km s−1, as inferred from single Gaussian model- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 We selected these sources within the equatorial area of 180 deg2 covered by GAMA DR3 as well as by the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz FIRST VLA survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Among the 632 spectroscopic sources from GAMA, we further limited ourselves to the subsample of 39 galaxies with available 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz fluxes from the FIRST VLA survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Multiple spectra are often available in the GAMA DR3 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We then inspected each of the available spectra and discarded those galaxies where the evidence of emission lines was dubious, and thus the associated fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This analysis yields 21 GAMA DR3 radio-sources at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' For them, we assigned unique WISE counterparts using a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 arcsec search radius, as in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' By combining these galaxies, denoted hereafter with the pre- fix G, with the 21 radio sources with SDSS spectra, denoted in- stead with the prefix RS, our final sample comprises 42 sources that we consider hereafter for the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The main prop- erties of this sample of galaxies are listed in Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Radio and infrared properties We now investigate the low frequency radio luminosities and the infrared colors of our sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' To this aim, similarly to previous studies (Condon 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Chiaberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2014), we first assumed a power-law for the radio spectrum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', S ν ∝ ν−α, where S ν is the radio flux density at the observer frequency ν and the spectral index α is fixed to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We then converted the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz VLA radio fluxes S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz into rest frame 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz luminosity densities as follows: L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz = 4π S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz DL(z)2 (1 + z)α−1 , (1) where DL is the luminosity distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Figure 1 (top) displays our sources in the L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' redshift plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' They all have L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz ≳ 3 × 1030 erg s−1 Hz−1 typical of radio-loud AGN, while purely starburst galaxies have lower L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz < 1030 erg s−1 Hz−1 (Chiaberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Furthermore, the majority (71%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', 30/42) of our sources have high radio powers, greater than L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz = 2 × 1032 erg s−1 Hz−1, which we use to distinguish between Low Luminosity Radio Sources (LLRS) from High Luminosity Radio Sources (HLRS), similarly to what has been done in previous studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Chiaberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' As the radio galaxy population has a bimodal distribution in radio power, it is worth mentioning that the adopted LLRS/HLRS lu- minosity threshold corresponds to the fiducial radio power which fairly separates FR I from FR II radio galaxies (Fanaroff & Riley 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Zirbel 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Furthermore, as a result of the Malmquist bias associated with the VLA FIRST flux limit of ∼ 1 mJy, the fraction of HLRS increases with redshift and reaches unity at z > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Figure 1 (bottom) shows instead the sources in our sample in the WISE color-color diagram, where sources are distinguished using the color-based classification by Jarrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2017), as highlighted in the Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Interestingly, our sample populates 4 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='gama-survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='org/dr3/ 3 High Luminosity Radio Sources 34 zH 33 s [erg Lv 32 Log 31 Low Luminosity Radio Sources 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='8 1 2 3 4 5 Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 AGN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 V2-W3[mag] ULIRGs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 Spheroids Intermediate disks Starbursts & LIRGs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 W1-W2[mag]R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' : Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 only three regions in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The majority (28/42) of our sources are classified as AGN, based on WISE colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This is not surprising as they have been selected as distant and powerful radio sources at z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Based on WISE colors the remaining sources are fairly equally distributed between the intermediate disk (9/42) and starburst (5/42) classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Furthermore, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 1 (top) the vast majority of z > 1 sources have WISE infrared colors consistent with AGN contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' They also show high 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz radio luminosities typical of radio-loud quasars (QSOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The majority (22/42, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', 52%) of our sources are in fact classified as quasars in the NED database, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', with counterparts in the 2dF–SDSS LRG And QSO (2SLAQ, Croom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2009) catalog, or with X-ray con- terparts (XMM, Rosen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2016), as outlined in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Black hole, jet, accretion, and stellar properties 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Black Hole masses One of the main goals of this work is to investigate the co- evolution of central black-holes with the host galaxies of the radio-loud AGN in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' To this aim we estimated black hole masses using the widely used Single Epoch (SE) method, that is particularly suited for distant Type 1 AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' According to this method, black hole masses MBH can be estimated under the assumption that the Broad Line Region (BLR) is in virial equilibrium, as follows: MBH = f RBLR∆V2 G , (2) where RBLR is the BLR radius, ∆V is the velocity of the BLR clouds that can be estimated from the broad emission line width, f is the virial coefficient that depends on the geometry and kine- matics of the BLR, and G is the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The SE method then uses the relation that exists between the BLR size and the AGN optical/ultraviolet continuum luminosity empiri- cally found from reverberation mapping (Peterson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Kaspi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Bentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2009), as well as the tight correla- tion between the continuum luminosity and that of broad emis- sion lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' With these considerations, the black hole mass can be expressed as: log � MBH M⊙ � = a + b log � L 1044 erg s−1 � + c log �FWHM km s−1 � , (3) where the coefficients a, b, and c are empirically calibrated against local AGNs with reverberation mapping masses or using different lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' L and FWHM are the line luminosity and width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In this work we use the coefficients obtained for Hα, Hβ, Mg ii, and C iv broad emission lines by Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2011) and Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These are widely used lines which are redshifted in the optical domain, depending on the redshift of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These lines indeed enable estimates of black hole masses over a wide range of redhifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Similarly to previous studies (Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2013), we used Hα, Hβ, and Mg ii for sources at z < 1, and to the Mg ii and C iv lines for sources at higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In the case where multiple broad emission lines are available for a given sources, we adopted the following order of preference: Hα, Hβ, Mg ii, and C iv(see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='Shen & Liu (2012) for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In Table 1 we report the coefficients used in this work, while in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2 we list the black hole masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' a b c Hα 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1 Hβ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='49 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 Mg ii 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 C iv 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 Table 1: Coefficients to estimate black hole masses using broad emission lines and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Values for Hβ, MgII, and CIV lines are from Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The coefficients for Hα come from Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2011) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Jet Power The sources in our sample are radio-loud AGN that are typi- cally characterized by jetted outflows which strongly emit at ra- dio wavelengths mainly via synchrotron emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' By studying jet properties such as its total power we will investigate the complex interplay between the jet, the black hole, and the gas accretion on it, which is commonly referred to as radio-mode AGN feedback (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Fabian 2012, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Following previous work by Willott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (1999), we esti- mate the jet power as: Q jet = 3 × 1045 ξ3/2 ������ L151 MHz 1035 erg s−1 Hz−1 sr−1 ������ 6/7 erg s−1 , (4) where L151 MHz is the extended total radio luminosity density at 151 MHz in the rest frame, and ξ is a factor ranging be- tween 10 and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In this work, we used an intermediate value ξ = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' To estimate L151 MHz we extrapolated the L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz lumi- nosity densities assuming an isotropic emission and a power law with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='8, as further described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The resulting jet powers are reported in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Eddington ratio As we want to link the gas accretion onto the black hole with the AGN properties we estimated the Eddington ratio, that is defined as η = Ldisc LEdd , (5) where Ldisc and LEdd are the disc and Eddington luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The latter can be expressed as: LEdd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='26 × 1038 � MBH M⊙ � erg s−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (6) To estimate the disc luminosity, we followed instead the pre- scriptions described in Celotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' First, we assume that BLR contributes to ∼ 10% of the total disc luminosity, that is Ldisc ≃ 10 LBLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' To estimate the BLR luminosity, we then used the line ratios reported in Francis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (1991), which are typical line luminosities of bright QSOs, normalized to that of the Lyα emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The BLR luminosity can therefore be estimated as : LBLR =< L∗ BLR > � i Li,obs � i L∗ i,est , (7) where Li,obs is the luminosity of the observed i-th line in the BLR and L∗ i,est is the line ratio of the i-th line presented in Francis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (1991) table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' With these prescriptions, the total normalized BLR luminosity is equal to < L∗ BLR >= L∗ Hα + L∗ Hβ + L∗ C iv + L∗ Mg ii + 4 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' : Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2: Examples of spectral Energy Distributions of the radio sources in our sample with black hole mass estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Data-points are from GALEX (brown dots), SDSS (red pentagons), DES (blue squares), WISE (green triangles), and IRAS (yellow upper limits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Dashed and solid lines are the best fit models for the stellar and dust components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' L∗ Lyα + L∗ Lyβ + L∗ Hγ + L∗ Al III + L∗ Si IV + L∗ C II + L∗ O I = 390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3, where L∗ Lyα = 100 is fixed as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The resulting Eddington ratios are reported in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' They are mostly in the range log η ∼ (−4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' −1), with a median = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='9, as typically found for Type 1 radio-loud AGN, but lower than those of Type 2 quasars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Kong & Ho 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' SED modeling The radio sources in our sample have a broad multi-wavelength photometric coverage, from the ultraviolet (UV) to the infrared (IR), which enables the determination of stellar masses and star formation rate (SFR) estimates via Spectral Energy Distribution (SED) modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' For the GAMA sources in our sample we considered the SED fits by Driver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2018) performed with MAGPHYS (da Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Photometric data include GALEX (Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Morrissey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2007) in the UV, SDSS (York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2000) in optical, as well as VISTA Kilo-degree Infrared Galaxy Survey (VIKING, Edge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2013), WISE (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2010), and Herschel- ATLAS (Eales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Valiante et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2016a) in the near- to far-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' For the sources in the DES SN deep fields available pho- tometry includes GALEX in the UV, ugriz (SDSS) and grizY (DES) magnitudes in the optical, WISE data in the near-IR, as well as IRAS upper limits in the far-IR, that we gathered as in Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2019), which we refer for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In this previous work we followed-up in molecular gas two radio sources in dense Mpc-scale environments at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 and z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6 within the DES SN deep fields, while in the present study we enlarge the sample to investigate the co-evolution of black holes with the radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Analogously to Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2019), we then performed fits to the SEDs using LePhare (Arnouts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Ilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Following the prescriptions provided for the LePhare code, we fitted the far-IR data separately to account for possi- ble dust emission, using the Chary & Elbaz (2001) library con- sisting of 105 templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The remaining photometric data-points at shorter wavelengths were fitted using the CE_NEW_MOD li- brary that consists of 66 galaxy templates based on linear inter- polation of the four original SEDs of Coleman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We then converted the rest frame (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0-1000) µm infrared (dust) lu- 5 Log(Λe/A) 3 4 5 6 7 12 RS 81 (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='779) G- 14 XXL-N 062 013 16 18 20 22 3 4 5 6 7 Log(Λobs/A)Log(Λe/A) 3 4 5 6 7 12 RS 83 (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='682) G- 14 PMN J0225-0536 16 18 20 22 3 4 5 6 7 Log(Λobs/A)Log(Λe/A) 3 4 5 6 12 RS 113 (z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='082) [G- 14 2SLAQJ024531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='53-002612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2 16 18 20 22 3 4 5 6 7 Log(Λobs/A)Log(入e/A) 3 4 5 6 7 12 RS 237 (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='953) 14 2SLAQJ024923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='20-005437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='7 16 18 20 22 3 4 5 6 7 Log(Λobs/A)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' : Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 3: SFR versus stellar mass plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Sources are color-coded ac- cording to their redshift, while the different symbols correspond to the different WISE classes (AGN as circles, starbursts as pen- tagons, squares for the rest), as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Upper limits to the SFRs are indicated with arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The diagonal lines correspond to the MS model prescriptions by Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2014) at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2, 1, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The red cross at the bottom right shows the typical un- certainties of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 dex for both SFRs and stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' minosity into an SFR estimate by using the Kennicutt (1998) re- lation, calibrated to a Chabrier (2003) initial mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The SEDs of four of our radio sources are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' RS 81 and 83 have prominent elliptical type emission in the optical do- main, while their IR emission is consistent with dust emission due to star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' They are indeed classified as intermediate disks based on WISE colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' RS 113 and 237 are WISE AGN and show steep SEDs at near-IR-to-optical wavelengths, which suggest that the emission is contaminated by non-thermal AGN emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Star formation rate vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' stellar mass Figure 3 displays the radio sources of our sample in the star for- mation rate (SFR) versus stellar mass (M⋆) plane, resulting from the SED fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The sources are color-coded according to the red- shift, while the different symbols correspond to the WISE clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Overall, the sources are massive, with log(M⋆/M⊙) ≃ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3− 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 (median=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1), which is in agreement with being radio- loud AGN, that are indeed typically hosted by massive ellipticals (Best et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Our galaxies also lie mostly along the star forming main sequence (MS), although with a large scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The mean specific SFR is sSFR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='44 Gyr−1, where the root mean square (rms) dispersion is reported as uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Galaxies at higher redshifts tend to have higher SFRs, in agreement with the MS model prescriptions (Speagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' However, as highlighted in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6, the fraction of sources with AGN contamination also increases with redshift, which may result in biased-high SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The latter may be the case where the optical-IR SED is steep, and thus the IR emission is likely dominated by the AGN contribution, more than star for- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' To overcome this limitation, we conservatively reconsid- ered the SFR estimates and assigned upper limits when the SFRs largely exceed 100 M⋆/yr or in the cases of steep spectrum SEDs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', as for RS 113 and 237 mentioned above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Namely, we con- sidered as steep spectra those AGN for which their optical-IR SED has a characteristic power-law behavior Fλ ∝ λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We ver- ified a posteriori that these radio sources are indeed mostly lo- cated in the upper part of the MS and are classified as WISE AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Comparison Sample To put the AGN in our sample into a context, we additionally considered a compilation of sources with available black hole and stellar mass estimates: 30 nearby galaxies from Häring & Rix (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Galaxy masses were derived by the authors through Jeans equation modeling or adopted from dynamical models in the litera- ture, while black hole masses are from Tremaine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2002) and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 35 nearby galaxies from sample of McConnell & Ma (2013), who expanded and revised available galaxy bulge masses and dynamical measurements of black hole masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 32 Type 1 AGN at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='9 from Cisternas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2011), drawn from the XMM-COSMOS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Available stellar masses are based on the modeling of HST images, taking into account both AGN and host galaxy contributions, while black hole masses are from Hβ (Trump et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 18 broad-line X-ray AGN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 < z < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2 in the Extended Chandra Deep Field-South Survey from Schramm & Silverman (2013) who estimated Mg ii-based black hole masses, as well as HST color-based stellar mass estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 78 radio-quite Type 1 AGN at z ≃ 1 − 2 from the COSMOS survey Merloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Stellar masses were determined via SED fitting, while black hole masses are based on the Mg ii emission lines of VIMOS/VLT spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 10 Type 1 AGN at 1 < z < 2 in COSMOS, from Jahnke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2009), who estimated HST color- based stellar masses, while virial black hole masses come from the spectroscopic COSMOS Magellan/IMACS and zCOSMOS surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 53 radio-quiet QSOs at z < 3 from Decarli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Virial black hole masses come from Hβ, Mg ii, and C iv emission lines, while stellar masses have been esti- mated by the authors assuming a stellar R-band mass-to-light ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Two SDSS luminous quasars at z ∼ 4 from Targett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Virial black hole mass estimates come from C iv emission, while stellar masses were estimated on the basis of Bruzual & Charlot (2003) evolutionary synthesis models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 9 distant z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 QSOs from Shields et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Black hole masses were derived from broad emission lines, while they used CO emission line widths to infer the dynamical bulge masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The 7 QSOs at z ≃ 6 from Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' They cal- culated the stellar mass as the difference between the bulge dynamical mass and the CO molecular gas mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' For these QSOs, we used the black hole masses adopted by the authors and estimated using the AGN continuum luminosity (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In addition to the sources listed above, a second group of galaxies that we use as a comparison is composed by powerful AGN with available estimates of the black hole mass, the jet power, and the Eddington ratio: 44 radio-loud AGN studied in Le et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2018), at redshifts z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2, thus lower than those of the radio sources in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These sources have estimates of the jet powers (Le et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2018) and of their black hole masses (Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Balmaverde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 6 Z 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 log[SFR/(M o /yr)] z=1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 log(M* /M o)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' : Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 208 γ-ray Fermi blazars at 0 < z < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1 from Xiong & Zhang (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Virial black-hole mass estimates mostly come from different broad emission lines, and the rest from scaling rela- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Jet powers Qjet are mostly from Nemmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2012), and were estimated using the correlation between the ex- tended radio emission and the jet power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Alternatively, Xiong & Zhang (2014) calculated Qjet using the scaling relation provided by Nemmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2012) between the γ-ray lumi- nosity and the kinetic power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 146 radio-loud QSOs at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1 < z < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5, from Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2006), classified as flat spectrum (54%) or steep spectrum radio quasars (46%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The black hole virial masses come from Hβ, Mg ii, or C iv emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The jet power were calculated by the authors using low-frequency radio emission, follow- ing Punsly (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These sources outlined above are radio-loud AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' However we verified that none of them is in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' While these stud- ies investigated the black hole and jet properties of large sam- ples of radio sources, they did not characterize their infrared- to-optical SEDs, as done here for our smaller sample of radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 4: Black hole vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' stellar mass scatter plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Filled red dots correspond to our sample of radio sources, while those in com- parison sample are shown as open symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Scaling relations are overlaid (Sani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' DeGraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Häring & Rix 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The legend at the right displays the adopted color code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The red cross at the top left shows the typical uncertain- ties ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 dex and ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 dex for the black hole and stellar masses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Results In this section we report different scaling relations for the radio sources in our sample including black hole and stellar masses, jet powers, Eddington ratios, and the redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We also include as a comparison the sources from the literature as outlined in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 4 as well as scaling relations derived in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Black hole versus stellar mass relation We start by considering black hole and stellar masses and their relative evolution with redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Figure 4 displays the black hole mass (MBH) versus the stellar mass (M⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Interestingly, our radio-loud sources nicely follow the trend previously observed for different samples of both local galaxies and distant AGN, overplotted in the Figure, as well as those inferred by the scaling relations, which also reported (Sani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Häring & Rix 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' DeGraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In particular, our sources densely populate the high log(MBH/M⊙) ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1 − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 and high log(M⋆/M⊙) ≃ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2 − 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 4, which is in agreement with the fact that radio- loud AGN are almost invariably associated with the most mas- sive galaxies and black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Interestingly, a substantial frac- tion of our sources 9/42 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', 21%) have black hole masses log(MBH/M⊙) > 9 well above the scaling relations for both local (Häring & Rix 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Sani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2011) and distant sources at the median resdhift z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6 of our sample (DeGraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This behaviour suggests that the growth of black hole masses in radio loud AGN largely occurs at early z > 1 epochs, while the early stellar mass assembly may not be equally effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Substantial growth of the stellar mass may take place even at lower red- shifts, in order to flatten the observed MBH-M⋆ scaling relation by z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Previous studies indeed suggested that massive ellip- ticals may indeed double their stellar mass between z = 1 and z = 0 (Ilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Lidman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We further investigate this evolutionary scenario in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5, which shows the MBH/M⋆ ratio as a function of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The large majority (36/42, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', 86%) of our radio sources have MBH/M⋆ ratios which are similar to those of AGN in the com- parison sample at similar redshifts, and are in agreement with model prescriptions by McLure et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2006), which are over- plotted as dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' It is worth mentioning that our sample of radio sources is flux limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' However, we expect the Malmquist bias to have a marginal impact on the (MBH/M⋆) ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Indeed, both MBH and M⋆ scale with the BLR line and the infrared-to-optical lumi- nosities, respectively, and therefore have a similar dependence on redshift, via the luminosity distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Furthermore, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5, at fixed redshift the MBH/M⋆ ratios of both our ra- dio sources and those in the comparison sample span a broad range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Similarly, when plotting M⋆ and MBH against redshift, separately, we did not find any clear trend, as indeed the points are scattered, at fixed redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These findings suggest that any possible Malmquist bias likely has a sub dominant effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' There are, however, five clear outliers among our radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' RS 197 at the highest redshift z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='79 has a low log(MBH/M⋆) = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2, well below the expected range of val- ues, according to the McLure et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2006) model prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Furthermore, a substantial fraction (5 sources, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', 12%) of our radio loud AGN, namely RS 214, RG 237, G 537618, G 721940, and G 746605, at redshifts between z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='37 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='95, have high log MBH/M⋆ ratios in the range ∼ (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='69;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0), well above the model predictions displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5, as well as higher than the ratios found in AGN at similar redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Indeed, in the red- shift range z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='8 spanned by our radio sources, there are only 3/186 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6%) AGN with log MBH/M⋆ > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='69 in our comparison sample, while the proportion is significantly higher (12%) for our radio-loud AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These results suggest that 7 11 10 - Sani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2011) 口 9 Haring & Rix(2004) DeGraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2015) Haring & Rix(2004) Shields et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2006) Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010) 0 Cisternas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010) 口 Merloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010) 8 0 Jahnke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2009) 口 Schramm & Silverman(2013) Decarli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010ab) McConnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2013) Targett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2012) Nesvadba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010) This work 10 11 12R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' : Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5: Black hole to stellar mass ratio as a function of redshift for our sample of radio sources (filled red dots) and for the galaxies in our comparison sample (open symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The dashed black lines correspond to the evolutionary model described by McLure et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2006), along with the associated 1σ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The color coding is reported in the legend at the bottom right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The error bars to the left of the legend show the typical ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 dex uncertainty in log(MBH/M⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' a non-negligible fraction of radio-loud AGN may experience a different stellar mass assembly path than radio-quiet AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We stress that these five radio sources are a subsample of the 9 out- liers of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 4, discussed above, and have high S/N line fluxes in Hβ or Mg ii, which yielded robust MBH estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The only exception is represented by G 721940, for which the Hβ emis- sion and associated FWHM are at lower S/N∼ 2, as highlighted in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The excess of overmassive SMBHs in radio-loud AGN sug- gests that their stellar and SMBH mass built up is regulated by their large scale radio jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' A possible scenario is that SMBHs of the sub-population with high MBH/M⋆ are mature, that is, their mass has been effectively assembled already by redshift z = 1, via accretion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Delvecchio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' On the other hand their stellar mass growth may have not occurred as effectively as in the overall AGN population, plausibly because of radio- mode AGN feedback (Fabian 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' While the accretion of hot gas onto the SMBH sustains the AGN activity and the SMBH growth, the large scale radio jets may prevent accretion and cool- ing of the inter-galactic medium gas, which is ultimately respon- sible for the stellar mass assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Altogether, we suggest that radio-mode AGN feedback results in the observed high values for MBH/M⋆ in radio-loud AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In order to investigate further this scenario, in the next Sections we link accretion and jet properties to the black hole mass by considering both the jet power and Eddington ratio of our radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We stress that the usual MBH − Mbulge or MBH − σ relations typically refer to the central spheroid, and not to the total stellar mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Kormendy & Ho 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' However, our radio-loud AGN sample is composed in a large majority of early-type galaxies, where the spheroid constitutes most of the stellar mass, and this approximation is justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Furthermore, because of the potential AGN contamination to the SED, the stel- lar mass may be biased high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This implies that MBH/M⋆ ratios can be even higher than reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' By considering MBH/M⋆ ra- tios as lower limits we would have an even stronger discrepancy, in particular for the subsample of high MBH/M⋆ radio sources mentioned above, with respect to the model prescriptions and the comparison sample of distant AGN, at fixed resdshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' All these results seem to corroborate the scenario that SMBH growth is more rapid than stellar mass assembly, and this is particularly true for distant radio sources, in comparison to the overall AGN population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Jet power, black hole mass, and accretion As mentioned in the previous sections, mechanical radio-mode AGN feedback can regulate the cooling of hot gas in the inter- galactic medium, and thus the stellar mass growth of the host galaxy itself as well as the accretion onto the central SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' To better understand the interplay between jet, black hole, and accretion properties, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 6 we show the jet power Qjet (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2), plotted against the black hole mass MBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The ra- dio sources of our sample are highlighted, while we also over- plot the comparison sources outlined in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 4 (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Balmaverde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Xiong & Zhang 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Our radio-loud AGN densely populate the upper right region of the Qjet-MBH plane, which is occupied by sources with high values of both the black hole mass (MBH ≳ 108M⊙) and the jet power (Qjet ≳ 1043 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Sources in the comparison sam- ple similarly occupy this region, while they extend as well to lower values of MBH (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Xiong & Zhang 2014) and jet power (Balmaverde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These results suggest that the distant radio-loud AGN, quite independently of the redshift, are almost invariably associated with massive black holes and powerful radio jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This is in agreement with the tight connec- tion existing between black hole accretion and jet production in powerful radio-loud AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Ghisellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Sbarrato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Inoue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Furthermore, HLRSs are characterized by a jet power that is typically higher with respect to LLRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' As discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6, these two classes have indeed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz rest frame luminosity densities typical of FR I and FR II radio galaxies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' As Qjet increases with the radio luminosity den- sity (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 4), high luminosity radio sources have higher Qjet val- ues than low luminosity ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Furthermore, the two classes of LLRGs and HLRGs are also delimited in the Qjet-MBH plane by the relation found in previous studies (Wu & Cao 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Chen 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 )60/ 中 口 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 0 VO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 - ★ ★ McLure et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2006) 口 Merloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010) ☆ McConnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2013) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 - Haring & Rix(2004) Jahnke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2009) Targett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2012) Shields et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2006) Schramm & Silverman (2013) Nesvadba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010) Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010) Decarli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010ab) This work Cisternas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2010) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 0 3 4 5 6 ZR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' : Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2015), originally used to distinguish between FR I and FR II radio galaxy populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The clear separation of LLRGs and HLRGs in the Qjet vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' MBH plane can be explained by com- bining both the MBH vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Mbulge relation for elliptical galaxies and the relation between Qjet and the host galaxy optical luminosity (Ledlow & Owen 1996) that separates the FR I and FR II radio galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The combination of these two relations also implies the observed spread of our sources in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Indeed, we did not find any significant correlation (as measured with the Spearman test) between Qjet and MBH for our radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Figure 7 displays instead the jet power, plotted against the Eddington ratio η (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3) for both our radio sources and the galaxies in the comparison sample (Xiong & Zhang 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Higher accretion rates favor more powerful jets to be launched by the central engine, as indeed the jet power increases with increasing Eddington ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' For our sample of ra- dio sources we find that the two quantities are well correlated at a level of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='9-σ (p − value = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='30 × 10−3), by means of the Spearman test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' No clear distinction in terms of η is found be- tween the two classes of low and high luminosity radio sources, which are distinguished in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' However, as pointed out in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 our radio sources have, on average, an Eddington ra- tio of log η = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This value is typical of radiatively effi- cient accretion disks, such as the Shakura & Sunyaev (1973) optically thick and geometrically thin accretion disk, which is commonly invoked to explain the optical-ultraviolet emission in Type I AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Ghisellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We can then estimate the accretion rate ˙M = Ldisc/(ϵ c2), where ϵ is the mass-to-light conversion efficiency for which we adopt the standard value ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1, typical of radiatively efficient disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' For our radio sources we obtain a median (mean) accre- tion rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='16 M⊙ yr−1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6 M⊙ yr−1), which corresponds to a substantial SMBH mass growth of ∆MBH = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6 × 106 M⊙, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 × 106 M⊙), over an AGN duty cycle with typical duration of ∼ 107 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Altogether, the fact that the SMBHs of the radio sources in our sample accrete at a sub-Edddington rate, irrespectively of their redshift, suggests that most of their mass has been likely built up at earlier epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Furthermore, while on one hand the observed accretion state sustains both the nuclear activity and the SMBH growth at sub-parsec scales, on the other hand it also ultimately favors the persistence of large scale radio jets, which may prevent the host galaxy to accrete gas at kilo-parsec scales and thus form stars effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This radio-mode AGN feedback may be responsible for the presence of overmassive SMBHs in our sample of radio-loud AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Indeed, it is worth mentioning that the five z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='37 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='95 radio sources with high MBH/M⋆ ratios, discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1, accrete a sub-Eddington rate of η ∼ 1%, while, on average, they have a normal jet power Qjet ∼ 1044 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Radio-loud AGN and their environments A substantial fraction (17 ± 5)% of our radio sources have high MBH/M⋆ ratios (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1), and may be early type galaxies at the center of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' For these galaxies the stellar mass as- sembly may have been halted, with reduced star formation ac- tivity, as typically found in cluster core ellipticals, while their black holes continue to grow via accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This interpretation is consistent with earlier studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Trujillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2014) as well as with the substantial fraction (19%) of radio sources in our sample with low SFR < 5 M⊙/yr, while many others have SFR upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' It is indeed known that cluster core early type Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 6: Jet power versus black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The diagonal dashed region represents the model reported found in previous studies (Wu & Cao 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2015) that distinguishes between FRI and FRII radio galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Sources from our sample are dis- tinguished between high luminosity and low-luminosity radio sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Sources from our comparison sample are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We refer to the legend for the color code adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 7: Plot of the jet power versus the Eddington ratio (η) of the associated black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In addition to our sample the data found in Xiong & Zhang (2014) and Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2006) are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' galaxies tend to be outliers in the MBH vs Mbulge relation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', McConnell & Ma 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' A famous example of a possibly over- massive black hole is the case of NGC 1277, a lenticular galaxy in the Perseus cluster (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', van den Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Emsellem 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Scharwächter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Motivated by these studies we looked in the literature for clusters around the radio sources in our sample, search- ing by coordinates in the NASA/IPAC Extragalactic Database (NED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' NED includes several catalogs of clusters identified in wide field surveys (Goto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Koester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' McConnachie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Durret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2011, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Radovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Rykoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Oguri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Our search yielded three matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Radio sources RS 49, G 372455, and G 748815 are in the cores (at cluster-centric distances ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 Mpc) of the clusters [LIK2015] J034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='16359-04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='73395 (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='89, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2015), WHL J090325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6+011215 (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='31, Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Wen & Han 2015), and HSCS J142538+002320 (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='33, Oguri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2018), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These clusters have redshifts that are 9 47 ☆★ ☆☆☆ ☆ ☆ ☆ 口 ☆ 46 ☆ ☆ 口 口 口 口 口 中 中 口 口 45 ☆ M ☆ 电 ☆ Qjet erg/s 口 Φ 中 口 口 ☆ ☆ ★ )60/ 口 44 口 43 ☆ ★ 8 Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2015) 口 Xiong & Zhang (2014) 42 Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2006) Balmaverde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2008) This work(High Radio Luminosity) Thiswork(LowRadio Luminosity) 41 7 8 MBH 9 10 11 10 MoXiong & Zhang (2014) 47 ☆ Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=" (2006) ☆ ★ ☆☆ This work(Low Radio Luminosity) ★ This work(High Radio Luminosity) ★ ★ ★ 46 ★★ 口 口 口 口 'erg/s 口 45 口 口 )60/ 口 ☆ 口 口 口 ★ 口 44 8 : 口 ☆ 口口 43 42 6 4 0 2 log(n)R." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' : Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 consistent with those of the radio sources as well as richness- based masses M200 ∼ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='9 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0) × 1014 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These are there- fore moderately massive clusters at intermediate-to-high red- shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The three associated radio sources have instead moder- ately overmassive black holes log(MBH/M⊙) ≃ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3, in particular in comparison to the stellar masses of the systems −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='65 ≲ log(MBH/M⋆) ≲ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='24 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These results support the above mentioned interpretation that the cluster environments tend to prevent the stellar mass assem- bly of cluster early-type galaxies, resulting in observed over- massive black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Nevertheless, it is worth mentioning that only three radio sources of our sample are found in clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' However, this is not surprising as clusters at higher redshifts (z ≳ 1) or with lower masses M200 ≲ 1 × 1014 M⊙ typical of rich groups are more difficult to find with current surveys and observational facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' It is thus possible that additional galax- ies are hosted in clusters, as distant radio sources are often found in dense mega-parsec scale environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Galametz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Castignani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Malavasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Golden-Marx et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Moravec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Discussion and Conclusions In this work we have investigated the evolution of distant radio- loud active galactic nuclei (AGN), as well their co-evolution with their host galaxies and their super massive black holes (SMBHs) at their center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' To this aim we have built a sam- ple of 42 radio-loud AGN, with spectroscopic redshift between z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='8, by cross matching the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz VLA FIRST point source catalog with available infrared-to-optical spectro- photometric surveys including SDSS and DES in optical, WISE in infrared, and the Galaxy And Mass Assembly (GAMA) spec- troscopic survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' As we are interested in assessing the SMBH masses, the 42 galaxies have been further selected by requiring broad emission lines in Hα, Hβ, Mg ii, or C iv, with full width at half maximum FWHM > 1000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Thanks to the available multi-wavelength photometry we modeled the spectral energy distributions (SEDs) of the sources in the sample, and then de- rived estimates to the stellar mass (M⋆) and the star formation rate (SFR) for all sources, while for GAMA sources we took them from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We find that the 42 radio sources are broadly consistent with the star forming main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' For all sources we then estimated i) the black hole mass MBH, based on single-epoch broad-line region spectra, ii) the black hole to stellar mass ratio MBH/M⋆, iii) the jet power Qjet, on the basis of the low frequency radio continuum emission, and iv) the Eddington ratio η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Although samples of distant AGN with SMBH mass estimates are rapidly growing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Shaw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Dabhade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Rakshit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Gloudemans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2022), the present study still represents one of the first where all the above quantities are de- rived simultaneously for a single sample of distant radio-loud AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Our radio sources have log(MBH/M⊙) ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1 − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 and high log(M⋆/M⊙) ≃ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2 − 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0, which is in agreement with the fact that radio-loud AGN are almost invariably associated with the most massive galaxies and black holes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Best et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Chiaberge & Marconi 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' While overall our sources follow the expected trends previously found in the literature, a sub- stantial fraction of our sources 9/42 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', 21%) have black hole masses log(MBH/M⊙) > 9 well above the values predicted by the scaling relations (Häring & Rix 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Sani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' DeGraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' In particular, five sources out of the nine (12% of the full radio source sample) are clearly overmassive outliers, hav- ing MBH/M⋆ > 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This fraction is remarkably higher than that of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6% found for AGN at similar redshifts from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These overmassive SMBHs are thus the high-z counterparts of low-z overmassive SMBHs found in previous studies of nearby early type galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', McConnell & Ma 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Trujillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Our results imply that the growth of black hole masses in at least a substantial fraction of radio loud AGN largely occurs at early epochs, while the early stellar mass assembly may not be so efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This population of radio-loud AGN with high MBH/M⋆ ratios have likely experienced a different stellar mass growth than other types of AGN, and we further investigated this scenario in terms of additional complementary probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Following early studies on nearby galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', McConnell & Ma 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Trujillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2014), we found that three of our radio-loud AGN with moderately overmassive SMBHs are hosted in clusters from the literature, while clusters and groups around the majority of the remaining radio-loud AGN will likely be detected with forthcoming surveys such as Euclid (Euclid Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' These results suggest that the cluster environments tend to prevent the stellar mass assembly of cluster early-type galaxies, possibly via radio-mode AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Concerning instead the nuclear accretion and jet properties, we found that the SMBHs of the radio sources in our sample accrete, on average, at a sub-Eddington rate (η ∼ 1%), where higher accretion rates favor more powerful jets to be launched by the central engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We also find that high jet powers (Qjet ≳ 1045 erg s−1) are invariably associated with high radio luminosity sources (L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz > 2 × 1032 erg s−1 Hz−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Altogether, the ob- served accretion state sustains both the nuclear activity and the SMBH growth at sub-parsec scales, while it ultimately favors the persistence of large scale radio jets, which may prevent the host galaxy to accrete gas at kilo-parsec scales and thus form stars effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Radio-mode AGN feedback may be responsible for the presence of overmassive SMBHs in our sample of radio-loud AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Targeted observations of the ionized and the molecular gas are nevertheless needed to further investigate the proposed radio- mode AGN feedback scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Future studies on larger and higher-redshift samples of radio-loud AGN will become pos- sible with the advent of forthcoming radio-to-optical surveys such as The Vera Rubin telescope and Euclid in infrared-optical, SKA in radio, as well as its pathfinders and precursors (LOFAR, ASKAP, and MeerKAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We thank the anonymous referee for helpful comments which contributed to improve the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' GC acknowledges the support from the grants ASI n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2018-23-HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 and PRIN-MIUR 2017 WSCC32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We thank Christophe Benoist for helpful discussion about the exploitation of DES data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This publication makes use of data products from the Wide- field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, and NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' WISE and NEOWISE are funded by the National Aeronautics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This research has made use of the NASA/IPAC Extragalactic Database (NED), which is op- erated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' This project used public archival data from the Dark Energy Survey (DES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Funding for the DES Projects has been provided by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Department of Energy, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' National Science Foundation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the Ministry of Science and Education of Spain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the Science and Technology FacilitiesCouncil of the United Kingdom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the Higher Education Funding Council for England,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the Kavli Institute of Cosmological Physics at the University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the Center for Cosmology and Astro-Particle Physics at the Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Financiadora de Estudos 10 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' : Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The Collaborating Institutions are Argonne National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the University of California at Santa Cruz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Centro de Investigaciones Energéticas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Medioambientales y Tecnológicas-Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the DES-Brazil Consortium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the University of Edinburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the Eidgenössische Technische Hochschule (ETH) Zürich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Fermi National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the Institut de Ciències de l’Espai (IEEC/CSIC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the Institut de Física d’Altes Energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the Ludwig-Maximilians Universität München and the associated Excellence Cluster Universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the National Optical Astronomy Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the University of Nottingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the OzDES Membership Consortium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the University of Pennsylvania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' SLAC National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Stanford University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' the University of Sussex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' and Texas A&M University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Based in part on observa- tions at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2006, AJ, 132, 2127 Kaspi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Brandt, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Maoz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2007, ApJ, 659, 997 Kennicutt, Robert C.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2007, ApJ, 660, 239 Kong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' & Ho, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2018, ApJ, 859, 116 Kormendy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Bender, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Magorrian, J.' metadata={'source': 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687, 848 Wen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' & Han, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2015, ApJ, 807, 178 Wen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Han, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', Mainzer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2010, AJ, 140, 1868 Wu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' & Cao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2008, ApJ, 687, 156 Xiong, D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 2000, AJ, 120, 1579 Zirbel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 1996, ApJ, 473, 713 Appendix A: Tables We report below some Tables summarizing the properties of the radio sources in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 12 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' : Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 ID RA Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' z log L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz erg s−1 Hz−1 log(Ldust/L⊙) log(M⋆/M⊙) SFR WISE class Type Name hh:mm:ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='ss dd:mm:ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='ss (M⊙/yr) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) RS 16 02:23:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='94 06:37:22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='217 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='48 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='40 32 AGN Radio Source SDSS J022334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='90-063722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='9 RS 49 02:16:40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='74 04:44:04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='875 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='65 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='95 <4801 AGN QSO FBQS J0216-0444 RS 52 02:19:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='89 05:18:05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='207 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='04 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='34 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='73 <235 AGN QSO 3XMM J021938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='8-051805 RS 60 02:19:54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='63 05:49:22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='322 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='08 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='13 AGN QSO SDSS J021954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='62-054922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2 RS 62 02:22:47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='91 04:33:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='635 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='02 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='43 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='02 <289 AGN QSO XXL-N 027_020 RS 79 02:22:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='95 05:18:15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='756 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='32 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='48 <225 AGN QSO PKS 0220-055 RS 81 02:26:07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='42 05:32:09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='779 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='53 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='76 36 Intermediate disk X-ray source XXL-N 062_013 RS 82 02:25:56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='39 05:34:51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='879 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='75 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='25 <605 AGN QSO XXL-N 062_009 RS 83 02:25:05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='12 05:36:47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='682 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='58 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='41 Intermediate disk QSO PMN J0225-0536 RS 104 02:27:12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='99 04:46:36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='981 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='62 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='15 <448 AGN X-ray source SPIRE 13430 RS 113 02:45:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='53-002612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2 RS 151 02:27:40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='56 04:02:51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='603 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='07 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='69 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='54 <5265 AGN QSO XXL-N 044_070 RS 159 02:29:15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='79 04:42:15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='074 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='29 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='28 <21 AGN QSO 3XMM J022915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='7-044216 RS 177 02:51:56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='32 +00:57:06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} 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02:47:06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='66 +00:23:18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='363 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='03 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='67 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='94 <50 AGN QSO FBQS J0247+0023 RS 197 02:46:16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='61 +00:19:53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='791 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='02 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='61 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='11 44 Starburst/LIRG QSO SDSS J024616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='60+001953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6 RS 205 02:53:40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='94 +00:11:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='683 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='73 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='23 <577 AGN QSO LBQS 0251-0001 RS 206 02:48:54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='80 +00:10:53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='145 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='92 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='05 <904 AGN QSO 2SLAQ J024854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='80+001053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='9 RS 214 02:50:48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='66 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='02 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='35 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='80 <238 AGN Radio Source NVSS J142533+010739 G 714228 14:31:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='49 01:14:56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} 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+page_content='2 G 720847 08:47:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='80 01:30:01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='417 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='45 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='11 30 AGN QSO WISEA J084702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='78+013001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='5 G 721940 14:21:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='03 02:13:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='640 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='89 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='94 <75 Intermediate disk G SDSS J142130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='60+021308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='9 G 745066 14:51:22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='48 00:33:41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='377 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='06 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='81 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='72 45 AGN QSO WISEA J145122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='47-003341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='0 G 746605 12:21:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='95 00:07:33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='364 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='10 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='08 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='70 <131 AGN G SDSS J122103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='51-000749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1 G 748144 11:39:54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='20 00:13:47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='589 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='03 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='09 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='67 <109 AGN Radio Source GAMA J113952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='95+001348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6 G 748815 14:25:45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='91 00:22:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='326 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='31 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='87 14 AGN QSO WISEA J142545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='90+002242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='7 G 804203 09:20:53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='32 00:03:53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='506 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='03 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='93 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='79 41 Starburst/LIRG G SDSS J092053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='32+000353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='9 G 835899 08:42:16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='99 01:09:17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='762 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='06 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='19 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='83 <70 AGN G WISEA J084217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='25+010834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='6 G 887308 14:37:01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='00 01:03:49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='547 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='03 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='47 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 Intermediate disk G SDSS J143702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='15-010357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='2 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='1: Properties of the radio sources in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Column description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (1) galaxy ID;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2-3) RA and Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' coordinates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (4) spectroscopic redshift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='4 GHz rest frame luminosity density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (6-7) SED-based dust luminosity and stellar mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (8) star formation rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (9) WISE color-based class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (10-11) source type and name as found in the NED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 13 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Poitevineau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' : Black hole and galaxy co-evolution in radio-loud AGN at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='3 − 4 ID z Line FWHM log(Lline/(erg s−1)) log(MBH/M⊙) log(Qjet/(erg s−1)) log(LBLR/(erg s−1)) log η (103 km s−1) (1) (2) (3) (4) (5) (6) (7) (8) (9) RS 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='22 Mg ii 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='42 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='07 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='12 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='81 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='60 RS 49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='88 Hβ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='11 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='30± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='04 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='69 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='34 RS 52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='21 Mg ii 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='43 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='14 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='13 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='94 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='48 RS 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='32 Hα 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='07 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='04 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='22 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='80 RS 62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='64 Mg ii 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='11 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='08 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='90 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='07 RS 79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='76 Mg ii 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='16 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='03 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='08 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='60 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='30 RS 81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='78 Mg ii 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='47 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='07 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='14 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='18 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='60 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='30 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='24 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='83 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='55 RS 83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='68 Hβ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='03 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='67 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='14 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='23 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='70 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='57 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='80 RS 104 0.' metadata={'source': 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(1) (2) (3) (4) (5) (6) (7) (8) (9) G 714133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='56 Hβ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='20 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='19 a 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content='74 ± 0.' metadata={'source': 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of the radio sources in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Column description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (1) source ID;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (2) spectroscopic redshift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (3-5) broad emission line, FWHM, and line luminosity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (6) single-epoch SMBH mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (7) jet power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (8) BLR luminosity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' (9) Eddington ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' Sources marked with the symbol a in column (4) have uncertain FWHM errors from GAMA, when the lines were fitted with a single Gaussian component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' To overcome this limitation we considered the GAMA fits that include both narrow and broad components of the emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' We then assumed a FWHM relative error equal to that resulting from the fit to the broad component of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} +page_content=' 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E4T4oBgHgl3EQfow0v/content/2301.05186v1.pdf'} diff --git a/cNAyT4oBgHgl3EQfXPfd/vector_store/index.faiss b/cNAyT4oBgHgl3EQfXPfd/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e38f52bc46b40720476e6774f75cfd8050c555f5 --- /dev/null +++ b/cNAyT4oBgHgl3EQfXPfd/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8308a2718af13af678180f8efde9529f168c28d305e18bafe2b73909a2677837 +size 3080237 diff --git a/cNAyT4oBgHgl3EQfXPfd/vector_store/index.pkl b/cNAyT4oBgHgl3EQfXPfd/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..1e04fccc26a4d6a287329a14317831f80a948a75 --- 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Under review. +NEURAL RADIANCE FIELD CODEBOOKS +Matthew Wallingford1, Aditya Kusupati1, Alex Fang1, Vivek Ramanujan1, +Aniruddha Kembhavi2, Roozbeh Mottaghi2, Ali Farhadi1 +1University of Washington, 2PRIOR, Allen Institute for AI +ABSTRACT +Compositional representations of the world are a promising step towards enabling +high-level scene understanding and efficient transfer to downstream tasks. Learning +such representations for complex scenes and tasks remains an open challenge. To- +wards this goal, we introduce Neural Radiance Field Codebooks (NRC), a scalable +method for learning object-centric representations through novel view reconstruc- +tion. NRC learns to reconstruct scenes from novel views using a dictionary of +object codes which are decoded through a volumetric renderer. This enables the +discovery of reoccurring visual and geometric patterns across scenes which are +transferable to downstream tasks. We show that NRC representations transfer well +to object navigation in THOR, outperforming 2D and 3D representation learning +methods by 3.1% success rate. We demonstrate that our approach is able to perform +unsupervised segmentation for more complex synthetic (THOR) and real scenes +(NYU Depth) better than prior methods (29% relative improvement). Finally, we +show that NRC improves on the task of depth ordering by 5.5% accuracy in THOR. +1 +INTRODUCTION +Parsing the world at the abstraction of objects is a key characteristic of human perception and +reasoning (Rosch et al., 1976; Johnson et al., 2003). Such object-centric representations enable +us to infer attributes such as geometry, affordances, and physical properties of objects solely from +perception (Spelke, 1990). For example, upon perceiving a cup for the first time one can easily infer +how to grasp it, know that it is designed for holding liquid, and estimate the force needed to lift it. +Learning such models of the world without explicit supervision remains an open challenge. +Unsupervised decomposition of the visual world into objects has been a long-standing challenge (Shi +& Malik, 2000). More recent work focuses on reconstructing images from sparse encodings as an +objective for learning object-centric representations (Burgess et al., 2019; Greff et al., 2019; Locatello +et al., 2020; Lin et al., 2020; Monnier et al., 2021; Smirnov et al., 2021). The intuition is that +object encodings which map closely to the underlying structure of the data should provide the most +accurate reconstruction given a limited encoding size. Such methods have shown to be effective at +decomposing 2D games and simple synthetic scenes into their parts. However, they rely solely on +color cues and do not scale to more complex datasets (Karazija et al., 2021; Papa et al., 2022). +Advances in neural rendering (Mildenhall et al., 2021; Yang et al., 2021) have enabled learning +geometric representations of objects from 2D images. Recent work has leveraged scene reconstruction +from different views as a source of supervision for learning object-centric representations (Stelzner +et al., 2021; Yu et al., 2021b; Sajjadi et al., 2022a; Smith et al., 2022). However, such methods have +a few key limitations. The computational cost of rendering scenes grows linearly with the number +of objects which inhibits scaling to more complex datasets. Additionally, the number of objects per +scene is fixed and fails to consider variable scene complexity. Finally, objects are decomposed on a +per scene basis, therefore semantic and geometric information is not shared across object categories. +With this in consideration we introduce, Neural Radiance Codebooks (NRC). NRC learns a codebook +of object categories which are composed to explain the appearance of 3D scenes from multiple views. +By reconstructing scenes from different views NRC captures reoccurring geometric and visual +patterns to form object categories. This learned representation can be used for segmentation as well +as geometry-based tasks such as object navigation and depth ordering. Furthermore, NRC resolves +the limitations of current 3D object-centric methods. First, NRC’s method for assigning object +1 +arXiv:2301.04101v1 [cs.CV] 10 Jan 2023 + +Preprint. Under review. +Code 1 +(Couch) +Code 3 +(Fridge) +Code 2 +(Floor) +Figure 1: Visualization of learned codes. The NRC codebook encodes reoccurring geometric and +visual patterns. In the top row, couches of differing appearance are grouped by geometric structure. +In the middle row, different textured floors are categorized based on their shared planar geometry. In +the bottom row, NRC learns correspondences between fridges from different views and scenes. +codes to regions of the image enables constant rendering compute whereas that of other methods +scales with number of objects. Second, we introduce a novel mechanism for differentiably adding +new categories which allows the codebook to scale with the complexity of the data. Last, modeling +intra-category variation in conjunction with the codebook enables sharing of semantic and geometric +object information across scenes. +We evaluate NRC on unsupervised segmentation, object navigation, and depth ordering. For seg- +mentation on indoor scenes from ProcTHOR (Deitke et al., 2022) we show 29.4% relative ARI +improvement compared to current 3D object-centric methods (Stelzner et al., 2021; Yu et al., 2021b). +On real-world images (NYU Depth (Silberman et al., 2012)) we show promising qualitative results +(Figure 3) and 29% relative improvement. For object navigation and depth ordering, where geo- +metric understanding is relevant, we observe 3.1% improvement in navigation success rate 5.5% +improvement in depth ordering accuracy over comparable self-supervised and object-centric methods. +Interestingly, we find qualitative evidence that the learned codes categorize objects by both visual +appearance and geometric structure (Figure 1). +2 +RELATED WORK +Object-Centric Learning +Object-centric learning aims to build compositional models of the world +from building blocks which share meaningful properties and regularities across scenes. Prior works +2 + +HPreprint. Under review. +such as MONet (Burgess et al., 2019), IODINE (Greff et al., 2019), Slot Attention (Locatello et al., +2020), and Monnier et al. (2021) have demonstrated the potential for disentangling objects from +images. Other work has shown the ability to decompose videos (Kabra et al., 2021; Kipf et al., 2021). +In particular, Marionette (Smirnov et al., 2021) learns a shared dictionary for decomposing scenes of +2D sprites. We draw inspiration from MarioNette for learning codebooks, but differ in that we model +the image formation process and intra-code variation, and dynamically add codes to our dictionary. +3D Object-Centric Learning +Recent work has shown novel view reconstruction to be a promising +approach for disentangling object representations. uORF (Yu et al., 2021b) and ObSuRF (Stelzner +et al., 2021) combine Slot Attention with Neural Radiance Fields (Mildenhall et al., 2021) to +decompose scenes. COLF (Smith et al., 2022) replaces the volumetric renderer with light fields +to improve computational efficiency. NeRF-SOS Fan et al. (2022) uses contrastive loss for both +geometry and appearance to perform object segmentation. SRT (Sajjadi et al., 2022b) encodes scenes +into a set of latent vectors which are used to condition a light field. OSRT (Sajjadi et al., 2022a) +extends SRT by explicitly assigning regions of the image to latent vectors. NeSF Vora et al. (2021) +learns to perform 3D object segmentation using NeRF with 2D supervision. Although great progress +has been made, these methods are limited to synthetic and relatively simple scenes. Our work differs +from previous 3D object-centric works in that we learn reoccurring object codes across scenes and +explicitly localize the learned codes. Additionally, our method can model an unbounded number of +objects per scene compared to prior work which fixes this hyper-parameter a priori. We show that our +approach generalizes to more complex synthetic and real-world scenes. +Neural Rendering +Advances in neural rendering, in particular Neural Radiance Fields +(NeRF) (Mildenhall et al., 2021), have enabled a host of new applications (Jang & Agapito, 2021; +Mildenhall et al., 2022; Pumarola et al., 2021; Park et al., 2021; Lazova et al., 2022; Niemeyer & +Geiger, 2021; Zhi et al., 2021). NeRF differentiably renders novel views of a scene by optimizing a +continuous volumetric scene function given a sparse set of input views. Original formulation of NeRF +learned one representation for each scene; other works (Yu et al., 2021a; Jain et al., 2021; Kosiorek +et al., 2021) showed conditioning NeRFs on images enables generation of novel views of new scenes. +Dictionary/Codebook Learning +Dictionary (codebook) learning (Olshausen & Field, 1997) in- +volves learning of a specific set of atoms or codes that potentially form a basis and span the input +space through sparse combinations. Codebooks have been widely used for generative and discrimina- +tive tasks across vision (Elad & Aharon, 2006; Mairal et al., 2008), NLP (Mcauliffe & Blei, 2007) +and signal processing (Huang & Aviyente, 2006). Learning sparse representations based on codes +enables large-scale methods which rely on latent representations. More recently, codebooks have +been shown to be crucial in scaling discrete representation learning (Van Den Oord et al., 2017; +Kusupati et al., 2021). Marionette (Smirnov et al., 2021) is an object-centric representation learning +method that relies on codebooks, unlike most other methods that are developed around set latent +representations (Sajjadi et al., 2022a;b; Locatello et al., 2020; Yu et al., 2021b). Object-centric +codebooks help in semantic grounding for transfer between category instances and are important for +large-scale representation learning across diverse scenes and objects. +3 +METHOD +Our goal is to discover object categories without supervision, learn priors over their geometry and +visual appearance, and model the variation between instances belonging to each group. Given +multiple views of a scene, the objective is to explain all views of the scene given a set of object-codes. +This learned decomposition can be used for segmentation and other downstream tasks that require +semantic and geometric understanding such as depth ordering and object-navigation. +Figure 2 illustrates the training pipeline. We begin by processing the image through a convolutional +network to obtain a spatial feature map. The feature map is then projected to a novel view using +the relative camera matrix. Feature vectors from each respective region of the image are assigned +to categorical latent codes from the finite-size codebook. The object codes and feature vectors are +passed to a convolutional network which transform the categorical codes to fine-grained instance +codes. A volumetric renderer is then conditioned on the instance code, view direction, and positional +encodings to render each region of the scene from the novel view. The rendered image from the +3 + +Preprint. Under review. +Input View (𝐼!) + − +! +! +Codebook +𝑙!, 𝑙" … , 𝑙# +Code Assignment +" 𝑙" · Τ +𝑒||$!%&" +#,%||& +∑ +𝑒||$'%&" +#,%||& +' +()* +' +")* +Encoder +Novel View (𝐼!′) +𝐺! +Variation Module +MLP +𝑓! +",$ +Render +(𝑹𝑮𝑩𝝈) +𝒍𝒊 +0.5 +𝜖 +𝐻! +𝐹! +Positional +Encoding +Figure 2: An overview of NRC. We learn a set of shared codes for decomposing scenes into objects. +Each point in the scene is assigned one of n latent codes from the codebook. The variation module +models the intra-code variation between objects by perturbing the code in latent space. A conditional +NERF model renders the scene and is compared to the ground truth novel view for supervision. +novel view is compared to the ground truth using L2 pixel loss. The categorical codes, assignment +mechanism, and volumetric renderer are learned jointly in an end-to-end fashion. +Image Encoding and Camera Projection +Given an input frame and novel image, Ih, I′ +h ∈ +R3×H×W respectively from scene Sh, we first encode Ih into a spatial feature map, fh ∈ +Rd×H/k×W/k, using a convolutional network, Fθ. We project each point, (x, y, z), in world co- +ordinates of the novel view to camera coordinates in the input frame, (x, y), using the relative camera +pose. Given (x, y), we select the spatial feature f x,y +h +∈ Rd from the patch that contains the projected +coordinates. The spatial features vectors are then passed to the next stage where they are assigned to +categorical object-codes. +Assigning + Learning Codes +Our goal is to jointly learn a shared set of object categories and +priors about their appearance and geometry. By mapping the spatial features from a continuous vector +space to a discrete, finite set of codes the model is incentivized to find reoccurring patterns in the +images. +Given the features for a point in the novel view, f x,y +h +, we assign a code, l∗, chosen from the shared +codebook, L. We do so with an arg max 1-nearest-neighbors during inference: +l∗(x, y) ← +(STE) +arg max +li; i∈[k] +e−∥li−f x,y +h +∥2 +�k +j=1 e−∥lj−f x,y +h +∥2 +(1) +The nearest-neighbor assignment used during inference is a non-differentiable operation therefore +propagating gradients to the encoder and codebook would not be possible. To enable learning of the +codebook elements we use a softmax relaxation of nearest-neighbors during the backward pass in +conjunction with the straight-through-estimator (STE) Bengio et al. (2013): +l∗ +back(x, y) ← +e−∥li−f x,y +h +∥2 +�k +j=1 e−∥lj−f x,y +h +∥2 +(2) +Adding Categorical Codes +The number of codes should depend on the complexity of the scenes +they model. Learning when to add new codes is non-trivial because the number and selection of +codes is discrete and non-differentiable. To circumvent this problem, we use a series of step functions +with a straight-through-estimator (STE) to sequentially add elements to the codebook. Each code +li ∈ L is gated according to the following: +4 + +Preprint. Under review. +li ← T +� +σ(s − i2/λ), 1 +2 +� +· li; T (a, t) := +� +1, +a > t +0, +a ≤ t +(3) +T (.) is a binarization function in the forward pass and lets the gradients pass through using STE in +the back pass. σ(·) is the sigmoid function, λ is a scaling hyperparameter, and s is a learnable scoring +parameter whose magnitude is correlated with the overall capacity (number of codes) required to +model the scenes accurately. A new code li is added when s exceeds the threshold i2/λ. New codes +are initialized using a standard normal distribution. Throughout training we keep k + 1 total codes +where k is the current number of learnable codes. The extra code is used by the straight-through- +estimator to optimize for s on the backward pass. This formulation can be viewed as the discrete +analog of a gaussian prior over the number of elements, k in the codebook: P(k) = e +−k2/λ. +Modeling Intra-Code Variation +Once a categorical code has been assigned to a region in the +novel frame, the model must account for variation across instances. We model this variation in +latent space using an encoder that takes in both the spatial feature, f x,y +h +, and the categorical code +l∗(x, y). We rescale the norm of the variation vector by ϵ, a hyperparameter, to ensure the instance +and categorical codes are close in latent space. The instance code is formulated as the following: +l∗ +instance(x, y) = l∗(x, y) + ϵ · +Gθ′([l∗(x, y), f x,y +h +]) +||Gθ′([l∗(x, y), f x,y +h +])||2 +. +(4) +We concatenate f x,y +h +and l∗(x, y) as input to the variation module, Gθ′, which we model as a 3- +layer convolutional network. Gθ′ provides a d dimensional perturbation vector which models the +intra-category variation and transforms the categorical code to an instance level code. +Decoding and Rendering +Given the localized instance codes for a scene, we render it in the novel +view and compare with the ground truth using L2 pixel loss. Intuitively, object categories which +encode geometric and visual patterns should render the scene more accurately from novel views. Each +region of the scene is rendered using an MLP conditioned on the instance codes and the volumetric +rendering equation following the convention of NeRF Mildenhall et al. (2021): +Hˆθ(l∗ +instance(x, y), p, d) = (c, σ), +(5) +Here p = (x, y, z) is a coordinate in the scene, d ∈ R3 is a view direction, c is the RGB value at p +in the direction of d and σ is the volume density at that point. Recall that (x, y, z) corresponds to +(x, y) in the input frame Ih. We can project (x, y, z) into the camera coordinates of the novel view I′ +h +to get (x′, y′). This pixel (x′, y′) in the novel view corresponds to (x, y) in the input frame, meaning +they represent the same point in world coordinates. To get an RGB value for (x, y), we use volume +rendering along the ray from camera view Ih into the scene, given by +ˆC(r) = +� tf +tn +T(t) · σ(t) · c(t) · dt, +(6) +where T(t) = exp +� +− +� t +tn σ(s) · ds +� +models absorbance and tn and tf are the near and far field. +Given a target view with pose P, the ray to the target camera is given by r(t) = o + t · d where d is a +unit direction vector which passes through (x, y). The volume rendering for a particular pixel occurs +along this ray. Let d′ be the direction associated with the novel view (x′, y′) and r′(t) = o + t · d′. +The pixel intensity at (x′, y′) is given by ˆC′ = ˆC(r′) and our final loss is +L(Ih, I′ +h, x, y) = ∥ˆC′ − I′ +h(x′, y′)∥2 + s, +(7) +where I′ +h(x′, y′) is the ground-truth pixel value at (x′, y′). We penalize the scoring parameter, s, +from section 3 in the loss to encourage learning a minimal number of codes. +NRC for Downstream Tasks +Once the encoder, codebook, and MLP have been trained, we +evaluate the learned representation on various downstream tasks. To perform segmentation, we +process each image through the trained encoder, Fθ, to obtain the spatial feature map. Each feature +5 + +Preprint. Under review. +Input Image +NRC +Ground Truth +Figure 3: Unsupervised segmentation of real-world images. NRC segments scenes that have +significant object category overlap with ProcTHOR. We show the first results for object-centric +unsupervised segmentation of real-world scenes. +vector, fx,y in the spatial map is assigned to the nearest categorical code, l⋆ in the learned codebook. +The categorical codes are then designated to the corresponding pixel to obtain a segmentation mask. +Traditionally in object navigation, frames from the embodied agent are processed by a frozen, +pretrained network. The resulting feature vector is then passed to a policy network which chooses an +action. To assess the utility of the NRC representation, we replace the pretrained network with the +NRC encoder and codebook. We process each frame to obtain instance codes for each region of the +image which are then fed to the policy network. +Depth ordering task consists of predicting which of two objects is closer to the camera. To perform +depth ordering with NRC we predict a segmentation mask and depth map. To predict the depth map +we condition the trained MLP on the instance codes & predict the density, σ, along a given ray. We +estimate the transmittance to predict the depth following the method of Yu et al. (2021a). Depth map +and segmentation mask are combined to predict the average distance of each object from the camera. +4 +EXPERIMENTS +We evaluate our decomposition and representations on several downstream tasks: unsupervised +segmentation (real and synthetic), object navigation, and depth ordering. NRC shows improvement +over baseline methods on all three tasks. Prior works in object-centric learning have focused on +unsupervised segmentation for measuring the quality of their decomposition. We show that NRC +representations are also effective for downstream applications that require geometric and semantic +understanding of scenes such as object navigation and depth ordering. +4.1 +DATASETS +ProcTHOR & RoboTHOR +THOR (Kolve et al., 2017) consists of interactive home environments +built in the Unity game engine. We benchmark on the task of object navigation in RoboTHOR (Deitke +et al., 2020), a variant of the THOR environment aimed at sim2real benchmarking. Object navigation +consists of an agent moving through different scenes to locate specified objects. RoboTHOR consists +of 89 indoor scenes split between train, validation, and test. ProcTHOR (Deitke et al., 2022) consists +of procedurally generated indoor scenes similar to RoboTHOR. Examples of THOR scenes can be +found in Appendix C. +6 + +Preprint. Under review. +CLEVR-3D +CLEVR-3D (Johnson et al., 2017) is a synthetic dataset consisting of geometric +primitives from multiple views and is used for unsupervised segmentation. Following the convention +of Stelzner et al. (2021), we test on the first 500 scenes of the validation set and report foreground- +adjusted random index (FG-ARI). Adjusted random index (ARI) Yeung & Ruzzo (2001) measures +the agreement between two clusterings and is a standard metric for unsupervised segmentation. In +our case the two clusterings are the predicted and ground truth segmentations. Foreground adjusted +random index only measures the ARI for pixels belonging to foreground objects. For comparison +to prior works, we consider segmentations at both the class and instance level to be correct for +CLEVR-3D, ProcTHOR, and NYU Depth. Further details can be found in Appendix C. +NYU Depth +The NYU Depth Dataset (Silberman et al., 2012) consists of images from real-world +indoor scenes accompanied by depth and segmentation maps. Methods are trained on the ProcThor +dataset then evaluated on NYU Depth for segmentation. We chose NYU Depth because it has object +categories and scene layouts that are similar to THOR. We report the adjusted random index (ARI). +Table 1: Segmentation results (ARI) for NRC and comparable methods. We find that for more +complex datasets, ProcTHOR and NYU Depth, NRC outperforms other methods. +Method +ProcTHOR (ARI) +NYU Depth (ARI) +CLEVR-3D (FG-ARI) +MarioNette +.127 +.035 +- +uORF +.193 +.115 +.962 +ObSuRF +.228 +.141 +.978 +NRC +.295 +.182 +.977 +4.2 +UNSUPERVISED SEGMENTATION +Experimental Setup +We evaluate NRC, ObSuRF, uORF, and MarioNette for unsupervised seg- +mentation on ProcTHOR, CLEVR-3D, and NYU Depth. We compare with MarioNette because it +uses a similar code mechanism for reconstruction. We report FG-ARI on CLEVR-3D for comparison +to prior works and ARI on the other datasets. For NYU Depth evaluation we use the representations +trained on ProcTHOR and only consider classes that are seen in the training dataset. +Results +We find that for NYU Depth and ProcTHOR which have more complex layouts and object +diversity, NRC significantly outperforms other methods (Table 1). Figure 1 shows examples of the +object codes learned by ProcTHOR and Figure 3 shows segmentation examples of real-world images. +To our knowledge, this is the first object-centric method which has shown unsupervised segmentation +results for complex real-world images. We find that NRC categorizes similar objects across scenes +based on both geometry and visual appearance. In the top row of Figure 1, we find that couches of +similar shape are assigned to the same code despite differing visual appearance which indicates that +NRC codes capture geometric similarity. In the same figure, we show further examples where floors +of different texture are categorized by the same code. The improved segmentation performance and +geometric categorization of objects indicates that NRC can leverage more than simple color cues +which has been significant limitation for object-centric learning. +4.3 +OBJECT NAVIGATION +Experimental Setup +We design the object navigation experiments in THOR to understand how +well the learned representations transfer from observational data to embodied navigation (Anderson +et al., 2018; Batra et al., 2020). Object navigation consists of an embodied agent with the goal of +moving through indoor scenes to specified objects. The agent can rotate its camera and move in +discrete directions. At each step, agent is fed with the current RGB frame relayed by the camera. +For the representation learning component of the experiment we collect observational video data +from a heuristic planner (Appendix B), which walks through procedurally generated ProcTHOR +scenes. In total, the dataset consists of 1.5 million video frames from 500 indoor scenes. For further +dataset details and example videos see Appendix C. +After training on ProcThor videos, we freeze the visual representations following standard prac- +tice (Khandelwal et al., 2022). We train a policy using DD-PPO (Wijmans et al., 2019) for 200M +7 + +Preprint. Under review. +Table 2: Results for object navigation on RoboTHOR object navigation. Visual representations are +trained on observations from 500 scenes of ProcThor. A policy is learned on top of the frozen visual +representations by training on the object navigation task in RoboTHOR training scenes. The results +are obtained by evaluating on RoboTHOR test scenes. +Method +Success Rate (%) +SPL +uORF (Yu et al., 2021b) +31.3 +.146 +ImageNet Pretraining +33.4 +.150 +ObSuRF (Stelzner et al., 2021) +38.9 +.167 +Video MoCo (Feichtenhofer et al., 2021) +43.9 +.184 +EmbCLIP (Khandelwal et al., 2022) +47.0 +.200 +NRC (Ours) +50.1 +.239 +steps on the training set of RoboTHOR then evaluate on the test set. We report success rate (SR) +and success weighted by path length (SPL). Success is defined as the agent signaling the stop action +within 1 meter of the goal object with it in the view. SPL is defined as 1 +N +�N +i=1 Si +ℓi +max(pi,ℓi), where li +is the shortest possible path, pi is the taken path, & Si is the binary indicator of success for episode i. +We compare with the following baselines: ObSuRF, uORF, Video MoCo (Feichtenhofer et al., 2021), +and EmbedCLIP (Khandelwal et al., 2022). ObSuRF and uORF are 3D, object-centric methods, and +Video MoCo is a contrastive video representation learning method. We include Video MoCo for +comparison as it was designed for large-scale, discriminative tasks, while ObSuRF and uORF were +primarily intended for segmentation. See Appendix B for further implementation details of baselines. +Results +Table 2 shows the performance of NRC and baselines on RoboTHOR object navigation. +NRC outperforms the best baseline by 3% in success rate (SR) and by a 20% relative improvement +in SPL. These performance gains indicate that the geometrically-aware NRC representation provide +an advantage over traditional representations such as EmbCLIP and Video MoCo. Another key +observation is that NRC has at least 18.8% improvement in SR and relative SPL over the recent +object-centric learning methods, uORF and ObSuRF. In particular, NRC performs better than ObSuRF +and uORF when navigating near furniture and other immovable objects (see supplemental for object +navigation videos). We hypothesize this is due to NRC’s more precise localization of objects. +4.4 +DEPTH ORDERING +Experimental Setup +We evaluate NRC on the task of ordering objects based on their depth from +the camera. Understanding the relative depth of objects requires both geometric and semantic +understanding of a scene. For this task we evaluate on the ProcTHOR test dataset which provides +dense depth and segmentation maps. Following the convention of Ehsani et al. (2018), we determine +ground truth depth of each object by computing the mean depth of all pixels associated with its +ground truth segmentation mask. +For evaluation, we select pairs of objects in a scene and the goal is to predict which object is closer. +We take the segmentation that has the largest IoU with the ground truth mask as the predicted object +mask. To determine the predicted object depth, we compute the mean predicted depth of each pixel +associated with the predicted object mask. All representations are trained on the ProcTHOR dataset +and evaluated on the ProcTHOR test set. In total we evaluate 2,000 object pairs. +Table 3: We compare depth ordering results on RoboTHOR with other geometrically-aware represen- +tations. Given pairs of objects in the scene, the model must infer which object is closer. We report the +accuracy as the number of correct orderings over the total number of object pairs. +Method +Depth Order Acc. (%) +uORF (Yu et al., 2021b) +13.5 +ObSuRF (Stelzner et al., 2021) +18.3 +NRC (Ours) +23.8 +8 + +Preprint. Under review. +Results +Depth ordering requires accurate segmentation and depth estimation. Due to NRC’s +stronger segmentation performance and better depth estimation, we see 5.5% and 10.3% depth +ordering accuracy compared to ObSuRF and uORF respectively. The fine-grained localization of +categorical latent codes in NRC allows for better depth ordering over existing object-centric methods. +In particular, the other object-centric methods tend to assign object instances to the background which +leads to large errors in estimating the depth of objects (Figure 5). +4.5 +ABLATION STUDY +Table 4: Ablation study for modeling the intra-code variation and learning the number of codes +evaluated on unsupervised segmentation in ProcTHOR. Default fixed number of codes is set to 25. +Method + (Ablation) +ProcTHOR (ARI) +NRC +.182 +NRC + Learned # of Codes +.197 +NRC + Variation Module +.284 +NRC + Variation Module + Learned # of Codes +.295 +We present an ablation study on the ProcTHOR dataset to determine the effect of the variation module +and learnable codebook size on unsupervised segmentation performance. Quantitative results can +be found in Table 4. We observe that performance improves by ∼ 9% when intra-class variation is +explicitly modeled. Intuitively, allowing for small variation between instances of the same category +should lead to better representations and allow for greater expressiveness. +We also find that learning the number of codes moderately improves performance. However, if +number of codes is found via hyper-parameter tuning, performance is matched (Table 8). Nonetheless, +differentiably learning the codebook size avoids computationally expensive hyper-parameter tuning. +5 +LIMITATIONS +Novel view reconstruction requires camera pose, which is not available for most images and videos. +Some datasets such as Ego4D (Grauman et al., 2022) provide data from inertial measurement units +that can be used to approximate camera pose, although this approach is prone to drift. +An incorrect assumption that NRC and most object-centric prior works make is that scenes are static. +However, it rarely is the case that scenes are free of movement due to the physical dynamics of our +world. Recently, Kipf et al. (2021); Pumarola et al. (2021) made strides in learning representations +from dynamic scenes. +Although NRC is relatively efficient compared to the other NeRF based methods, the NeRF sampling +procedure is compute and memory intensive. Sajjadi et al. (2022a) and Smith et al. (2022) leverage +object-centric light fields to reduce memory and compute costs. The efficiency improvement from +modeling scenes as light fields is orthogonal to NRC and can be combined. +A final challenge inherent to novel view reconstruction is finding appropriate corresponding frames +of videos. For example, if two subsequent frames differ by a 60° rotation of the camera, then most of +the scene in the subsequent frame will be completely new. Therefore, constructing the content in the +novel view is ill-posed. Pairing frames with overlapping frustums is a potential solution, although the +content of the scene may not be contained in the intersecting volume of the frustums. +6 +CONCLUSION +Compositional, object-centric understanding of the world is a fundamental characteristic of human +vision and such representations have the potential to enable high-level reasoning and efficient transfer +to downstream tasks. Towards this goal, we presented Neural Radiance Field Codebooks (NRC), +a new approach for learning geometry-aware, object-centric representations through novel view +reconstruction. By jointly learning a shared dictionary of object codes through a differentiable renderer +and explicitly localizing object codes within the scene, NRC finds reoccurring geometric and visual +similarities to form objects categories. Through experiments, we show that NRC representations +improve performance on object navigation and depth ordering compared to strong baselines by 3.1% +success rate and 5.5% accuracy respectively. Additionally, we find our method is capable of scaling +9 + +Preprint. Under review. +to complex scenes with more objects and greater diversity. 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In general performance degrades (Table 5), though NRC +performance decreases less compared to uORF and ObSuRF. +Image +Ground Truth +Predicted +Figure 4: Segmentation examples of cluttered scenes. Further segmentation examples of cluttered, +real-world scenes from NYU Depth. NRC scales better to cluttered scenes compared to other methods +as the rendering and memory cost is constant in the number of objects. +A.2 +MATTERPORT-3D AND GIBSON EXPERIMENTS +We train NRC on scenes from MatterPort3D. Qualitative examples can be found in figure 6 and +quantitative results can be found in Table 6. We find NRC performs slightly worse on MatterPort3D +compared to on THOR likely due to the greater visual complexity. This is reinforced by the evidence +that the learned codebook size is 67 whereas for THOR it is on average ∼ 53. A greater number of +learned codes indicates that NRC requires more expressivity to accurately reconstruct Matterport3D +scenes. For a given scene we find corresponding frames by filtering for images within 20 degrees +viewing angle of each other. +We evaluate NRC and ObSuRF on the Habitat Point Navigation Challenge. We train NRC and +ObSuRF on data collected from random walks through the Habitat environment. We use a ResNet-50 +initialized with a MoCo backbone pretrained on ImageNet. We report success rate, shortest-path +length, and distance to goal. Qualitative examples can be found in the supplemental. We find that +NRC outperforms ObSuRF and ImageNet pretraining. Qualitatively we find that NRC is better at +navigating around objects compared to ObSuRF. +14 + +Preprint. Under review. +Figure 5: Predicted object depth maps for depth ordering. The predicted object masks are +overlayed with the predicted depth map. We average the per pixel depth for each predicted object +mask to infer which object is closer. NRC’s object localization leads to sharp object masks and less +error when predicting average object depth. +B +IMPLEMENTATION DETAILS +B.1 +NRC +For the segmentation experiments we use the modified ResNet34 used by Yu et al. (2021b). For the +decoder we use a 3-layer with hidden dimension of 512. We train with a learning rate of 1e − 4 +with the ADAM optimizer. We set the near field to .4 and far field to 5.5. We train from scratch, +and from the video MoCo model trained on ProcTHOR. We found starting from the pretrained +initialization reduced the number of training epochs required for convergence. For the model from +scratch we train for 300 epochs. For the model from MoCo initialization we train for 100 epochs. +While training we use a sliding window of 5 frames to determine corresponding images in ProcTHOR. +By corresponding images we mean the image given to the model and the ground truth novel view +image. Given a sliding window of n frames we randomly select 2 images from this interval. We +ablate over 2, 5, and 10 frames. We found 5 images led to the best segmentation results and therefore +used that window size for object-navigation. +For object-navigation we use a ResNet50 for fair comparison to CLIP and Video MoCo. We train +wit the same hyper-parameters as discussed above for segmentation with the only change being the +architecture. We train for 200M steps using PPO with default hyper-parameters to match Khandelwal +et al. (2022) in RoboTHOR. +Table 5: Performance on cluttered scenes of NYU Depth. We filter for scenes with 5 or more +objects and report the ARI of NRC and comparable methods. NRC outperforms ObSuRF and uORF +particularly on cluttered scenes, as it can support an unbounded number of objects. +Method +NYU Depth (Hard) +uORF +.046 +ObSuRF +.092 +NRC +.139 +15 + +Preprint. Under review. +Image +Ground Truth +Predicted +Figure 6: Segmentation examples for MatterPort3D. Segmentation examples on MatterPort3D. +We train NRC on ∼ 190,000 images of MatterPort3d and learn 67 categorical codes. This increase +in the learned number of codes is likely due to the increased visual complexity of MatterPort3D +compared to THOR. We find similar segmentation results to those on THOR and NYU Depth. +B.2 +UORF +We use the implementation available at https://github.com/KovenYu/uORF We train on +the training set of ProcTHOR which contains ∼ 1.5 million images. We train three different variations +of the uORF no GAN model: from scratch, fine-tune with encoder and decoder pre-trained on +CLEVR-567, and fine-tune with all three components pre-trained on CLEVR-567. For variations that +train the slot attention component from scratch, we try setting the number of slots to 8 (default) and +16. We train for 10 epochs when loading from a pre-trained model, and for 15 epochs when training +from scratch; roughly half of the epochs are coarse epochs, and we set percept_in to roughly one +third of the epochs. We train for less epochs because our dataset is much larger. Our images within +scene batch size is 4, and we turn off shuffling to ensure that batches contain images that are close +together within the scene. We pass in the fixed locality parameter, but also set the decoder’s locality +parameter to false. To compensate between differences between our dataset and CLEVR-567, we +tune nss scale, object scale, near plane, and far plane. We train models on a single NVIDIA A40. +B.3 +OBSURF +We use the official implementation provided by Stelzner et al. (2021) at https://github.com/ +stelzner/obsurf. We trained 3 variations of the ObSuRF model: onfrom scratch, from CLEVR- +Table 6: Unsupervised segmentation performance of NRC and ObSuRF on MatterPort3D. We +find NRC performs slightly worse on MatterPort3D compared to on THOR likely due to the greater +visual complexity. This is reinforced by the observation that NRC learns 67 codes for MatterPort3D +whereas 53 are learned for THOR. +Method +MatterPort3D +ObSuRF +.178 +NRC +.237 +16 + +Preprint. Under review. +Table 7: Habitat Point Navigation Challenge. The performance of various visual encoders on the +point navigation challenge. We find that NRC outperforms other 3D-object centric methods due to its +ability to scale to more objects and visually complex environments. +Method +SPL +SR +Goal Dist +ImageNet Pretrain +.82 +.94 +.73 +ObSuRF +.77 +.84 +.81 +NRC +.85 +.96 +.68 +3D pretrained weights, and from MultiShapeNet pretrained weights. For the pretrained models we +lowered the learning rate by a factor of 10 to 1e−5. For training from scratch we used the default +learning rate of 1e − 4. For the number of slots we tried the default number, 5, and 10. We report the +results of the best model which was from scratch with 10 slots. For object navigation experiments +in RoboTHOR, we used the model with the highest ARI which was training from scratch with 10 +slots. We learn a policy network comprised of a 1-layer GRU with 512 hidden units which maps to a +6 dimensional logit and scalar which are used for the actor-critic. We train with DD-PPO for 200M +steps. For the encoder, we use a ResNet-50 architecture. +To construct the feature vector fed to the GRU we use the output of the ResNet-50 which is 2048×7×7. +We pass this through a 2-convolutional layers to sample down to 32 × 7 × 7 then concatenate this +with the trainable goal state embedding matrix which is 32x7x7. This visual feature vector of shape +64 × 7 × 7 is then passed through another 2-layer convolutional network to be of size 32 × 7 × 7. +B.4 +OSRT +We do not compare with OSRT Sajjadi et al. (2022a) on data sets that were not evaluated in the +original paper. We contacted the authors, but a public implementation has not been released yet at the +time of this work. +C +DATASETS +C.1 +PROCTHOR +The ProcTHOR dataset Deitke et al. (2022) consists of 10,000 procedurally generated indoor rooms +in the THOR environment. For our dataset we collect 10 video sequences of 300 actions from 500 +randomly chosen training scenes. Each actions consisting of either rotation or stepping in a specified +direction. Actions are determined by a heuristic planner which moves throughout the scene. In total +we collect 1.5 millions image frames. See Figure 7 for a sample of the dataset. +For evaluation of ARI we consider the 20 largest objects, floor, and ceiling. uORF and ObSuRF +decompose scenes into 5-10 objects and therefore do not handle scenes with many small objects. +We evaluate ARI only on the 20 largest objects in THOR which are the following: {ArmChair, +Bathtub, BathtubBasin, Bed, Cabinet, Drawer, Dresser, Chair, CounterTop, Curtains, Desk, Desktop, +CoffeeTable, DiningTable, SideTable, Floor, Fridge, Television, TVStand, Toilet.} +C.2 +CLEVR-3D +CLEVR-3D (Johnson et al., 2017) is a synthetic dataset which consists of geometric primitives placed +on a monochrome background. Following the convention of (Stelzner et al., 2021) we evaluate on +this benchmark for unsupervised object segmentation. Following convention we evaluate on the first +500 scenes of the validation set and report foreground - adjusted random index. For class level labels +we consider objects with the shape to be of the same class. +17 + +Preprint. Under review. +Figure 7: Snippets of Videos from ProcTHOR dataset used for training NRC. +Table 8: Performance of NRC with a fixed codebook of various sizes on ProcTHOR. +Codebook Size +ProcTHOR (ARI) +10 +.247 +20 +.256 +30 +.266 +40 +.279 +50 +.286 +60 +.293 +70 +.288 +80 +.295 +90 +.279 +100 +.273 +18 + +Preprint. Under review. +Predicted +Novel View +Input View +Figure 8: Reconstruction examples from NRC. For scenes and objects that are closer to the training +distribution, reconstruction quality is significantly better. +C.3 +INSTANCE LEVEL ACCURACY +For comparison to the instance level classification performed by ObSuRF and uORF, we filtered for +images in ProcTHOR which have one object of each class. In this setting, class and instance level +classification are equivalent. +Table 9: The size of the learned codebook averaged over 5 runs on each dataset. +Codebook Size +CLEVR-3D +15.6 ± 1.2 +ProcTHOR +53.8 ± 2.4 +19 + diff --git a/cNE2T4oBgHgl3EQfwgg2/content/tmp_files/load_file.txt b/cNE2T4oBgHgl3EQfwgg2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..82ecf3166bafb4186128ffd61a270a7c72c60802 --- /dev/null +++ b/cNE2T4oBgHgl3EQfwgg2/content/tmp_files/load_file.txt @@ -0,0 +1,806 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf,len=805 +page_content='Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NEURAL RADIANCE FIELD CODEBOOKS Matthew Wallingford1, Aditya Kusupati1, Alex Fang1, Vivek Ramanujan1, Aniruddha Kembhavi2, Roozbeh Mottaghi2, Ali Farhadi1 1University of Washington, 2PRIOR, Allen Institute for AI ABSTRACT Compositional representations of the world are a promising step towards enabling high-level scene understanding and efficient transfer to downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Learning such representations for complex scenes and tasks remains an open challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' To- wards this goal, we introduce Neural Radiance Field Codebooks (NRC), a scalable method for learning object-centric representations through novel view reconstruc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NRC learns to reconstruct scenes from novel views using a dictionary of object codes which are decoded through a volumetric renderer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' This enables the discovery of reoccurring visual and geometric patterns across scenes which are transferable to downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We show that NRC representations transfer well to object navigation in THOR, outperforming 2D and 3D representation learning methods by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='1% success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We demonstrate that our approach is able to perform unsupervised segmentation for more complex synthetic (THOR) and real scenes (NYU Depth) better than prior methods (29% relative improvement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Finally, we show that NRC improves on the task of depth ordering by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='5% accuracy in THOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 1 INTRODUCTION Parsing the world at the abstraction of objects is a key characteristic of human perception and reasoning (Rosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Such object-centric representations enable us to infer attributes such as geometry, affordances, and physical properties of objects solely from perception (Spelke, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For example, upon perceiving a cup for the first time one can easily infer how to grasp it, know that it is designed for holding liquid, and estimate the force needed to lift it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Learning such models of the world without explicit supervision remains an open challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Unsupervised decomposition of the visual world into objects has been a long-standing challenge (Shi & Malik, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' More recent work focuses on reconstructing images from sparse encodings as an objective for learning object-centric representations (Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Greff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Monnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Smirnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The intuition is that object encodings which map closely to the underlying structure of the data should provide the most accurate reconstruction given a limited encoding size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Such methods have shown to be effective at decomposing 2D games and simple synthetic scenes into their parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' However, they rely solely on color cues and do not scale to more complex datasets (Karazija et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Papa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Advances in neural rendering (Mildenhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021) have enabled learning geometric representations of objects from 2D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Recent work has leveraged scene reconstruction from different views as a source of supervision for learning object-centric representations (Stelzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Sajjadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' However, such methods have a few key limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The computational cost of rendering scenes grows linearly with the number of objects which inhibits scaling to more complex datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Additionally, the number of objects per scene is fixed and fails to consider variable scene complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Finally, objects are decomposed on a per scene basis, therefore semantic and geometric information is not shared across object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' With this in consideration we introduce, Neural Radiance Codebooks (NRC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NRC learns a codebook of object categories which are composed to explain the appearance of 3D scenes from multiple views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' By reconstructing scenes from different views NRC captures reoccurring geometric and visual patterns to form object categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' This learned representation can be used for segmentation as well as geometry-based tasks such as object navigation and depth ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Furthermore, NRC resolves the limitations of current 3D object-centric methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' First, NRC’s method for assigning object 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='04101v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='CV] 10 Jan 2023 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Code 1 (Couch) Code 3 (Fridge) Code 2 (Floor) Figure 1: Visualization of learned codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The NRC codebook encodes reoccurring geometric and visual patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In the top row, couches of differing appearance are grouped by geometric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In the middle row, different textured floors are categorized based on their shared planar geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In the bottom row, NRC learns correspondences between fridges from different views and scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' codes to regions of the image enables constant rendering compute whereas that of other methods scales with number of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Second, we introduce a novel mechanism for differentiably adding new categories which allows the codebook to scale with the complexity of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Last, modeling intra-category variation in conjunction with the codebook enables sharing of semantic and geometric object information across scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We evaluate NRC on unsupervised segmentation, object navigation, and depth ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For seg- mentation on indoor scenes from ProcTHOR (Deitke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022) we show 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='4% relative ARI improvement compared to current 3D object-centric methods (Stelzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' On real-world images (NYU Depth (Silberman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2012)) we show promising qualitative results (Figure 3) and 29% relative improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For object navigation and depth ordering, where geo- metric understanding is relevant, we observe 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='1% improvement in navigation success rate 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='5% improvement in depth ordering accuracy over comparable self-supervised and object-centric methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Interestingly, we find qualitative evidence that the learned codes categorize objects by both visual appearance and geometric structure (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 2 RELATED WORK Object-Centric Learning Object-centric learning aims to build compositional models of the world from building blocks which share meaningful properties and regularities across scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Prior works 2 HPreprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' such as MONet (Burgess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2019), IODINE (Greff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2019), Slot Attention (Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2020), and Monnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2021) have demonstrated the potential for disentangling objects from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Other work has shown the ability to decompose videos (Kabra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Kipf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In particular, Marionette (Smirnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021) learns a shared dictionary for decomposing scenes of 2D sprites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We draw inspiration from MarioNette for learning codebooks, but differ in that we model the image formation process and intra-code variation, and dynamically add codes to our dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 3D Object-Centric Learning Recent work has shown novel view reconstruction to be a promising approach for disentangling object representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' uORF (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021b) and ObSuRF (Stelzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021) combine Slot Attention with Neural Radiance Fields (Mildenhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021) to decompose scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' COLF (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022) replaces the volumetric renderer with light fields to improve computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NeRF-SOS Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2022) uses contrastive loss for both geometry and appearance to perform object segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' SRT (Sajjadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022b) encodes scenes into a set of latent vectors which are used to condition a light field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' OSRT (Sajjadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022a) extends SRT by explicitly assigning regions of the image to latent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NeSF Vora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2021) learns to perform 3D object segmentation using NeRF with 2D supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Although great progress has been made, these methods are limited to synthetic and relatively simple scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Our work differs from previous 3D object-centric works in that we learn reoccurring object codes across scenes and explicitly localize the learned codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Additionally, our method can model an unbounded number of objects per scene compared to prior work which fixes this hyper-parameter a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We show that our approach generalizes to more complex synthetic and real-world scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Neural Rendering Advances in neural rendering, in particular Neural Radiance Fields (NeRF) (Mildenhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021), have enabled a host of new applications (Jang & Agapito, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Mildenhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Pumarola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Lazova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Niemeyer & Geiger, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Zhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NeRF differentiably renders novel views of a scene by optimizing a continuous volumetric scene function given a sparse set of input views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Original formulation of NeRF learned one representation for each scene;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' other works (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Jain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Kosiorek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021) showed conditioning NeRFs on images enables generation of novel views of new scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Dictionary/Codebook Learning Dictionary (codebook) learning (Olshausen & Field, 1997) in- volves learning of a specific set of atoms or codes that potentially form a basis and span the input space through sparse combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Codebooks have been widely used for generative and discrimina- tive tasks across vision (Elad & Aharon, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Mairal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2008), NLP (Mcauliffe & Blei, 2007) and signal processing (Huang & Aviyente, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Learning sparse representations based on codes enables large-scale methods which rely on latent representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' More recently, codebooks have been shown to be crucial in scaling discrete representation learning (Van Den Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Kusupati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Marionette (Smirnov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021) is an object-centric representation learning method that relies on codebooks, unlike most other methods that are developed around set latent representations (Sajjadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Locatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Object-centric codebooks help in semantic grounding for transfer between category instances and are important for large-scale representation learning across diverse scenes and objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 3 METHOD Our goal is to discover object categories without supervision, learn priors over their geometry and visual appearance, and model the variation between instances belonging to each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Given multiple views of a scene, the objective is to explain all views of the scene given a set of object-codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' This learned decomposition can be used for segmentation and other downstream tasks that require semantic and geometric understanding such as depth ordering and object-navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Figure 2 illustrates the training pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We begin by processing the image through a convolutional network to obtain a spatial feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The feature map is then projected to a novel view using the relative camera matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Feature vectors from each respective region of the image are assigned to categorical latent codes from the finite-size codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The object codes and feature vectors are passed to a convolutional network which transform the categorical codes to fine-grained instance codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' A volumetric renderer is then conditioned on the instance code, view direction, and positional encodings to render each region of the scene from the novel view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The rendered image from the 3 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Input View (𝐼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=') − !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Codebook 𝑙!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 𝑙" … , 𝑙# Code Assignment " 𝑙" · Τ 𝑒||$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='%&" #,%||& ∑ 𝑒||$\'%&" #,%||& \' ()* \' ")* Encoder Novel View (𝐼!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='′) 𝐺!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Variation Module MLP 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' ",$ Render (𝑹𝑮𝑩𝝈) 𝒍𝒊 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='5 𝜖 𝐻!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 𝐹!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Positional Encoding Figure 2: An overview of NRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We learn a set of shared codes for decomposing scenes into objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Each point in the scene is assigned one of n latent codes from the codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The variation module models the intra-code variation between objects by perturbing the code in latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' A conditional NERF model renders the scene and is compared to the ground truth novel view for supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' novel view is compared to the ground truth using L2 pixel loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The categorical codes, assignment mechanism, and volumetric renderer are learned jointly in an end-to-end fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Image Encoding and Camera Projection Given an input frame and novel image, Ih, I′ h ∈ R3×H×W respectively from scene Sh, we first encode Ih into a spatial feature map, fh ∈ Rd×H/k×W/k, using a convolutional network, Fθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We project each point, (x, y, z), in world co- ordinates of the novel view to camera coordinates in the input frame, (x, y), using the relative camera pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Given (x, y), we select the spatial feature f x,y h ∈ Rd from the patch that contains the projected coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The spatial features vectors are then passed to the next stage where they are assigned to categorical object-codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Assigning + Learning Codes Our goal is to jointly learn a shared set of object categories and priors about their appearance and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' By mapping the spatial features from a continuous vector space to a discrete, finite set of codes the model is incentivized to find reoccurring patterns in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Given the features for a point in the novel view, f x,y h , we assign a code, l∗, chosen from the shared codebook, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We do so with an arg max 1-nearest-neighbors during inference: l∗(x, y) ← (STE) arg max li;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' i∈[k] e−∥li−f x,y h ∥2 �k j=1 e−∥lj−f x,y h ∥2 (1) The nearest-neighbor assignment used during inference is a non-differentiable operation therefore propagating gradients to the encoder and codebook would not be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' To enable learning of the codebook elements we use a softmax relaxation of nearest-neighbors during the backward pass in conjunction with the straight-through-estimator (STE) Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2013): l∗ back(x, y) ← e−∥li−f x,y h ∥2 �k j=1 e−∥lj−f x,y h ∥2 (2) Adding Categorical Codes The number of codes should depend on the complexity of the scenes they model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Learning when to add new codes is non-trivial because the number and selection of codes is discrete and non-differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' To circumvent this problem, we use a series of step functions with a straight-through-estimator (STE) to sequentially add elements to the codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Each code li ∈ L is gated according to the following: 4 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' li ← T � σ(s − i2/λ), 1 2 � li;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' T (a, t) := � 1, a > t 0, a ≤ t (3) T (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=') is a binarization function in the forward pass and lets the gradients pass through using STE in the back pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' σ(·) is the sigmoid function, λ is a scaling hyperparameter, and s is a learnable scoring parameter whose magnitude is correlated with the overall capacity (number of codes) required to model the scenes accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' A new code li is added when s exceeds the threshold i2/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' New codes are initialized using a standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Throughout training we keep k + 1 total codes where k is the current number of learnable codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The extra code is used by the straight-through- estimator to optimize for s on the backward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' This formulation can be viewed as the discrete analog of a gaussian prior over the number of elements, k in the codebook: P(k) = e −k2/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Modeling Intra-Code Variation Once a categorical code has been assigned to a region in the novel frame, the model must account for variation across instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We model this variation in latent space using an encoder that takes in both the spatial feature, f x,y h , and the categorical code l∗(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We rescale the norm of the variation vector by ϵ, a hyperparameter, to ensure the instance and categorical codes are close in latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The instance code is formulated as the following: l∗ instance(x, y) = l∗(x, y) + ϵ · Gθ′([l∗(x, y), f x,y h ]) ||Gθ′([l∗(x, y), f x,y h ])||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (4) We concatenate f x,y h and l∗(x, y) as input to the variation module, Gθ′, which we model as a 3- layer convolutional network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Gθ′ provides a d dimensional perturbation vector which models the intra-category variation and transforms the categorical code to an instance level code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Decoding and Rendering Given the localized instance codes for a scene, we render it in the novel view and compare with the ground truth using L2 pixel loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Intuitively, object categories which encode geometric and visual patterns should render the scene more accurately from novel views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Each region of the scene is rendered using an MLP conditioned on the instance codes and the volumetric rendering equation following the convention of NeRF Mildenhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2021): Hˆθ(l∗ instance(x, y), p, d) = (c, σ), (5) Here p = (x, y, z) is a coordinate in the scene, d ∈ R3 is a view direction, c is the RGB value at p in the direction of d and σ is the volume density at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Recall that (x, y, z) corresponds to (x, y) in the input frame Ih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We can project (x, y, z) into the camera coordinates of the novel view I′ h to get (x′, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' This pixel (x′, y′) in the novel view corresponds to (x, y) in the input frame, meaning they represent the same point in world coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' To get an RGB value for (x, y), we use volume rendering along the ray from camera view Ih into the scene, given by ˆC(r) = � tf tn T(t) · σ(t) · c(t) · dt, (6) where T(t) = exp � − � t tn σ(s) · ds � models absorbance and tn and tf are the near and far field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Given a target view with pose P, the ray to the target camera is given by r(t) = o + t · d where d is a unit direction vector which passes through (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The volume rendering for a particular pixel occurs along this ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Let d′ be the direction associated with the novel view (x′, y′) and r′(t) = o + t · d′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The pixel intensity at (x′, y′) is given by ˆC′ = ˆC(r′) and our final loss is L(Ih, I′ h, x, y) = ∥ˆC′ − I′ h(x′, y′)∥2 + s, (7) where I′ h(x′, y′) is the ground-truth pixel value at (x′, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We penalize the scoring parameter, s, from section 3 in the loss to encourage learning a minimal number of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NRC for Downstream Tasks Once the encoder, codebook, and MLP have been trained, we evaluate the learned representation on various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' To perform segmentation, we process each image through the trained encoder, Fθ, to obtain the spatial feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Each feature 5 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Input Image NRC Ground Truth Figure 3: Unsupervised segmentation of real-world images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NRC segments scenes that have significant object category overlap with ProcTHOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We show the first results for object-centric unsupervised segmentation of real-world scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' vector, fx,y in the spatial map is assigned to the nearest categorical code, l⋆ in the learned codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The categorical codes are then designated to the corresponding pixel to obtain a segmentation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Traditionally in object navigation, frames from the embodied agent are processed by a frozen, pretrained network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The resulting feature vector is then passed to a policy network which chooses an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' To assess the utility of the NRC representation, we replace the pretrained network with the NRC encoder and codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We process each frame to obtain instance codes for each region of the image which are then fed to the policy network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Depth ordering task consists of predicting which of two objects is closer to the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' To perform depth ordering with NRC we predict a segmentation mask and depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' To predict the depth map we condition the trained MLP on the instance codes & predict the density, σ, along a given ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We estimate the transmittance to predict the depth following the method of Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Depth map and segmentation mask are combined to predict the average distance of each object from the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 4 EXPERIMENTS We evaluate our decomposition and representations on several downstream tasks: unsupervised segmentation (real and synthetic), object navigation, and depth ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NRC shows improvement over baseline methods on all three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Prior works in object-centric learning have focused on unsupervised segmentation for measuring the quality of their decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We show that NRC representations are also effective for downstream applications that require geometric and semantic understanding of scenes such as object navigation and depth ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='1 DATASETS ProcTHOR & RoboTHOR THOR (Kolve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2017) consists of interactive home environments built in the Unity game engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We benchmark on the task of object navigation in RoboTHOR (Deitke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2020), a variant of the THOR environment aimed at sim2real benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Object navigation consists of an agent moving through different scenes to locate specified objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' RoboTHOR consists of 89 indoor scenes split between train, validation, and test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' ProcTHOR (Deitke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022) consists of procedurally generated indoor scenes similar to RoboTHOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Examples of THOR scenes can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 6 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' CLEVR-3D CLEVR-3D (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2017) is a synthetic dataset consisting of geometric primitives from multiple views and is used for unsupervised segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Following the convention of Stelzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2021), we test on the first 500 scenes of the validation set and report foreground- adjusted random index (FG-ARI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Adjusted random index (ARI) Yeung & Ruzzo (2001) measures the agreement between two clusterings and is a standard metric for unsupervised segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In our case the two clusterings are the predicted and ground truth segmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Foreground adjusted random index only measures the ARI for pixels belonging to foreground objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For comparison to prior works, we consider segmentations at both the class and instance level to be correct for CLEVR-3D, ProcTHOR, and NYU Depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Further details can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NYU Depth The NYU Depth Dataset (Silberman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2012) consists of images from real-world indoor scenes accompanied by depth and segmentation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Methods are trained on the ProcThor dataset then evaluated on NYU Depth for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We chose NYU Depth because it has object categories and scene layouts that are similar to THOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We report the adjusted random index (ARI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Table 1: Segmentation results (ARI) for NRC and comparable methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We find that for more complex datasets, ProcTHOR and NYU Depth, NRC outperforms other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Method ProcTHOR (ARI) NYU Depth (ARI) CLEVR-3D (FG-ARI) MarioNette .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='127 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='035 uORF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='193 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='115 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='962 ObSuRF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='228 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='141 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='978 NRC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='295 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='182 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='977 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='2 UNSUPERVISED SEGMENTATION Experimental Setup We evaluate NRC, ObSuRF, uORF, and MarioNette for unsupervised seg- mentation on ProcTHOR, CLEVR-3D, and NYU Depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We compare with MarioNette because it uses a similar code mechanism for reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We report FG-ARI on CLEVR-3D for comparison to prior works and ARI on the other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For NYU Depth evaluation we use the representations trained on ProcTHOR and only consider classes that are seen in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Results We find that for NYU Depth and ProcTHOR which have more complex layouts and object diversity, NRC significantly outperforms other methods (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Figure 1 shows examples of the object codes learned by ProcTHOR and Figure 3 shows segmentation examples of real-world images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' To our knowledge, this is the first object-centric method which has shown unsupervised segmentation results for complex real-world images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We find that NRC categorizes similar objects across scenes based on both geometry and visual appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In the top row of Figure 1, we find that couches of similar shape are assigned to the same code despite differing visual appearance which indicates that NRC codes capture geometric similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In the same figure, we show further examples where floors of different texture are categorized by the same code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The improved segmentation performance and geometric categorization of objects indicates that NRC can leverage more than simple color cues which has been significant limitation for object-centric learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='3 OBJECT NAVIGATION Experimental Setup We design the object navigation experiments in THOR to understand how well the learned representations transfer from observational data to embodied navigation (Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Batra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Object navigation consists of an embodied agent with the goal of moving through indoor scenes to specified objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The agent can rotate its camera and move in discrete directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' At each step, agent is fed with the current RGB frame relayed by the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For the representation learning component of the experiment we collect observational video data from a heuristic planner (Appendix B), which walks through procedurally generated ProcTHOR scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In total, the dataset consists of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='5 million video frames from 500 indoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For further dataset details and example videos see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' After training on ProcThor videos, we freeze the visual representations following standard prac- tice (Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We train a policy using DD-PPO (Wijmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2019) for 200M 7 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Table 2: Results for object navigation on RoboTHOR object navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Visual representations are trained on observations from 500 scenes of ProcThor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' A policy is learned on top of the frozen visual representations by training on the object navigation task in RoboTHOR training scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The results are obtained by evaluating on RoboTHOR test scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Method Success Rate (%) SPL uORF (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021b) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='146 ImageNet Pretraining 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='150 ObSuRF (Stelzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='167 Video MoCo (Feichtenhofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='184 EmbCLIP (Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='200 NRC (Ours) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='239 steps on the training set of RoboTHOR then evaluate on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We report success rate (SR) and success weighted by path length (SPL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Success is defined as the agent signaling the stop action within 1 meter of the goal object with it in the view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' SPL is defined as 1 N �N i=1 Si ℓi max(pi,ℓi), where li is the shortest possible path, pi is the taken path, & Si is the binary indicator of success for episode i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We compare with the following baselines: ObSuRF, uORF, Video MoCo (Feichtenhofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021), and EmbedCLIP (Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' ObSuRF and uORF are 3D, object-centric methods, and Video MoCo is a contrastive video representation learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We include Video MoCo for comparison as it was designed for large-scale, discriminative tasks, while ObSuRF and uORF were primarily intended for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' See Appendix B for further implementation details of baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Results Table 2 shows the performance of NRC and baselines on RoboTHOR object navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NRC outperforms the best baseline by 3% in success rate (SR) and by a 20% relative improvement in SPL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' These performance gains indicate that the geometrically-aware NRC representation provide an advantage over traditional representations such as EmbCLIP and Video MoCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Another key observation is that NRC has at least 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='8% improvement in SR and relative SPL over the recent object-centric learning methods, uORF and ObSuRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In particular, NRC performs better than ObSuRF and uORF when navigating near furniture and other immovable objects (see supplemental for object navigation videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We hypothesize this is due to NRC’s more precise localization of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='4 DEPTH ORDERING Experimental Setup We evaluate NRC on the task of ordering objects based on their depth from the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Understanding the relative depth of objects requires both geometric and semantic understanding of a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For this task we evaluate on the ProcTHOR test dataset which provides dense depth and segmentation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Following the convention of Ehsani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2018), we determine ground truth depth of each object by computing the mean depth of all pixels associated with its ground truth segmentation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For evaluation, we select pairs of objects in a scene and the goal is to predict which object is closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We take the segmentation that has the largest IoU with the ground truth mask as the predicted object mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' To determine the predicted object depth, we compute the mean predicted depth of each pixel associated with the predicted object mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' All representations are trained on the ProcTHOR dataset and evaluated on the ProcTHOR test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In total we evaluate 2,000 object pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Table 3: We compare depth ordering results on RoboTHOR with other geometrically-aware represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Given pairs of objects in the scene, the model must infer which object is closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We report the accuracy as the number of correct orderings over the total number of object pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Method Depth Order Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (%) uORF (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021b) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='5 ObSuRF (Stelzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='3 NRC (Ours) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='8 8 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Results Depth ordering requires accurate segmentation and depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Due to NRC’s stronger segmentation performance and better depth estimation, we see 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='5% and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='3% depth ordering accuracy compared to ObSuRF and uORF respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The fine-grained localization of categorical latent codes in NRC allows for better depth ordering over existing object-centric methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In particular, the other object-centric methods tend to assign object instances to the background which leads to large errors in estimating the depth of objects (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='5 ABLATION STUDY Table 4: Ablation study for modeling the intra-code variation and learning the number of codes evaluated on unsupervised segmentation in ProcTHOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Default fixed number of codes is set to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Method + (Ablation) ProcTHOR (ARI) NRC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='182 NRC + Learned # of Codes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='197 NRC + Variation Module .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='284 NRC + Variation Module + Learned # of Codes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='295 We present an ablation study on the ProcTHOR dataset to determine the effect of the variation module and learnable codebook size on unsupervised segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Quantitative results can be found in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We observe that performance improves by ∼ 9% when intra-class variation is explicitly modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Intuitively, allowing for small variation between instances of the same category should lead to better representations and allow for greater expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We also find that learning the number of codes moderately improves performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' However, if number of codes is found via hyper-parameter tuning, performance is matched (Table 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Nonetheless, differentiably learning the codebook size avoids computationally expensive hyper-parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 5 LIMITATIONS Novel view reconstruction requires camera pose, which is not available for most images and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Some datasets such as Ego4D (Grauman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2022) provide data from inertial measurement units that can be used to approximate camera pose, although this approach is prone to drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' An incorrect assumption that NRC and most object-centric prior works make is that scenes are static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' However, it rarely is the case that scenes are free of movement due to the physical dynamics of our world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Recently, Kipf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Pumarola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2021) made strides in learning representations from dynamic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Although NRC is relatively efficient compared to the other NeRF based methods, the NeRF sampling procedure is compute and memory intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Sajjadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2022a) and Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2022) leverage object-centric light fields to reduce memory and compute costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The efficiency improvement from modeling scenes as light fields is orthogonal to NRC and can be combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' A final challenge inherent to novel view reconstruction is finding appropriate corresponding frames of videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For example, if two subsequent frames differ by a 60° rotation of the camera, then most of the scene in the subsequent frame will be completely new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Therefore, constructing the content in the novel view is ill-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Pairing frames with overlapping frustums is a potential solution, although the content of the scene may not be contained in the intersecting volume of the frustums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 6 CONCLUSION Compositional, object-centric understanding of the world is a fundamental characteristic of human vision and such representations have the potential to enable high-level reasoning and efficient transfer to downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Towards this goal, we presented Neural Radiance Field Codebooks (NRC), a new approach for learning geometry-aware, object-centric representations through novel view reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' By jointly learning a shared dictionary of object codes through a differentiable renderer and explicitly localizing object codes within the scene, NRC finds reoccurring geometric and visual similarities to form objects categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Through experiments, we show that NRC representations improve performance on object navigation and depth ordering compared to strong baselines by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='1% success rate and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='5% accuracy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Additionally, we find our method is capable of scaling 9 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' to complex scenes with more objects and greater diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NRC shows relative ARI improvement over baselines for unsupervised segmentation by 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='4% on ProcTHOR and 29.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Cognitive science, 14(1):29–56, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Karl Stelzner, Kristian Kersting, and Adam R Kosiorek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Decomposing 3d scenes into objects via unsupervised volume segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='01148, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Aaron Van 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='13260, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 12 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Erik Wijmans, Abhishek Kadian, Ari Morcos, Stefan Lee, Irfan Essa, Devi Parikh, Manolis Savva, and Dhruv Batra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Dd-ppo: Learning near-perfect pointgoal navigators from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='5 billion frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' arXiv preprint arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='00357, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Bangbang Yang, Yinda Zhang, Yinghao Xu, Yijin Li, Han Zhou, Hujun Bao, Guofeng Zhang, and Zhaopeng Cui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Learning object-compositional neural radiance field for editable scene rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 13779–13788, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Ka Yee Yeung and Walter L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Ruzzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Details of the adjusted rand index and clustering algorithms supplement to the paper “ an empirical study on principal component analysis for clustering gene expression data ” ( to appear in bioinformatics ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Alex Yu, Vickie Ye, Matthew Tancik, and Angjoo Kanazawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' pixelnerf: Neural radiance fields from one or few images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 4578–4587, 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Hong-Xing Yu, Leonidas J Guibas, and Jiajun Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Unsupervised discovery of object radiance fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='07905, 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Shuaifeng Zhi, Tristan Laidlow, Stefan Leutenegger, and Andrew J Davison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In-place scene la- belling and understanding with implicit scene representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 15838–15847, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 13 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' A FURTHER EXPERIMENTS AND QUALITATIVE EXAMPLES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='1 MORE COMPLEX REAL-WORLD SCENES (NYU-DEPTH) We provide more segmentation examples of NRC on cluttered, real-world scenes (figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We find that NRC reasonably segments categories that overlap with those of THOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Additionally we evaluate NRC and comparable methods on more difficult scenes of NYU Depth to see how complexity of the scene affects performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Specifically we filter for scenes with 5 or more objects in them for evaluation on unsupervised segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In general performance degrades (Table 5), though NRC performance decreases less compared to uORF and ObSuRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Image Ground Truth Predicted Figure 4: Segmentation examples of cluttered scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Further segmentation examples of cluttered, real-world scenes from NYU Depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NRC scales better to cluttered scenes compared to other methods as the rendering and memory cost is constant in the number of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='2 MATTERPORT-3D AND GIBSON EXPERIMENTS We train NRC on scenes from MatterPort3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Qualitative examples can be found in figure 6 and quantitative results can be found in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We find NRC performs slightly worse on MatterPort3D compared to on THOR likely due to the greater visual complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' This is reinforced by the evidence that the learned codebook size is 67 whereas for THOR it is on average ∼ 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' A greater number of learned codes indicates that NRC requires more expressivity to accurately reconstruct Matterport3D scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For a given scene we find corresponding frames by filtering for images within 20 degrees viewing angle of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We evaluate NRC and ObSuRF on the Habitat Point Navigation Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We train NRC and ObSuRF on data collected from random walks through the Habitat environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We use a ResNet-50 initialized with a MoCo backbone pretrained on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We report success rate, shortest-path length, and distance to goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Qualitative examples can be found in the supplemental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We find that NRC outperforms ObSuRF and ImageNet pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Qualitatively we find that NRC is better at navigating around objects compared to ObSuRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 14 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Figure 5: Predicted object depth maps for depth ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The predicted object masks are overlayed with the predicted depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We average the per pixel depth for each predicted object mask to infer which object is closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NRC’s object localization leads to sharp object masks and less error when predicting average object depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' B IMPLEMENTATION DETAILS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='1 NRC For the segmentation experiments we use the modified ResNet34 used by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For the decoder we use a 3-layer with hidden dimension of 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We train with a learning rate of 1e − 4 with the ADAM optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We set the near field to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='4 and far field to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We train from scratch, and from the video MoCo model trained on ProcTHOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We found starting from the pretrained initialization reduced the number of training epochs required for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For the model from scratch we train for 300 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For the model from MoCo initialization we train for 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' While training we use a sliding window of 5 frames to determine corresponding images in ProcTHOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' By corresponding images we mean the image given to the model and the ground truth novel view image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Given a sliding window of n frames we randomly select 2 images from this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We ablate over 2, 5, and 10 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We found 5 images led to the best segmentation results and therefore used that window size for object-navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For object-navigation we use a ResNet50 for fair comparison to CLIP and Video MoCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We train wit the same hyper-parameters as discussed above for segmentation with the only change being the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We train for 200M steps using PPO with default hyper-parameters to match Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2022) in RoboTHOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Table 5: Performance on cluttered scenes of NYU Depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We filter for scenes with 5 or more objects and report the ARI of NRC and comparable methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' NRC outperforms ObSuRF and uORF particularly on cluttered scenes, as it can support an unbounded number of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Method NYU Depth (Hard) uORF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='046 ObSuRF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='092 NRC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='139 15 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Image Ground Truth Predicted Figure 6: Segmentation examples for MatterPort3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Segmentation examples on MatterPort3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We train NRC on ∼ 190,000 images of MatterPort3d and learn 67 categorical codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' This increase in the learned number of codes is likely due to the increased visual complexity of MatterPort3D compared to THOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We find similar segmentation results to those on THOR and NYU Depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='2 UORF We use the implementation available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='com/KovenYu/uORF We train on the training set of ProcTHOR which contains ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='5 million images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We train three different variations of the uORF no GAN model: from scratch, fine-tune with encoder and decoder pre-trained on CLEVR-567, and fine-tune with all three components pre-trained on CLEVR-567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For variations that train the slot attention component from scratch, we try setting the number of slots to 8 (default) and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We train for 10 epochs when loading from a pre-trained model, and for 15 epochs when training from scratch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' roughly half of the epochs are coarse epochs, and we set percept_in to roughly one third of the epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We train for less epochs because our dataset is much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Our images within scene batch size is 4, and we turn off shuffling to ensure that batches contain images that are close together within the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We pass in the fixed locality parameter, but also set the decoder’s locality parameter to false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' To compensate between differences between our dataset and CLEVR-567, we tune nss scale, object scale, near plane, and far plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We train models on a single NVIDIA A40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='3 OBSURF We use the official implementation provided by Stelzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2021) at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='com/ stelzner/obsurf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We trained 3 variations of the ObSuRF model: onfrom scratch, from CLEVR- Table 6: Unsupervised segmentation performance of NRC and ObSuRF on MatterPort3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We find NRC performs slightly worse on MatterPort3D compared to on THOR likely due to the greater visual complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' This is reinforced by the observation that NRC learns 67 codes for MatterPort3D whereas 53 are learned for THOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Method MatterPort3D ObSuRF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='178 NRC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='237 16 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Table 7: Habitat Point Navigation Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' The performance of various visual encoders on the point navigation challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We find that NRC outperforms other 3D-object centric methods due to its ability to scale to more objects and visually complex environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Method SPL SR Goal Dist ImageNet Pretrain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='82 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='94 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='73 ObSuRF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='77 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='84 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='81 NRC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='85 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='96 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='68 3D pretrained weights, and from MultiShapeNet pretrained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For the pretrained models we lowered the learning rate by a factor of 10 to 1e−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For training from scratch we used the default learning rate of 1e − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For the number of slots we tried the default number, 5, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We report the results of the best model which was from scratch with 10 slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For object navigation experiments in RoboTHOR, we used the model with the highest ARI which was training from scratch with 10 slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We learn a policy network comprised of a 1-layer GRU with 512 hidden units which maps to a 6 dimensional logit and scalar which are used for the actor-critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We train with DD-PPO for 200M steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For the encoder, we use a ResNet-50 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' To construct the feature vector fed to the GRU we use the output of the ResNet-50 which is 2048×7×7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We pass this through a 2-convolutional layers to sample down to 32 × 7 × 7 then concatenate this with the trainable goal state embedding matrix which is 32x7x7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' This visual feature vector of shape 64 × 7 × 7 is then passed through another 2-layer convolutional network to be of size 32 × 7 × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='4 OSRT We do not compare with OSRT Sajjadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2022a) on data sets that were not evaluated in the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We contacted the authors, but a public implementation has not been released yet at the time of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' C DATASETS C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='1 PROCTHOR The ProcTHOR dataset Deitke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' (2022) consists of 10,000 procedurally generated indoor rooms in the THOR environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For our dataset we collect 10 video sequences of 300 actions from 500 randomly chosen training scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Each actions consisting of either rotation or stepping in a specified direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Actions are determined by a heuristic planner which moves throughout the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In total we collect 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='5 millions image frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' See Figure 7 for a sample of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For evaluation of ARI we consider the 20 largest objects, floor, and ceiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' uORF and ObSuRF decompose scenes into 5-10 objects and therefore do not handle scenes with many small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' We evaluate ARI only on the 20 largest objects in THOR which are the following: {ArmChair, Bathtub, BathtubBasin, Bed, Cabinet, Drawer, Dresser, Chair, CounterTop, Curtains, Desk, Desktop, CoffeeTable, DiningTable, SideTable, Floor, Fridge, Television, TVStand, Toilet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='} C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='2 CLEVR-3D CLEVR-3D (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2017) is a synthetic dataset which consists of geometric primitives placed on a monochrome background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Following the convention of (Stelzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=', 2021) we evaluate on this benchmark for unsupervised object segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Following convention we evaluate on the first 500 scenes of the validation set and report foreground - adjusted random index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For class level labels we consider objects with the shape to be of the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' 17 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Figure 7: Snippets of Videos from ProcTHOR dataset used for training NRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Table 8: Performance of NRC with a fixed codebook of various sizes on ProcTHOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Codebook Size ProcTHOR (ARI) 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='247 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='256 30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='266 40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='279 50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='286 60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='293 70 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='288 80 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='295 90 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='279 100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='273 18 Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Predicted Novel View Input View Figure 8: Reconstruction examples from NRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' For scenes and objects that are closer to the training distribution, reconstruction quality is significantly better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='3 INSTANCE LEVEL ACCURACY For comparison to the instance level classification performed by ObSuRF and uORF, we filtered for images in ProcTHOR which have one object of each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' In this setting, class and instance level classification are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Table 9: The size of the learned codebook averaged over 5 runs on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content=' Codebook Size CLEVR-3D 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='2 ProcTHOR 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} +page_content='4 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE2T4oBgHgl3EQfwgg2/content/2301.04101v1.pdf'} diff --git a/cNFST4oBgHgl3EQfDTjC/vector_store/index.faiss b/cNFST4oBgHgl3EQfDTjC/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..2779871ffe8b787335f42f9f159a6e8d1b36fb3f --- /dev/null +++ b/cNFST4oBgHgl3EQfDTjC/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:990a788421a7d9f5928e5b7199bf84c01a6d2dc0bfb9b3b80850143dbb1f97b5 +size 4915245 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file mode 100644 index 0000000000000000000000000000000000000000..791a31d8d232fb2f2125289dca03537cfe0db7b5 --- /dev/null +++ b/fdE4T4oBgHgl3EQfqg0X/content/tmp_files/2301.05200v1.pdf.txt @@ -0,0 +1,2984 @@ +Charged pion electric polarizability from four-point functions in lattice QCD +Frank X. Lee,1, ∗ Andrei Alexandru,1, 2, † Chris Culver,3, ‡ and Walter Wilcox4, § +1Physics Department, The George Washington University, Washington, DC 20052, USA +2Department of Physics, University of Maryland, College Park, MD 20742, USA +3Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom +4Department of Physics, Baylor University, Waco, Texas 76798, USA +Polarizabilities reveal valuable information on the internal structure of hadrons in terms of charge +and current distributions. For neutral hadrons, the standard approach is the background field method. +But for a charged hadron, its acceleration under the the applied field complicates the isolation of the +polarization energy. In this work, we explore an alternative method based on four-point functions +in lattice QCD. The approach offers a transparent picture on how polarizabilities arise from quark +and gluon interactions. We carry out a proof-of-concept simulation on the electric polarizability +of a charged pion, using quenched Wilson action on a 243 × 48 lattice at β = 6.0 with pion mass +from 1100 to 370 MeV. We show in detail the evaluation and analysis of the four-point correlation +functions and report results on charge radius and electric polarizability. Our results from connected +diagrams suggest that charged pion αE is due to a large cancellation between elastic and inelastic +contributions, leaving a small and positive value that has a relatively mild pion mass dependence. +I. +INTRODUCTION +Understanding electromagnetic polarizabilities has been +a long-term goal of lattice QCD. The challenge in the +effort lies in the need to apply both QCD and QED princi- +ples. The standard approach to compute polarizabilities is +the background field method which has been widely used +for dipole polarizabilities [1–19]. Methods to study higher- +order polarizabilities have also been proposed [20–23] in +this approach. Although such calculations are relatively +straightforward, requiring only energy shifts from two- +point functions, there are a number of unique challenges. +First, since weak fields are needed, the energy shift in- +volved is very small relative to the mass of the hadron (on +the order of one part in a million depending on the field +strength). This challenge has been successfully overcome +by relying on statistical correlations with or without the +field. Second, there is the issue of discontinuities across +the boundaries when applying a uniform field on a pe- +riodic lattice. This has been largely resolved by using +quantized values for the fields, or Dirichlet boundary con- +ditions. Third and more importantly, a charged hadron +accelerates in an electric field and exhibits Landau levels +in a magnetic field. Such motions are unrelated to polar- +izability and must be disentangled from the deformation +energy on which the polarizabilities are defined. For this +reason, most calculations have focused on neutral hadrons. +For charged hadrons, what happens is that the two-point +correlator does not develop single exponential behavior +at large times. In Ref. [24], a relativistic propagator for +a charged scalar is used to demonstrate how to fit such +lattice data for charged pions and kaons. This approach +is improved recently in Ref. [25] with an effective charged +∗ fxlee@gwu.edu +† aalexan@gwu.edu +‡ C.Culver@liverpool.ac.uk +§ walter wilcox@baylor.edu +scalar propagator exactly matching the lattice being used +to generate the lattice QCD data. A new fitting procedure +is proposed where a χ2-function that utilizes information +in both the real and imaginary parts of the correlator +while remaining invariant under gauge transformations of +the background field. For magnetic polarizability, a field- +dependent quark-propagator eigenmode projector is used +to filter out the effects of Landau levels [26, 27]. These +special techniques for charged particles involve fairly com- +plicated analysis to treat the collective motion of the +system in order to isolate the polarizabilities. +In this work, we explore an alternative approach based +on four-point functions in lattice QCD. Instead of back- +ground fields, electromagnetic currents couple to quark +fields to induce interactions to all orders. It is a general +approach that treats neutral and charged particles on +equal footing, but particularly suited for charged parti- +cles. The trade-off is an increased computational demand +of four-point functions. Although four-point functions +have been applied to study various aspects of hadron +structure [28–33], not too much attention has been paid +to its potential application for polarizabilities. We know +of two such studies from a long time ago [34, 35], a recent +calculation on the pion [36], and preliminary one on the +proton [37]. A reexamination of the formalism in Ref. [35] +is recently carried out in Ref. [38] for both electric and +magnetic polarizabilities of a charged pion and a proton. +Experimentally, polarizabilities are primarily studied by +low-energy Compton scattering. Theoretically, a variety +of methods have been employed to describe the physics +involved, from dispersion relations [39–42], to chiral per- +turbation theory (ChPT) [43–45] or chiral effective field +theory (EFT) [46, 47]. Reviews on hadron polarizabilities +can be found in Refs. [43, 47, 48]. +The presentation is organized as follows. In Sec. II we +outline the methodology to extract polarizabilities, using +the electric polarizability of a charged pion as an example. +In Sec. III we detail our notations and algorithms used +to evaluate the four-point functions, including how the +arXiv:2301.05200v1 [hep-lat] 12 Jan 2023 + +2 +Sequential-Source Technique (SST) can be applied in this +context. In Sec. IV we show our analysis procedure and +results from a proof-of-concept simulation. In Sec. V we +give concluding remarks and an outlook. Some technical +details are put in the Appendices. +II. +METHODOLOGY +In Ref. [38], a formula is derived for electric polarizabil- +ity of a charged pion, +απ +E = α r2 +E +3mπ ++ lim +q→0 +2αa +q 2 +� ∞ +0 +dt +� +Q44(q, t) − Qelas +44 (q, t) +� +. (1) +Here α = 1/137 is the fine structure constant and a the +lattice spacing. The first term in the formula involves the +charge radius and pion mass (we will refer to this term +as the elastic contribution). The second term has the +elastic contribution Qelas +44 +subtracted from the total (we +will refer to this term as the inelastic contribution). The +formula will be used in discrete Euclidean spacetime but +we keep the Euclidean time axis continuous for notational +convenience. Special kinematics (called zero-momentum +Breit frame) are employed in the formula to mimic low- +energy Compton scattering. The process is illustrated in +Fig 1, where the initial (p1) and final (p2) pions are at +rest and the photons have purely spacelike momentum, +p1 = (0, mπ), q1 = (q, 0), q2 = (−q, 0), p2 = (0, mπ). +(2) +FIG. 1. Four-point function for charged pion polarizabilities +under the zero-momentum Breit frame. Time flows from right +to left and the four momentum conservation is expressed as +p2 = q2 + q1 + p1. +The Q44 is defined as the µ = 4 = ν component of the +Fourier transforms, +Qµν(q, t2, t1) ≡ +� +x2,x1 +e−iq·x2eiq·x1Pµν(x2, x1, t3, t2, t1, t0), +(3) +where Pµν is a four-point function defined in position +space (Ω denotes the vacuum), +Pµν(x2, x1, t3, t2, t1, t0) ≡ +� +x3,x0 Ω|ψ(x3) : jL +µ (x2)jL +ν (x1) : ψ†(x0)|Ω +� +x3,x0 Ω|ψ(x3)ψ†(x0)|Ω +. +(4) +Here ψ is the interpolating field of the pion and jL +µ the +lattice version of the electromagnetic current density. The +two-point function in the denominator is for normaliza- +tion. Normal ordering is used to include the required +subtraction of vacuum expectation values (VEV) on the +lattice. The sums over x0 and x3 enforce zero-momentum +pions at the source (t0) and sink (t3). The two currents +are inserted at t1 and t2 with two possibilities of time or- +dering implied in the normal ordering. The field operators +for ψ and jL +µ used in this work, along with conservation +properties of Q44 at q = 0, are given in Appendix A. +To see the structure of the four-point function in Eq.(4), +we insert a complete set of states in the numerator (twice) +and in the denominator (once). When the times are well +separated (defined by the time limits t3 ≫ t1,2 ≫ t0) the +correlator is dominated by the ground state, +Pµν(x2, x1, t3, t2, t1, t0) → +N 2 +s | π(0)|ψ(0)|Ω |2e−mπt3 π(0) : jL +µ (x2)jL +ν (x1) : |π(0) +N 2s | π(0)|ψ(0)|Ω |2e−mπt3 +→ π(0)| : jL +µ (x2)jL +ν (x1) : |π(0) += π(0)|TjL +µ (x2)jL +ν (x1)|π(0) − Ω|TjL +µ (x2)jL +ν (x1)|Ω . +(5) +Here Ns = NxNyNz is the number of spatial sites on the +lattice. The role of the two-point function as normaliza- +tion and the inclusion of VEV subtraction is evident in +the limit. +Assuming time separation t = t2 − t1 > 0 and inserting +a complete set of intermediate states, the diagonal com- +ponent of Qµν develops the time dependence in the same +limits, +Qµµ(q, t) = +N 2 +s +� +n +| π(0)|jL +µ (0)|n(q) |2e−a(En−mπ)t +− N 2 +s +� +n +| Ω|jL +µ (0)|n(q) |2e−aEnt. +(6) +At large time separations, it is dominated by the elastic +contribution (n = π term in the first sum), +Qelas +µµ (q, t) ≡ N 2 +s | π(0)|jL +µ (0)|π(q) |2e−a(Eπ−mπ)t. +(7) +We see that the elastic piece in the four-point function has +information on the form factor of the pion through the +amplitude squared. The form factor Fπ can be determined +from Q44 at large time separations, +Qelas +44 (q, t) = (Eπ + mπ)2 +4Eπmπ +F 2 +π(q2) e−a(Eπ−mπ)t. +(8) +The charge radius +r2 +E +in the formula can then be ex- +tracted from Fπ. A salient feature here is that the elastic +contribution in four-point functions is positive definite. +Aside from the charge radius term in Eq. (1), αE is +proportional to the difference in the areas under the Q44 +and Qelas +44 +curves. It is this difference that is responsible +for the sign of απ +E. On a finite lattice the time integral does + +-q +q +0 +03 +FIG. 2. Skeleton diagrams of a four-point function contributing +to polarizabilities of a meson: (a) connected insertion: differ- +ent flavor, (b) connected insertion: same flavor, (c) connected +insertion: same flavor Z-graph, (d) disconnected insertion: +single loop, double current, (e) disconnected insertion: single +loop, (f) disconnected insertion: double loop. In each diagram, +flavor permutations are assumed as well as gluon lines that +connect the quark lines. The zero-momentum pion interpolat- +ing fields are represented by vertical bars (wall sources). Time +flows from right to left. +not really extend to ∞, but are limited to the available +time slices between the two current insertions. In practice, +one should check if the largest time separation is enough +to establish the elastic limit. Equivalent directions for q +can be used to improve the signal-to-noise ratio. Note +that απ +E has the expected physical unit of a3 (fm3) since +1/q 2 scales like a2 and Q44 and t are dimensionless in our +notation. +III. +CORRELATION FUNCTIONS +In this section, we detail how to simulate Eq.(4) and its +Fourier transform Eq.(3) at the quark level. Wick contrac- +tions of quark-antiquark pairs in the unsubtracted part +lead to topologically distinct quark-line diagrams shown +in Fig. 2. The raw correlation functions can be found in +Appendix B. Diagrams a, b, and c are connected. Dia- +gram d has a loop that is disconnected from the hadron, +but connected between the two currents. Diagrams e has +one disconnected loop and diagram f has two such loops. +Furthermore, diagrams d, e and f must have associated +VEV subtracted. However, if conserved lattice current +density is used, there is no need for subtraction in diagram +e since the VEV vanishes in the configuration average [49]. +In this work, we focus on the connected contributions (di- +agrams a,b,c). The disconnected contributions (diagrams +d,e,f) are more challenging and are left for future work. +In particular, we will explain how to use the sequential +source technique (SST) to simplify the evaluations. +A. +Two-point functions +First, we show how to evaluate the two-point function +in Eq.(4) which serves as normalization for the four-point +functions. It has the following Wick contraction using the +interpolating operator in Eq.(A1), +� +x3,x0 +Ω|ψ(x3)ψ†(x0)|Ω += +� +x3,x0 +Tr +s,c +� +γ5Sd(x0, x3)γ5Su(x3, x0) +� +, +(9) +where Sq denotes a quark propagator that carries the +full space-time and spin and color information between +two points1. The double sum projects to zero momen- +tum both as the source x0 and the sink x3 as required +by the special kinematics. The full evaluation involves +essentially all-to-all propagation which is computation- +ally prohibitive. Instead, we employ wall sources without +gauge fixing as an approximation, with the expectation +that gauge-dependent contributions to the final observ- +ables will vanish in the configuration average [30, 50]. +Only terms in the double sum where the quarks are at +the same location form the signal, the rest contribute to +noise. Details of our implementation of the wall source +can be found in Appendix C. +1 In this work, all correlation functions in such expressions are +understood as path integral expectation values in lattice QCD. +They are evaluated as averages over gauge configurations in Monte +Carlo simulations. + +(a) +(b) +(c) +X +(d) +(e) +(f)4 +If we insert the wall at time slice t0 and project to zero momentum at t3 in Eq.(9), we have +� +x3,x0 +Ω|ψ(x3)ψ†(x0)|Ω = Tr +s,c +� +WT γ5Sd(x0, x3)γ5Su(x3, x0)W +� += Tr +s,c +� +WT γ5P(t0)M −1 +d P(t3)T γ5P(t3)M −1 +u P(t0)T W +� +. +(10) +The symbols W and P(t) are defined in Appendix C. We introduce two zero-momentum quark propagators called a1 +and a2 emanating from the walls at t0 and t3, respectively, +V (q) +a1 ≡ M −1 +q +P(t0)T W, +V (q) +a2 ≡ M −1 +q +P(t3)T W. +(11) +We use “V” to emphasize that the wall-to-point quark propagators so defined are column vectors in the (x, s, c) space. +Using a1, the two-point function can be written as, +� +x3,x0 +Ω|ψ(x3)ψ†(x0)|Ω = Tr +s,c +�� +P(t3)γ5V (d) +a1 +�†� +P(t3)γ5V (u) +a1 +�� += Tr +s,c +�� +P(t3)V (d) +a1 +�†� +P(t3)V (u) +a1 +�� +(Type 1) +(12) +In the last step the γ5-hermiticity of M −1 +q +is used to eliminate γ5. Similarly, if we insert the wall at time slice t3 and +project to zero momentum at t0, we get in terms of a2, +� +x3,x0 +Ω|ψ(x3)ψ†(x0)|Ω = Tr +s,c +�� +P(t0)γ5V (u) +a2 +�†� +P(t0)γ5V (d) +a2 +�� += Tr +s,c +�� +P(t0)V (u) +a2 +�†� +P(t0)V (d) +a2 +�� +(Type 2) +(13) +If we insert two walls, one at t0, one at t3, we obtain additional expressions, +� +x3,x0 +Ω|ψ(x3)ψ†(x0)|Ω = Tr +s,c +� +WT γ5Sd(x0, x3)WWT γ5Su(x3, x0)W +� += Tr +s,c +�� +WT P(t3)V (d) +a1 +�†� +WT P(t3)V (u) +a1 +�� += Tr +s,c +�� +WT P(t0)V (u) +a2 +�†� +WT P(t0)V (d) +a2 +�� +(Type 3) +(14) +The expressions in the above three equations (which we denote as Type 1, 2, 3 as indicated) are different estimators +of the wall-to-wall two-point function with zero momentum for both initial and final pions. They are expected to +approach the same value in the limit of infinite number of configurations. In the following, we use our notation to +evaluate the connected four-point functions in Fig. 2. +B. +Four-point functions +We start with local (or point) current insertions of four-point functions which have relatively simple Wick contractions. +The results in this work will be based on conserved (or point-split) currents which avoids the issue of computing the +renormalization constant ZV for vector currents. Below we detail how to evaluate the connected contributions using +both local and conserved currents. +1. +Diagram a (different flavor) +There are two terms, d4 and d2 in Eq.(B4), that are contributing to the connected part of diagram a. They are +characterized by the charge factor quq ¯ +d = 2/9. The two terms are related by a flavor permutation (1 ↔ 2 switch). +Under isospin symmetry in u and d quarks, the two terms have equal contributions. Including the Fourier transforms +and setting µ = 4 = ν for electric polarizability, the correlation function can be written as2, +˜Q(a,P C) +44 += −4 +9Z2 +V κ2 Tr +s,c +� +γ5S(t0, t2)γ4e−iqS(t2, t3)γ5S(t3, t1)γ4eiqS(t1, t0) +� +. +(15) +2 We use Qµν for normalized correlation functions as defined in Eq.(3) and Eq.(4), and tilded ˜Qµν for unnormalized, i.e., without the +denominator Eq.(4). + +5 +We evaluate the correlation function by inserting two walls, one at t0 and one at t3, +˜Q(a,P C) +44 +(q, t1, t2) = −4 +9Z2 +V κ2 Tr +s,c +� +WT γ5S(t0, t2)e−iqγ4S(t2, t3)WWT γ5S(t3, t1)eiqγ4S(t1, t0)W +� += −4 +9Z2 +V κ2 Tr +s,c +� +WT γ5P(t0)M −1 +q +P(t2)T e−iqγ4P(t2)M −1 +q +P(t3)T W +WT γ5P(t3)M −1 +q +P(t1)T eiqγ4P(t1)M −1 +q +P(t0)T W +� +. +(16) +The notation makes it clear that all spatial sums are automatically incorporated into the matrix multiplications. Using +the Va1 and Va2 propagators defined in Eq.(11) and the γ5-hermiticity of M −1, the final expression for diagram a can +be written as, +˜Q(a,P C) +44 +(q, t1, t2) = 4 +9Z2 +V κ2 Tr +s,c +�� +[P(t2)Va2]† γ5γ4eiqP(t2)Va1 +�†� +[P(t1)Va2]† γ5γ4eiqP(t1)Va1 +�� +. +(17) +There is an overall sign change from taking the dagger. The first parenthesis corresponds to the current insertion at t2 +on one of the quark lines in the pion; the second parenthesis the current insertion at t1 on the other quark line. Both +t1 and t2 are free to vary between t0 and t3. +In the case of conserved current, there are 8 terms contributing to diagram a in Eq.(B6). Their sum under isospin +symmetry, along with the Fourier transforms and wall-source insertions, can be written in similar form, +˜Q(a,P S) +44 +(q, t1, t2) = 1 +9κ2� +d16 + d18 + d20 + d22 + d8 + d10 + d12 + d14 +� += 4 +9κ2 Tr +s,c +�� +[P(t2)Va2]† γ5(1 − γ4)eiqU4(t2, t2 + 1)P(t2 + 1)Va1 − [P(t2 + 1)Va2]† γ5(1 + γ4)U † +4(t2 + 1, t2)eiqP(t2)Va1 +�† +� +[P(t1)Va2]† γ5(1 − γ4)eiqU4(t1, t1 + 1)P(t1 + 1)Va1 − [P(t1 + 1)Va2]† γ5(1 + γ4)U † +4(t1 + 1, t1)eiqP(t1)Va1 +�� +, +(18) +with local current replaced by its point-split form in the parentheses. +2. +Diagram b (same flavor) and SST +For local current, there are 2 terms, d1 and d7 in Eq.(B4), that are contributing to the connected part of same-flavor +correlations. They are characterized by the charge factors ququ = 4/9 or q ¯ +dq ¯ +d = 1/9. The d1 diagram is clock-wise +propagation t0 → t3 → t2 → t1 → t0 where the two currents couple to the same u quark, while the d7 diagram is counter +clock-wise propagation t0 → t1 → t2 → t3 → t0 where the two currents couple to the same d quark. Under isospin +symmetry, the total contribution from uu and dd correlations has a total charge factor of 4/9 + 1/9 = 5/9. +Including the Fourier transforms, setting µ = 4 = ν for electric polarizability, and inserting the wall sources, the +correlation function can be written as, +˜Q(b,P C) +44 += 5 +9Z2 +V κ2 Tr +s,c +� +WT γ5S(t0, t1)γ4eiqS(t1, t2)γ4e−iqS(t2, t3)WWT γ5S(t3, t0)W +� +. +(19) +This expression involves numerous quark propagators: t0 and t3 are fixed, but t1 and t2 are free to vary. To cut down +the computational cost, we fix the current at t1. Then only one new inversion between t1 and t2 is required. Since the +current insertions take place between the hadron source (t0) and sink (t3), a method called SST (Sequential Source +Technique) can be employed for the propagator. To see how SST arises in this context, we first define the product that +involves t0 → t3 → t2 propagation as, +γ4e−iqS(t2, t3)WWT γ5S(t3, t0)W = γ4e−iqP(t2)M −1 +q +P(t3)T WWT γ5P(t3)M −1 +q +P(t0)T W += γ4e−iqP(t2)Va2WT P(t3)γ5Va1, +(20) +which is built directly from the two previously-computed propagators Va1 and Va2 along with other factors. This does + +6 +not require a new inversion. Next, we define the rest in Eq.(19) as, +WT γ5S(t0, t1)γ4eiqS(t1, t2) = WT γ5P(t0)M −1 +q +P(t1)T γ4eiqP(t1)M −1 +q +P(t2)T += +� +P(t2)γ5M −1 +q +γ5P(t1)T γ4e−iqP(t1)γ5M −1 +q +γ5P(t0)T γ5W +�† += − +� +P(t2)γ5M −1 +q +P(t1)T γ4e−iqP(t1)Va1 +�† += − +� +P(t2)γ5V (4,P C) +a3 +�† +, +(21) +where we have introduced a SST propagator called a3 (specialized to µ = 4 here), +V (µ,P C) +a3 +(q) ≡ M −1 +q +P(t1)T � +γµe−iqP(t1)Va1 +� +. +(22) +This expression indicates that V (4,P C) +a3 +can be obtained by a standard inversion Mx = b with a “spatially extended +source” b = +� +γ4e−iqP(t1)Va1 +� at t1. This source is constructed from a previously defined quark propagator Va1 and the +current insertion, hence the name “sequential source”. Using (a1, a2) and the newly-defined propagator a3, the final +expression for diagram b takes the form, +˜Q(b,P C) +44 +(q, t2) = −5 +9Z2 +V κ2 Tr +s,c +� � +P(t2)γ5V (4,P C) +a3 +(q) +�† +γ4e−iqP(t2)Va2WT P(t3)γ5Va1 +� +. +(23) +Fig. 3 is a schematic depiction of how the propagators form the full correlation function in Eq.(23). +FIG. 3. Diagram (b) in terms of quark propagators: one part is Va1 to the pion wall at t3, then Va2 to the current insertion at +t2; the other is a SST propagator Va3 (red) built from Va1 and the current insertion at t1. +For conserved current, there are 8 terms contributing to diagram b in Eq.(B6). +Following the same procedure as for +point current, the final expression for diagram b from point-split current can be written as, +˜Q(b,P S) +44 +(q, t2) = 1 +9κ2� +d1 + d3 + d5 + d7 + d25 + d31 + d37 + d43 +� += −5 +9κ2 Tr +s,c +� +[P(t2)γ5V (4,P S) +a3 +(q)]†(1 − γ4)e−iqU4(t2, t2 + 1)P(t2 + 1)Va2WT P(t3)γ5Va1 +− [P(t2 + 1)γ5V (4,P S) +a3 +(q)]†(1 + γ4)U † +4(t2 + 1, t2)e−iqP(t2)Va2WT P(t3)γ5Va1 +� +, +(24) +where a new inversion is needed for the SST propagator, +V (4,P S) +a3 +(q) ≡ M −1 +q +� +P T (t1)(1 − γ4)e−iqU4(t1, t1 + 1)P(t1 + 1)Va1 − P T (t1 + 1)(1 + γ4)U † +4(t1 + 1, t1)e−iqP(t1)Va1 +� +. +(25) +This is the point-split version of Eq.(22) with µ = 4. Since the current is split in the t direction, U4 and U † +4 commute +with e−iq in these two equations. + +a3 +a2 +a1 +t2 +ti +t3 +a1 +to7 +3. +Diagram c (same flavor Z-graph) and SST +For local current, there are 2 terms, d0 and d9 in Eq.(B4), that are contributing to the connected part of same-flavor +correlations. They are characterized by the same charge factors ququ = 4/9 or q ¯ +dq ¯ +d = 1/9. The d0 diagram is a clock-wise +propagation t0 → t3 → t1 → t2 → t0 where the two currents couple to the u quark, while the d9 diagram is a counter +clock-wise propagation t0 → t2 → t1 → t3 → t0 where the two currents couple to the d quark. They are essentially the +Z-graph of diagram b with the current insertions 1 and 2 switched, whose correlation function can be written as, +˜Q(c,P C) +44 += 5 +9Z2 +V κ2 Tr +s,c +� +γ5WT S(t0, t2)γ4e−iqS(t2, t1)γ4eiqS(t1, t3)WWT γ5S(t3, t0)W +� +. +(26) +First we isolate the t3 → t1 → t2 propagation, +S(t2, t1)γ4eiqS(t1, t3)W = P(t2)M −1 +q +P(t1)T γ4eiqP(t1)M −1 +q +P(t3)T W += P(t2)M −1 +q +P(t1)T γ4eiqP(t1)Va2 ≡ P(t2)V (4,P C) +a4 +(q), +(27) +where a new SST propagator is introduced (specialized to µ = 4 here), +V (µ,P C) +a4 +(q) ≡ M −1 +q +P(t1)T � +γµeiqP(t1)Va2 +� +. +(28) +Using a1 and a4, the final expression for diagram c using point current takes the form, +˜Q(c,P C) +44 +(q, t2) = 5 +9Z2 +V κ2 Tr +s,c +� � +γ4eiqP(t2)γ5Va1 +�† +P(t2)V (4,P C) +a4 +(q)WT P(t3)γ5Va1 +� +. +(29) +Fig. 4 is a schematic depiction of how the propagators form this correlation function. +FIG. 4. Diagram (c) in terms of quark propagators: a1 from t0 to t3, SST quark propagator a4 (red) with sequential source +built from a2 and current insertion at t1, and a1 from t2 to t0. This is the Z-graph of diagram b. +For conserved current, there are 8 terms contributing to diagram c in Eq.(B6). +Following a similar procedure as for +local current, the final expression for diagram (c) from point-split current can be written as, +˜Q(c,P S) +44 +(q, t2) = 1 +9κ2� +d0 + d2 + d4 + d6 + d27 + d33 + d39 + d45 +� += 5 +9κ2 Tr +s,c +� +[P(t2)γ5Va1]†(1 − γ4)e−iqU4(t2, t2 + 1)P(t2 + 1)V (4,P S) +a4 +(q)WT P(t3)γ5Va1 +−[P(t2 + 1)γ5Va1]†(1 + γ4)U † +4(t2 + 1, t2)e−iqP(t2)V (4,P S) +a4 +(q)WT P(t3)γ5Va1 +� +, +(30) +where +V (4,P S) +a4 +(q) ≡ M −1 +q +� +P T (t1)(1 − γ4)eiqU4(t1, t1 + 1)P(t1 + 1)Va2 − P(t1 + 1)T (1 + γ4)U † +4(t1 + 1, t1)eiqP(t1)Va2 +� +. +(31) +Compare to Eq.(25) for diagram b, this expression has a2 instead of a1, q instead of −q, and no γ5. +The total connected contribution to the polarizabilities +in Eq.(1) is simply the sum of the individual normalized +terms in Fig. 2, +Q44(q, t2, t1) = Q(a) +44 + Q(b) +44 + Q(c) +44 , +(32) + +a2 +a4 +a1 +X +t2 +t1 +t3 +a1 +to8 +for either point current or conserved current. The charge +factors and flavor-equivalent contributions have been in- +cluded in each diagram. +IV. +SIMULATION DETAILS AND RESULTS +Having laid out the methodology and detailed the +correlations functions, we now discuss how to numeri- +cally evaluate them in a Monte Carlo simulation in or- +der to extract the polarizability. As a proof-of-principle +test, we use quenched Wilson action with β = 6.0 and +κ = 0.1520, 0.1543, 0.1555, 0.1565 on the lattice 243 × 48. +The pion mass corresponding to the kappas will be deter- +mined in our simulation. We analyzed 500 configurations +for κ = 0.1520 and 1000 configurations each for rest of +the kappas. The scale of this action has been determined +in Ref. [51], with inverse lattice spacing 1/a = 2.312 GeV +and kappa critical κc = 0.15708. It also gives the pion +mass as a function of kappa, +(mπa)2 = 2.09 × 1 +2 +� 1 +κ − 1 +κc +� +, +(33) +which will be compared with the measured mπ. Dirichlet +(or open) boundary condition is imposed in the time +direction, while periodic boundary conditions are used in +spatial dimensions. The pion source is placed at t0 = 7 +and sink at t3 = 42 (time is labeled from 1 to 48). One +current is inserted at a fixed time t1, while the other +current t2 is free to vary. We use integers {nx, ny, nz} to +label the discrete momentum on the lattice, +q = +�2πnx +Lx , 2πny +Ly , 2πnz +Lz +� +, +nx, ny, nz = 0, ±1, ±2, · · · , +(34) +and +consider +five +different +combinations +{0, 0, 0}, {0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {0, 0, 2}. +In +order +to evaluate the connected diagrams, we need four +inversions of the quark matrix with varying sources: +two wall-sourced propagators Va1 and Va2, and two +SST propagators Va3(q) and Va4(q) at a fixed q. So the +count for five momenta is 2 + 2 × 5 = 12 per kappa per +configuration. It takes longer to do the inversions for +larger kappas due to critical slowing down. +A. +Raw correlation functions +First, we discuss how to determine pion mass from the +various two-point functions in Sec. III A. In Fig. 5 we +show the wall-to-wall pion correlations based on Eq.(12) +(Type 1) and Eq.(13) (Type 2) at κ = 0.1555. Type 1 only +depends on the a1 quark propagator originating from the +wall source at t0 = 7. Instead of ending at fixed t3 = 42, +we allow it to vary in the entire range of t on the lattice. +One can visualize it as a moving wall sink. In this way, +we get to observe a plateau in the effective mass function +0 +10 +20 +30 +40 +-1 +0 +1 +2 +3 +4 +5 +t +Log10 of Two-point Functions +10 +20 +30 +40 +0.20 +0.25 +0.30 +0.35 +0.40 +t +Effective mass of Two-point Functions +FIG. 5. +Moving sink zero-momentum pion correlator Type 1 +(blue) and Type 2 (orange) and their effective mass functions +at mπ = 600 MeV. They are constructed from either a1 or +a2 quark propagators as explained in the text. The vertical +gridlines indicate the three fixed time points in the study. +These functions can be used to extract the pion mass in single- +exponential fashion. The value at t3 = 42 in Type 1 or at +t0 = 7 in Type 2 can be used for normalization of four-point +functions. +which we use to extract the mass. Similarly, Type 2 only +depends on the a2 quark propagator originating from +the wall source at t3 = 42. Instead of ending at fixed +t0 = 7, we allow it to vary in the entire range of t on the +lattice. We flip the sign of its effective mass function so a +direct comparison of the plateaus for the two types can +be made. We use Type 1 with a varying sink to extract +pion and rho masses at the four kappa values. We obtain +approximately 1100, 800, 600, and 370 MeV for pion mass +at κ = 0.1520, 0.1543, 0.1555, 0.1565, respectively. These +values agree well with those predicted from the relation +in Eq.(33). From this point on, we will refer to pion mass +rather than kappa values. The rho meson is considered in +this work to judge the efficacy of vector meson dominance +in form factor extraction. +The more precise numbers +for mπ and mρ with uncertainties will be given in the +summary table at the end (Table I). Another benefit of +plotting the Type 1 and Type 2 correlators with a varying +sink is we get to see the limited “window of opportunity” +in the effective mass where ground state dominates. This +is the window in which we study the current-current +correlations. We utilize this information to fix one of +the two currents in the four-point function calculation +so it mainly couple to the zero-momentum ground state. + +9 +0 +10 +20 +30 +40 +-0.2 +0.0 +0.2 +0.4 +0.6 +t2 +Diagram c +-0.2 +0.0 +0.2 +0.4 +0.6 +Diagram b +{0, 0, 0} +{0, 0, 1} +{0, 1, 1} +{1, 1, 1} +{0, 0, 2} +-0.2 +0.0 +0.2 +0.4 +0.6 +Four-point function Q44(q,t1,t2) +Diagram a +20 +25 +30 +35 +40 +0.0 +0.2 +0.4 +0.6 +t2 +0.0 +0.2 +0.4 +0.6 +0.0 +0.2 +0.4 +0.6 +Effective Mass of Q44(q,t1,t2) +FIG. 6. +Normalized four-point functions (left panel) and their effective mass functions (right panel) from the connected +diagrams as a function of current separation at mπ = 600 MeV. The q = 0 results serve as a check of current conservation. The +results for non-zero q between t2 = 18 and t2 = 41 will become the basis for our analysis. The vertical gridlines indicate the +pion walls (t0 = 7 and t3 = 42) and the fixed current insertion (t1 = 18). The horizontal gridlines in the effective mass functions +indicate the value of Eπ − mπ where the continuum dispersion relation Eπ = +� +q2 + m2π is used. +Having examined the plots, we settle on t1 = 18, 18, 18, 14 +for mπ = 1100, 800, 600, 370 MeV, respectively. +Next, we discuss normalization constant for four-point +functions. This is the zero-momentum wall-to-wall two- +point function in the denominator of Eq.(4). We have +three options, corresponding to the three types in Eq.(12), +Eq.(13), and Eq.(14). Type 1 normalization constant is +simply the special value at t = t3 = 42 in the blue curve +of Fig. 5, and Type 2 the special value at t = t0 = 7 in the +orange curve of Fig. 5. Type 3 normalization constant is +computed separately. The three types are not expected to +agree configuration by configuration since they originate +from different wall sources, but they should approach +the same value in the configuration average within statis- +tics. We found the numerical values 0.4683(6), 0.4672(6), +0.468(7), from Type1, Type2, and Type 3, respectively, at +this pion mass. We see that Type 3 has larger statistical +uncertainties than in Type 1 and Type 2. This is expected +since Type 3 is constructed from two wall sources, while +the other two from one. We will Type 3 as normalization +for the reason to be discussed below. +Having determined the two-point functions, we present +in Fig. 6 the raw normalized four-point functions Q44 at +five different values of momentum q and at mπ = 600 MeV. +For comparison purposes, all points in Q44 are displayed +on the same linear scale. For the effective mass function +ln Q44(t)/Q44(t+1), only points between the pion walls are +displayed for clarity. The results are based on conserved +currents and only the connected diagrams a, b and c. +There are a number of interesting features in these plots. +First, the results for q = 0 confirms the current con- +servation property discussed in Eq.(A9). Basically, for +conserved current, we expect the ratio of four-point func- +tion to two-point function to approach the charge factor +quq ¯ +d + qdq¯u = 4/9 for diagram a in the isospin limit, inde- +pendent of current insertion points t1 and t2. For diagrams +b and c, the factor is ququ + q ¯ +dq ¯ +d = 5/9. Indeed, this is +confirmed in all three diagrams (black dots). In diagram +a, current conservation is limited between t2 = 7 (on the +pion wall source) and t2 = 41 (one step inside the pion +wall sink) because the two currents independently cou- +ple to two different quarks in this range. In diagram b, +where they couple to the same quark, current conserva- +tion emerges only starting from t2 = 19. In diagram c, +it is limited between t2 = 7 and t2 = 17 because it is the +Z-graph of b (different time-ordering). If diagrams b and + +10 +0 +200 +400 +600 +800 +0.0 +0.2 +0.4 +0.6 +0.8 +Configuration No. +Type 3 +0.0 +0.2 +0.4 +0.6 +0.8 +Type 2 +0.0 +0.2 +0.4 +0.6 +0.8 +4pt, 2pt, and their ratios +Type 1 +FIG. 7. +Statistical fluctuations are shown in the unnormalized +four-point function (red), three types of two-point functions +(black), and their ratios (blue) at 20 randomly-selected config- +urations. For this figure, Diagram (a) at q = 0 and mπ = 600 +MeV is used as an example. Neighboring points are connected +by straight lines to facilitate visualization. The faint horizontal +gridline indicates the expected ratio 4/9 for this diagram and +conserved currents. +c are added, then current conservation extends to the +whole range, just like diagram a, except for the special +point of t1 = t2 to be discussed below. Outside the regions +of current conservation, the q = 0 signal is exactly zero, +while the q ̸= 0 signal gradually goes to zero towards the +Dirichlet wall. +Second, we found that although we have three options +for two-point functions to be used as normalization, they +have different statistical fluctuations. +This is demon- +strated in Fig. 7 where we plot the three types for a +select few configurations out of the 1000, using diagram +(a) at zero momentum and a fixed time slice in the con- +served region (7 < t2 < 41) as an example. For each type, +we plot separately the unnormalized four-point function, +two-point function, and their ratios. We see that the +ratio from Type 3 gives the expected value (4/9) exactly +whereas Type 1 and Type 2 fluctuate around it. The +reason is that Type 3, despite being more noisy than +Type 1 and Type 2, is exactly correlated with the four- +point function configuration by configuration, both being +constructed from the same two wall sources. We rely on +this perfect correlation in Type 3 to serve as a strong +numerical validation that the wall sources and the con- +served currents are correctly implemented in our study. +At nonzero momentum (q ̸= 0), however, we found that +all three normalization types produce comparable statisti- +cal uncertainties for the normalized four-point functions. +Fig. 6 is plotted using Type 3 normalization. +Third, the special point of t1 = t2 is regular in diagram +a, but gives irregular results in diagram b and c for all +values of q. This is the contact term in the discussion +surrounding Eq.(A9). We avoid this point in our analysis. +Fourth, we observe that the results about t1 = 18 in +diagram b and c are mirror images of each other, simply +due to the fact that they are from the two different time +orderings of the same diagram. In principle, this property +could be exploited to reduce the cost of simulations. In +this study, however, we computed all three diagrams +separately, and add them between t1 = 19 and t3 = 41 as +the signal. We also note in passing that the Q44 signal +in diagram c is negative definite whereas it is positive +definite in diagrams a and b. +Finally, the effective mass function of Q44 for diagram b +approaches the value of Eπ − mπ at large separation times +between t1 and t2. This is an indication that the four- +point function for diagram b is dominated by the elastic +contribution with a fall-off rate of Eπ − mπ according +to Eq.(7). +The same is true for diagram a, although +deviations are slightly larger at higher momentum. The +situation for diagram c, however, is completely different. +The fall-off rates approach high above their respective +Eπ−mπ values, suggesting they are dominated by inelastic +contributions. In other words, the intermediate state is +not a pion, but some four-quark state at higher mass and +energy. +We also used local current as a guide to develop our for- +malism and algorithms. If we take the four-point function +ratio of local current (PC) to conserved current (PS) at +q = 0, we expect ˜Q(P S) +44 +/ ˜Q(P C) +44 +→ Z2 +V . This is confirmed for +all three diagrams and we obtain an estimate of ZV ∼ 1.47, +consistent with literature [52]. Since our results are based +exclusively on conserved current, we will not discuss local +current further. +B. +Elastic form factor +The formula for electric polarizability in Eq.(1) involves +the charge radius rE and the elastic contribution Qelas +44 , +both of which can be extracted from the large-time behav- +ior of four-point functions Q44. According to Eq.(8), Qelas +44 +is expected to exhibit single-exponential behavior with a +fall-off rate of Eπ − mπ. The form factor Fπ is contained +in the amplitude of this fall-off. Based on the discussion +about Fig. 6, diagrams a and b have the expected fall-off +whereas diagram c does not. As far as elastic contribution +is concerned, we can drop diagram c and focus only on +diagrams a and b. This improves the form factor analysis + +11 +{0, 0, 1} +{0, 1, 1} +{1, 1, 1} +{0, 0, 2} +0 +5 +10 +15 +20 +25 +-3 +-2 +-1 +0 +t=t2-t1 +Log10 of Q44 +(ab)(q,t) +0 +5 +10 +15 +20 +25 +0.0 +0.1 +0.2 +0.3 +0.4 +t=t2-t1 +Effective Mass of Q44 +(ab)(q,t) +FIG. 8. +Normalized four-point functions from diagrams a +and b in log plot and their effective mass functions at different +values of q and mπ = 600 MeV. They are plotted as time +separations t = t2 − t1 between the two currents relative to +fixed t1 = 18. The horizontal gridlines in the effective mass +are Eπ − mπ using continuum dispersion relation for Eπ with +measured mπ. These functions are used to extract the elastic +contributions Qelas +44 . +by eliminating the inelastic ‘contamination’ from diagram +c. It can be regarded as a form of optimization in the +analysis. Fig.8 shows an example of the four-point func- +tions Qab +44 including only diagrams a and b, along with +their effective mass functions. We focus in the region of +signal between t1 and t3 and plot them as a function of +time separation t = t2 −t1 between the two currents. Note +that we exclude the t = 0 point from the analysis due to +contact terms, as discussed earlier. We see that there is a +region where the effective mass functions coincide with +the Eπ −mπ gridlines, indicating that Qab +44 is dominated by +elastic contributions. The agreement is better at smaller +momentum values. The signal at large times is noisy and +increasingly so at higher momentum. We also see the +effect of the Dirichlet wall which forces the effective mass +to curve down. In this context, the inclusion of diagram +c would push the elastic limit into larger times where the +signal is lost. To account for possible violation of the +continuum dispersion relation, we perform a fit to the +functional form of Qelas +44 +in Eq.(8), treating both {Fπ, Eπ} +as free parameters with mπ fixed at the measured values +from two-point functions. Details of the fits at all four +pion passes are given in Table II in Appendix D. From +this table, we observe that the Eπ from the fit largely +agrees with that from the continuum dispersion relation. +Deviations become more apparent at higher momentum. +0.0 +0.5 +1.0 +1.5 +0.4 +0.6 +0.8 +1.0 +q2 (GeV2) +Elastic Form Factor Fπ (q2) +mπ=370 MeV +0.4 +0.6 +0.8 +1.0 +mπ=600 MeV +0.4 +0.6 +0.8 +1.0 +mπ800 MeV +0.4 +0.6 +0.8 +1.0 +mπ=1100 MeV +FIG. 9. +Pion elastic form factors extracted from four-point +functions. The red data points are the measured values in +Table II. The green solid line is a fit to the z-expansion in +Eq. (36). The green dashed line is a fit to the monopole form +in Eq. (35). The blue dashed line is the same monopole form +plotted with the measured rho mass, and the black solid line +with the physical rho mass. +After the form factor data are obtained, we fit them to +the monopole form, +Fπ(q2) = +1 +1 + q2/m2 +V +, +(35) +which is the well-known vector meson dominance (VMD) +commonly considered in pion form factor studies. The +results are illustrated in Fig. 9. We see that the monopole +form does not fit the data very well, especially at higher +momentum and lower pion mass. For this reason, we also + +12 +considered the z-expansion parametrization [53] +Fπ(q2) = 1 + +kmax +� +k=1 +ak zk, +where z ≡ +√tcut − t − √tcut − t0 +√tcut − t + √tcut − t0 +and t = −q2, tcut = 4m2 +π, +(36) +where ak are free parameters and tcut is the two-pion +production threshold. We take t0 = 0 so the form goes +through Fπ(0) = 1 by construction. Using this form, we +can find a good fit with kmax = 3 in all cases. For compari- +son, we also plot the monopole function with the measured +rho mass mρ and the physical rho mass of mphys +ρ += 0.77 +GeV. We observe significant differences between the fitted +monopole form (mV ) and the VMD form (mρ). The dif- +ference grows with increasing momentum and decreasing +pion mass. Similar behavior has been observed in previ- +ous studies [49, 54]. Once the functional form of form +factor is determined, the charge radius is obtained by +r2 +E += −6dFπ(q2) +dq2 +��� +q2→0. +(37) +Their values in physical units are put in Table I. +0 +2 +4 +6 +8 +10 +12 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +t=t2-t1 +q= {0, 0, 2} +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +q= {1, 1, 1} +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +q= {0, 1, 1} +Q44 +Q44 +elas +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Four-point function +q= {0, 0, 1} +FIG. 10. +Total Q44 and elastic Qelas +44 +at different values of q at +mπ = 600 MeV. The shaded area, +� +dt +� +Q44(q, t)−Qelas +44 (q, t) +� +, +is the signal contributing to polarizability. +C. +Electric polarizability +Having obtained the elastic contribution Qelas +44 , we now +turn to the extraction of αE from Eq.(1). In Fig. 10 we +show separately the total contribution Q44 (from all three +diagrams) and Qelas +44 +as a function of current separation +t = t2 − t1. W use mπ = 600 MeV as an example; the +graphs at the other pion masses look similar. Note that +although Qelas +44 +is obtained in the large time region, the +subtraction is done in the whole region according to the +functional form in Eq.(8). Most of the contribution is in +the small time region where inelastic contributions are +significant. We observe that Qelas +44 +is consistently larger +than Q44, suggesting that the inelastic term in the formula +is negative. The time integral is simply the negative of +the shaded area between the two curves. One detail to +notice is that the curves include the t = 0 point which has +unphysical contributions in Q44 as mentioned earlier. We +would normally avoid this point and only start the integral +from t = 1. However, as one can see, the chunk of area +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +-1.2 +-1.0 +-0.8 +-0.6 +-0.4 +-0.2 +q2(GeV2) +αE inelastic (10-4 fm3) +mπ=370 MeV +-0.8 +-0.6 +-0.4 +-0.2 +mπ=600 MeV +-0.6 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +mπ=800 MeV +-0.30 +-0.25 +-0.20 +-0.15 +-0.10 +-0.05 +0.00 +mπ=1100 MeV +FIG. 11. +Extrapolation of the second term (inelastic) in +Eq. (1) to q2 = 0 in physical units. +The red points are +based on the shaded areas in Fig. 10. The blue curve is a +quadratic extrapolation using all points. The green curve is +a linear extrapolation based on the two smallest q2 values +with straight lines connecting all the points. The black points +indicate the extrapolated values contributing to αE. + +13 +TABLE I. Summary of results in physical units from two-point and four-point functions. Two sets of results are given: one +based on charge radius from z-expansion fits, one from monopole fits. The total αE is chirally extrapolated to the physical point. +This value, in conjunction with the elastic αE derived from charge radius physical pion mass by PDG, yields the prediction for +the inelastic αE at the physical point. All αE values are in units of 10−4 fm3. +κ=0.1520 +κ=0.1543 +κ=0.1555 +κ=0.1565 +physical point +mπ (MeV) +1104.7 ± 1.2 +795.0 ± 1.1 +596.8 ± 1.4 +367.7 ± 2.2 +138.000 +mρ (MeV) +1273.1 ± 2.5 +1047.3 ± 3.4 +930. ± 7. +830. ± 17. +770.000 +r2 +E +(fm2) (z-expansion) +0.144 ± 0.004 +0.195 ± 0.007 +0.258 ± 0.005 +0.305 ± 0.016 +0.434 ± 0.005 +r2 +E +(fm2) (monopole) +0.1215 ± 0.0009 0.1798 ± 0.0022 0.1951 ± 0.0011 0.1993 ± 0.0023 +0.434 ± 0.005 +αE elastic (z-expansion) +0.626 ± 0.016 +1.17 ± 0.04 +2.07 ± 0.04 +3.98 ± 0.21 +15.08 ± 0.18 +αE elastic (monopole) +0.527 ± 0.004 +1.084 ± 0.013 +1.567 ± 0.010 +2.599 ± 0.033 +15.08 ± 0.18 +αE inelastic (z-expansion) +−0.296 ± 0.008 +−0.567 ± 0.008 +−0.866 ± 0.006 +−1.136 ± 0.012 +−10.7 ± 0.5 +αE inelastic (monopole) +−0.296 ± 0.008 +−0.567 ± 0.008 +−0.866 ± 0.006 +−1.136 ± 0.012 +−13.02 ± 0.22 +αE total (z-expansion) +0.331 ± 0.018 +0.61 ± 0.04 +1.21 ± 0.04 +2.84 ± 0.21 +4.3 ± 0.5 +αE total (monopole) +0.232 ± 0.009 +0.517 ± 0.016 +0.701 ± 0.011 +1.463 ± 0.035 +2.06 ± 0.28 +between t = 0 and t = 1 is the largest piece in the integral. +To include this contribution, we linearly extrapolated the +Q44 term back to t = 0 using the two points at t = 1 and +t = 2. This will incur a systematic effect on the order +of O(a2) since the error itself is order of O(a). As the +continuum limit is approached, the systematic effect will +vanish (the chunk will shrink to zero). There is no issue +to include this point in Qelas +44 +using its functional form. +The entire second term (prefactor and time integral) +is a function of momentum. Since αE is a static prop- +erty, we extrapolate it to q2 = 0 smoothly. To assess the +systematic effect of this extrapolation, we consider two +fitting forms, one is a + bx + cx2 (x = q2) using all data +points, the other a simple linear extrapolation using the +two lowest points. The results are shown in Fig. 11 for all +pion masses. One observes a difference between the two +that decreases with decreasing pion mass. The difference +in the extrapolation is a systematic effect in the analysis. +We will use the linearly extrapolated values to determine +αE. Finally, we assemble the two terms in the formula +in Eq.(1) to obtain αE in physical units. We summarize +all of the results measured in this study in Table I. We +include two set of results, one based on monopole, the +other on z-expansion. At each pion mass the elastic term +makes a positive contribution, whereas the inelastic term +makes a negative and smaller contribution, resulting in +a positive and relatively small value in the total. To see +how the trend continues, we include the physical point +in the following way. We take the total values for αE +and perform a smooth extrapolation to the physical point +using a + bmπ + cm2 +π form. This is done for both sets. +The extrapolated values αE = 4.3 ± 0.5 or αE = 2.06 ± 0.28 +can be compared to known values for charged pion αE. +PDG [55] quotes a value αE = 2.0 ± 0.6 ± 0.7 from ex- +periment with large uncertainties. Chiral perturbation +theory (ChPT) [56] gives αE = 2.8. Our values are compa- +rable. We also attempted to extrapolate using the 1/mπ +form expected from ChPT. This form does not fit our +data well which is not surprising since some of our pion +masses lie beyond the region of validity for ChPT. To get +a sense of individual contributions at the physical point, +we take the PDG value +r2 +E += 0.435(5) fm2 and physical +mπ to arrive at the elastic αE value of 15.08(18). Then +the inelastic values αE = −10.7(5) or αE = −13.0(2) can +be inferred from the total and the elastic. We should +mention that our values are consistent with the inelastic +contribution obtained in another lattice study [36] near +physical pion mass. It employs a formula derived from a +different method but has a similar structure. +In any event, a physical picture starts to emerge from +our results. In the approach to the physical point, the +elastic contribution grows positive strongly; at the same +time the inelastic contribution grows negative strongly; +the total is relatively small and positive and has mild pion +mass dependence. This picture is displayed in Fig. 12. +★ +★ +elastic +inelastic +total +0.2 +0.4 +0.6 +0.8 +1.0 +-10 +-5 +0 +5 +10 +15 +mπ (GeV) +Charged Pion αE (10-4 fm3) +FIG. 12. +Pion mass dependence of electric polarizability of a +charged pion from four-point functions in lattice QCD. The +elastic and inelastic contributions correspond to the two terms +in the formula in Eq.(1). Two sets of results are displayed: +one based on charge radius from z-expansion fits (solid lines), +one from monopole fits (dashed lines). The green star is the +known value from chiral perturbation theory. + +14 +V. +SUMMARY AND OUTLOOK +We investigated the feasibility of using four-point func- +tions in lattice QCD to extract charged pion electric polar- +izability. The approach is based on low-energy Compton +scattering tensor constructed with quark and gluon fields +in Euclidean spacetime [38]. The central object is the +formula given in Eq.(1) which consists of two terms. One +is an elastic contribution involving charge radius +r2 +E +and pion mass. The other an inelastic contribution in the +form of a subtracted time integral. In addition to four- +point functions, it requires two-point functions for pion +mass and normalization, but not three-point functions. +The elastic contribution can be obtained from the same +four-point function in the elastic limit. +We laid out a detailed formalism and notation using +standard Wilson fermion as a baseline. +Although we +use both local current and conserved current on the lat- +tice to develop and test the formalism, our results are +based on conserved current on the lattice. It sidesteps the +renormalization issue (ZV = 1), but comes with increased +complexity in implementation. To apply the special kine- +matics (zero-momentum Breit frame) in the formula, we +employ wall sources without gauge-fixing for the creation +and annihilation of pions. We show how to construct +the four-point functions using SST quark propagation, +develop efficient algorithms for numerical evaluation, and +use a high-performance implementation [57]. +We carried out a proof-of-concept simulation using +quenched Wilson action with pion mass ranging from +1100 to 370 MeV. We only considered the connected +contributions in this work. +We discussed three types +wall-to-wall two-point functions for normalization. We +found a perfect correlation between the four-point function +Q44(q2 = 0) and Type 3 two-point function imposed by +current conservation, configuration by configuration. This +property provides a strong check of our implementation. +The analysis procedure used to determine αE in Eq.(1) +involves multiple steps which we summarize here. 1) Fit +Type 1 two-point function to obtain mπ (and mρ). 2) +Fit four-point function Q(ab) +44 +from diagrams a and b to +Qelas +44 +at large times for elastic form factor Fπ. 3) Fit +Fπ data to a functional form (monopole or z-expansion), +then extract charge radius +r2 +E . 4) Perform subtraction +Q(abc) +44 +(q)−Qelas +44 (q) at small times using all three diagrams +a,b,c. Do the time integration. Extrapolate back to t = 0 +to include the missing chunk due to contact terms. 5) +Extrapolate the inelastic term to q2 = 0 to obtain the +static limit, then assemble everything in physical units +for αE. 6) Extrapolate the total αE in pion mass to the +physical point. +The final results reveal a clear physical picture for +charged pion αE: it is the result of a large cancellation be- +tween the elastic and inelastic contributions. Individually, +each contribution has strong pion mass dependence in the +approach to chiral limit, but the total has a small posi- +tive value with only a mild pion mass dependence. The +simulation also demonstrates that the four-point function +methodology can be a viable alternative to the background +method for polarizabilities of charged hadrons. We cau- +tion that the picture is subject to a number of systematic +effects at this stage, such as the quenched approxima- +tion, finite-volume effects, and disconnected loops. Aside +from these effects, the largest source of uncertainty in the +present analysis is in the form factor fitting. We observe +significant differences between monopole and z-expansion. +Although the uncertainty does not alter the picture quali- +tatively, it matters for quantitative comparisons. This is +an open issue that warrants further study. +Going forward, the investigation can proceed in multiple +directions. +First, the quenched approximation should +be removed by employing dynamical fermions. +Work +is underway to use our collection of two-flavor nHYP- +clover ensembles [58] which have been successfully used +in a number of physics projects. They have smaller pion +masses (about 315 MeV and 227 MeV) that can be used +to check the expected chiral behavior. The elongated +geometries in these ensembles offer a cost-effective way +of studying finite-volume effects and reaching smaller q +values. It would be interesting to see how the charge +radius is affected by the change of action. +Second, a +simulation of charged pion magnetic polarizability (βM) is +straightforward. The formula has been derived in Ref. [38]. +One just needs to replace Q44 with Q11 in the formalism. +It would be interesting to check the well-known prediction +αE + βM ≈ 0 from chiral perturbation theory. Third, the +disconnected contributions should be included. This is a +challenging task. Although disconnected loops generally +give relatively smaller contributions than connected ones, +they must be dealt with for a complete picture from lattice +QCD. Fourth, the methodology can be equally applied +to neutral particles (for example π0 and the neutron). +The advantage it offers over the background field method +is the natural treatment of disconnected loops (or sea +quarks) [4, 5]. Our ultimate target is the proton for which +a formula is also available [38]. 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D 46, 1109–1122 (1992), arXiv:hep- +lat/9205015. +[62] Andrei Alexandru, Manfried Faber, Ivan Horv´ath, and +Keh-Fei Liu, “Lattice qcd at finite density via a new +canonical approach,” Physical Review D 72 (2005), +10.1103/physrevd.72.114513. + +17 +Appendix A: Operators and current conservation +To evaluate Eq.(4) in lattice QCD, we use standard annihilation (ψ) and creation (ψ†) operators for a charged pion, +ψπ+(x) = ¯d(x)γ5u(x), ψ† +π+(x) = −¯u(x)γ5d(x). +(A1) +We also consider rho meson two-point functions constructed from, +ψρ(x)i = ¯d(x)γiu(x), +i = 1, 2, 3, +(A2) +and average over the spatial directions. For Wilson fermions, the Dirac operator Mq = ̸D + mq takes the standard form +for a single quark flavor labeled by q, +Mq = 1 − κq +� +µ +� +(1 − γµ)Uµ + (1 + γµ)U † +µ +� +, +(A3) +where κq = 1/(2mq + 4) is the hopping parameter and mq the bare quark mass. +For current operators, we consider two options. One is the lattice point current built from up and down quark fields, +j(P C) +µ +≡ ZV κ +� +qu¯uγµu + qd ¯dγµd +� +. +(A4) +The factor κ here is to account for the quark-field rescaling ψ → +√ +2κψ in Wilson fermions. The factor 2 is canceled by +the 1/2 factor in the definition of the vector current 1 +2 ¯ψγµψ. The charge factors are qu = 2/3 and qd = −1/3 where the +resulting e2 = α ≈ 1/137 in the four-point function has been absorbed in the definition of απ +E. The advantage of this +operator is that it leads to simple correlation functions. The drawback is that the renormalization constant for the +vector current (ZV ) has to be determined. +We also consider conserved vector current on the lattice (ZV ≡ 1) which can be derived by the Noether procedure. +For the Wilson fermion action S = ¯ψqMqψq built from the matrix in Eq.(A3), the simplest way [59] is to substitute the +gauge fields by +Uµ(x) → Uµ(x)eiqqvq +µ, +(A5) +and differentiate with respect to the external vector field vq +µ, then take vq +µ → 0. The result is the point-split form +j(q,P S) +µ +(x) = −i δS +δvq +µ +���� +vq +µ→0 += −qqκq +� ¯ψq(x)(1 − γµ)Uµ(x)ψq(x + ˆµ) − ¯ +ψq(x + ˆµ)(1 + γµ)U † +µ(x)ψq(x) +� +. +(A6) +The phase factor −i is explained in Ref. [60]. An alternative method [61, 62] is through a local transformation on +the quark fields, ψ → e−iω(x)ψ, and do variation +δS +δ(∆µω) on the finite difference ∆µω = ω(x + ˆµ) − ω(x). For two quark +flavors (u and d), we have +j(P S) +µ +(x) = quκu +� +− ¯u(x)(1 − γµ)Uµ(x)u(x + ˆµ) + ¯u(x + ˆµ)(1 + γµ)U † +µ(x)u(x) +� ++ qdκd +� +− ¯d(x)(1 − γµ)Uµ(x)d(x + ˆµ) + ¯d(x + ˆµ)(1 + γµ)U † +µ(x)d(x) +� +. +(A7) +The conserved current for nhyp fermion has the same form, except the gauge links are nhyp-smeared. Although +conserved currents explicitly involve gauge fields and lead to more complicated correlation functions, they have the +advantage of circumventing the renormalization issue. +Just like current conservation guarantees the normalization condition in three-point functions, +� +x1 +Ω|ψ(x) j(q,P S) +4 +(x1) ψ†(0)|Ω = qq Ω|ψ(x)ψ†(0)|Ω , +(A8) +a similar condition holds in fount-point functions, +� +x2,x1 +Ω|ψ(x) j(q2,P S) +4 +(x2) j(q1,P S) +4 +(x1) ψ†(0)|Ω = q1q2 Ω|ψ(x)ψ†(0)|Ω . +(A9) +In physical terms, the charge overlap at q = 0 on the left-hand-side is effectively reconstructing the two-point function. +Each charge density is spread over all spatial sites on the lattice. By summing over x1 and x2 at zero momentum, we +recover the total charge factor from each insertion, regardless of the time points of the insertions. +There is a subtle +issue with four-point functions. If the two currents couple to different quark lines (q1 ̸= q2), the conservation is for all +combinations of t1 and t2 between source and sink, including t1 = t2. If they couple to the same quark line (q1 = q2), +the conservation is only true for t1 ̸= t2. The point t1 = t2 introduces unwanted contact terms on the lattice and is +avoided. The issue is a lattice artifact; in the continuum, the contact interaction is regular and well-defined. The +conservation property in Eq.(A9) is used to validate the four-point diagrams in this work. + +18 +Appendix B: Wick contractions +Here we give the unnormalized correlation functions in Eq.(3) by contracting out all quark-antiqurk pairs. +1. +Local current +For point current (PC), using Eq.(A1) and Eq.(A4), the full correlation function has 20 diagrams, +˜Q(P C) +µν +(q, t3, t2, t1, t0) = +� +x2,x1 +e−iq·x2eiq·x1 � +x3,x0 +⟨Ω|ψπ+(x3, t3)j(P C) +µ +(x2, t2)j(P C) +ν +(x1, t1)ψ† +π+(x0, t0)|Ω⟩ +≡ Z2 +V κ2 +9 +19 +� +i=0 +di(q, t3, t2, t1, t0), +(B1) +where +dA +10 = −2 tr +� +Su(t1, t3)γ5Sd(t3, t2)γµe−iqSd(t2, t0)γ5Su(t0, t1)γνeiq� +dA-bwd +7 += −2 tr +� +Su(t2, t3)γ5Sd(t3, t1)γνeiqSd(t1, t0)γ5Su(t0, t2)γµe−iq� +dB +5 = 4 tr +� +Su(t2, t3)γ5Sd(t3, t0)γ5Su(t0, t1)γνeiqSu(t1, t2)γµe−iq� +dB-bwd +15 += 1 tr +� +Su(t0, t3)γ5Sd(t3, t2)γµe−iqSd(t2, t1)γνeiqSd(t1, t0)γ5 +� +dC +1 = 4 tr +� +Su(t1, t3)γ5Sd(t3, t0)γ5Su(t0, t2)γµe−iqSu(t2, t1)γνeiq� +dC-bwd +17 += 1 tr +� +Su(t0, t3)γ5Sd(t3, t1)γνeiqSd(t1, t2)γµe−iqSd(t2, t0)γ5 +� +dD +0 = −4 tr [Su(t0, t3)γ5Sd(t3, t0)γ5] tr +� +Su(t1, t2)γµe−iqSu(t2, t1)γνeiq� +dD +18 = −1 tr [Su(t0, t3)γ5Sd(t3, t0)γ5] tr +� +Sd(t1, t2)γµe−iqSd(t2, t1)γνeiq� +dEl +4 = −4 tr +� +Su(t1, t3)γ5Sd(t3, t0)γ5Su(t0, t1)γνeiq� +tr +� +Su(t2, t2)γµe−iq� +dEl +13 = 2 tr +� +Su(t1, t3)γ5Sd(t3, t0)γ5Su(t0, t1)γνeiq� +tr +� +Sd(t2, t2)γµe−iq� +dEl-bwd +6 += 2 tr +� +Su(t0, t3)γ5Sd(t3, t1)γνeiqSd(t1, t0)γ5 +� +tr +� +Su(t2, t2)γµe−iq� +dEl-bwd +14 += −1 tr +� +Su(t0, t3)γ5Sd(t3, t1)γνeiqSd(t1, t0)γ5 +� +tr +� +Sd(t2, t2)γµe−iq� +dEr +2 += −4 tr +� +Su(t2, t3)γ5Sd(t3, t0)γ5Su(t0, t2)γµe−iq� +tr +� +Su(t1, t1)γνeiq� +dEr +8 += 2 tr +� +Su(t2, t3)γ5Sd(t3, t0)γ5Su(t0, t2)γµe−iq� +tr +� +Sd(t1, t1)γνeiq� +dEr-bwd +11 += 2 tr +� +Su(t0, t3)γ5Sd(t3, t2)γµe−iqSd(t2, t0)γ5 +� +tr +� +Su(t1, t1)γνeiq� +dEr-bwd +16 += −1 tr +� +Su(t0, t3)γ5Sd(t3, t2)γµe−iqSd(t2, t0)γ5 +� +tr +� +Sd(t1, t1)γνeiq� +dF +3 = 4 tr [Su(t0, t3)γ5Sd(t3, t0)γ5] tr +� +Su(t2, t2)γµe−iq� +tr +� +Su(t1, t1)γνeiq� +dF +9 = −2 tr [Su(t0, t3)γ5Sd(t3, t0)γ5] tr +� +Su(t2, t2)γµe−iq� +tr +� +Sd(t1, t1)γνeiq� +dF +12 = −2 tr [Su(t0, t3)γ5Sd(t3, t0)γ5] tr +� +Sd(t2, t2)γµe−iq� +tr +� +Su(t1, t1)γνeiq� +dF +19 = 1 tr [Su(t0, t3)γ5Sd(t3, t0)γ5] tr +� +Sd(t2, t2)γµe−iq� +tr +� +Sd(t1, t1)γνeiq� +(B2) +We use a matrix notation that highlights time dependence. The trace is over spin and color. The momentum factor is +defined by a diagonal matrix, +[e±iq]s,c,x;s′,c′,x′ ≡ δss′δcc′δx,x′e±iq·x. +(B3) +The spatial sums over (x2, x1, x3, x0) are implicit in the matrix multiplications. We use S(t2, t1) to denote a quark +propagator from t1 to t2 (from right to left), obtained from the inverse of quark matrix M with a source Mx = b, see + +19 +Eq.(C11). The terms are grouped into six distinct topological diagrams depicted in Fig 2, labeled by superscripts on +di. If isospin limit (κu = κd = κ) is taken, we get 12 diagrams (first six connected, the rest disconnected), +dA +4 = −2 tr +� +S(t1, t3)γ5S(t3, t2)γµe−iqS(t2, t0)γ5S(t0, t1)γνeiq� +dA-bwd +2 += −2 tr +� +S(t2, t3)γ5S(t3, t1)γνeiqS(t1, t0)γ5S(t0, t2)γµe−iq� +dB +1 = 4 tr +� +S(t2, t3)γ5S(t3, t0)γ5S(t0, t1)γνeiqS(t1, t2)γµe−iq� +dB-bwd +7 += 1 tr +� +S(t0, t3)γ5S(t3, t2)γµe−iqS(t2, t1)γνeiqS(t1, t0)γ5 +� +dC +0 = 4 tr +� +S(t1, t3)γ5S(t3, t0)γ5S(t0, t2)γµe−iqS(t2, t1)γνeiq� +dC-bwd +9 += 1 tr +� +S(t0, t3)γ5S(t3, t1)γνeiqS(t1, t2)γµe−iqS(t2, t0)γ5 +� +dD +10 = −5 tr [S(t0, t3)γ5S(t3, t0)γ5] tr +� +S(t1, t2)γµe−iqS(t2, t1)γνeiq� +dEl +5 = −2 tr +� +S(t1, t3)γ5S(t3, t0)γ5S(t0, t1)γνeiq� +tr +� +S(t2, t2)γµe−iq� +dEl-bwd +6 += 1 tr +� +S(t0, t3)γ5S(t3, t1)γνeiqS(t1, t0)γ5 +� +tr +� +S(t2, t2)γµe−iq� +dEr +3 = −2 tr +� +S(t2, t3)γ5S(t3, t0)γ5S(t0, t2)γµe−iq� +tr +� +S(t1, t1)γνeiq� +dEr-bwd +8 += 1 tr +� +S(t0, t3)γ5S(t3, t2)γµe−iqS(t2, t0)γ5 +� +tr +� +S(t1, t1)γνeiq� +dF +11 = 1 tr [S(t0, t3)γ5S(t3, t0)γ5] tr +� +S(t2, t2)γµe−iq� +tr +� +S(t1, t1)γνeiq� +(B4) +2. +Conserved current +For point-split current (PS), using Eq.(A1) and Eq.(A7), Wick contraction yields 80 diagrams (not shown here) if u +and d are distinct. If isospin limit is taken, there are 48 diagrams which we express as, +˜Q(P S) +µν +(q, t3, t2, t1, t0) = +� +x2,x1 +e−iq·x2eiq·x1 � +x3,x0 +⟨Ω|ψπ+(x3, t3)j(P S) +µ +(x2, t2)j(P S) +ν +(x1, t1)ψ† +π+(x0, t0)|Ω⟩ +≡ κ2 +9 +47 +� +i=0 +di(q, t3, t2, t1, t0). +(B5) + +20 +The 24 connected diagrams are given by, +dA +16 = −2 tr +� +S(t1 + ˆν4, t3)(γ5)S(t3, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4)S(t2 + ˆµ4, t0)(γ5)S(t0, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +dA +18 = 2 tr +� +S(t1 + ˆν4, t3)(γ5)S(t3, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iqS(t2, t0)(γ5)S(t0, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +dA +20 = 2 tr +� +S(t1, t3)(γ5)S(t3, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4)S(t2 + ˆµ4, t0)(γ5)S(t0, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +dA +22 = −2 tr +� +S(t1, t3)(γ5)S(t3, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iqS(t2, t0)(γ5)S(t0, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +dA-bwd +8 += −2 tr +� +S(t2 + ˆµ4, t3)(γ5)S(t3, t1)(1 − γν)eiqUν(t1, t1 + ˆν4)S(t1 + ˆν4, t0)(γ5)S(t0, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4) +� +dA-bwd +10 += 2 tr +� +S(t2, t3)(γ5)S(t3, t1)(1 − γν)eiqUν(t1, t1 + ˆν4)S(t1 + ˆν4, t0)(γ5)S(t0, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iq� +dA-bwd +12 += 2 tr +� +S(t2 + ˆµ4, t3)(γ5)S(t3, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiqS(t1, t0)(γ5)S(t0, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4) +� +dA-bwd +14 += −2 tr +� +S(t2, t3)(γ5)S(t3, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiqS(t1, t0)(γ5)S(t0, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iq� +dB +1 = 4 tr +� +S(t2 + ˆµ4, t3)(γ5)S(t3, t0)(γ5)S(t0, t1)(1 − γν)eiqUν(t1, t1 + ˆν4)S(t1 + ˆν4, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4) +� +dB +3 = −4 tr +� +S(t2, t3)(γ5)S(t3, t0)(γ5)S(t0, t1)(1 − γν)eiqUν(t1, t1 + ˆν4)S(t1 + ˆν4, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iq� +dB +5 = −4 tr +� +S(t2 + ˆµ4, t3)(γ5)S(t3, t0)(γ5)S(t0, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiqS(t1, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4) +� +dB +7 = 4 tr +� +S(t2, t3)(γ5)S(t3, t0)(γ5)S(t0, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiqS(t1, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iq� +dB-bwd +25 += 1 tr +� +S(t0, t3)(γ5)S(t3, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4)S(t2 + ˆµ4, t1)(1 − γν)eiqUν(t1, t1 + ˆν4)S(t1 + ˆν4, t0)(γ5) +� +dB-bwd +31 += −1 tr +� +S(t0, t3)(γ5)S(t3, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iqS(t2, t1)(1 − γν)eiqUν(t1, t1 + ˆν4)S(t1 + ˆν4, t0)(γ5) +� +dB-bwd +37 += −1 tr +� +S(t0, t3)(γ5)S(t3, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4)S(t2 + ˆµ4, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiqS(t1, t0)(γ5) +� +dB-bwd +43 += 1 tr +� +S(t0, t3)(γ5)S(t3, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iqS(t2, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiqS(t1, t0)(γ5) +� +dC +0 = 4 tr +� +S(t1 + ˆν4, t3)(γ5)S(t3, t0)(γ5)S(t0, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4)S(t2 + ˆµ4, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +dC +2 = −4 tr +� +S(t1 + ˆν4, t3)(γ5)S(t3, t0)(γ5)S(t0, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iqS(t2, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +dC +4 = −4 tr +� +S(t1, t3)(γ5)S(t3, t0)(γ5)S(t0, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4)S(t2 + ˆµ4, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +dC +6 = 4 tr +� +S(t1, t3)(γ5)S(t3, t0)(γ5)S(t0, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iqS(t2, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +dC-bwd +27 += 1 tr +� +S(t0, t3)(γ5)S(t3, t1)(1 − γν)eiqUν(t1, t1 + ˆν4)S(t1 + ˆν4, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4)S(t2 + ˆµ4, t0)(γ5) +� +dC-bwd +33 += −1 tr +� +S(t0, t3)(γ5)S(t3, t1)(1 − γν)eiqUν(t1, t1 + ˆν4)S(t1 + ˆν4, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iqS(t2, t0)(γ5) +� +dC-bwd +39 += −1 tr +� +S(t0, t3)(γ5)S(t3, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiqS(t1, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4)S(t2 + ˆµ4, t0)(γ5) +� +dC-bwd +45 += 1 tr +� +S(t0, t3)(γ5)S(t3, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiqS(t1, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iqS(t2, t0)(γ5) +� +(B6) + +21 +The 24 disconnected diagrams are given by, +dD +28 = −5 tr [S(t0, t3)(γ5)S(t3, t0)(γ5)] tr +� +S(t1 + ˆν4, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4)S(t2 + ˆµ4, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +dD +34 = 5 tr [S(t0, t3)(γ5)S(t3, t0)(γ5)] tr +� +S(t1 + ˆν4, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iqS(t2, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +dD +40 = 5 tr [S(t0, t3)(γ5)S(t3, t0)(γ5)] tr +� +S(t1, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4)S(t2 + ˆµ4, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +dD +46 = −5 tr [S(t0, t3)(γ5)S(t3, t0)(γ5)] tr +� +S(t1, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iqS(t2, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +dEl +17 = −2 tr +� +S(t1 + ˆν4, t3)(γ5)S(t3, t0)(γ5)S(t0, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +tr +� +S(t2 + ˆµ4, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4) +� +dEl +19 = 2 tr +� +S(t1 + ˆν4, t3)(γ5)S(t3, t0)(γ5)S(t0, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +tr +� +S(t2, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iq� +dEl +21 = 2 tr +� +S(t1, t3)(γ5)S(t3, t0)(γ5)S(t0, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +tr +� +S(t2 + ˆµ4, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4) +� +dEl +23 = −2 tr +� +S(t1, t3)(γ5)S(t3, t0)(γ5)S(t0, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +tr +� +S(t2, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iq� +dEl-bwd +24 += 1 tr +� +S(t0, t3)(γ5)S(t3, t1)(1 − γν)eiqUν(t1, t1 + ˆν4)S(t1 + ˆν4, t0)(γ5) +� +tr +� +S(t2 + ˆµ4, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4) +� +dEl-bwd +30 += −1 tr +� +S(t0, t3)(γ5)S(t3, t1)(1 − γν)eiqUν(t1, t1 + ˆν4)S(t1 + ˆν4, t0)(γ5) +� +tr +� +S(t2, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iq� +dEl-bwd +36 += −1 tr +� +S(t0, t3)(γ5)S(t3, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiqS(t1, t0)(γ5) +� +tr +� +S(t2 + ˆµ4, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4) +� +dEl-bwd +42 += 1 tr +� +S(t0, t3)(γ5)S(t3, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiqS(t1, t0)(γ5) +� +tr +� +S(t2, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iq� +dEr +9 += −2 tr +� +S(t2 + ˆµ4, t3)(γ5)S(t3, t0)(γ5)S(t0, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4) +� +tr +� +S(t1 + ˆν4, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +dEr +11 = 2 tr +� +S(t2, t3)(γ5)S(t3, t0)(γ5)S(t0, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iq� +tr +� +S(t1 + ˆν4, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +dEr +13 = 2 tr +� +S(t2 + ˆµ4, t3)(γ5)S(t3, t0)(γ5)S(t0, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4) +� +tr +� +S(t1, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +dEr +15 = −2 tr +� +S(t2, t3)(γ5)S(t3, t0)(γ5)S(t0, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iq� +tr +� +S(t1, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +dEr-bwd +26 += 1 tr +� +S(t0, t3)(γ5)S(t3, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4)S(t2 + ˆµ4, t0)(γ5) +� +tr +� +S(t1 + ˆν4, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +dEr-bwd +32 += −1 tr +� +S(t0, t3)(γ5)S(t3, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iqS(t2, t0)(γ5) +� +tr +� +S(t1 + ˆν4, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +dEr-bwd +38 += −1 tr +� +S(t0, t3)(γ5)S(t3, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4)S(t2 + ˆµ4, t0)(γ5) +� +tr +� +S(t1, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +dEr-bwd +44 += 1 tr +� +S(t0, t3)(γ5)S(t3, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iqS(t2, t0)(γ5) +� +tr +� +S(t1, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +dF +29 = 1 tr [S(t0, t3)(γ5)S(t3, t0)(γ5)] tr +� +S(t2 + ˆµ4, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4) +� +tr +� +S(t1 + ˆν4, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +dF +35 = −1 tr [S(t0, t3)(γ5)S(t3, t0)(γ5)] tr +� +S(t2, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iq� +tr +� +S(t1 + ˆν4, t1)(1 − γν)eiqUν(t1, t1 + ˆν4) +� +dF +41 = −1 tr [S(t0, t3)(γ5)S(t3, t0)(γ5)] tr +� +S(t2 + ˆµ4, t2)(1 − γµ)e−iqUµ(t2, t2 + ˆµ4) +� +tr +� +S(t1, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +dF +47 = 1 tr [S(t0, t3)(γ5)S(t3, t0)(γ5)] tr +� +S(t2, t2 + ˆµ4)(1 + γµ)U † +µ(t2 + ˆµ4, t2)e−iq� +tr +� +S(t1, t1 + ˆν4)(1 + γν)U † +ν(t1 + ˆν4, t1)eiq� +(B7) +The shifted quark propagators have the following meaning depending on whether the current is split in time or spatial +directions, for example, +S(t3, t2 + ˆµ4) ≡ +� +S(t3, t2 + 1) = P(t3)M −1P(t2 + 1)T , +if µ = 4 +S(t3, t2) = P(t3)M −1P(t2)T , +if µ ̸= 4, +(B8) +where the projector P(t) is defined in Eq.(C9). The associated gauge links have the meaning, +Uµ(t2, t2 + ˆµ4) ≡ +� +U4(t2, t2 + 1), +if µ = 4 +Uµ(t2, t2), +if µ ̸= 4, +U † +µ(t2 + ˆµ4, t2) ≡ +� +U † +4(t2 + 1, t2), +if µ = 4 +U † +µ(t2, t2), +if µ ̸= 4, +(B9) + +22 +where the gauge links are defined in Eq.(C6). So the split in time is explicitly carried in both the propagators and +gauge links, whereas the split in space is only implicitly carried in the gauge links. Note the placement of e±iq in +relation to U and U †. They do not commute when the currents are split in spatial directions. +Appendix C: Wall source implementation +We introduce a rigorous matrix notation to elucidate the implementation of wall sources. We define wall sources as +a vector in spatial coordinates, diagonal in spin and color, +[W]s,c,x;s′,c′ ≡ δss′δcc′. +(C1) +That is, all spatial entries of the real part are set to 1, imaginary part to zero. It can be placed at any time slice. +Under a gauge transformation G, the gauge average is +G(t)WWT G(t)† +G = 1x,s,c, +where [G(t)W]x = G(t, x)1s,c. +(C2) +More explicitly, +� +G(t)WWT G(t)† +G +� +x,y = +1 +|G| +� +DG G(t, x)1spinG(t, y)†1spin = δx,y1s1c. +(C3) +We insert the wall source in between a pair of quark propagators in the path integral by the following steps, only +highlighting the time dependence in S to keep the notation simple, +� +DUP(U) Tr +x,s,c +� +. . . S[U](t′, t)S[U](t, t′′) . . . +� += +� +DUP(U) Tr +x,s,c +� +. . . S[U](t′, t) 1x,s,c S[U](t, t′′) . . . +� += +1 +|G| +� +DG +� +DUP(U) Tr +x,s,c +� +. . . S[U](t′, t)G(t)WWTG(t)†S[U](t, t′′) . . . +� += +1 +|G| +� +DG +� +DUP(UG) Tr +x,s,c +� +. . . S[UG](t′, t)G(t)WWTG(t)†S[UG](t, t′′) . . . +� += +1 +|G| +� +DG +� +DUP(U) Tr +x,s,c +� +. . . S[U](t′, t)WWTS[U](t, t′′) . . . +� += +� +DUP(U) Tr +x,s,c +� +. . . S[U](t′, t)WWTS[U](t, t′′) . . . +� += +� +DUP(U) Tr +s,c +� +WTS[U](t, t′′) . . . S[U](t′, t)W +� +. +(C4) +In the last step, we use the cyclic property of trace Tr AB = Tr BA. We also used the property that under a gauge +transformation Uµ → (UG)µ ≡ GUµG†, the propagator transforms as, +S[UG](t, t′) = G(t)S[U](t, t′)G(t′)†. +(C5) +More explicitly, the gauge links are, +(Uµ)x,t;x′,t′ = δ(x,t),(x′,t′)−µUµ(x, t)1s, +(C6) +and its gauge transformation is +(GUµG†)x,y = G(x)[Uµ]x,yG(y)† = G(x)δx,y−µUµ(x)G(y)† = δx,y−µG(x)Uµ(x)G(x + µ)†. +(C7) +Note that we will use +Uµ(t, t′) = P(t)UµP(t′)T +and +U † +µ(t, t′) = P(t)U † +µP(t′)T. +(C8) + +23 +Here P(t) is defined as projection to a time slice (not to be confused with the weighting factor P(U) in the path +integral in Eq.(C4)), +[P(tp)]s,c,x;s′,c′,t′,x′ ≡ δtp,t′δss′δcc′δx,x′, +(C9) +which is diagonal in spin, color, and space. When we take the dagger of Uµ(t, t′), we need to switch the time arguments +since +[Uµ(t, t′)]† = [P(t)UµP(t′)T]† = P(t′)U † +µP(t)T = U † +µ(t′, t) . +(C10) +Operationally, a quark propagator can be written in terms of the inverse of the quark matrix as, +S(t, t′) ≡ P(t)M −1 +q P(t′)T. +(C11) +For Wilson-type fermions, Mq satisfies the γ5-hermiticity relation +M † +q = γ5Mqγ5, +� +M −1 +q +�† = γ5M −1 +q γ5. +(C12) +Examples on how to use the notation to calculate two-point and four-point correlation functions are discussed in +Sec. III. +Appendix D: Form factor from four-point functions +TABLE II. Pion form factor Fπ(q2) from four-point functions. An example of the data to be fitted is given in Fig. 8. The +fit form is in Eq.(8) with Fπ and Eπ treated as free parameters and mπ taken from the measured value. For comparison, +the Eπ from the continuum dispersion relation is provided with the same mπ values. +The four columns correspond to +q = {0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {0, 0, 2} from left to right. +mπ=1100 MeV +Fπ +0.8209 ± 0.0023 0.7213 ± 0.0023 +0.650 ± 0.004 +0.604 ± 0.005 +Eπ fit +1.2556 ± 0.0016 1.4021 ± 0.0027 +1.530 ± 0.004 +1.644 ± 0.006 +Eπ continuum 1.2597 ± 0.0010 1.3976 ± 0.0009 1.5230 ± 0.0009 1.6389 ± 0.0008 +Fit range +{7,9} +{6,8} +{7,10} +{7,12} +χ2/dof +2.00 +1.40 +2.70 +1.90 +mπ=800 MeV +Fπ +0.7677 ± 0.0027 +0.646 ± 0.006 +0.568 ± 0.010 +0.552 ± 0.011 +Eπ fit +0.9967 ± 0.0020 +1.163 ± 0.005 +1.308 ± 0.010 +1.463 ± 0.013 +Eπ continuum 0.9992 ± 0.0009 1.1682 ± 0.0007 1.3157 ± 0.0007 1.4483 ± 0.0006 +Fit range +{9,13} +{10,17} +{10,17} +{9,14} +χ2/dof +1.40 +1.40 +1.10 +0.72 +mπ=600 MeV +Fπ +0.7412 ± 0.0015 0.6360 ± 0.0025 +0.583 ± 0.004 +0.525 ± 0.012 +Eπ fit +0.8508 ± 0.0017 +1.050 ± 0.004 +1.231 ± 0.007 +1.354 ± 0.016 +Eπ continuum 0.8500 ± 0.0010 1.0435 ± 0.0008 1.2063 ± 0.0007 1.3497 ± 0.0006 +Fit range +{4,13} +{6,15} +{6,9} +{8,13} +χ2/dof +0.52 +1.30 +1.50 +0.39 +mπ=360 MeV +Fπ +0.720 ± 0.004 +0.616 ± 0.005 +0.554 ± 0.006 +0.530 ± 0.008 +Eπ fit +0.695 ± 0.005 +0.911 ± 0.009 +1.076 ± 0.013 +1.258 ± 0.018 +Eπ continuum 0.7082 ± 0.0011 0.9316 ± 0.0009 1.1110 ± 0.0007 1.2652 ± 0.0006 +Fit range +{6,11} +{6,11} +{6,11} +{6,11} +χ2/dof +0.68 +0.34 +0.68 +1.00 + diff --git a/fdE4T4oBgHgl3EQfqg0X/content/tmp_files/load_file.txt b/fdE4T4oBgHgl3EQfqg0X/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d79ab9c53347fae8cfeb3088efc736506b54c67b --- /dev/null +++ b/fdE4T4oBgHgl3EQfqg0X/content/tmp_files/load_file.txt @@ -0,0 +1,1910 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf,len=1909 +page_content='Charged pion electric polarizability from four-point functions in lattice QCD Frank X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Lee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' ∗ Andrei Alexandru,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' † Chris Culver,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' ‡ and Walter Wilcox4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' § 1Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The George Washington University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' DC 20052,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' USA 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' University of Maryland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' College Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' MD 20742,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' USA 3Department of Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' University of Liverpool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Liverpool L69 7ZL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' United Kingdom 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Baylor University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Waco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Texas 76798,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' USA Polarizabilities reveal valuable information on the internal structure of hadrons in terms of charge and current distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For neutral hadrons, the standard approach is the background field method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' But for a charged hadron, its acceleration under the the applied field complicates the isolation of the polarization energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In this work, we explore an alternative method based on four-point functions in lattice QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The approach offers a transparent picture on how polarizabilities arise from quark and gluon interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We carry out a proof-of-concept simulation on the electric polarizability of a charged pion, using quenched Wilson action on a 243 × 48 lattice at β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 with pion mass from 1100 to 370 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We show in detail the evaluation and analysis of the four-point correlation functions and report results on charge radius and electric polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Our results from connected diagrams suggest that charged pion αE is due to a large cancellation between elastic and inelastic contributions, leaving a small and positive value that has a relatively mild pion mass dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' INTRODUCTION Understanding electromagnetic polarizabilities has been a long-term goal of lattice QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The challenge in the effort lies in the need to apply both QCD and QED princi- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The standard approach to compute polarizabilities is the background field method which has been widely used for dipole polarizabilities [1–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Methods to study higher- order polarizabilities have also been proposed [20–23] in this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Although such calculations are relatively straightforward, requiring only energy shifts from two- point functions, there are a number of unique challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' First, since weak fields are needed, the energy shift in- volved is very small relative to the mass of the hadron (on the order of one part in a million depending on the field strength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This challenge has been successfully overcome by relying on statistical correlations with or without the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Second, there is the issue of discontinuities across the boundaries when applying a uniform field on a pe- riodic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This has been largely resolved by using quantized values for the fields, or Dirichlet boundary con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Third and more importantly, a charged hadron accelerates in an electric field and exhibits Landau levels in a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Such motions are unrelated to polar- izability and must be disentangled from the deformation energy on which the polarizabilities are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For this reason, most calculations have focused on neutral hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For charged hadrons, what happens is that the two-point correlator does not develop single exponential behavior at large times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' [24], a relativistic propagator for a charged scalar is used to demonstrate how to fit such lattice data for charged pions and kaons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This approach is improved recently in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' [25] with an effective charged ∗ fxlee@gwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='edu † aalexan@gwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='edu ‡ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='Culver@liverpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='uk § walter wilcox@baylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='edu scalar propagator exactly matching the lattice being used to generate the lattice QCD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' A new fitting procedure is proposed where a χ2-function that utilizes information in both the real and imaginary parts of the correlator while remaining invariant under gauge transformations of the background field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For magnetic polarizability, a field- dependent quark-propagator eigenmode projector is used to filter out the effects of Landau levels [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' These special techniques for charged particles involve fairly com- plicated analysis to treat the collective motion of the system in order to isolate the polarizabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In this work, we explore an alternative approach based on four-point functions in lattice QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Instead of back- ground fields, electromagnetic currents couple to quark fields to induce interactions to all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' It is a general approach that treats neutral and charged particles on equal footing, but particularly suited for charged parti- cles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The trade-off is an increased computational demand of four-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Although four-point functions have been applied to study various aspects of hadron structure [28–33], not too much attention has been paid to its potential application for polarizabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We know of two such studies from a long time ago [34, 35], a recent calculation on the pion [36], and preliminary one on the proton [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' A reexamination of the formalism in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' [35] is recently carried out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' [38] for both electric and magnetic polarizabilities of a charged pion and a proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Experimentally, polarizabilities are primarily studied by low-energy Compton scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Theoretically, a variety of methods have been employed to describe the physics involved, from dispersion relations [39–42], to chiral per- turbation theory (ChPT) [43–45] or chiral effective field theory (EFT) [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Reviews on hadron polarizabilities can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' [43, 47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The presentation is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' II we outline the methodology to extract polarizabilities, using the electric polarizability of a charged pion as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' III we detail our notations and algorithms used to evaluate the four-point functions, including how the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='05200v1 [hep-lat] 12 Jan 2023 2 Sequential-Source Technique (SST) can be applied in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' IV we show our analysis procedure and results from a proof-of-concept simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' V we give concluding remarks and an outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Some technical details are put in the Appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' METHODOLOGY In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' [38], a formula is derived for electric polarizabil- ity of a charged pion, απ E = α r2 E 3mπ + lim q→0 2αa q 2 � ∞ 0 dt � Q44(q, t) − Qelas 44 (q, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (1) Here α = 1/137 is the fine structure constant and a the lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The first term in the formula involves the charge radius and pion mass (we will refer to this term as the elastic contribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The second term has the elastic contribution Qelas 44 subtracted from the total (we will refer to this term as the inelastic contribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The formula will be used in discrete Euclidean spacetime but we keep the Euclidean time axis continuous for notational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Special kinematics (called zero-momentum Breit frame) are employed in the formula to mimic low- energy Compton scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The process is illustrated in Fig 1, where the initial (p1) and final (p2) pions are at rest and the photons have purely spacelike momentum, p1 = (0, mπ), q1 = (q, 0), q2 = (−q, 0), p2 = (0, mπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Four-point function for charged pion polarizabilities under the zero-momentum Breit frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Time flows from right to left and the four momentum conservation is expressed as p2 = q2 + q1 + p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The Q44 is defined as the µ = 4 = ν component of the Fourier transforms, Qµν(q, t2, t1) ≡ � x2,x1 e−iq·x2eiq·x1Pµν(x2, x1, t3, t2, t1, t0), (3) where Pµν is a four-point function defined in position space (Ω denotes the vacuum), Pµν(x2, x1, t3, t2, t1, t0) ≡ � x3,x0 Ω|ψ(x3) : jL µ (x2)jL ν (x1) : ψ†(x0)|Ω � x3,x0 Ω|ψ(x3)ψ†(x0)|Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (4) Here ψ is the interpolating field of the pion and jL µ the lattice version of the electromagnetic current density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The two-point function in the denominator is for normaliza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Normal ordering is used to include the required subtraction of vacuum expectation values (VEV) on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The sums over x0 and x3 enforce zero-momentum pions at the source (t0) and sink (t3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The two currents are inserted at t1 and t2 with two possibilities of time or- dering implied in the normal ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The field operators for ψ and jL µ used in this work, along with conservation properties of Q44 at q = 0, are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' To see the structure of the four-point function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (4), we insert a complete set of states in the numerator (twice) and in the denominator (once).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' When the times are well separated (defined by the time limits t3 ≫ t1,2 ≫ t0) the correlator is dominated by the ground state, Pµν(x2, x1, t3, t2, t1, t0) → N 2 s | π(0)|ψ(0)|Ω |2e−mπt3 π(0) : jL µ (x2)jL ν (x1) : |π(0) N 2s | π(0)|ψ(0)|Ω |2e−mπt3 → π(0)| : jL µ (x2)jL ν (x1) : |π(0) = π(0)|TjL µ (x2)jL ν (x1)|π(0) − Ω|TjL µ (x2)jL ν (x1)|Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (5) Here Ns = NxNyNz is the number of spatial sites on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The role of the two-point function as normaliza- tion and the inclusion of VEV subtraction is evident in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Assuming time separation t = t2 − t1 > 0 and inserting a complete set of intermediate states, the diagonal com- ponent of Qµν develops the time dependence in the same limits, Qµµ(q, t) = N 2 s � n | π(0)|jL µ (0)|n(q) |2e−a(En−mπ)t − N 2 s � n | Ω|jL µ (0)|n(q) |2e−aEnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (6) At large time separations, it is dominated by the elastic contribution (n = π term in the first sum), Qelas µµ (q, t) ≡ N 2 s | π(0)|jL µ (0)|π(q) |2e−a(Eπ−mπ)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (7) We see that the elastic piece in the four-point function has information on the form factor of the pion through the amplitude squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The form factor Fπ can be determined from Q44 at large time separations, Qelas 44 (q, t) = (Eπ + mπ)2 4Eπmπ F 2 π(q2) e−a(Eπ−mπ)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (8) The charge radius r2 E in the formula can then be ex- tracted from Fπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' A salient feature here is that the elastic contribution in four-point functions is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Aside from the charge radius term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (1), αE is proportional to the difference in the areas under the Q44 and Qelas 44 curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' It is this difference that is responsible for the sign of απ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' On a finite lattice the time integral does q q 0 03 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Skeleton diagrams of a four-point function contributing to polarizabilities of a meson: (a) connected insertion: differ- ent flavor, (b) connected insertion: same flavor, (c) connected insertion: same flavor Z-graph, (d) disconnected insertion: single loop, double current, (e) disconnected insertion: single loop, (f) disconnected insertion: double loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In each diagram, flavor permutations are assumed as well as gluon lines that connect the quark lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The zero-momentum pion interpolat- ing fields are represented by vertical bars (wall sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Time flows from right to left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' not really extend to ∞, but are limited to the available time slices between the two current insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In practice, one should check if the largest time separation is enough to establish the elastic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Equivalent directions for q can be used to improve the signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Note that απ E has the expected physical unit of a3 (fm3) since 1/q 2 scales like a2 and Q44 and t are dimensionless in our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' CORRELATION FUNCTIONS In this section, we detail how to simulate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (4) and its Fourier transform Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (3) at the quark level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Wick contrac- tions of quark-antiquark pairs in the unsubtracted part lead to topologically distinct quark-line diagrams shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The raw correlation functions can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Diagrams a, b, and c are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Dia- gram d has a loop that is disconnected from the hadron, but connected between the two currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Diagrams e has one disconnected loop and diagram f has two such loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Furthermore, diagrams d, e and f must have associated VEV subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' However, if conserved lattice current density is used, there is no need for subtraction in diagram e since the VEV vanishes in the configuration average [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In this work, we focus on the connected contributions (di- agrams a,b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The disconnected contributions (diagrams d,e,f) are more challenging and are left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In particular, we will explain how to use the sequential source technique (SST) to simplify the evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Two-point functions First, we show how to evaluate the two-point function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (4) which serves as normalization for the four-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' It has the following Wick contraction using the interpolating operator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (A1), � x3,x0 Ω|ψ(x3)ψ†(x0)|Ω = � x3,x0 Tr s,c � γ5Sd(x0, x3)γ5Su(x3, x0) � , (9) where Sq denotes a quark propagator that carries the full space-time and spin and color information between two points1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The double sum projects to zero momen- tum both as the source x0 and the sink x3 as required by the special kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The full evaluation involves essentially all-to-all propagation which is computation- ally prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Instead, we employ wall sources without gauge fixing as an approximation, with the expectation that gauge-dependent contributions to the final observ- ables will vanish in the configuration average [30, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Only terms in the double sum where the quarks are at the same location form the signal, the rest contribute to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Details of our implementation of the wall source can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 1 In this work, all correlation functions in such expressions are understood as path integral expectation values in lattice QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' They are evaluated as averages over gauge configurations in Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (a) (b) (c) X (d) (e) (f)4 If we insert the wall at time slice t0 and project to zero momentum at t3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (9), we have � x3,x0 Ω|ψ(x3)ψ†(x0)|Ω = Tr s,c � WT γ5Sd(x0, x3)γ5Su(x3, x0)W � = Tr s,c � WT γ5P(t0)M −1 d P(t3)T γ5P(t3)M −1 u P(t0)T W � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (10) The symbols W and P(t) are defined in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We introduce two zero-momentum quark propagators called a1 and a2 emanating from the walls at t0 and t3, respectively, V (q) a1 ≡ M −1 q P(t0)T W, V (q) a2 ≡ M −1 q P(t3)T W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (11) We use “V” to emphasize that the wall-to-point quark propagators so defined are column vectors in the (x, s, c) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Using a1, the two-point function can be written as, � x3,x0 Ω|ψ(x3)ψ†(x0)|Ω = Tr s,c �� P(t3)γ5V (d) a1 �†� P(t3)γ5V (u) a1 �� = Tr s,c �� P(t3)V (d) a1 �†� P(t3)V (u) a1 �� (Type 1) (12) In the last step the γ5-hermiticity of M −1 q is used to eliminate γ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' if we insert the wall at time slice t3 and project to zero momentum at t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' we get in terms of a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' � x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='x0 Ω|ψ(x3)ψ†(x0)|Ω = Tr s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='c �� P(t0)γ5V (u) a2 �†� P(t0)γ5V (d) a2 �� = Tr s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='c �� P(t0)V (u) a2 �†� P(t0)V (d) a2 �� (Type 2) (13) If we insert two walls,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' one at t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' one at t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' we obtain additional expressions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' � x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='x0 Ω|ψ(x3)ψ†(x0)|Ω = Tr s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='c � WT γ5Sd(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' x3)WWT γ5Su(x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' x0)W � = Tr s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='c �� WT P(t3)V (d) a1 �†� WT P(t3)V (u) a1 �� = Tr s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='c �� WT P(t0)V (u) a2 �†� WT P(t0)V (d) a2 �� (Type 3) (14) The expressions in the above three equations (which we denote as Type 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 3 as indicated) are different estimators of the wall-to-wall two-point function with zero momentum for both initial and final pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' They are expected to approach the same value in the limit of infinite number of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In the following, we use our notation to evaluate the connected four-point functions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Four-point functions We start with local (or point) current insertions of four-point functions which have relatively simple Wick contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The results in this work will be based on conserved (or point-split) currents which avoids the issue of computing the renormalization constant ZV for vector currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Below we detail how to evaluate the connected contributions using both local and conserved currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Diagram a (different flavor) There are two terms, d4 and d2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (B4), that are contributing to the connected part of diagram a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' They are characterized by the charge factor quq ¯ d = 2/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The two terms are related by a flavor permutation (1 ↔ 2 switch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Under isospin symmetry in u and d quarks, the two terms have equal contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Including the Fourier transforms and setting µ = 4 = ν for electric polarizability, the correlation function can be written as2, ˜Q(a,P C) 44 = −4 9Z2 V κ2 Tr s,c � γ5S(t0, t2)γ4e−iqS(t2, t3)γ5S(t3, t1)γ4eiqS(t1, t0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (15) 2 We use Qµν for normalized correlation functions as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (4), and tilded ˜Qµν for unnormalized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=', without the denominator Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 5 We evaluate the correlation function by inserting two walls, one at t0 and one at t3, ˜Q(a,P C) 44 (q, t1, t2) = −4 9Z2 V κ2 Tr s,c � WT γ5S(t0, t2)e−iqγ4S(t2, t3)WWT γ5S(t3, t1)eiqγ4S(t1, t0)W � = −4 9Z2 V κ2 Tr s,c � WT γ5P(t0)M −1 q P(t2)T e−iqγ4P(t2)M −1 q P(t3)T W WT γ5P(t3)M −1 q P(t1)T eiqγ4P(t1)M −1 q P(t0)T W � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (16) The notation makes it clear that all spatial sums are automatically incorporated into the matrix multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Using the Va1 and Va2 propagators defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (11) and the γ5-hermiticity of M −1, the final expression for diagram a can be written as, ˜Q(a,P C) 44 (q, t1, t2) = 4 9Z2 V κ2 Tr s,c �� [P(t2)Va2]† γ5γ4eiqP(t2)Va1 �†� [P(t1)Va2]† γ5γ4eiqP(t1)Va1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (17) There is an overall sign change from taking the dagger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The first parenthesis corresponds to the current insertion at t2 on one of the quark lines in the pion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' the second parenthesis the current insertion at t1 on the other quark line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Both t1 and t2 are free to vary between t0 and t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In the case of conserved current, there are 8 terms contributing to diagram a in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='(B6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Their sum under isospin symmetry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' along with the Fourier transforms and wall-source insertions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' can be written in similar form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' ˜Q(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='P S) 44 (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2) = 1 9κ2� d16 + d18 + d20 + d22 + d8 + d10 + d12 + d14 � = 4 9κ2 Tr s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='c �� [P(t2)Va2]† γ5(1 − γ4)eiqU4(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + 1)P(t2 + 1)Va1 − [P(t2 + 1)Va2]† γ5(1 + γ4)U † 4(t2 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)eiqP(t2)Va1 �† � [P(t1)Va2]† γ5(1 − γ4)eiqU4(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + 1)P(t1 + 1)Va1 − [P(t1 + 1)Va2]† γ5(1 + γ4)U † 4(t1 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiqP(t1)Va1 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (18) with local current replaced by its point-split form in the parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Diagram b (same flavor) and SST For local current, there are 2 terms, d1 and d7 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (B4), that are contributing to the connected part of same-flavor correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' They are characterized by the charge factors ququ = 4/9 or q ¯ dq ¯ d = 1/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The d1 diagram is clock-wise propagation t0 → t3 → t2 → t1 → t0 where the two currents couple to the same u quark, while the d7 diagram is counter clock-wise propagation t0 → t1 → t2 → t3 → t0 where the two currents couple to the same d quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Under isospin symmetry, the total contribution from uu and dd correlations has a total charge factor of 4/9 + 1/9 = 5/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Including the Fourier transforms, setting µ = 4 = ν for electric polarizability, and inserting the wall sources, the correlation function can be written as, ˜Q(b,P C) 44 = 5 9Z2 V κ2 Tr s,c � WT γ5S(t0, t1)γ4eiqS(t1, t2)γ4e−iqS(t2, t3)WWT γ5S(t3, t0)W � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (19) This expression involves numerous quark propagators: t0 and t3 are fixed, but t1 and t2 are free to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' To cut down the computational cost, we fix the current at t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Then only one new inversion between t1 and t2 is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Since the current insertions take place between the hadron source (t0) and sink (t3), a method called SST (Sequential Source Technique) can be employed for the propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' To see how SST arises in this context, we first define the product that involves t0 → t3 → t2 propagation as, γ4e−iqS(t2, t3)WWT γ5S(t3, t0)W = γ4e−iqP(t2)M −1 q P(t3)T WWT γ5P(t3)M −1 q P(t0)T W = γ4e−iqP(t2)Va2WT P(t3)γ5Va1, (20) which is built directly from the two previously-computed propagators Va1 and Va2 along with other factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This does 6 not require a new inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Next, we define the rest in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (19) as, WT γ5S(t0, t1)γ4eiqS(t1, t2) = WT γ5P(t0)M −1 q P(t1)T γ4eiqP(t1)M −1 q P(t2)T = � P(t2)γ5M −1 q γ5P(t1)T γ4e−iqP(t1)γ5M −1 q γ5P(t0)T γ5W �† = − � P(t2)γ5M −1 q P(t1)T γ4e−iqP(t1)Va1 �† = − � P(t2)γ5V (4,P C) a3 �† , (21) where we have introduced a SST propagator called a3 (specialized to µ = 4 here), V (µ,P C) a3 (q) ≡ M −1 q P(t1)T � γµe−iqP(t1)Va1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (22) This expression indicates that V (4,P C) a3 can be obtained by a standard inversion Mx = b with a “spatially extended source” b = � γ4e−iqP(t1)Va1 � at t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This source is constructed from a previously defined quark propagator Va1 and the current insertion, hence the name “sequential source”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Using (a1, a2) and the newly-defined propagator a3, the final expression for diagram b takes the form, ˜Q(b,P C) 44 (q, t2) = −5 9Z2 V κ2 Tr s,c � � P(t2)γ5V (4,P C) a3 (q) �† γ4e−iqP(t2)Va2WT P(t3)γ5Va1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (23) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 3 is a schematic depiction of how the propagators form the full correlation function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Diagram (b) in terms of quark propagators: one part is Va1 to the pion wall at t3, then Va2 to the current insertion at t2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' the other is a SST propagator Va3 (red) built from Va1 and the current insertion at t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For conserved current, there are 8 terms contributing to diagram b in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (B6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Following the same procedure as for point current,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' the final expression for diagram b from point-split current can be written as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' ˜Q(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='P S) 44 (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2) = 1 9κ2� d1 + d3 + d5 + d7 + d25 + d31 + d37 + d43 � = −5 9κ2 Tr s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='c � [P(t2)γ5V (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='P S) a3 (q)]†(1 − γ4)e−iqU4(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + 1)P(t2 + 1)Va2WT P(t3)γ5Va1 − [P(t2 + 1)γ5V (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='P S) a3 (q)]†(1 + γ4)U † 4(t2 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqP(t2)Va2WT P(t3)γ5Va1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (24) where a new inversion is needed for the SST propagator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' V (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='P S) a3 (q) ≡ M −1 q � P T (t1)(1 − γ4)e−iqU4(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + 1)P(t1 + 1)Va1 − P T (t1 + 1)(1 + γ4)U † 4(t1 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)e−iqP(t1)Va1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (25) This is the point-split version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (22) with µ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Since the current is split in the t direction, U4 and U † 4 commute with e−iq in these two equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' a3 a2 a1 t2 ti t3 a1 to7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Diagram c (same flavor Z-graph) and SST For local current, there are 2 terms, d0 and d9 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (B4), that are contributing to the connected part of same-flavor correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' They are characterized by the same charge factors ququ = 4/9 or q ¯ dq ¯ d = 1/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The d0 diagram is a clock-wise propagation t0 → t3 → t1 → t2 → t0 where the two currents couple to the u quark, while the d9 diagram is a counter clock-wise propagation t0 → t2 → t1 → t3 → t0 where the two currents couple to the d quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' They are essentially the Z-graph of diagram b with the current insertions 1 and 2 switched, whose correlation function can be written as, ˜Q(c,P C) 44 = 5 9Z2 V κ2 Tr s,c � γ5WT S(t0, t2)γ4e−iqS(t2, t1)γ4eiqS(t1, t3)WWT γ5S(t3, t0)W � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (26) First we isolate the t3 → t1 → t2 propagation, S(t2, t1)γ4eiqS(t1, t3)W = P(t2)M −1 q P(t1)T γ4eiqP(t1)M −1 q P(t3)T W = P(t2)M −1 q P(t1)T γ4eiqP(t1)Va2 ≡ P(t2)V (4,P C) a4 (q), (27) where a new SST propagator is introduced (specialized to µ = 4 here), V (µ,P C) a4 (q) ≡ M −1 q P(t1)T � γµeiqP(t1)Va2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (28) Using a1 and a4, the final expression for diagram c using point current takes the form, ˜Q(c,P C) 44 (q, t2) = 5 9Z2 V κ2 Tr s,c � � γ4eiqP(t2)γ5Va1 �† P(t2)V (4,P C) a4 (q)WT P(t3)γ5Va1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (29) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 4 is a schematic depiction of how the propagators form this correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Diagram (c) in terms of quark propagators: a1 from t0 to t3, SST quark propagator a4 (red) with sequential source built from a2 and current insertion at t1, and a1 from t2 to t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This is the Z-graph of diagram b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For conserved current, there are 8 terms contributing to diagram c in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (B6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Following a similar procedure as for local current,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' the final expression for diagram (c) from point-split current can be written as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' ˜Q(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='P S) 44 (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2) = 1 9κ2� d0 + d2 + d4 + d6 + d27 + d33 + d39 + d45 � = 5 9κ2 Tr s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='c � [P(t2)γ5Va1]†(1 − γ4)e−iqU4(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + 1)P(t2 + 1)V (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='P S) a4 (q)WT P(t3)γ5Va1 −[P(t2 + 1)γ5Va1]†(1 + γ4)U † 4(t2 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqP(t2)V (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='P S) a4 (q)WT P(t3)γ5Va1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (30) where V (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='P S) a4 (q) ≡ M −1 q � P T (t1)(1 − γ4)eiqU4(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + 1)P(t1 + 1)Va2 − P(t1 + 1)T (1 + γ4)U † 4(t1 + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiqP(t1)Va2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (31) Compare to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (25) for diagram b, this expression has a2 instead of a1, q instead of −q, and no γ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The total connected contribution to the polarizabilities in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (1) is simply the sum of the individual normalized terms in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 2, Q44(q, t2, t1) = Q(a) 44 + Q(b) 44 + Q(c) 44 , (32) a2 a4 a1 X t2 t1 t3 a1 to8 for either point current or conserved current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The charge factors and flavor-equivalent contributions have been in- cluded in each diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' SIMULATION DETAILS AND RESULTS Having laid out the methodology and detailed the correlations functions, we now discuss how to numeri- cally evaluate them in a Monte Carlo simulation in or- der to extract the polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' As a proof-of-principle test, we use quenched Wilson action with β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 and κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1520, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1543, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1555, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1565 on the lattice 243 × 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The pion mass corresponding to the kappas will be deter- mined in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We analyzed 500 configurations for κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1520 and 1000 configurations each for rest of the kappas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The scale of this action has been determined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' [51], with inverse lattice spacing 1/a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='312 GeV and kappa critical κc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='15708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' It also gives the pion mass as a function of kappa, (mπa)2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='09 × 1 2 � 1 κ − 1 κc � , (33) which will be compared with the measured mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Dirichlet (or open) boundary condition is imposed in the time direction, while periodic boundary conditions are used in spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The pion source is placed at t0 = 7 and sink at t3 = 42 (time is labeled from 1 to 48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' One current is inserted at a fixed time t1, while the other current t2 is free to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We use integers {nx, ny, nz} to label the discrete momentum on the lattice, q = �2πnx Lx , 2πny Ly , 2πnz Lz � , nx, ny, nz = 0, ±1, ±2, · · · , (34) and consider five different combinations {0, 0, 0}, {0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {0, 0, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In order to evaluate the connected diagrams, we need four inversions of the quark matrix with varying sources: two wall-sourced propagators Va1 and Va2, and two SST propagators Va3(q) and Va4(q) at a fixed q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' So the count for five momenta is 2 + 2 × 5 = 12 per kappa per configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' It takes longer to do the inversions for larger kappas due to critical slowing down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Raw correlation functions First, we discuss how to determine pion mass from the various two-point functions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 5 we show the wall-to-wall pion correlations based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (12) (Type 1) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (13) (Type 2) at κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Type 1 only depends on the a1 quark propagator originating from the wall source at t0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Instead of ending at fixed t3 = 42, we allow it to vary in the entire range of t on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' One can visualize it as a moving wall sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In this way, we get to observe a plateau in the effective mass function 0 10 20 30 40 1 0 1 2 3 4 5 t Log10 of Two-point Functions 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='40 t Effective mass of Two-point Functions FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Moving sink zero-momentum pion correlator Type 1 (blue) and Type 2 (orange) and their effective mass functions at mπ = 600 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' They are constructed from either a1 or a2 quark propagators as explained in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The vertical gridlines indicate the three fixed time points in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' These functions can be used to extract the pion mass in single- exponential fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The value at t3 = 42 in Type 1 or at t0 = 7 in Type 2 can be used for normalization of four-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' which we use to extract the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Similarly, Type 2 only depends on the a2 quark propagator originating from the wall source at t3 = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Instead of ending at fixed t0 = 7, we allow it to vary in the entire range of t on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We flip the sign of its effective mass function so a direct comparison of the plateaus for the two types can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We use Type 1 with a varying sink to extract pion and rho masses at the four kappa values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We obtain approximately 1100, 800, 600, and 370 MeV for pion mass at κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1520, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1543, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1555, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1565, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' These values agree well with those predicted from the relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='(33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' From this point on, we will refer to pion mass rather than kappa values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The rho meson is considered in this work to judge the efficacy of vector meson dominance in form factor extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The more precise numbers for mπ and mρ with uncertainties will be given in the summary table at the end (Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Another benefit of plotting the Type 1 and Type 2 correlators with a varying sink is we get to see the limited “window of opportunity” in the effective mass where ground state dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This is the window in which we study the current-current correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We utilize this information to fix one of the two currents in the four-point function calculation so it mainly couple to the zero-momentum ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 9 0 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 t2 Diagram c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 Diagram b {0, 0, 0} {0, 0, 1} {0, 1, 1} {1, 1, 1} {0, 0, 2} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 Four-point function Q44(q,t1,t2) Diagram a 20 25 30 35 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 t2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 Effective Mass of Q44(q,t1,t2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Normalized four-point functions (left panel) and their effective mass functions (right panel) from the connected diagrams as a function of current separation at mπ = 600 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The q = 0 results serve as a check of current conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The results for non-zero q between t2 = 18 and t2 = 41 will become the basis for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The vertical gridlines indicate the pion walls (t0 = 7 and t3 = 42) and the fixed current insertion (t1 = 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The horizontal gridlines in the effective mass functions indicate the value of Eπ − mπ where the continuum dispersion relation Eπ = � q2 + m2π is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Having examined the plots, we settle on t1 = 18, 18, 18, 14 for mπ = 1100, 800, 600, 370 MeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Next, we discuss normalization constant for four-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This is the zero-momentum wall-to-wall two- point function in the denominator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We have three options, corresponding to the three types in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (12), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (13), and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='(14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Type 1 normalization constant is simply the special value at t = t3 = 42 in the blue curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 5, and Type 2 the special value at t = t0 = 7 in the orange curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Type 3 normalization constant is computed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The three types are not expected to agree configuration by configuration since they originate from different wall sources, but they should approach the same value in the configuration average within statis- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We found the numerical values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4683(6), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4672(6), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='468(7), from Type1, Type2, and Type 3, respectively, at this pion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We see that Type 3 has larger statistical uncertainties than in Type 1 and Type 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This is expected since Type 3 is constructed from two wall sources, while the other two from one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We will Type 3 as normalization for the reason to be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Having determined the two-point functions, we present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 6 the raw normalized four-point functions Q44 at five different values of momentum q and at mπ = 600 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For comparison purposes, all points in Q44 are displayed on the same linear scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For the effective mass function ln Q44(t)/Q44(t+1), only points between the pion walls are displayed for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The results are based on conserved currents and only the connected diagrams a, b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' There are a number of interesting features in these plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' First, the results for q = 0 confirms the current con- servation property discussed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='(A9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Basically, for conserved current, we expect the ratio of four-point func- tion to two-point function to approach the charge factor quq ¯ d + qdq¯u = 4/9 for diagram a in the isospin limit, inde- pendent of current insertion points t1 and t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For diagrams b and c, the factor is ququ + q ¯ dq ¯ d = 5/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Indeed, this is confirmed in all three diagrams (black dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In diagram a, current conservation is limited between t2 = 7 (on the pion wall source) and t2 = 41 (one step inside the pion wall sink) because the two currents independently cou- ple to two different quarks in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In diagram b, where they couple to the same quark, current conserva- tion emerges only starting from t2 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In diagram c, it is limited between t2 = 7 and t2 = 17 because it is the Z-graph of b (different time-ordering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' If diagrams b and 10 0 200 400 600 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 Configuration No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Type 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 Type 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 4pt, 2pt, and their ratios Type 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Statistical fluctuations are shown in the unnormalized four-point function (red), three types of two-point functions (black), and their ratios (blue) at 20 randomly-selected config- urations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For this figure, Diagram (a) at q = 0 and mπ = 600 MeV is used as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Neighboring points are connected by straight lines to facilitate visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The faint horizontal gridline indicates the expected ratio 4/9 for this diagram and conserved currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' c are added, then current conservation extends to the whole range, just like diagram a, except for the special point of t1 = t2 to be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Outside the regions of current conservation, the q = 0 signal is exactly zero, while the q ̸= 0 signal gradually goes to zero towards the Dirichlet wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Second, we found that although we have three options for two-point functions to be used as normalization, they have different statistical fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This is demon- strated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 7 where we plot the three types for a select few configurations out of the 1000, using diagram (a) at zero momentum and a fixed time slice in the con- served region (7 < t2 < 41) as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For each type, we plot separately the unnormalized four-point function, two-point function, and their ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We see that the ratio from Type 3 gives the expected value (4/9) exactly whereas Type 1 and Type 2 fluctuate around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The reason is that Type 3, despite being more noisy than Type 1 and Type 2, is exactly correlated with the four- point function configuration by configuration, both being constructed from the same two wall sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We rely on this perfect correlation in Type 3 to serve as a strong numerical validation that the wall sources and the con- served currents are correctly implemented in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' At nonzero momentum (q ̸= 0), however, we found that all three normalization types produce comparable statisti- cal uncertainties for the normalized four-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 6 is plotted using Type 3 normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Third, the special point of t1 = t2 is regular in diagram a, but gives irregular results in diagram b and c for all values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This is the contact term in the discussion surrounding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='(A9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We avoid this point in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Fourth, we observe that the results about t1 = 18 in diagram b and c are mirror images of each other, simply due to the fact that they are from the two different time orderings of the same diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In principle, this property could be exploited to reduce the cost of simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In this study, however, we computed all three diagrams separately, and add them between t1 = 19 and t3 = 41 as the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We also note in passing that the Q44 signal in diagram c is negative definite whereas it is positive definite in diagrams a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Finally, the effective mass function of Q44 for diagram b approaches the value of Eπ − mπ at large separation times between t1 and t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This is an indication that the four- point function for diagram b is dominated by the elastic contribution with a fall-off rate of Eπ − mπ according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The same is true for diagram a, although deviations are slightly larger at higher momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The situation for diagram c, however, is completely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The fall-off rates approach high above their respective Eπ−mπ values, suggesting they are dominated by inelastic contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In other words, the intermediate state is not a pion, but some four-quark state at higher mass and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We also used local current as a guide to develop our for- malism and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' If we take the four-point function ratio of local current (PC) to conserved current (PS) at q = 0, we expect ˜Q(P S) 44 / ˜Q(P C) 44 → Z2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This is confirmed for all three diagrams and we obtain an estimate of ZV ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='47, consistent with literature [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Since our results are based exclusively on conserved current, we will not discuss local current further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Elastic form factor The formula for electric polarizability in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (1) involves the charge radius rE and the elastic contribution Qelas 44 , both of which can be extracted from the large-time behav- ior of four-point functions Q44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (8), Qelas 44 is expected to exhibit single-exponential behavior with a fall-off rate of Eπ − mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The form factor Fπ is contained in the amplitude of this fall-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Based on the discussion about Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 6, diagrams a and b have the expected fall-off whereas diagram c does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' As far as elastic contribution is concerned, we can drop diagram c and focus only on diagrams a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This improves the form factor analysis 11 {0, 0, 1} {0, 1, 1} {1, 1, 1} {0, 0, 2} 0 5 10 15 20 25 3 2 1 0 t=t2-t1 Log10 of Q44 (ab)(q,t) 0 5 10 15 20 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 t=t2-t1 Effective Mass of Q44 (ab)(q,t) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Normalized four-point functions from diagrams a and b in log plot and their effective mass functions at different values of q and mπ = 600 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' They are plotted as time separations t = t2 − t1 between the two currents relative to fixed t1 = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The horizontal gridlines in the effective mass are Eπ − mπ using continuum dispersion relation for Eπ with measured mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' These functions are used to extract the elastic contributions Qelas 44 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' by eliminating the inelastic ‘contamination’ from diagram c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' It can be regarded as a form of optimization in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 shows an example of the four-point func- tions Qab 44 including only diagrams a and b, along with their effective mass functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We focus in the region of signal between t1 and t3 and plot them as a function of time separation t = t2 −t1 between the two currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Note that we exclude the t = 0 point from the analysis due to contact terms, as discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We see that there is a region where the effective mass functions coincide with the Eπ −mπ gridlines, indicating that Qab 44 is dominated by elastic contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The agreement is better at smaller momentum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The signal at large times is noisy and increasingly so at higher momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We also see the effect of the Dirichlet wall which forces the effective mass to curve down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In this context, the inclusion of diagram c would push the elastic limit into larger times where the signal is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' To account for possible violation of the continuum dispersion relation, we perform a fit to the functional form of Qelas 44 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (8), treating both {Fπ, Eπ} as free parameters with mπ fixed at the measured values from two-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Details of the fits at all four pion passes are given in Table II in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' From this table, we observe that the Eπ from the fit largely agrees with that from the continuum dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Deviations become more apparent at higher momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 q2 (GeV2) Elastic Form Factor Fπ (q2) mπ=370 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 mπ=600 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 mπ800 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 mπ=1100 MeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Pion elastic form factors extracted from four-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The red data points are the measured values in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The green solid line is a fit to the z-expansion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The green dashed line is a fit to the monopole form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The blue dashed line is the same monopole form plotted with the measured rho mass, and the black solid line with the physical rho mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' After the form factor data are obtained, we fit them to the monopole form, Fπ(q2) = 1 1 + q2/m2 V , (35) which is the well-known vector meson dominance (VMD) commonly considered in pion form factor studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The results are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We see that the monopole form does not fit the data very well, especially at higher momentum and lower pion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For this reason, we also 12 considered the z-expansion parametrization [53] Fπ(q2) = 1 + kmax � k=1 ak zk, where z ≡ √tcut − t − √tcut − t0 √tcut − t + √tcut − t0 and t = −q2, tcut = 4m2 π, (36) where ak are free parameters and tcut is the two-pion production threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We take t0 = 0 so the form goes through Fπ(0) = 1 by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Using this form, we can find a good fit with kmax = 3 in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For compari- son, we also plot the monopole function with the measured rho mass mρ and the physical rho mass of mphys ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='77 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We observe significant differences between the fitted monopole form (mV ) and the VMD form (mρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The dif- ference grows with increasing momentum and decreasing pion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Similar behavior has been observed in previ- ous studies [49, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Once the functional form of form factor is determined, the charge radius is obtained by r2 E = −6dFπ(q2) dq2 ��� q2→0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (37) Their values in physical units are put in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 0 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 t=t2-t1 q= {0, 0, 2} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 q= {1, 1, 1} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 q= {0, 1, 1} Q44 Q44 elas 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 Four-point function q= {0, 0, 1} FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Total Q44 and elastic Qelas 44 at different values of q at mπ = 600 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The shaded area, � dt � Q44(q, t)−Qelas 44 (q, t) � , is the signal contributing to polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Electric polarizability Having obtained the elastic contribution Qelas 44 , we now turn to the extraction of αE from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 10 we show separately the total contribution Q44 (from all three diagrams) and Qelas 44 as a function of current separation t = t2 − t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' W use mπ = 600 MeV as an example;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' the graphs at the other pion masses look similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Note that although Qelas 44 is obtained in the large time region, the subtraction is done in the whole region according to the functional form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='(8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Most of the contribution is in the small time region where inelastic contributions are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We observe that Qelas 44 is consistently larger than Q44, suggesting that the inelastic term in the formula is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The time integral is simply the negative of the shaded area between the two curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' One detail to notice is that the curves include the t = 0 point which has unphysical contributions in Q44 as mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We would normally avoid this point and only start the integral from t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' However, as one can see, the chunk of area 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 q2(GeV2) αE inelastic (10-4 fm3) mπ=370 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 mπ=600 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1 mπ=800 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='00 mπ=1100 MeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Extrapolation of the second term (inelastic) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (1) to q2 = 0 in physical units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The red points are based on the shaded areas in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The blue curve is a quadratic extrapolation using all points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The green curve is a linear extrapolation based on the two smallest q2 values with straight lines connecting all the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The black points indicate the extrapolated values contributing to αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 13 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Summary of results in physical units from two-point and four-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Two sets of results are given: one based on charge radius from z-expansion fits, one from monopole fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The total αE is chirally extrapolated to the physical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This value, in conjunction with the elastic αE derived from charge radius physical pion mass by PDG, yields the prediction for the inelastic αE at the physical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' All αE values are in units of 10−4 fm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1520 κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1543 κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1555 κ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1565 physical point mπ (MeV) 1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1 596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='000 mρ (MeV) 1273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='5 1047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='000 r2 E (fm2) (z-expansion) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='144 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='195 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='258 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='305 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='434 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='005 r2 E (fm2) (monopole) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1215 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1798 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1951 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1993 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='434 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='005 αE elastic (z-expansion) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='626 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='21 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='18 αE elastic (monopole) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='527 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='084 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='013 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='567 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='010 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='599 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='033 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='18 αE inelastic (z-expansion) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='296 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='567 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='866 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='006 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='136 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='012 −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='5 αE inelastic (monopole) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='296 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='567 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='866 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='006 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='136 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='012 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='22 αE total (z-expansion) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='331 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='5 αE total (monopole) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='232 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='517 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='701 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='463 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='035 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='28 between t = 0 and t = 1 is the largest piece in the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' To include this contribution, we linearly extrapolated the Q44 term back to t = 0 using the two points at t = 1 and t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This will incur a systematic effect on the order of O(a2) since the error itself is order of O(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' As the continuum limit is approached, the systematic effect will vanish (the chunk will shrink to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' There is no issue to include this point in Qelas 44 using its functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The entire second term (prefactor and time integral) is a function of momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Since αE is a static prop- erty, we extrapolate it to q2 = 0 smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' To assess the systematic effect of this extrapolation, we consider two fitting forms, one is a + bx + cx2 (x = q2) using all data points, the other a simple linear extrapolation using the two lowest points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 11 for all pion masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' One observes a difference between the two that decreases with decreasing pion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The difference in the extrapolation is a systematic effect in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We will use the linearly extrapolated values to determine αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Finally, we assemble the two terms in the formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (1) to obtain αE in physical units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We summarize all of the results measured in this study in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We include two set of results, one based on monopole, the other on z-expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' At each pion mass the elastic term makes a positive contribution, whereas the inelastic term makes a negative and smaller contribution, resulting in a positive and relatively small value in the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' To see how the trend continues, we include the physical point in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We take the total values for αE and perform a smooth extrapolation to the physical point using a + bmπ + cm2 π form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This is done for both sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The extrapolated values αE = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='5 or αE = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='28 can be compared to known values for charged pion αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' PDG [55] quotes a value αE = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='7 from ex- periment with large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Chiral perturbation theory (ChPT) [56] gives αE = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Our values are compa- rable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We also attempted to extrapolate using the 1/mπ form expected from ChPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This form does not fit our data well which is not surprising since some of our pion masses lie beyond the region of validity for ChPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' To get a sense of individual contributions at the physical point, we take the PDG value r2 E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='435(5) fm2 and physical mπ to arrive at the elastic αE value of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='08(18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Then the inelastic values αE = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='7(5) or αE = −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0(2) can be inferred from the total and the elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We should mention that our values are consistent with the inelastic contribution obtained in another lattice study [36] near physical pion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' It employs a formula derived from a different method but has a similar structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In any event, a physical picture starts to emerge from our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In the approach to the physical point, the elastic contribution grows positive strongly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' at the same time the inelastic contribution grows negative strongly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' the total is relatively small and positive and has mild pion mass dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This picture is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' ★ ★ elastic inelastic total 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0 10 5 0 5 10 15 mπ (GeV) Charged Pion αE (10-4 fm3) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Pion mass dependence of electric polarizability of a charged pion from four-point functions in lattice QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The elastic and inelastic contributions correspond to the two terms in the formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Two sets of results are displayed: one based on charge radius from z-expansion fits (solid lines), one from monopole fits (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The green star is the known value from chiral perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 14 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' SUMMARY AND OUTLOOK We investigated the feasibility of using four-point func- tions in lattice QCD to extract charged pion electric polar- izability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The approach is based on low-energy Compton scattering tensor constructed with quark and gluon fields in Euclidean spacetime [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The central object is the formula given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (1) which consists of two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' One is an elastic contribution involving charge radius r2 E and pion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The other an inelastic contribution in the form of a subtracted time integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' In addition to four- point functions, it requires two-point functions for pion mass and normalization, but not three-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The elastic contribution can be obtained from the same four-point function in the elastic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We laid out a detailed formalism and notation using standard Wilson fermion as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Although we use both local current and conserved current on the lat- tice to develop and test the formalism, our results are based on conserved current on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' It sidesteps the renormalization issue (ZV = 1), but comes with increased complexity in implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' To apply the special kine- matics (zero-momentum Breit frame) in the formula, we employ wall sources without gauge-fixing for the creation and annihilation of pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We show how to construct the four-point functions using SST quark propagation, develop efficient algorithms for numerical evaluation, and use a high-performance implementation [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We carried out a proof-of-concept simulation using quenched Wilson action with pion mass ranging from 1100 to 370 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We only considered the connected contributions in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We discussed three types wall-to-wall two-point functions for normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We found a perfect correlation between the four-point function Q44(q2 = 0) and Type 3 two-point function imposed by current conservation, configuration by configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This property provides a strong check of our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The analysis procedure used to determine αE in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (1) involves multiple steps which we summarize here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 1) Fit Type 1 two-point function to obtain mπ (and mρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 2) Fit four-point function Q(ab) 44 from diagrams a and b to Qelas 44 at large times for elastic form factor Fπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 3) Fit Fπ data to a functional form (monopole or z-expansion), then extract charge radius r2 E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 4) Perform subtraction Q(abc) 44 (q)−Qelas 44 (q) at small times using all three diagrams a,b,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Do the time integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Extrapolate back to t = 0 to include the missing chunk due to contact terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 5) Extrapolate the inelastic term to q2 = 0 to obtain the static limit, then assemble everything in physical units for αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 6) Extrapolate the total αE in pion mass to the physical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The final results reveal a clear physical picture for charged pion αE: it is the result of a large cancellation be- tween the elastic and inelastic contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Individually, each contribution has strong pion mass dependence in the approach to chiral limit, but the total has a small posi- tive value with only a mild pion mass dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The simulation also demonstrates that the four-point function methodology can be a viable alternative to the background method for polarizabilities of charged hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We cau- tion that the picture is subject to a number of systematic effects at this stage, such as the quenched approxima- tion, finite-volume effects, and disconnected loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Aside from these effects, the largest source of uncertainty in the present analysis is in the form factor fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We observe significant differences between monopole and z-expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Although the uncertainty does not alter the picture quali- tatively, it matters for quantitative comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This is an open issue that warrants further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Going forward, the investigation can proceed in multiple directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' First, the quenched approximation should be removed by employing dynamical fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Work is underway to use our collection of two-flavor nHYP- clover ensembles [58] which have been successfully used in a number of physics projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' They have smaller pion masses (about 315 MeV and 227 MeV) that can be used to check the expected chiral behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The elongated geometries in these ensembles offer a cost-effective way of studying finite-volume effects and reaching smaller q values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' It would be interesting to see how the charge radius is affected by the change of action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Second, a simulation of charged pion magnetic polarizability (βM) is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The formula has been derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' One just needs to replace Q44 with Q11 in the formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' It would be interesting to check the well-known prediction αE + βM ≈ 0 from chiral perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Third, the disconnected contributions should be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' This is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Although disconnected loops generally give relatively smaller contributions than connected ones, they must be dealt with for a complete picture from lattice QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Fourth, the methodology can be equally applied to neutral particles (for example π0 and the neutron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The advantage it offers over the background field method is the natural treatment of disconnected loops (or sea quarks) [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Our ultimate target is the proton for which a formula is also available [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' A first-principles-based calculation of its polarizabilities will be a valuable addition to the Compton scattering effort in nuclear physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported in part by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Department of Energy under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' DE-FG02-95ER40907 (FL, AA) and UK Research and Innovation grant MR/S015418/1 (CC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' AA would like to acknowledge support from Uni- versity of Maryland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' WW would like to acknowledge support from the Baylor College of Arts and Sciences SRA program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 15 [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' D 46, 1109–1122 (1992), arXiv:hep- lat/9205015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' [62] Andrei Alexandru, Manfried Faber, Ivan Horv´ath, and Keh-Fei Liu, “Lattice qcd at finite density via a new canonical approach,” Physical Review D 72 (2005), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1103/physrevd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='114513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 17 Appendix A: Operators and current conservation To evaluate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (4) in lattice QCD, we use standard annihilation (ψ) and creation (ψ†) operators for a charged pion, ψπ+(x) = ¯d(x)γ5u(x), ψ† π+(x) = −¯u(x)γ5d(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (A1) We also consider rho meson two-point functions constructed from, ψρ(x)i = ¯d(x)γiu(x), i = 1, 2, 3, (A2) and average over the spatial directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For Wilson fermions, the Dirac operator Mq = ̸D + mq takes the standard form for a single quark flavor labeled by q, Mq = 1 − κq � µ � (1 − γµ)Uµ + (1 + γµ)U † µ � , (A3) where κq = 1/(2mq + 4) is the hopping parameter and mq the bare quark mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For current operators, we consider two options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' One is the lattice point current built from up and down quark fields, j(P C) µ ≡ ZV κ � qu¯uγµu + qd ¯dγµd � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (A4) The factor κ here is to account for the quark-field rescaling ψ → √ 2κψ in Wilson fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The factor 2 is canceled by the 1/2 factor in the definition of the vector current 1 2 ¯ψγµψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The charge factors are qu = 2/3 and qd = −1/3 where the resulting e2 = α ≈ 1/137 in the four-point function has been absorbed in the definition of απ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The advantage of this operator is that it leads to simple correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The drawback is that the renormalization constant for the vector current (ZV ) has to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We also consider conserved vector current on the lattice (ZV ≡ 1) which can be derived by the Noether procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For the Wilson fermion action S = ¯ψqMqψq built from the matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (A3), the simplest way [59] is to substitute the gauge fields by Uµ(x) → Uµ(x)eiqqvq µ, (A5) and differentiate with respect to the external vector field vq µ, then take vq µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The result is the point-split form j(q,P S) µ (x) = −i δS δvq µ ���� vq µ→0 = −qqκq � ¯ψq(x)(1 − γµ)Uµ(x)ψq(x + ˆµ) − ¯ ψq(x + ˆµ)(1 + γµ)U † µ(x)ψq(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (A6) The phase factor −i is explained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' An alternative method [61, 62] is through a local transformation on the quark fields, ψ → e−iω(x)ψ, and do variation δS δ(∆µω) on the finite difference ∆µω = ω(x + ˆµ) − ω(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For two quark flavors (u and d), we have j(P S) µ (x) = quκu � − ¯u(x)(1 − γµ)Uµ(x)u(x + ˆµ) + ¯u(x + ˆµ)(1 + γµ)U † µ(x)u(x) � + qdκd � − ¯d(x)(1 − γµ)Uµ(x)d(x + ˆµ) + ¯d(x + ˆµ)(1 + γµ)U † µ(x)d(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (A7) The conserved current for nhyp fermion has the same form, except the gauge links are nhyp-smeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Although conserved currents explicitly involve gauge fields and lead to more complicated correlation functions, they have the advantage of circumventing the renormalization issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Just like current conservation guarantees the normalization condition in three-point functions, � x1 Ω|ψ(x) j(q,P S) 4 (x1) ψ†(0)|Ω = qq Ω|ψ(x)ψ†(0)|Ω , (A8) a similar condition holds in fount-point functions, � x2,x1 Ω|ψ(x) j(q2,P S) 4 (x2) j(q1,P S) 4 (x1) ψ†(0)|Ω = q1q2 Ω|ψ(x)ψ†(0)|Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (A9) In physical terms, the charge overlap at q = 0 on the left-hand-side is effectively reconstructing the two-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Each charge density is spread over all spatial sites on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' By summing over x1 and x2 at zero momentum, we recover the total charge factor from each insertion, regardless of the time points of the insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' There is a subtle issue with four-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' If the two currents couple to different quark lines (q1 ̸= q2), the conservation is for all combinations of t1 and t2 between source and sink, including t1 = t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' If they couple to the same quark line (q1 = q2), the conservation is only true for t1 ̸= t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The point t1 = t2 introduces unwanted contact terms on the lattice and is avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The issue is a lattice artifact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' in the continuum, the contact interaction is regular and well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The conservation property in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (A9) is used to validate the four-point diagrams in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 18 Appendix B: Wick contractions Here we give the unnormalized correlation functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (3) by contracting out all quark-antiqurk pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Local current For point current (PC), using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (A1) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (A4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' the full correlation function has 20 diagrams,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' ˜Q(P C) µν (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0) = � x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='x1 e−iq·x2eiq·x1 � x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='x0 ⟨Ω|ψπ+(x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)j(P C) µ (x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)j(P C) ν (x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)ψ† π+(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)|Ω⟩ ≡ Z2 V κ2 9 19 � i=0 di(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (B1) where dA 10 = −2 tr � Su(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqSd(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dA-bwd 7 = −2 tr � Su(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiqSd(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� dB 5 = 4 tr � Su(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiqSu(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� dB-bwd 15 = 1 tr � Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqSd(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiqSd(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5 � dC 1 = 4 tr � Su(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqSu(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dC-bwd 17 = 1 tr � Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiqSd(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqSd(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5 � dD 0 = −4 tr [Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5] tr � Su(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqSu(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dD 18 = −1 tr [Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5] tr � Sd(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqSd(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dEl 4 = −4 tr � Su(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� tr � Su(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� dEl 13 = 2 tr � Su(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� tr � Sd(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� dEl-bwd 6 = 2 tr � Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiqSd(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5 � tr � Su(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� dEl-bwd 14 = −1 tr � Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiqSd(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5 � tr � Sd(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� dEr 2 = −4 tr � Su(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� tr � Su(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dEr 8 = 2 tr � Su(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� tr � Sd(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dEr-bwd 11 = 2 tr � Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqSd(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5 � tr � Su(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dEr-bwd 16 = −1 tr � Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqSd(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5 � tr � Sd(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dF 3 = 4 tr [Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5] tr � Su(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� tr � Su(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dF 9 = −2 tr [Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5] tr � Su(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� tr � Sd(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dF 12 = −2 tr [Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5] tr � Sd(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� tr � Su(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dF 19 = 1 tr [Su(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5Sd(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5] tr � Sd(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� tr � Sd(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� (B2) We use a matrix notation that highlights time dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The trace is over spin and color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The momentum factor is defined by a diagonal matrix, [e±iq]s,c,x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='s′,c′,x′ ≡ δss′δcc′δx,x′e±iq·x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (B3) The spatial sums over (x2, x1, x3, x0) are implicit in the matrix multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We use S(t2, t1) to denote a quark propagator from t1 to t2 (from right to left), obtained from the inverse of quark matrix M with a source Mx = b, see 19 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='(C11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The terms are grouped into six distinct topological diagrams depicted in Fig 2, labeled by superscripts on di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' If isospin limit (κu = κd = κ) is taken,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' we get 12 diagrams (first six connected,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' the rest disconnected),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' dA 4 = −2 tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dA-bwd 2 = −2 tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� dB 1 = 4 tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� dB-bwd 7 = 1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5 � dC 0 = 4 tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dC-bwd 9 = 1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5 � dD 10 = −5 tr [S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5] tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dEl 5 = −2 tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� dEl-bwd 6 = 1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5 � tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� dEr 3 = −2 tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dEr-bwd 8 = 1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5 � tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� dF 11 = 1 tr [S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)γ5S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)γ5] tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)γµe−iq� tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)γνeiq� (B4) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Conserved current For point-split current (PS), using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (A1) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (A7), Wick contraction yields 80 diagrams (not shown here) if u and d are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' If isospin limit is taken, there are 48 diagrams which we express as, ˜Q(P S) µν (q, t3, t2, t1, t0) = � x2,x1 e−iq·x2eiq·x1 � x3,x0 ⟨Ω|ψπ+(x3, t3)j(P S) µ (x2, t2)j(P S) ν (x1, t1)ψ† π+(x0, t0)|Ω⟩ ≡ κ2 9 47 � i=0 di(q, t3, t2, t1, t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (B5) 20 The 24 connected diagrams are given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' dA 16 = −2 tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � dA 18 = 2 tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � dA 20 = 2 tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� dA 22 = −2 tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� dA-bwd 8 = −2 tr � S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) � dA-bwd 10 = 2 tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iq� dA-bwd 12 = 2 tr � S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) � dA-bwd 14 = −2 tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iq� dB 1 = 4 tr � S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) � dB 3 = −4 tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iq� dB 5 = −4 tr � S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) � dB 7 = 4 tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iq� dB-bwd 25 = 1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � dB-bwd 31 = −1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � dB-bwd 37 = −1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � dB-bwd 43 = 1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � dC 0 = 4 tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � dC 2 = −4 tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � dC 4 = −4 tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� dC 6 = 4 tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� dC-bwd 27 = 1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � dC-bwd 33 = −1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � dC-bwd 39 = −1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � dC-bwd 45 = 1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � (B6) 21 The 24 disconnected diagrams are given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' dD 28 = −5 tr [S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)] tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � dD 34 = 5 tr [S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)] tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � dD 40 = 5 tr [S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)] tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� dD 46 = −5 tr [S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)] tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� dEl 17 = −2 tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � tr � S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) � dEl 19 = 2 tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iq� dEl 21 = 2 tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� tr � S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) � dEl 23 = −2 tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iq� dEl-bwd 24 = 1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � tr � S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) � dEl-bwd 30 = −1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iq� dEl-bwd 36 = −1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � tr � S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) � dEl-bwd 42 = 1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiqS(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iq� dEr 9 = −2 tr � S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) � tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � dEr 11 = 2 tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iq� tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � dEr 13 = 2 tr � S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) � tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� dEr 15 = −2 tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iq� tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� dEr-bwd 26 = 1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � dEr-bwd 32 = −1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � dEr-bwd 38 = −1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� dEr-bwd 44 = 1 tr � S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iqS(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5) � tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� dF 29 = 1 tr [S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)] tr � S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) � tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � dF 35 = −1 tr [S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)] tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iq� tr � S(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)(1 − γν)eiqUν(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4) � dF 41 = −1 tr [S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)] tr � S(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)(1 − γµ)e−iqUµ(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) � tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� dF 47 = 1 tr [S(t0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t3)(γ5)S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t0)(γ5)] tr � S(t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4)(1 + γµ)U † µ(t2 + ˆµ4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2)e−iq� tr � S(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1 + ˆν4)(1 + γν)U † ν(t1 + ˆν4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t1)eiq� (B7) The shifted quark propagators have the following meaning depending on whether the current is split in time or spatial directions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + ˆµ4) ≡ � S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2 + 1) = P(t3)M −1P(t2 + 1)T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' if µ = 4 S(t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' t2) = P(t3)M −1P(t2)T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' if µ ̸= 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (B8) where the projector P(t) is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='(C9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The associated gauge links have the meaning, Uµ(t2, t2 + ˆµ4) ≡ � U4(t2, t2 + 1), if µ = 4 Uµ(t2, t2), if µ ̸= 4, U † µ(t2 + ˆµ4, t2) ≡ � U † 4(t2 + 1, t2), if µ = 4 U † µ(t2, t2), if µ ̸= 4, (B9) 22 where the gauge links are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='(C6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' So the split in time is explicitly carried in both the propagators and gauge links, whereas the split in space is only implicitly carried in the gauge links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Note the placement of e±iq in relation to U and U †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' They do not commute when the currents are split in spatial directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Appendix C: Wall source implementation We introduce a rigorous matrix notation to elucidate the implementation of wall sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We define wall sources as a vector in spatial coordinates, diagonal in spin and color, [W]s,c,x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='s′,c′ ≡ δss′δcc′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (C1) That is, all spatial entries of the real part are set to 1, imaginary part to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' It can be placed at any time slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Under a gauge transformation G, the gauge average is G(t)WWT G(t)† G = 1x,s,c, where [G(t)W]x = G(t, x)1s,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (C2) More explicitly, � G(t)WWT G(t)† G � x,y = 1 |G| � DG G(t, x)1spinG(t, y)†1spin = δx,y1s1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (C3) We insert the wall source in between a pair of quark propagators in the path integral by the following steps, only highlighting the time dependence in S to keep the notation simple, � DUP(U) Tr x,s,c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' S[U](t′, t)S[U](t, t′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' � = � DUP(U) Tr x,s,c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' S[U](t′, t) 1x,s,c S[U](t, t′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' � = 1 |G| � DG � DUP(U) Tr x,s,c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' S[U](t′, t)G(t)WWTG(t)†S[U](t, t′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' � = 1 |G| � DG � DUP(UG) Tr x,s,c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' S[UG](t′, t)G(t)WWTG(t)†S[UG](t, t′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' � = 1 |G| � DG � DUP(U) Tr x,s,c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' S[U](t′, t)WWTS[U](t, t′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' � = � DUP(U) Tr x,s,c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' S[U](t′, t)WWTS[U](t, t′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' � = � DUP(U) Tr s,c � WTS[U](t, t′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' S[U](t′, t)W � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (C4) In the last step, we use the cyclic property of trace Tr AB = Tr BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' We also used the property that under a gauge transformation Uµ → (UG)µ ≡ GUµG†, the propagator transforms as, S[UG](t, t′) = G(t)S[U](t, t′)G(t′)†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (C5) More explicitly, the gauge links are, (Uµ)x,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='x′,t′ = δ(x,t),(x′,t′)−µUµ(x, t)1s, (C6) and its gauge transformation is (GUµG†)x,y = G(x)[Uµ]x,yG(y)† = G(x)δx,y−µUµ(x)G(y)† = δx,y−µG(x)Uµ(x)G(x + µ)†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (C7) Note that we will use Uµ(t, t′) = P(t)UµP(t′)T and U † µ(t, t′) = P(t)U † µP(t′)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (C8) 23 Here P(t) is defined as projection to a time slice (not to be confused with the weighting factor P(U) in the path integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (C4)), [P(tp)]s,c,x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='s′,c′,t′,x′ ≡ δtp,t′δss′δcc′δx,x′, (C9) which is diagonal in spin, color, and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' When we take the dagger of Uµ(t, t′), we need to switch the time arguments since [Uµ(t, t′)]† = [P(t)UµP(t′)T]† = P(t′)U † µP(t)T = U † µ(t′, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (C10) Operationally, a quark propagator can be written in terms of the inverse of the quark matrix as, S(t, t′) ≡ P(t)M −1 q P(t′)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (C11) For Wilson-type fermions, Mq satisfies the γ5-hermiticity relation M † q = γ5Mqγ5, � M −1 q �† = γ5M −1 q γ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (C12) Examples on how to use the notation to calculate two-point and four-point correlation functions are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Appendix D: Form factor from four-point functions TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' Pion form factor Fπ(q2) from four-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' An example of the data to be fitted is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The fit form is in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' (8) with Fπ and Eπ treated as free parameters and mπ taken from the measured value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' For comparison, the Eπ from the continuum dispersion relation is provided with the same mπ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' The four columns correspond to q = {0, 0, 1}, {0, 1, 1}, {1, 1, 1}, {0, 0, 2} from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content=' mπ=1100 MeV Fπ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8209 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='7213 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='650 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='604 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='005 Eπ fit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2556 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4021 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0027 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='530 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='644 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='006 Eπ continuum 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2597 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3976 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0009 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='5230 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0009 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6389 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0008 Fit range {7,9} {6,8} {7,10} {7,12} χ2/dof 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='90 mπ=800 MeV Fπ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='7677 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='646 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='568 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='552 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='011 Eπ fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='9967 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='163 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='308 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='463 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='013 Eπ continuum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='9992 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0009 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1682 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0007 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3157 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0007 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='4483 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0006 Fit range {9,13} {10,17} {10,17} {9,14} χ2/dof 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='72 mπ=600 MeV Fπ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='7412 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='6360 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='583 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='525 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='012 Eπ fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8508 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0017 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='050 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='231 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='007 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='354 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='016 Eπ continuum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='8500 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0435 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0008 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2063 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0007 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='3497 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0006 Fit range {4,13} {6,15} {6,9} {8,13} χ2/dof 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='39 mπ=360 MeV Fπ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='720 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='616 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='554 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='530 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='008 Eπ fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='695 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='911 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='009 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='076 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='013 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='258 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='018 Eπ continuum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='7082 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='9316 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0009 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='1110 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0007 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='2652 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='0006 Fit range {6,11} {6,11} {6,11} {6,11} χ2/dof 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} +page_content='00' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf'} diff --git 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+ing min-max problems which are being widely used in a variety of areas. To improve +the applicability and efficiency of PDHG for different application scenarios, we fo- +cus on the preconditioned PDHG (PrePDHG) algorithm, which is a framework +covering PDHG, alternating direction method of multipliers (ADMM), and other +methods. We give the optimal convergence condition of PrePDHG in the sense +that the key parameters in the condition can not be further improved, which fills +the theoretical gap in the-state-of-art convergence results of PrePDHG, and ob- +tain the ergodic and non-ergodic sublinear convergence rates of PrePDHG. The +theoretical analysis is achieved by establishing the equivalence between PrePDHG +and indefinite proximal ADMM. Besides, we discuss various choices of the proxi- +mal matrices in PrePDHG and derive some interesting results. For example, the +convergence condition of diagonal PrePDHG is improved to be tight, the dual step- +size of the balanced augmented Lagrangian method can be enlarged to 4/3 from +Xingju Cai is supported by the NSFC grants 12131004 and 11871279. Bo Jiang is supported +by the NSFC grant 11971239 and the Natural Science Foundation of the Higher Education +Institutions of Jiangsu Province (21KJA110002). Deren Han is supported by the NSFC grants +2021YFA1003600, 12126603. +Yumin Ma +School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, +210023, P.R. China. +E-mail: mayumin@nufe.edu.cn +Xingju Cai +School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing +Normal University, Nanjing 210023, P.R. China. +E-mail: caixingju@njnu.edu.cn +Bo Jiang +School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing +Normal University, Nanjing 210023, P.R. China. +E-mail: jiangbo@njnu.edu.cn +Deren Han (Corresponding author) +LMIB, School of Mathematical Sciences, Beihang University, Beijing 100191, P.R. China. +E-mail: handr@buaa.edu.cn + +2 +Y. Ma, X. Cai, B. Jiang & D. Han +1, and a balanced augmented Lagrangian method with symmetric Gauss-Seidel +iterations is also explored. Numerical results on the matrix game, projection onto +the Birkhoff polytope, earth mover’s distance, and CT reconstruction verify the +effectiveness and superiority of PrePDHG. +Keywords Preconditioned PDHG · Indefinite proximal ADMM · Tight +convergence condition · Enhanced balanced ALM +Mathematics Subject Classification (2020) 90C08 · 90C25 · 90C47 +1 Introduction +In this paper, we consider the convex-concave min-max problem: +min +x∈Rn max +y∈Rm L(x, y) := f(x) + ⟨Kx, y⟩ − g∗(y), +(PD) +where K ∈ Rm×n, f : Rn → (−∞, +∞], and g : Rm → (−∞, +∞] are proper +closed convex functions, g∗ is the convex conjugate of g, i.e., g∗(y) = supz∈Rm{⟨z, y⟩− +g(z)}. Here ⟨·, ·⟩ denotes the standard inner product. The primal and dual formu- +lations of problem (PD) are, respectively, given as +min +x∈Rn f(x) + g(Kx) +(P) +and +min +y∈Rm f ∗(−KTy) + g∗(y). +(D) +Such problems have wide applications in matrix completion [4], image denoising +[7,44], compressed sensing [20], earth mover’s distance [33], computer vision [41], +CT reconstruction [45], magnetic resonance imaging [46], robust face recognition +[47] and image restoration [50], etc. +An efficient method to solve (PD) is the primal-dual hybrid gradient (PDHG) +algorithm which was originally proposed by Zhu and Chan [50] and further de- +veloped by Chambolle and Pock [7]. The recursion of the PDHG for (PD) reads +as: +PDHG procedure for (PD): Let τ > 0 and σ > 0. For given (xk, yk), the new +iterate (xk+1, yk+1) is generated by: + + + + + + + +xk+1 = argmin +x∈Rn +f(x) + +� +Kx, yk� ++ 1 +2τ +���x − xk��� +2 +, +(1.1a) +yk+1 = argmin +y∈Rm +g∗(y) − +� +K(2xk+1 − xk), y +� ++ 1 +2σ ∥y − yk∥2. +(1.1b) +Here, ∥ · ∥ means the vector ℓ2 norm. In (1.1), τ, σ > 0 are the primal and dual +stepsize parameters, respectively. Chambolle and Pock [7] and He and Yuan [24] +established the convergence of PDHG under the condition τσ∥K∥2 < 1, in which +∥K∥ is the spectral norm of the matrix K. This condition is improved to τσ∥K∥2 ≤ +1 by Condat [13] and further enhanced to +τσ∥K∥2 < 4 +3 +(1.2) + +Understanding PrePDHG: a view of indefinite proximal ADMM +3 +very recently by He et al. [21] for a special case of (PD), i.e., g∗(y) = ⟨b, y⟩ and +b ∈ Rm, other than general g∗(·). Under the condition (1.2), the convergence of +PDHG (1.1) for (PD) with general g∗(·) is established in [30] and [35]. For more +results about the convergence of PDHG, readers can refer to [6,23,28,29]. +As observed in [40] that for cases when ∥K∥ may not be estimated easily, or +it might be very large, the practical convergence of the PDHG (1.1) significantly +slows down. To overcome this issue, we are concerned in this paper with a general +algorithm, i.e., the preconditioned PDHG (PrePDHG), which is given as: +PrePDHG procedure for (PD): Let M1 ∈ Rn×n and M2 ∈ Rm×m be sym- +metric matrices. For given (xk, yk), the new iterate (xk+1, yk+1) is generated +by: + + + + + + + +xk+1 = argmin +x∈Rn +f(x) + +� +Kx, yk� ++ 1 +2 +���x − xk��� +2 +M1 +, +(1.3a) +yk+1 = argmin +y∈Rm +g∗(y) − +� +K(2xk+1 − xk), y +� ++ 1 +2∥y − yk∥2 +M2. +(1.3b) +Here, ∥z∥2 +M1 = ⟨z, M1z⟩ for a vector z ∈ Rn. Obviously, the PrePDHG (1.3) re- +duces to the PDHG (1.1) by taking M1 = τ −1In and M2 = σ−1Im. More impor- +tantly, by taking other specific forms of M1 and M2, the framework of PrePDHG +can take several other algorithms as special cases, see Section 4 for more details. +The PrePDHG is first proposed by Pock and Chambolle [40]1. They established +the convergence of the PrePDHG (1.3) under the condition +� +M1 −KT +−K +M2 +� +≻ 0, +(1.4) +see [40, Theorem 1]. For a symmetric positive matrix M, denote M − 1 +2 as the square +root of M −1, namely, M −1/2M −1/2 = M −1, then condition (1.4) is equivalent to +(see Lemma 2.2) +M1 ≻ 0, +M2 ≻ 0, +���M +− 1 +2 +2 +KM +− 1 +2 +1 +��� < 1. +(1.5) +Besides, [40] also proposed a family of diagonal preconditioners for M1 and M2, +which make the subproblems easier to solve and guarantee the convergence of the +algorithm. From the point view of an indefinite proximal point algorithm, Jiang +et al. [27] showed that the condition (1.5) can be improved to +M1 + 1 +2Σf ≻ 0, M2 + 1 +2Σg∗ ≻ 0, +���� +� +M2 + 1 +2Σg∗ +�− 1 +2 K +� +M1 + 1 +2Σf +�− 1 +2 +���� < 1, +where Σf and Σg∗ are symmetric semidefinite matrices related to f and g∗ (see +(2.1)). +Since min-max problems are equivalent to constrained or composite optimiza- +tion problems under certain conditions, some literatures focus on understanding +1 The setting in [40] is for the general finite-dimensional vector space other than the Eu- +clidean space. For simplicity of presentation, we focus on the Euclidean space. However, our +results in this paper can be easily extended to the general finite-dimensional vector space. + +4 +Y. Ma, X. Cai, B. Jiang & D. Han +PDHG and PrePDHG from various perspectives. For example, the equivalence be- +tween PDHG and linearized alternating direction method of multipliers (ADMM) +is discussed in [14, 39]. Similarly, Liu et al. [36] established the equivalence be- +tween PrePDHG for (PD) and positive semidefinite proximal ADMM (sPADMM) +for an equivalent problem of (D). Based on the equivalence and the convergence +analysis of the first-order primal-dual algorithm in [8], Liu et al. [36] established +the ergodic convergence result (but without sequence convergence) of PrePDHG +under the condition +� +M1 −KT +−K +M2 +� +⪰ 0, +(1.6) +and also considered some inexact versions of PrePDHG. Note that a similar condi- +tion of (1.6) is extended for infinite dimensional Hilbert space in [3]. Very recently, +under condition (1.6), Jiang and Vandenberghe [31] showed convergence of iterates +for Bregman PDHG, of which PrePDHG is a special case. +As mentioned above, when M1 = τ −1In and M2 = σ−1Im, the PrePDHG +(1.3) reduces to the original PDHG (1.1). However, the convergence condition +(1.6) degrades into τσ∥K∥2 ≤ 1 other than (1.2). This raises a natural question: +can we obtain a tighter convergence condition of PrePDHG to fill this gap? +Motivated by [36], we intend to investigate PrePDHG from the perspective of +proximal ADMM. A known result is that indefinite proximal ADMM (iPADMM), +with weaker convergence conditions, outperforms positive semidefinite proximal +ADMM (sPADMM) [5, 10, 11, 17, 18, 22, 32, 37, 49]. In this paper, we restudy the +PrePDHG (1.3) from the point view of iPADMM other than sPADMM as done +in [36] and give positive answers to the above question. The main contributions of +this paper are as follows: +Firstly, we establish the equivalence between PrePDHG for (PD) and iPADMM +for an equivalent problem of (P). Based on the equivalence, we improve the con- +vergence condition (1.5) of the PrePDHG to +� 4 +3 +� +M1 + 1 +2Σf +� +KT +K +M2 +� +≻ 0, +which can be rewritten as (see Lemma 2.2) +M1 + 1 +2Σf ≻ 0, +M2 ≻ 0, +����M +− 1 +2 +2 +K +� +M1 + 1 +2Σf +�− 1 +2 +���� +2 +< 4 +3. +(1.7) +Note that (1.7) is exactly (1.2) when PrePDHG reduces to the original PDHG and +Σf is taken as a zero matrix. Some counter-examples are given in Section 3.3 to +illustrate that condition (1.7) is tight in the sense that the constants 4/3 and 1/2 +can not be replaced by any larger numbers, namely, the inequality sign “<” can +not be replaced by “≤”. +Secondly, we establish the ergodic and non-ergodic sublinear convergence rate +results of the PrePDHG both in the sense of the KKT residual and the function +value residual. To the best of our knowledge, the sublinear convergence rate based +on the KKT residual is new for PDHG-like methods since the existing results +mainly focus on the function value residual. And for the function value residual +measurement, our sublinear rate result is the first non-ergodic result since the +existing results are all ergodic. The numerical experiments in Section 5 show that +the KKT residual is more practical than the function value residual. + +Understanding PrePDHG: a view of indefinite proximal ADMM +5 +Thirdly, we discuss some practical choices of M1 and M2 and get some in- +teresting results. For example, condition (1.2) is tight for PDHG (1.1); the sharp +range of parameters for diagonal PrePDHG is given, and the dual stepsize of the +balanced ALM (BALM) [25] can be enlarged to 4/3 from 1, and we rename it +an enhanced BALM (eBALM). Besides, we explore the eBALM with symmetric +Gauss-Seidel iterations (eBALM-sGS), which can be understood as a special case +of PrePDHG. +Finally, we perform four groups of numerical experiments on solving the ma- +trix game, projection onto the Birkhoff polytope, earth mover’s distance, and CT +reconstruction problems. We choose proper M1 and M2 and the numerical results +verify the effectiveness of the choices of M1 and M2 and the superiority of the +PrePDHG (with tighter convergence condition). +This paper is organized as follows. Some notations and preliminaries are pre- +sented in Section 2. In Section 3, we first establish the equivalence between PrePDHG +and iPADMM and then develop the global convergence of PrePDHG from the +iPADMM point of view. The existing convergence condition of PrePDHG is im- +proved to be tight, as shown by counter-examples. Then, the sublinear convergence +rate of the PrePDHG is obtained. We revisit the choices of M1 and M2 in Section 4 +and get some new results. In Section 5, we perform numerical experiments on four +practical problems to verify the effectiveness of the PrePDHG. Some concluding +remarks are made in Section 6. +2 Notations and Preliminaries +We use ∥x∥1, ∥x∥, and ∥x∥∞ to denote the ℓ1, ℓ2 and ℓ∞ norm of the vector x +respectively, and ∥A∥ to denote the spectral norm of the matrix A. We use vec(A) +to denote a vector formulated by stacking the columns of A one by one, from +first to last. We slightly abuse the notation ∥x∥2 +M := ⟨x, Mx⟩ as long as M is +symmetric. When M is symmetric positive semidefinite, we use M +1 +2 to represent +the square root of M, namely, M +1 +2 M +1 +2 = M. For symmetric matrices A and B, +A ⪰ (≻) B means that A − B is positive semidefinite (positive definite). For a +symmetric matrix P ∈ Rn×n, we can always decompose it as +P = P+ − P− +with P+, P− ⪰ 0. We name this decomposition a DC decomposition of P. Note +that the DC decomposition of a symmetric matrix is not unique. +We adopt some standard notations in convex analysis; see [43] for instance. +The distance from a point x to a nonempty convex closed set S ⊆ Rn is denoted +as dist(x, S) = miny∈S ∥y − x∥. For any proper closed convex function f : Rn → +(−∞, +∞] and ¯x ∈ domf := {x ∈ Rn | f(x) < +∞}, the subdifferential at ¯x is +defined as ∂f(¯x) := {ξ ∈ Rn | f(x) ≥ f(¯x) + ⟨ξ, x − ¯x⟩ , ∀x ∈ Rn}, in which any +ξ is a subgradient at ¯x. Moreover, there exists a symmetric positive semidefinite +matrix Σf such that for all x1, x2 ∈ Rn and ξ1 ∈ ∂f(x1), ξ2 ∈ ∂f(x2), +⟨ξ1 − ξ2, x1 − x2⟩ ≥ ∥x1 − x2∥2 +Σf . +(2.1) +For any proper closed convex function f, the convex conjugate of f is defined as +f ∗(y) := supx∈Rn{⟨x, y⟩ − f(x)}, and we have +y ∈ ∂f(x) ⇔ x ∈ ∂f ∗(y). +(2.2) + +6 +Y. Ma, X. Cai, B. Jiang & D. Han +Given a symmetric matrix M with M + Σf ≻ 0, we define the generalized +proximal operator as +proxM +f (x) := argmin +z∈Rn +f(z) + 1 +2∥z − x∥2 +M. +(2.3) +If M = τ −1In for some τ > 0, we simply denote proxτf (x) := proxM +f (x). Let +˜f(·) := f(·) − 1 +2∥ · ∥2 +Σf . Observing that +f(z) + 1 +2∥z − x∥2 +M = ˜f(z) + 1 +2 +���z − (M + Σf)−1Mx +��� +2 +M+Σf ++ 1 +2∥x∥2 +M +− 1 +2∥(M + Σf)−1Mx∥2 +M+Σf , +we have an equivalent characterization of proxM +f (x) as +proxM +f (x) = proxM+Σf +˜ +f +� +(M + Σf)−1 Mx +� +. +(2.4) +We now present a generalization of Moreau’s identity, see [12, Theorem 1 (ii)] +or [2, Lemma 3.3], which is very useful in our analysis. +Lemma 2.1 Let f : Rn → (−∞, +∞] be a proper closed convex function. Suppose +M ≻ 0, then we have +x = proxM +f (x) + M −1proxM−1 +f∗ +(Mx), +∀x ∈ Rn. +In the following lemma, the equivalence between (1.4) and (1.5) is established. +In [40], the authors proved that (1.5) implies (1.4). Here we present a simple proof +of the equivalence based on the well-known Schur complement. +Lemma 2.2 Let M1 ∈ Rn×n, M2 ∈ Rm×m be symmetric matrices. Then (1.4) is +equivalent to (1.5). +Proof By [48, Theorem 1.12], we know that (1.4) is equivalent to +M1 ≻ 0, +M2 ≻ 0, +M1 − KTM −1 +2 +K ≻ 0. +Since M1 ≻ 0 and M2 ≻ 0, we have +M1 − KTM −1 +2 +K ≻ 0 ⇐⇒ In − M +− 1 +2 +1 +KTM +− 1 +2 +2 +M +− 1 +2 +2 +KM +− 1 +2 +1 +≻ 0 +⇐⇒ ∥M +− 1 +2 +2 +KM +− 1 +2 +1 +∥ < 1. +The proof is completed. +⊓⊔ +Throughout this paper, we assume that problem (PD) has a saddle point +(x⋆, y⋆), which satisfies the optimality condition +L(x⋆, y) ≤ L(x⋆, y⋆) ≤ L(x, y⋆), +∀x ∈ Rn, +∀y ∈ Rm +(2.5) +and the KKT-type optimality condition +0 ∈ ∂f(x⋆) + KTy⋆, +0 ∈ ∂g∗(y⋆) − Kx⋆. +(2.6) + +Understanding PrePDHG: a view of indefinite proximal ADMM +7 +Such x⋆ and y⋆ are also optimal for (P) and (D), respectively. Define the KKT +residual mapping R : Rn × Rm → R as +R(x, y) = max +� +dist(0, ∂f(x) + KTy), dist(0, ∂g∗(y) − Kx) +� +. +(2.7) +Clearly, we have the following equivalent characterization of the optimality condi- +tion. +Proposition 2.1 The KKT-type optimality condition (2.6) holds if and only if +R(x⋆, y⋆) = 0. +Based on this, we define the ǫ-solution of problem (PD) as follows. +Definition 2.1 Given ǫ ≥ 0, a pair (x, y) is called an ǫ-solution of problem (PD) +if R(x, y) ≤ ǫ. +Note that the KKT residual (2.7) may be difficult or expensive to calculate +since it involves computing the distance of a point to a convex set. However, in +some practical circumstances, the upper bound of R(x, y) in (2.7) could be easily +obtained; see the discussion in Remark 3.1 and Remark 4.5. +In the rest of this section, we present the existing convergence and sublinear +convergence rate results of iPADMM developed in [17], which are key to the con- +vergence analysis of PrePDHG. Note that the algorithm in [17] is more general +and takes iPADMM as a special case. Here we display the corresponding results +of iPADMM. +Consider the convex minimization problem with linear constraints and a sep- +arable objective function +min +x∈Rn1,y∈Rn2 θ1(x) + θ2(y) +s.t. +Ax + By = 0, +(2.8) +where A ∈ Rm×n1, B ∈ Rm×n2, θ1 : Rn1 → (−∞, +∞], and θ2 : Rn2 → (−∞, +∞] +are proper closed convex functions. The augmented Lagrangian function of (2.8) +is defined by: +Lβ(x, y, λ) = θ1(x) + θ2(y) − ⟨λ, Ax + By⟩ + β +2 ∥Ax + By∥2, +where λ is the corresponding Lagrange multiplier of the linear constraints and +β > 0 is a penalty parameter. The iPADMM for (2.8) in [17] is given as: +iPADMM procedure for (2.8): Choose the symmetric indefinite matrices S +and T. For given (xk, yk, λk), the new iterate (xk+1, yk+1, λk+1) is generated by: + + + + + + + + + + + + + +xk+1 = argmin +x∈X +Lβ(x, yk, λk) + 1 +2∥x − xk∥2 +S, +(2.9a) +yk+1 = argmin +y∈Y +Lβ(xk+1, y, λk) + 1 +2∥y − yk∥2 +T , +(2.9b) +λk+1 = λk − β(Axk+1 + Byk+1). +(2.9c) +Let Σ1 and Σ2 be the symmetric positive semidefinite matrices related to θ1 +and θ2, respectively; see (2.1) for details. The sequence {(xk, yk, λk)} is denoted +as {wk}. Now we present the convergence results of iPADMM. + +8 +Y. Ma, X. Cai, B. Jiang & D. Han +Lemma 2.3 +[17, Theorem 3.2] Let the sequence {wk} be generated by iPADMM +(2.9). If the proximal terms S and T are chosen such that +S + 1 +2Σ1 ⪰ 0, +S + 1 +2Σ1 + βATA ≻ 0 +(2.10) +and +T + Σ2 + βBTB ≻ 0, +T + 1 +2Σ2 + κ1(−2T− + Σ2) + κ2βBTB ≻ 0, +(2.11) +where κ1 = 1, κ2 ∈ (0, 3 +4), and T− comes from one DC decomposition of T, then +{wk} converges to an optimal solution of (2.8). +Lemma 2.4 +[17, Theorem 4.1] Let the sequence {wk} be generated by iPADMM +(2.9). If the proximal terms S and T are chosen such that (2.10) and (2.11) hold, +and +S + 1 +2Σ1 ⪰ c +2Σ1, +(2.12) +with c > 0, then we have +min +1≤i≤k ∥wi − wi+1∥2 +ˆ +G = o(1/k), +in which +ˆG = + + +S + Σ1 +T + Σ2 + βBTB +1 +β Im + + . +Lemma 2.5 +[17, Theorem 4.2] Let the sequence {wk} be generated by iPADMM +(2.9). If the proximal terms S and T are chosen such that (2.10), (2.11), and (2.12) +hold, and T + 1 +2Σ2 ≻ 0, then we have +∥wk − wk+1∥2 +ˆ +G = o(1/k), +where ˆG is defined in Lemma 2.4. +3 The Preconditioned PDHG and its Convergence +We first present the PrePDHG with practical stopping criterion for convex-concave +min-max optimization (PD) in Algorithm 1. We shall first establish an equivalence +between PrePDHG and iPADMM, which is key to analyzing the algorithm, in +Section 3.1 and deduce the global convergence of Algorithm 1 in Section 3.2. +Section 3.3 provides counter-examples to show the tightness of condition (1.7). The +sublinear convergence rate in both ergodic and non-ergodic sense is investigated +in Section 3.4. +The PrePDHG is given in Algorithm 1. Note that the stopping criterion R(xk+1, yk+1) ≤ +ǫ can be replaced by R(xk+1, yk) ≤ ǫ. + +Understanding PrePDHG: a view of indefinite proximal ADMM +9 +Algorithm 1: PrePDHG: Preconditioned PDHG for solving (PD). +1 Initialization: Choose the initial points x0 ∈ Rn, y0 ∈ Rm, and set the tolerance +ǫ ≥ 0. Choose the matrices M1 ∈ Rn×n and M2 ∈ Rm×m satisfying (1.7). +2 for k = 0, 1, . . . , do +3 +Update xk+1 and yk+1 as follows: + + + + + + + + + +xk+1 = argmin +x∈Rn +f(x) + +� +Kx, yk� ++ 1 +2 +���x − xk��� +2 +M1 +, +(3.1a) +yk+1 = argmin +y∈Rm +g∗(y) − +� +K(2xk+1 − xk), y +� ++ 1 +2 ∥y − yk∥2 +M2, +(3.1b) +4 +if R(xk+1, yk+1) ≤ ǫ then +5 +break. +6 +end +7 end +Remark 3.1 By (3.24) and (3.26), we have R(xk+1, yk+1) ≤ ˆR(xk+1, yk+1) with +ˆR(xk+1, yk+1) := max +� ���KT(yk+1 − yk) − M1(xk+1 − xk) +��� , +���K(xk+1 − xk) − M2(yk+1 − yk) +��� +� +, +which can be easily computed. Therefore, if R(xk+1, yk+1) is difficult to com- +pute, we can use the stopping criterion ˆR(xk+1, yk+1) ≤ ǫ. Similarly, by the first +inequalities in (3.30) and (3.31), we can also replace R(xk+1, yk) by its upper +bound as +ˆR(xk+1, yk) := max +� ���M1(xk+1 − xk) +��� , +���K(xk − xk−1) + K(xk − xk+1) − M2(yk − yk−1) +��� +� +. +Note that for some special cases, such as g∗ is a linear function, a more compact +upper bound of R(xk+1, yk+1) or R(xk+1, yk) can be obtained, see Remark 4.5 +for instance. +3.1 Equivalence of PrePDHG and iPADMM +We first show that PrePDHG (3.1) can be understood as an iPADMM applied on +the equivalent formulation of problem (P): +min +x∈Rn, u∈Rm g(u) + f(x) +s.t. +M +− 1 +2 +2 +(Kx − u) = 0, +(P1) +where M2 ≻ 0. Let +L1(u, x, λ) = g(u) + f(x) + +� +λ, M +− 1 +2 +2 +(Kx − u) +� ++ 1 +2∥Kx − u∥2 +M−1 +2 + +10 +Y. Ma, X. Cai, B. Jiang & D. Han +be the augmented Lagrangian function of problem (P1), where λ is the correspond- +ing Lagrange multiplier of the linear constraints. Given the initial points x0 ∈ Rn +and λ0 ∈ Rm, the main iterations of the iPADMM are given as + + + + + + + + + + + + + +uk+1 = argmin +u∈Rm L1(u, xk, λk), +(3.2a) +xk+1 = argmin +x∈Rn L1(uk+1, x, λk) + 1 +2∥x − xk∥2 +M1−KTM−1 +2 +K, +(3.2b) +λk+1 = λk + M +− 1 +2 +2 +(Kxk+1 − uk+1), +(3.2c) +where the proximal matrix M1 − KTM −1 +2 +K could be indefinite. Note that in +(3.2), there is only an additional proximal term in the second subproblem. Using +the notations of (2.3) and (2.4), we can equivalently formulate (3.2) as + + + + + + + + + + + + + + + + + + + +uk+1 = proxM−1 +2 +g +� +M +1 +2 +2 λk + Kxk� +, +(3.3a) +xk+1 = proxM1+Σf +˜ +f +� +(M1 + Σf)−1M1xk − +(3.3b) +(M1 + Σf)−1KTM −1 +2 +� +M +1 +2 +2 λk + Kxk − uk+1�� +, +(3.3c) +λk+1 = λk + M +− 1 +2 +2 +(Kxk+1 − uk+1), +(3.3d) +where ˜f(·) := f(·) − 1 +2∥ · ∥2 +Σf is defined in Section 2. Similarly, we can reformulate +the iterations of PrePDHG (3.1) as + + + + + +xk+1 = proxM1+Σf +˜ +f +� +(M1 + Σf)−1M1xk − (M1 + Σ1)−1KTyk� +, +(3.4a) +yk+1 = proxM2 +g∗ +� +yk + M −1 +2 +K(2xk+1 − xk) +� +. +(3.4b) +We are now ready to deduce the equivalence between PrePDHG (3.1) and +iPADMM (3.2). +Lemma 3.1 PrePDHG (3.1) (or (3.4)) and iPADMM (3.2) (or (3.3)) are equiv- +alent in the sense that the sequence generated by either algorithm can explicitly +recover the sequence generated by the other. +Proof Let the sequence {(uk, xk, λk)} be generated by iPADMM (3.3) with initial +points x0 ∈ Rn and λ0 ∈ Rm. Consider the transform +yk = M −1 +2 +� +M +1 +2 +2 λk + Kxk − uk+1� +. +(3.5) +First, substituting (3.5) into (3.3c) yields (3.4a). By Lemma 2.1, we have from +(3.3a) that +M +1 +2 +2 λk + Kxk = uk+1 + M2proxM2 +g∗ +� +M +− 1 +2 +2 +λk + M −1 +2 +Kxk� +, +which with the transform (3.5) implies yk = proxM2 +g∗ +� +M +− 1 +2 +2 +λk + M −1 +2 +Kxk� +. This +also tells +yk+1 = proxM2 +g∗ +� +M +− 1 +2 +2 +λk+1 + M −1 +2 +Kxk+1� +. +(3.6) + +Understanding PrePDHG: a view of indefinite proximal ADMM +11 +Besides, with (3.3d) and (3.5), we have M +− 1 +2 +2 +λk+1 = M −1 +2 +K(xk+1 − xk) + yk. +Substituting this relation into (3.6) yields (3.4b). Now we can conclude that the +sequence {(xk, yk)} is exactly the sequence generated by PrePDHG (3.4) with +initial points x0 and y0 = M −1 +2 +(M +1 +2 +2 λ0 + Kx0 − u1). +On the other hand, let the sequence {(xk, yk)} be generated by PrePDHG (3.4) +with given initial points x0 ∈ Rn and y0 ∈ Rm. Consider the transforms +λk+1 = M +− 1 +2 +2 +K(xk+1 − xk) + M +1 +2 +2 yk, +uk+1 = M +1 +2 +2 λk + Kxk − M2yk. +Using a similar argument, we can show that {(uk, xk, λk)} is exactly the same +sequence generated by iPADMM (3.3) and the initial points of x and λ are taken +as x0 and M +− 1 +2 +2 +K(x1 − x0) + M +1 +2 +2 y0, respectively. We omit the details for brevity. +The proof is completed. +⊓⊔ +⊓⊔ +Remark 3.2 If M1 = KTM −1 +2 +K, then iPADMM (3.2) reduces to the classical +ADMM [15,16]. In this case, PrePDHG (3.1) is equivalent to the classical ADMM. +Based on the key observation that PrePDHG and iPADMM are equivalent, +we next investigate the convergence of PrePDHG (3.1), namely, Algorithm 1, via +the well-established convergence results of iPADMM; see [11, 17, 22, 32, 37, 49] for +instance. Here, we mainly use the global and sublinear convergence rate results +developed in [17]. +It should be mentioned that Liu et al. [36] also showed that PrePDHG (3.1) +is equivalent to a proximal ADMM applied on the equivalent formulation of dual +problem (D) as: +min +y∈Rm, v∈Rn g∗(y) + f ∗(v) +s.t. +M +− 1 +2 +1 +(KTy + v) = 0, +where they require (1.6) holds. The recursion of the proximal ADMM therein is +given as + + + + + + + + + + + + + +yk+1 = argmin +y∈Rm +�L1(y, vk, λk) + 1 +2∥y − yk∥M2−KM−1 +1 +KT, +(3.7a) +vk+1 = argmin +v∈Rn +�L1(yk+1, v, λk), +(3.7b) +˜λk+1 = ˜λk + M +− 1 +2 +1 +(KTyk+1 + vk+1), +(3.7c) +where +�L1(y, v, ˜λ) = g∗(y) + f ∗(v) + +� +λ, M +− 1 +2 +1 +(KTy + v) +� ++ 1 +2∥KTy + v∥2 +M−1 +1 , +in which ˜λ is the corresponding Lagrange multiplier of the linear constraints. A +main difference between (3.2) and (3.7) lies in that the proximal term of (3.2) is +in the second subproblem other than in the first subproblem as done by (3.7). It +is this key point that makes our condition on M1 and M2 weaker than that in [36] +since the iPADMM can always allow more indefiniteness of the proximal term in +the second subproblem other than that in the first subproblem. + +12 +Y. Ma, X. Cai, B. Jiang & D. Han +3.2 Global Convergence +It is clear that condition (1.6) implies M1 − KTM −1 +2 +K ⪰ 0, which further means +that the proximal matrix in (3.2b) is positive semidefinite. However, the well- +explored convergence results of iPADMM tell that the proximal matrix M1 − +KTM −1 +2 +K could be indefinite. Therefore, we could further improve the convergence +condition of PrePDHG from the perspective of iPADMM. +Lemma 3.2 Suppose condition (1.7) holds, that is, +M1 + 1 +2Σf ≻ 0, +M2 ≻ 0, +����M +− 1 +2 +2 +K +� +M1 + 1 +2Σf +�− 1 +2 +���� +2 +< 4 +3. +Then the sequence generated by iPADMM (3.2) converges to an optimal solution +of (P1). +Proof Let the sequence {(uk, xk, λk)} be generated by iPADMM (3.2). In problem +(2.8) and Lemma 2.3, we take n1 := m, n2 := n, θ1 := g, θ2 := f, A := M −1/2 +2 +, +B := −M −1/2 +2 +K, β = 1, S := 0, T := M1 − KTM −1 +2 +K, Σ1 := 0, Σ2 := Σf, +and the parameter κ2 := 1 − ρ with ρ ∈ ( 1 +4, 1). Then we immediately know that +{(uk, xk, λk)} converges to an optimal solution of (P1) as long as there exists a +DC decomposition of M1 − KTM −1 +2 +K and ρ ∈ (1/4,1) such that +M1 +Σf ≻ 0, H := M1 + 3 +2Σf −2 +� +M1 − KTM −1 +2 +K +� +− −ρKTM −1 +2 +K ≻ 0. (3.8) +Now we only need to show the correctness of (3.8) under the condition (1.7). +If M1−KTM −1 +2 +K ⪰ 0, we can take its DC decomposition as (M1−KTM −1 +2 +K)+ = +M1 − KTM −1 +2 +K and (M1 − KTM −1 +2 +K)− = 0. Hence, for any ρ ∈ ( 1 +4, 1), we have +H = M1 + 3 +2Σf − ρKTM −1 +2 +K = ρ(M1 − KTM −1 +2 +K) + (1 − ρ)M1 + 3 +2Σf ≻ 0, +where the last inequality is due to M1 + 1 +2Σf ≻ 0 which comes from (1.7). +Now, suppose M1 − KTM −1 +2 +K ̸⪰ 0. By (1.7) and the Schur complement theo- +rem, we have 4 +3 +� +M1 + 1 +2Σf +� +≻ KTM −1 +2 +K, namely, +M1 − KTM −1 +2 +K + 1 +3 (M1 + 2Σf) ≻ 0. +Let +M := (M1 + 2Σf)− 1 +2 +� +M1 − KTM −1 +2 +K +� +(M1 + 2Σf)− 1 +2 , +(3.9) +then obviously we have M + 1 +3In ≻ 0. Set M = UΣU T be the eigenvalue decompo- +sition of M with U TU = UU T = In and the diagonal matrix Σ = diag(σ1, . . . , σn) +with σ1 ≥ · · · ≥ σp ≥ 0 > σp+1 ≥ · · · ≥ σn. Then we have σn ∈ (− 1 +3, 0). +Consider a DC decomposition of M as M+ = U max(0, Σ)U T and M− = +U max(0, −Σ)U T, where the max-operator max(·, ·) takes the maximum of the +two matrices entry-wisely. It is clear that M− ≺ |σn|In. Recalling (3.9), we thus +obtain a DC decomposition of M1 − KTM −1 +2 +K as +(M1 − KTM −1 +2 +K)+ = (M1 + 2Σf) +1 +2 M+ (M1 + 2Σf) +1 +2 + +Understanding PrePDHG: a view of indefinite proximal ADMM +13 +and +(M1 − KTM −1 +2 +K)− = (M1 + 2Σf) +1 +2 M− (M1 + 2Σf) +1 +2 . +(3.10) +Choosing ρ = 1−2|σn| +1+|σn| ∈ ( 1 +4, 1), with (3.10) and M− ≺ |σn|In, we thus have +(2 + ρ)(M1 − KTM −1 +2 +K)− ≺ (2 + ρ)|σn|(M1 + 2Σf) = (1 − ρ)(M1 + 2Σf) +⪯ (1 − ρ)M1 + 3 +2Σf + ρ�M1 − KTM −1 +2 +K� ++. +Substituting +� +M1−KTM −1 +2 +K +� ++ = M1−KTM −1 +2 +K + +� +M1−KTM −1 +2 +K +� +− into the +above assertion, by some easy calculations, we get (3.8). The proof is completed. +⊓⊔ +Now we are ready to establish the convergence of PrePDHG (Algorithm 1). +Theorem 3.1 Let {(xk, yk)} be the sequence generated by Algorithm 1 with ǫ = 0 +and M1, M2 satisfying (1.7). Then {(xk, yk)} converges to a saddle point of (PD). +Proof Let the sequence {(uk, xk, λk)} be generated by iPADMM (3.2). Since M1 +and M2 satisfy (1.7), we know from Lemma 3.2 that {(uk, xk, λk)} converges to +an optimal solution (u⋆, x⋆, λ⋆) of (P1), namely, +0 ∈ ∂f(x⋆) + KTM +− 1 +2 +2 +λ⋆, +0 ∈ ∂g(u⋆) − M +− 1 +2 +2 +λ⋆, +Kx⋆ − u⋆ = 0. +(3.11) +Recalling the transform (3.5), we know from the proof of Lemma 3.1 that {(xk, yk)} +is exactly the sequence generated by PrePDHG (3.4) with x0 and y0 = M −1 +2 +(M +1 +2 +2 λ0+ +Kx0−u1). Since {(uk, xk, λk)} converges to (u⋆, x⋆, λ⋆), we know from (3.11) that +xk → x⋆ and yk → y⋆ := M +− 1 +2 +2 +λ⋆ and +0 ∈ ∂f(x⋆) + KTy⋆, +0 ∈ ∂g(Kx⋆) − y⋆, +which with the fact that g is proper closed convex and (2.2) shows +0 ∈ ∂f(x⋆) + KTy⋆, +0 ∈ ∂g∗(y⋆) − Kx⋆. +This means that (x⋆, y⋆) is a saddle point of (PD). The proof is completed. +⊓⊔ +3.3 Tightness of Condition (1.7) +We first claim that condition (1.7) is tight in the sense that the constant “4/3” +can not be replaced by any number larger than it, namely, the sign “<” can not +be improved to “≤”. +Lemma 3.3 Let {(xk, yk)} be the sequence generated by Algorithm 1 with ǫ = 0. +Suppose condition (1.7) is replaced by +M1 + 1 +2Σf ≻ 0, +M2 ≻ 0, +����M +− 1 +2 +2 +K +� +M1 + 1 +2Σf +�− 1 +2 +���� +2 +≤ ρ1. +(3.12) +(a). If ρ1 ∈ (0, 4/3), then {(xk, yk)} converges to a saddle point of (PD). +(b). If ρ1 ≥ 4/3, then {(xk, yk)} is not necessarily convergent. + +14 +Y. Ma, X. Cai, B. Jiang & D. Han +Proof The assertion of (a) comes from Theorem 3.1 and the fact that (1.7) is true +if (3.12) holds for any fixed ρ1 ∈ (0, 4/3). To prove (b), consider a simple instance +of problem (PD) as +min +x∈R max +y∈R xy. +(3.13) +Note that such an example is a special case of the one in [35, Section 3.2] by +setting n = m = 1 and A = 1 therein. It is easy to see (3.13) has a unique +saddle point (0, 0). For this problem, Σf = 0, K = 1, and M1, M2 take the form of +M1 = 1/τ, M2 = 1/σ with τ, σ > 0. In this case, condition (3.12) becomes τ, σ > 0 +and τσ ≤ ρ1. We next show that if +τ > 0, σ > 0, τσ = 4 +3, +�x0 +y0 +� +̸∈ S := +� +a +�2 +σ +� +: a ∈ R +� +, +then the sequence generated by PrePDHG diverges, which is enough to finish the +proof. +Specifically, by some easy calculations, the PrePDHG recursion (3.1) for prob- +lem (3.13) reads as +� +xk+1 = xk − τyk, +yk+1 = σxk + (1 − 2τσ) yk, +which can be reformulated as +�xk+1 +yk+1 +� += G +�xk +yk +� +with +G := +�1 +−τ +σ 1 − 2τσ +� +. +(3.14) +Since τσ = 4/3, it is easy to verify that the two eigenvalues of G is −1 and 1/3 +and +G = V +�1/3 0 +0 +−1 +� +V −1 +with +V = +�2/σ τ/2 +1 +1 +� +. +(3.15) +We have from (3.14) and (3.15) that +�xk+1 +yk+1 +� += Gk+1 +�x0 +y0 +� += V +�3−k +0 +0 +(−1)k +� +V −1 +�x0 +y0 +� +. +It is obvious that +��xk+1 +yk+1 +�� +is convergent +⇐⇒ V −1 +�x0 +y0 +� += +�a +0 +� +for some a ∈ R ⇐⇒ +�x0 +y0 +� +∈ S. +Hence, if +�x0 +y0 +� +̸∈ S, then +��xk+1 +yk+1 +�� +diverges and certainly will not converge to +�0 +0 +� +. The proof is completed. +⊓⊔ +We next claim that condition (1.7) is tight in the sense that the constant “1/2” +can not be replaced by any number larger than it. + +Understanding PrePDHG: a view of indefinite proximal ADMM +15 +Lemma 3.4 Let {(xk, yk)} be the sequence generated by Algorithm 1 with ǫ = 0. +Suppose condition (1.7) is replaced by +M1 + ρ2Σf ≻ 0, +M2 ≻ 0, +���M +− 1 +2 +2 +K (M1 + ρ2Σf)− 1 +2 +��� +2 +< 4 +3. +(3.16) +(a). If ρ2 ∈ (0, 1/2], then {(xk, yk)} converges to a saddle point of (PD). +(b). If ρ2 > 1/2, then {(xk, yk)} is not necessarily convergent. +Proof The assertion of (a) comes from Theorem 3.1 and the fact that (1.7) is true +if (3.16) holds for any fixed ρ2 ∈ (0, 1/2]. To prove (b), consider a simple instance +of problem (PD) as +min +x∈R max +y∈R +1 +2x2 + xy. +(3.17) +It is easy to see (3.17) has a unique saddle point (0, 0). For this problem, Σf = +1, K = 1, and M1, M2 take the form of M1 = 1/τ,M2 = 1/σ with 1/τ + ρ2 > +0, σ > 0. In this case, condition (3.16) becomes 0 < σ < 4 +3(1/τ + ρ2). We only +need to show that for any ρ2 ∈ (1/2,1] and ρ3 ∈ (1/2,ρ2) if +0 < σ = 4 +3(1/τ + ρ3) < 4 +3(1/τ + ρ2), +then the sequence generated by PrePDHG is not necessarily convergent. +First, it is not hard to verify that the PrePDHG recursion (3.1) for problem +(3.17) reads as + + + +xk+1 = (xk − τyk)/(1 + τ), +yk+1 = +� +σ(1 − τ)xk + (1 + τ − 2τσ) yk� +/(1 + τ), +which can be reformulated as +�xk+1 +yk+1 +� += �G +�xk +yk +� +with +�G := +1 +1 + τ +� +1 +−τ +σ(1 − τ) 1 + τ − 2τσ +� +. +(3.18) +The characteristic polynomial of �G is given as +p(µ) = µ2 − +1 +1 + τ +� +τ − 2(1 + 4ρ3τ) +3 +� +µ − 1 + 4ρ3τ +3(1 + τ) . +Noting 1+τ +τ +≥ 1 +τ + ρ2 > 0, we have +p(−1) = 2(1 − 2ρ3)τ +1 + τ +< 0, +which tells that at least one eigenvalue of �G is less than −1, i.e., ∥ �G∥ > 1. There- +fore, the sequence {(xk, yk)} generated by (3.18) is not necessarily convergent. +The proof is completed. +⊓⊔ + +16 +Y. Ma, X. Cai, B. Jiang & D. Han +3.4 Sublinear Convergence Rate +We now investigate the sublinear convergence rate of PrePDHG (Algorithm 1). +Theorem 3.2 Let {(xk, yk)} be the sequence generated by Algorithm 1 with ǫ = 0 +and M1, M2 satisfying (1.7). Then we have +min +1≤k≤t R(xk, yk) = o +� 1 +√ +t +� +and +min +1≤k≤t R(xk, yk−1) = o +� 1 +√ +t +� +. +(3.19) +Moreover, if condition (1.7) is replaced by +M1 + 1 +2Σf ≻ 0, +M2 ≻ 0, +����M +− 1 +2 +2 +K +� +M1 + 1 +2Σf +�− 1 +2 +���� < 1, +(3.20) +then we have +R(xt, yt) = o +� 1 +√ +t +� +and +R(xt, yt−1) = o +� 1 +√ +t +� +, +∀t ≥ 1. +(3.21) +Proof First, let us bound the KKT residual Rk+1 := R(xk+1, yk+1) and Rk+1/2 := +R(xk+1, yk) for k ≥ 0. From the optimality condition of (3.1a), we have +− M1(xk+1 − xk) ∈ ∂f(xk+1) + KTyk, +(3.22) +which implies +KT(yk+1 − yk) − M1(xk+1 − xk) ∈ ∂f(xk+1) + KTyk+1 +(3.23) +and thus +dist +� +0, ∂f(xk+1) + KTyk+1� +≤ +���KT(yk+1 − yk) − M1(xk+1 − xk) +��� . +(3.24) +Similarly, using the optimality condition of (3.1b), we have +K(xk+1 − xk) − M2(yk+1 − yk) ∈ ∂g∗(yk+1) − Kxk+1 +(3.25) +and thus +dist +� +0, ∂g∗(yk+1) − Kxk+1� +≤ +���K(xk+1 − xk) − M2(yk+1 − yk) +��� . +(3.26) +Let +ˆRk+1 = +���KT(yk+1 − yk) − M1(xk+1 − xk) +��� ++ +���K(xk+1 − xk) − M2(yk+1 − yk) +��� . +By the Cauchy-Schwarz inequality, we have +ˆRk+1 ≤ (∥K∥ + ∥M1∥) ∥xk+1 − xk∥ + ∥K∥ · ∥yk+1 − yk∥ + ∥M2(yk+1 − yk)∥. +Since M2 ≻ 0, for any z ∈ Rm, we have ⟨z, M2z⟩ ≥ λmin(M2) ⟨z, z⟩ and ∥M2z∥2 = +⟨z, M2M2z⟩ ≤ ∥M2∥ ⟨z, M2z⟩. Therefore, we have ∥yk+1−yk∥ ≤ +1 +√ +λmin(M2)∥yk+1− +yk∥M2 and ∥M2(yk+1 −yk)∥ ≤ +� +∥M2∥∥yk+1 −yk∥M2. Then we immediately have +ˆRk+1 ≤ c1∥xk+1 − xk∥ + c2∥yk+1 − yk∥M2, +(3.27) + +Understanding PrePDHG: a view of indefinite proximal ADMM +17 +where the constants +c1 = ∥K∥ + ∥M1∥, +c2 = +∥K∥ +� +λmin(M2) ++ +� +∥M2∥. +(3.28) +By the definition (2.7) of R(x, y), it is not hard to obtain from (3.24), (3.26), and +(3.27) that +Rk+1 ≤ ˆRk+1 ≤ c1∥xk+1 − xk∥ + c2∥yk+1 − yk∥M2, +(3.29) +Using (3.25) for k := k−1, we have K(xk−xk−1)−M2(yk−yk−1) ∈ ∂g∗(yk)−Kxk +and thus +K(xk − xk−1) + K(xk − xk+1) − M2(yk − yk−1) ∈ ∂g∗(yk) − Kxk+1. +Hence, we have +dist +� +0, ∂g∗(yk) − Kxk+1� +≤ +���K(xk − xk−1) + K(xk − xk+1) − M2(yk − yk−1) +��� +≤ ∥K∥(∥xk − xk−1∥ + ∥xk − xk+1∥) + +� +∥M2∥∥yk − yk−1∥M2, +(3.30) +where the second inequality uses ∥M2(yk − yk−1)∥ ≤ +� +∥M2∥∥yk − yk−1∥M2. On +the other hand, (3.22) implies +dist(0, ∂f(xk+1) + KTyk) ≤ ∥M1(xk+1 − xk)∥ ≤ ∥M1∥∥xk+1 − xk∥. +(3.31) +Combining (3.30) and (3.31) together, and by the definition (2.7) of R(x, y), we +have +Rk+1/2 ≤ c1∥xk+1 − xk∥ + ∥K∥ · ∥xk − xk−1∥ + +� +∥M2∥∥yk − yk−1∥M2. (3.32) +Second, we estimate the upper bound of ∥yk+1 −yk∥M2. From (3.2c) and (3.5), +we have +M +1 +2 +2 yk = λk + M +− 1 +2 +2 +(Kxk − uk+1) = λk+1 + M +− 1 +2 +2 +K(xk − xk+1), +which again with (3.5) for k := k + 1 yields +M +1 +2 +2 (yk+1 − yk) = M +− 1 +2 +2 +(Kxk+2 − uk+2) − M +− 1 +2 +2 +K(xk − xk+1) ++ M +− 1 +2 +2 +K(xk+1 − xk+2). +(3.33) +Condition (1.7) or (3.20) tells KTM −1 +2 +K ≺ 4 +3 +� +M1 + 1 +2Σf +� +. Thus, for any z ∈ Rn, +we have +���M +− 1 +2 +2 +Kz +��� = +���M −1 +2 +Kz +��� +M2 += ∥z∥KT M−1 +2 +K ≤ +2 +√ +3∥z∥M1+ 1 +2 Σf ≤ c3∥z∥, +(3.34) + +18 +Y. Ma, X. Cai, B. Jiang & D. Han +where c3 = 2 +� +λmax(M1 + 1 +2Σf)/3. Hence, noticing that ∥M 1/2 +2 +v∥ = ∥v∥M2 for +any v ∈ Rm, we have from (3.33) and (3.34) that +∥yk+1 − yk∥M2 ≤ +���M +− 1 +2 +2 +(Kxk+2 − uk+2) +��� ++ c3 +���xk+1 − xk�� + +��xk+1 − xk+2�� +� +. +(3.35) +Combining (3.29) and (3.35), we obtain +Rk+1 ≤ (c1 + c2c3) +��xk+1 − xk�� + c2c3 +��xk+1 − xk+2�� ++ c2 +���M +− 1 +2 +2 +(Kxk+2 − uk+2) +���. +(3.36) +Combining (3.32) and (3.35) with k := k − 1, we have +Rk+1/2 ≤ +� +c1 + +� +∥M2∥c3 +� ��xk+1 − xk�� + +� +∥K∥ + +� +∥M2∥c3 +� ��xk − xk−1�� ++ +� +∥M2∥ +��M +− 1 +2 +2 +(Kxk+1 − uk+1) +��. +(3.37) +Finally, similar to the proof of Theorem 3.1, if condition (1.7) holds, it is easy +to see that the conditions of Lemma 2.4 for iPADMM (3.2) are satisfied. Thus, by +applying Lemma 2.4, we have +min +0≤k≤t +� +∥xk − xk+1∥2 +M1+Σf + ∥M +− 1 +2 +2 +(Kxk+1 − uk+1)∥2� += o(1/t), +which with ∥xk − xk+1∥M1+Σf ≥ λmin(M1 + Σf)∥xk − xk+1∥, (3.36) and (3.37) +lead to (3.19). If condition (3.20) holds, it is easy to see that the conditions of +Lemma 2.5 for iPADMM (3.2) are satisfied. Thus, by applying Lemma 2.5, we +have +∥xt − xt+1∥2 +M1+Σf + ∥M +− 1 +2 +2 +(Kxt+1 − ut+1)∥2 = o(1/t), +which with (3.36) and (3.37) leads to (3.21). The proof is completed. +⊓⊔ +It is immediate to establish the iteration complexity of Algorithm 1. +Corollary 3.1 If ǫ > 0, then Algorithm 1 stops in O(1/ǫ2) iterations. +Revisiting the optimality condition (2.5), instead of using the KKT residual, +we can also measure the quality of approximate solution (ˆx, ˆy) by giving an upper +bound of the function value residual L(ˆx, y) − L(x, ˆy) for any x ∈ Rn and y ∈ Rm, +see [7,8,28,29,36,42] and the references therein for instance. However, the existing +results for PDHG and PrePDHG under condition (1.5) or (1.6) are all ergodic, +which always have the bound: +L(¯xt, y) − L(x, ¯yt) ≤ ϕ1(x, x0) + ϕ2(y, y0) +t +, +∀x ∈ Rn, ∀y ∈ Rm, +(3.38) +or +L(¯xt, y⋆) − L(x⋆, ¯yt) ≤ ϕ1(x⋆, x0) + ϕ2(y⋆, y0) +t +, +(3.39) +where ¯xt = 1 +t +�t +i=1 xi, ¯yt = 1 +t +�t +i=1 yi, and ϕ1(·, ·), ϕ2(·, ·) are some nonnegative +functions and (x⋆, y⋆) is a saddle point. +Here, we aim to investigate some non-ergodic results with the help of our +established bounds for the KKT residual. + +Understanding PrePDHG: a view of indefinite proximal ADMM +19 +Lemma 3.5 Let {(xk, yk)} be the sequence generated by Algorithm 1 with ǫ = 0 +and M1, M2 satisfying (1.7). Let (x∞, y∞) be the limit point of {(xk, yk)}. Define +a constant ¯c = supk≥0{∥xk − x∞∥ + ∥yk − y∞∥} and denote +k(t) := argmin +1≤k≤t +� +c1∥xk+1 − xk∥M1+ 1 +2 Σf + c2∥yk+1 − yk∥M2 +� +, +(3.40) +where c1 and c2 are defined (3.28). Then for t ≥ 1, +L(xk(t), y) − L(x, yk(t)) +≤ o(1/ +√ +t) (¯c + ∥x∞ − x∥ + ∥y∞ − y∥) , +∀x ∈ Rn, +∀y ∈ Rm +(3.41) +and +L(xk(t), y∞) − L(x∞, yk(t)) ≤ o(1/ +√ +t). +(3.42) +Moreover, if condition (1.7) is replaced by (3.20), then for any t ≥ 1 +L(xt, y) − L(x, yt) +≤ o(1/ +√ +t) (¯c + ∥x∞ − x∥ + ∥y∞ − y∥) , +∀x ∈ Rn, +∀y ∈ Rm +(3.43) +and +L(xt, y∞) − L(x∞, yt) ≤ o(1/ +√ +t). +(3.44) +Proof By the convexity of f(x) + +� +x, KTyk+1� +and (3.23), we have +f(xk+1) + +� +xk+1, KTyk+1� +− f(x) − +� +x, KTyk+1� +≤ +� +xk+1 − x, KT(yk+1 − yk) − M1(xk+1 − xk) +� +, +∀x ∈ Rn. +(3.45) +Similarly, by the convexity of g∗(y) − +� +y, Kxk+1� +and the optimality condition +(3.25), we have +� +g∗(yk+1) − +� +yk+1, Kxk+1�� +− +� +g∗(y) − +� +y, Kxk+1�� +≤ +� +yk+1 − y, K(xk+1 − xk) − M2(yk+1 − yk) +� +, +∀y ∈ Rm. +(3.46) +Summing up (3.45) and (3.46), for any x ∈ Rn and y ∈ Rm, we have +L(xk+1, y) − L(x, yk+1) +≤ +� +xk+1 − x, KT(yk+1 − yk) − M1(xk+1 − xk) +� ++ +� +yk+1 − y, K(xk+1 − xk) − M2(yk+1 − yk) +� +≤ +� +c1∥xk+1 − xk∥M1+ 1 +2 Σf + c2∥yk+1 − yk∥M2 +� � +∥xk+1 − x∥ + ∥yk+1 − y∥ +� +, +(3.47) +where the second inequality uses the Cauchy-Schwarz inequality and (3.27). +Suppose M1 and M2 satisfy (1.7). From the proof of Theorem 3.2, we know +that +min +1≤k≤t +� +c1∥xk+1 − xk∥M1+ 1 +2 Σf + c2∥yk+1 − yk∥M2 +� += o +� 1 +√ +t +� +. +(3.48) + +20 +Y. Ma, X. Cai, B. Jiang & D. Han +Note that for any k, by the Cauchy-Schwarz inequality and the definition of ¯c, we +have +∥xk+1 − x∥ + ∥yk+1 − y∥ +≤ ∥xk+1 − x∞∥ + ∥yk+1 − y∞∥ + ∥x∞ − x∥ + ∥y∞ − y∥ +≤ ¯c + ∥x∞ − x∥ + ∥y∞ − y∥, +(3.49) +which together with the definition of k(t) in (3.40), (3.47), and (3.48) implies +(3.41). +Suppose M1 and M2 satisfy (3.20). From the proof of Theorem 3.2, we know +that +c1∥xk+1 − xk∥M1+ 1 +2 Σf + c2∥yk+1 − yk∥M2 = o +� 1 +√ +k +� +, +which with (3.47) and (3.49) implies (3.43). +By (3.1), we know that (x∞, y∞) is a saddle point of (PD). Thus, (3.42) and +(3.44) follow directly from (3.41) and (3.43), respectively, by setting x = x∞ and +y = y∞. The proof is completed. +⊓⊔ +Some remarks on our results about the sublinear convergence rate of PrePDHG +are listed below. First, to the best of our knowledge, the sublinear rate based on +the KKT residual R(xk+1, yk+1) or R(xk+1, yk) is new for PDHG like methods +since the existing results mainly focus on (3.38). Compared with (3.38), the upper +bounds of the KKT residual R(xk+1, yk+1) or R(xk+1, yk) are always computable, +see Remark 3.1 ahead. Our sublinear rate result for the KKT residual also tells that +Algorithm 1 can return an ǫ-solution in O(1/ǫ2) iterations. Second, for the function +value residual measurement, our sublinear rate result is the first non-ergodic result +since the existing results are all ergodic, see [7,8,28,29] for instance. It should be +clear that our non-ergodic results are o(1/ +√ +t) while the existing ergodic results +are O(1/t) both under the condition that (x, y) is in a compact set. It remains +unknown whether the non-ergodic result can be improved to O(1/t). +To end this section, we briefly discuss a dual formulation of the PrePDHG +recursion (3.1) in the following remark. +Remark 3.3 In Section 2, we assume that problem (PD) has a saddle point, which +means that solving (PD) is equivalent to solving the following problem: +min +y∈Rm max +x∈Rn g∗(y) − ⟨y, Kx⟩ − f(x). +(3.50) +Using PrePDHG (3.1) to solve (3.50) and based on the symmetry of the primal and +dual variables between (PD) and (3.50) (the primal variable x in (PD) is the dual +variable in (3.50) and vice versa), we can obtain the other PrePDHG recursion, +which can also be used to solve (PD): + + + + + + + +xk+1 = argmin +x∈Rn +f(x) + +� +Kx, 2yk − yk−1� ++ 1 +2 +���x − xk��� +2 +Q1 +, +(3.51a) +yk+1 = argmin +y∈Rm +g∗(y) − +� +Kxk+1, y +� ++ 1 +2∥y − yk∥2 +Q2, +(3.51b) +where the symmetric matrices Q1 ∈ Rn×n and Q2 ∈ Rm×m satisfy +� +Q1 +−KT +−K 4 +3 +�Q2 + 1 +2Σg∗� +� +≻ 0. + +Understanding PrePDHG: a view of indefinite proximal ADMM +21 +Consider an equivalent formulation of problem (D) (note that (D) is also the primal +formulation of problem (3.50)) +min +z∈Rn, y∈Rm f ∗(z) + g∗(y) +s.t. +Q +− 1 +2 +1 +(z + KTy) = 0. +(D1) +The iPADMM recursion for (D1) is given as + + + + + + + + + + + + + +zk+1 = argmin +z∈Rn +¯L1(z, yk, λk), +(3.52a) +yk+1 = argmin +y∈Rm +¯L1(zk+1, y, λk) + 1 +2∥y − yk∥2 +Q2−KQ−1 +1 +KT, +(3.52b) +λk+1 = λk + Q +− 1 +2 +1 +(zk+1 + KTyk+1), +(3.52c) +where ¯L1(z, y, λ) = f ∗(z) + g∗(y) + +� +λ, Q−1/2 +1 +(z + KTy) +� ++ 1 +2∥z + KTy∥2 +Q−1 +1 +is +the augmented Lagrangian function of (D1). Using the same process in Lemma +3.1, we can show the equivalence between (3.51) and the iPADMM (3.52). The +convergence results of (3.51) can thus be established similar to that in Sections +3.2 and 3.4. We omit the details for brevity. +4 Revisit on the Choices of M1 and M2 +In this section, we revisit PrePDHG and discuss the choices of M1 and M2. Specif- +ically, with the choices in Sections 4.1 and 4.2, PrePDHG gives improved versions +of the original PDHG and PDHG with diagonal preconditioners, respectively. In +Section 4.3, we investigate the choice of M1 = τ −1In, M2 = γτKKT + P and its +extensions. In Section 4.4, we consider a special case when g∗(y) = ⟨b, y⟩ and dis- +cuss an enhanced BALM (eBALM) and an eBALM with symmetric Gauss-Seidel +iterations (eBALM-sGS). +4.1 M1 = τ −1In, M2 = σ−1Im +If the proximal operators of f and g∗ are both easy to compute, we can simply take +M1 = τ −1In, M2 = σ−1Im with τ, σ > 0. In this case, PrePDHG (3.4) reduces to +the original PDHG (1.1), which can be reformulated as + + + + + +xk+1 = proxτf +� +xk − τKTyk� +, +yk+1 = proxσg∗ +� +yk + σK(2xk+1 − xk) +� +, +(4.1) +where proxτf(x) is defined in Section 2. Define a constant λf +min := λmin(Σf) ≥ 0. +For such choices of M1 and M2, we have +����M +− 1 +2 +2 +K +� +M1 + 1 +2Σf +�− 1 +2 +���� +2 +≤ +σ∥K∥2 +1/τ + (1/2)λf +min += +τσ∥K∥2 +1 + (τ/2)λf +min +. + +22 +Y. Ma, X. Cai, B. Jiang & D. Han +To make condition (1.7) hold, we obtain the convergence condition of the PDHG +(1.1) or (4.1) as +τ, σ > 0, +τσ∥K∥2 < 4 +3 +� +1 + τλf +min +2 +� +, +(4.2) +which can imply (1.2). Besides, we also know from Lemma 3.3 and Lemma 3.4 +that (4.2) is tight in the sense that the constant 4/3 could not be enlarged. +Remark 4.1 If λf +min is not easy to estimate or f has no more property beyond +convexity, we can set λf +min as zero. Moreover, in the following part of this section, +to make the discussion precise, we choose Σf = 0 and λf +min = 0. We refer to +Section 5.2 for one exception, wherein there holds that Σf = In2 and λf +min = 1. +4.2 Diagonal M1 and M2 +If both f and g take the separable structures, namely, f(x) := �n +j=1 fj(xj), +g∗(y) = �m +i=1 g∗ +i (yi), and the proximal operators of fj and g∗ +i are all easy to +compute, we can consider the following choices of diagonal M1 and M2, which +were first proposed in [40]. +Proposition 4.1 For any α ∈ [0, 2] and γ1, γ2 > 0, let +M1 = γ1 diag(τ1, . . . , τn) with τj = δ + +m +� +i=1 +|Kij|2−α, j = 1, . . . , n, +(4.3) +M2 = γ2 diag(σ1, . . . , σm) with σi = δ + +n +� +j=1 +|Kij|α, i = 1, . . . , m, +(4.4) +where δ ≥ 0 is chosen such that τj, σi are positive. If γ1γ2 > 3 +4, then such M1 and +M2 satisfy (1.7). +Proof By [40, Lemma 2], we know ∥(M2/γ2)−1/2K(M1/γ1)−1/2∥ ≤ 1, which im- +plies that ∥M +− 1 +2 +2 +KM +− 1 +2 +1 +∥2 ≤ +1 +γ1γ2 < 4 +3. The proof is completed. +⊓⊔ +With choices (4.3) and (4.4), PrePDHG (3.4) becomes + + + + + +xk+1 +j += proxτjfj +� +xk +j − τj(KTyk)j +� +, j = 1, . . . , n, +yk+1 +i += proxσig∗ +i +� +yk +i + σi(K(2xk+1 − xk))i +� +, i = 1, . . . , m. +Remark 4.2 Taking γ1 = γ2 = 1 in (4.3) and (4.4) yields the diagonal precondi- +tioners in [40]. Lemma 3.3 tells that γ1γ2 > 3 +4 in Proposition 4.1 is tight in the +sense that “>” can not be improved to “≥”. + +Understanding PrePDHG: a view of indefinite proximal ADMM +23 +4.3 M1 = τ −1In, M2 = γτKKT + P and Extensions +Another choice is M1 = τ −1In and M2 = τKKT + θIm with τ, θ > 0, which was +proposed in [36]. Here, we consider a relaxed version of such choices. +Proposition 4.2 Let P ∈ Rm×m be a nonzero symmetric positive semidefinite +matrix such that KKT + P ≻ 0. Choose +M1 = τ −1In, +M2 = γτKKT + P, +τ > 0, +γ ≥ 3 +4, +(4.5) +then (1.7) holds. +Proof It is easy to see that M2 ≻ 0 from KKT +P ≻ 0 with KKT ⪰ 0 and P ⪰ 0. +Hence, We have +���M +− 1 +2 +2 +KM +− 1 +2 +1 +��� +2 += τλmax +� +KT � +γτKKT + P +�−1 +K +� += 1 +γ λmax +�� +γτKKT + P +�−1 +(γτKKT) +� +< 1 +γ λmax +�� +γτKKT + P +�−1 � +γτKKT + P +�� +≤ 4 +3, +where the first inequality is due to P ⪰ 0 but P ̸= 0, and the second one relies on +γ ≥ 3/4. The proof is completed. +⊓⊔ +Remark 4.3 Similar to (4.5), letting ˆP ∈ Rm×m be a nonzero symmetric positive +semidefinite matrix such that KTK + ˆP ≻ 0, we can choose +M1 = γσKTK + ˆP, +M2 = σ−1In, +σ > 0, +γ ≥ 3 +4 +(4.6) +such that condition (1.7) holds. Note that very recently Bai [1] considered (4.5) +and (4.6) with γ = 1 and symmetric positive definite P and ˆP. +In some problem, such as CT reconstruction in Section 5.4, g∗(y) takes a +separable structure as g∗(y) = g∗ +1(y1)+g∗ +2(y2) with y = +�y1 +y2 +� +, y1 ∈ Rm1, y2 ∈ Rm2, +in which the proximal of g1 takes a closed form solution. In this case, motivated +by [36], we can partition K as K = +�K1 +K2 +� +with K1 ∈ Rm1×n, K2 ∈ Rm2×n and +choose +M1 = 2γ +τ In, +M2 = +�σ−1Im1 +0 +0 +τK2KT +2 + P2 +� +with +τ, σ > 0. +(4.7) +We have the following result. +Proposition 4.3 Let P2 ∈ Rm2×m2 be a nonzero symmetric positive semidefinite +matrix such that K2KT +2 + P2 ≻ 0. Let τ, σ > 0. If (τσ∥K1∥2 + 1)/γ ≤ 8/3 and M1 +and M2 are chosen according to (4.7), then (1.7) holds. + +24 +Y. Ma, X. Cai, B. Jiang & D. Han +Proof It is easy to see that M2 ≻ 0 from K2KT +2 + P2 ≻ 0 with K2KT +2 ⪰ 0 and +P2 ⪰ 0. We thus have +���M +− 1 +2 +2 +KM +− 1 +2 +1 +��� +2 += τ +2γ λmax +� +σK1KT +1 + +� +τK2KT +2 + P2 +�−1/2 +K2 +� +τK2KT +2 + P2 +�−1/2� +≤ τ +2γ σ∥K1∥2 + 1 +2γ λmax +�� +τK2KT +2 + P2 +�−1 +(τK2KT +2 ) +� +< τσ∥K1∥2 + 1 +2γ +≤ 4 +3. +The proof is completed. +⊓⊔ +Remark 4.4 A particular choice in (4.7) is τ > 0, σ > 0, and τσ∥K1∥2 = 1. In this +case, Proposition 4.3 yields γ ≥ 3 +4. +4.4 A Special Case g∗ = ⟨b, y⟩ and Beyond +In this subsection, we mainly consider the case when g∗ is a linear function, for +which with choice (4.5), the y-subproblem in PrePDHG (3.4) can be efficiently +solved. Some more general cases of g∗ are also discussed at the end of this subsec- +tion. +Given a vector b ∈ Rm, we consider +g(y) = I{b} +and +g∗(y) = sup +z∈Rm{⟨z, y⟩ − g(z)} = ⟨b, y⟩ , +where I{b} is the indicator function of the singleton {b}. Hence, problem (PD) +becomes +min +x∈Rn max +y∈Rm L(x, y) := f(x) + ⟨y, Kx⟩ − ⟨b, y⟩ . +(4.8) +The recursions of PrePDHG (3.1) for (4.8) are given as + + + + + +xk+1 = proxτf +� +xk − τKTyk� +, +yk+1 = yk + (γτKKT + P)−1 � +K(2xk+1 − xk) − b +� +. +(4.9) +Remark 4.5 For g∗(y) = ⟨b, y⟩, compared with the results in Remark 3.1, we can +obtain more compact upper bounds of R(xk+1, yk+1) and R(xk+1, yk). By (3.24) +and (2.7), we have +R(xk+1, yk+1) ≤ max{∥KT(yk+1 − yk) − τ −1(xk+1 − xk)∥, ∥Kxk+1 − b∥}. +Besides, by (3.22) and (2.7), we have +R(xk+1, yk) ≤ max{∥τ −1(xk+1 − xk)∥, ∥Kxk+1 − b∥}. +(4.10) + +Understanding PrePDHG: a view of indefinite proximal ADMM +25 +We next consider two choices of P, where the y-subproblem in (4.9) is easy to +solve. +The first one is to choose γ = 1 and P = θIm for some θ > 0. Then (4.9) reduces +to the balanced ALM (BALM) [25] for solving the following convex optimization +problem +min +x∈Rn +f(x) +s.t. +Kx = b, +(4.11) +which corresponds to the primal formulation of (4.8) (see Sections 5.2 and 5.3 for +two instances of (4.11)). +BALM procedure: Let τ > 0 and θ > 0. For given (xk, yk), the new iterate +(xk+1, yk+1) is generated by: + + + + + +xk+1 = proxτf +� +xk − τKTyk� +, +yk+1 = yk + (τKKT + θIm)−1 � +K(2xk+1 − xk) − b +� +. +(4.12) +In [25], He and Yuan proved the convergence of BALM (4.12) in an elegant way +by using the framework of variational inequalities. Note that the parameters τ +and θ can be arbitrary positive constants. By applying the results in Section 3, +we obtain an enhanced BALM (eBALM), with global convergence and sublinear +convergence rate, as follows: +eBALM procedure: Let τ > 0, θ > 0 and γ ≥ 3/4. For given (xk, yk), the new +iterate (xk+1, yk+1) is generated by: + + + + + +xk+1 = proxτf +� +xk − τKTyk� +, +yk+1 = yk + γ−1(τKKT + θIm)−1 � +K(2xk+1 − xk) − b +� +. +(4.13) +Remark 4.6 Note that γ−1 is taken as 1 in (4.12) and can be any number in +(0, 4/3] in (4.13). Therefore, compared with BALM, the stepsize of y-subproblem +in eBALM can be enlarged to 4/3 from 1 Moreover, 4/3 is a tight upper bound of +γ−1 according to Lemma 3.3. +Next, we discuss the case when the inverse of the matrix in (4.13) does not take +a closed form or solving the corresponding linear system is difficult; see the earth +mover’s distance problem in Section 5.3 for instance. In this case, we can use the +block Gauss-Seidel method or the conjugate gradient method to inexactly solve the +corresponding linear system. However, the convergence issues are beyond the scope +of this paper, and we refer the readers to [28,29,36] and the reference therein for +some discussion on the inexact PDHG. As an alternative, we can adopt one block +symmetric Gauss-Seidel (sGS) iteration to solve the linear system inexactly. By the +sGS decomposition theorem developed by Li et al. [34], this approach corresponds +to taking P as a specific positive definite matrix in (4.9). More specifically, let +Q = γτKKT + θIm. Suppose that Q takes the block structure +Q = + + + +Q1,1 · · · Q1,s +... +... +... +QT +1,s · · · Qs,s + + + , + +26 +Y. Ma, X. Cai, B. Jiang & D. Han +where Qi,j ∈ Rmi×nj for 1 ≤ i, j ≤ s and Qi,i is symmetric positive definite and +its inverse is easy to compute. Note that if (γτKKT)i,i is positive definite, then θ +can be chosen to be zero. Let +U = + + + + + + +0 Q1,2 · · · +Q1,s +... +... +... Qs−1,s +0 + + + + + + +, +D = + + + + + +Q1,1 +Q2,2 +... +Qs,s + + + + + . +Suppose U ̸= 0, otherwise, the y-subproblem in (4.13) takes closed form solution +since the inverse of Qi,i is easy to compute. Taking ˜P = UD−1U T, by [34, Theorem +1], we have Q + ˜P = (D + U)D−1(D + U T) ≻ 0 and that (4.9) with +P = θIm + UD−1U T +is equivalent to the following procedure. +eBALM-sGS procedure: Let τ > 0, θ > 0, γ ≥ 3/4 and Q = γτKKT + θIm. +For given (xk, yk), the new iterate (xk+1, yk+1) is generated by: + + + + + + + + + + + + + + + + + + + + + + + + + + + +xk+1 = proxτf +� +xk − τKTyk� +, +¯bk+1 = K(2xk+1 − xk) − b, +¯yk+1 +i += yk +i + Q−1 +i,i +� +¯bk+1 +i +− +i−1 +� +j=1 +QT +j,iyk +j − +s +� +j=i+1 +Qi,j ¯yk+1 +j +� +, i = s, . . . , 2, +yk+1 +i += yk +i + Q−1 +i,i +� +¯bk+1 +i +− +i−1 +� +j=1 +QT +j,iyk+1 +j +− +s +� +j=i+1 +Qi,j¯yk+1 +j +� +, i = 1, . . . , s, +(4.14) +where ¯yk+1 +i +, yk+1 +i +∈ Rmi for 1 ≤ i ≤ s and yk+1 = +� +(yk+1 +1 +)T, · · · , (yk+1 +s +)T�T. +We name (4.14) as enhanced BALM with symmetric Gauss-Seidel iterations (eBALM- +sGS) for solving problem (4.11). By Proposition 4.2, Theorem 3.1 and Theorem +3.2, we have the following results. +Lemma 4.1 Suppose U ̸= 0. Let τ > 0, θ > 0, and γ ≥ 3/4. Then the sequence +{(xk, yk)} generated by eBALM-sGS (4.14) converges to an optimal solution of +problem (4.11). Moreover, for t ≥ 1, we have +min +1≤k≤t dist(0, ∂f(xk) + KTyk) = o +� 1 +√ +t +� +, +min +1≤k≤t dist(0, ∂f(xk) + KTyk−1) = o +� 1 +√ +t +� +, +and +min +1≤k≤t ∥Kxk − b∥ = o +� 1 +√ +t +� +. + +Understanding PrePDHG: a view of indefinite proximal ADMM +27 +If γ ≥ 1, θ > 0, the sublinear rate results are refined as +dist(0, ∂f(xt) + KTyt) = o +� 1 +√ +t +� +, +dist(0, ∂f(xt) + KTyt−1) = o +� 1 +√ +t +� +and +∥Kxt − b∥ = o +� 1 +√ +t +� +. +Remark 4.7 If the i-th block (γτKKT)i,i is positive definite for any 1 ≤ i ≤ s, +then θ > 0 in the above lemma becomes θ ≥ 0. +To end this subsection, we consider a more general scenario that g∗ takes the +block separable structure, i.e., y = +� +yT +1 , . . . , yT +s +�T and g∗(y) = �s +j=1 gj(yj). In +this case, the y-subproblem in PrePDHG (3.4) can be efficiently solved by cyclic +proximal block coordinate descent method, see [36] for more details. +5 Numerical Experiments +In this section, we present plenty of numerical results on the matrix game, pro- +jection onto the Birkhoff polytope, earth mover’s distance, and CT reconstruction +problems to verify the superiority of the larger range of the corresponding param- +eters in our PrePDHG. The codes are written in MATLAB (Release 2017b) and +run in macOS 10.15.4 on a MacBook Pro with a 2.9GHz Intel Core i7 processor +with 16GB memory. +5.1 Matrix Game +Let ∆n = {x ∈ Rn | �n +i=1 xi = 1, x ≥ 0} be the standard unit simplex in Rn. +Given a matrix K ∈ Rm×n, we consider the min-max matrix game +min +x∈∆n max +y∈∆m ⟨Kx, y⟩ . +(5.1) +This problem is a form of problem (PD) with f and g∗ chosen as the indicator +functions of ∆n and ∆m. The main iterations of PDHG (4.1) are thus given as + + + + + +xk+1 = Proj∆n +� +xk − τKTyk� +, +yk+1 = Proj∆m +� +yk + σK(2xk+1 − xk) +� +, +(5.2) +where Proj∆n(·) is the projection operator onto the simplex. For this problem, +λf +min = 0. By (4.2), the stepsizes σ > 0 and τ > 0 satisfy τσ∥K∥2 < 4/3. In +our numerical results, we consider τ = ˜τ/∥K∥ and σ = 1/(γ˜τ∥K∥) with γ ∈ +{1, 0.9, 0.85,0.751} (the requirement on γ is γ > 3/4). Note that γ = 1 corresponds +to the original PDHG method. +By Remark 3.1, we stop the algorithm when the iterations exceed 106 or +max{∥KT(yk+1 − yk) − τ −1(xk+1 − xk)∥, ∥K(xk+1 − xk) − σ−1(yk+1 − yk)∥} ≤ +10−5. The starting points are always chosen as x0 = +1 +n +�1, . . . , 1�T ∈ Rn and + +28 +Y. Ma, X. Cai, B. Jiang & D. Han +-0.7 +-0.65 +-0.6 +-0.55 +-0.5 +-0.45 +-0.4 +-0.35 +-0.3 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9 +2 +2.1 +2.2 +2.3 +iteration +104 +(a) Test 1 +-0.7 +-0.65 +-0.6 +-0.55 +-0.5 +-0.45 +-0.4 +-0.35 +-0.3 +4 +4.5 +5 +5.5 +6 +6.5 +7 +iteration +104 +(b) Test 2 +-1 +-0.95 +-0.9 +-0.85 +-0.8 +-0.75 +-0.7 +-0.65 +-0.6 +1.4 +1.6 +1.8 +2 +2.2 +2.4 +2.6 +2.8 +iteration +105 +(c) Test 3 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +4.4 +4.6 +4.8 +5 +5.2 +5.4 +5.6 +5.8 +6 +6.2 +iteration +104 +(d) Test 4 +-0.7 +-0.65 +-0.6 +-0.55 +-0.5 +-0.45 +-0.4 +-0.35 +-0.3 +-15 +-10 +-5 +0 +5 +10 +15 +20 +25 +30 +35 +ratio +(e) Test 1 +-0.7 +-0.65 +-0.6 +-0.55 +-0.5 +-0.45 +-0.4 +-0.35 +-0.3 +-5 +0 +5 +10 +15 +20 +25 +30 +ratio +(f) Test 2 +-1 +-0.95 +-0.9 +-0.85 +-0.8 +-0.75 +-0.7 +-0.65 +-0.6 +-10 +-5 +0 +5 +10 +15 +20 +25 +30 +35 +40 +ratio +(g) Test 3 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +0 +5 +10 +15 +20 +25 +ratio +(h) Test 4 +Fig. 1: Comparison of PDHG (5.2) with different values of γ for matrix game +problem (5.1). +y0 = +1 +m +�1, . . . , 1�T ∈ Rm. We follow the way in [9, 38] to generate the matrix +K. The corresponding Matlab commands are given as: i) m = 100; n = 100; A = +rand(m,n); ii) m = 100; n = 100; A = randn(m,n); iii) m = 500; n = 100; A = +10.*randn(m,n); iv) m = 1000; n = 2000; A = sprand(m,n,0.1). For each case, +we randomly generate the matrix K 20 times and report the average performance +of each algorithm. +We test a series of ˜τ ∈ 10a with a = [a1 : 0.01 : a1 + 0.4] with a1 = −0.7 +for Test 1 and Test 2, and a1 = −1.0 for Test 3 and a1 = −0.2 for Test 4. The +comparison results among different γ are reported in Figure 1, wherein the saved +ratio in terms of iteration number is defined as +ratio = +�iter − iter +iter +× 100 +� +%, +(5.3) +where the baseline iteration number “iter” is taken as the iteration number of +PDHG with γ = 1 and “iter” means the iteration number of PDHG with a chosen +γ. From these figures, we can see that PDHG with smaller γ always has better +performance than the classical PDHG with γ = 1, and for a large range of ˜τ, the +saved ratio is more than 20% for Tests 1-3 and is more than 15% for Test 4. We +also observe that the performance of PDHG with different γ might depend on the +choice of ˜τ. Therefore, to make a fair comparison, for PDHG with fixed γ, we take +the best ˜τ (in terms of the lowest iteration number), denoted by ˜τbest, from the set +10a. The comparison results are shown in Table 1. This table shows that PDHG +with smaller γ is still better than PDHG with γ = 1. For γ = 0.751, the saved +ratio is always more than 22%. Note that such improvement only needs to change +a parameter in the original PDHG without additional cost. + +Understanding PrePDHG: a view of indefinite proximal ADMM +29 +Table 1: Performance of PDHG (5.2) with best ˜τ for problem (5.1). In the table, +“a”, “b”, “c”, and “d” stands for PDHG (5.2) with γ = 1.0, γ = 0.90, γ = 0.85, +and γ = 0.751, respectively. +10 × log10(˜τbest) +time +iter +ratio % +Test +a +b +c +d +a +b +c +d +a +b +c +d +b +c +d +1 +-5.0 -4.7 -4.6 -4.0 1.6e-1 1.5e-1 1.5e-1 1.3e-1 17278 16350 15644 +14500 10.2 13.8 28.5 +2 +-4.7 -4.5 -4.4 -4.1 4.7e-1 4.4e-1 4.3e-1 4.0e-1 52040 49075 47458 +44458 14.5 17.3 27.5 +3 +-7.8 -7.9 -7.8 -7.5 2.9e1 2.6e1 2.5e1 2.3e1 202919 183475 173976 157250 12.6 14.3 36.3 +4 +-0.9 -0.8 -0.6 -0.3 1.7e1 1.6e1 1.6e1 1.4e1 56122 52226 50205 +45723 8.3 12.5 22.7 +5.2 Projection onto the Birkhoff Polytope +Given a matrix C ∈ Rn×n, computing its projection onto the Birkhoff polytope +can be formulated as +min +X∈Bn +1 +2∥X − C∥2 +F, +(5.4) +where Bn := {X ∈ Rn×n | Xen = en, XTen = en, X ≥ 0} with en ∈ Rn +being the all-one vector is known as the Birkhoff polytope. Problem (5.4) has +wide applications in solving the optimization problems involving permutations; +see [26, 34] and the references therein for more details. Let x = vec(X), problem +(5.4) can be seen as a special instance of (4.11) with f(x) = 1 +2∥x − vec(C)∥2 + +IX with X = Rn2 ++ and IX being the indicator function of the set X , and K = +�eT +n ⊗ In +In ⊗ eT +n +� +, b = e2n, where ⊗ is the Kronecker product. For such K, we have +∥K∥2 = 2n (see [21] for instance) and +� +KKT + θI2n +�−1 += +1 +n + θ I2n + +1 +2nθ + θ2 +� +n +n+θeneT +n +−eneT +−eneT +n +n +n+θ eneT +n +� +, +θ > 0. +We consider two particular choices of PrePDHG (4.9). The first one is eBALM +(4.13), whose main iterations are given as: + + + + + + + + + + + + + +Xk+1 = +1 +1 + τ Proj+ +� +Xk + τC − τ +� +yk +1eT +n + en(yk +2)T�� +, +ak+1 = eT +n(2Xk+1 − Xk)en + n + θ, +yk+1 = yk + +1 +γτ(n + θ) +� (2Xk+1 − Xk)en +(2Xk+1 − Xk)Ten +� +− +ak+1 +γτ(n + θ)(2n + θ)e2n, +(5.5) +where Proj+(·) is the projection operator over Rn×n ++ +and θ is taken as 10−4, yk +1 ∈ +Rn is the vector formulated by the first n components of yk and yk +2 ∈ Rn is the +vector formulated by the last n components of yk. The second one is PDHG (4.1) +with main iterations given as: + + + + + + + +Xk+1 = +1 +1 + τ Proj+ +� +Xk + τC − τ +� +yk +1eT +n + en(yk +2)T�� +, +yk+1 = yk + σ +� (2Xk+1 − Xk)en +(2Xk+1 − Xk)Ten +� +− σe2n. +(5.6) + +30 +Y. Ma, X. Cai, B. Jiang & D. Han +Table 2: Performance of eBALM (5.5) and PDHG (5.6) with best ˜τ for problem +(5.4). In the table, “a” and “b” stands for PDHG (5.6) with γ = 1.0 and γ = +0.751 +1+τ/2, +respectively; “c” and “d” stands for eBALM (5.5) with γ = 1.0 and γ = +0.75 +1+τ/2, +respectively. +log10(˜τbest) +time +iter +ratio % +n +a +b +c +d +a +b +c +d +a +b +c +d +b +c +d +200 0.22 0.29 0.37 0.44 1.1e-1 8.6e-2 7.5e-2 6.2e-2 +471 398 336 280 27.0 28.6 40.5 +400 0.24 0.31 0.39 0.46 2.6e-1 2.2e-1 1.8e-1 1.9e-1 +676 574 485 408 29.0 28.3 39.6 +600 0.23 0.30 0.38 0.45 5.5e-1 4.4e-1 3.7e-1 3.1e-1 +835 714 598 506 28.7 28.4 39.4 +800 0.23 0.30 0.38 0.45 9.6e-1 8.2e-1 7.0e-1 5.9e-1 1068 913 769 652 32.0 28.0 39.0 +Note that for problem (5.4), we have Σf = In2. By Lemma 3.2, we know that +the parameters τ > 0 and γ > 0 in (5.5) should satisfy γ ≥ +0.75 +1+τ/2. In our numerical +results, we consider τ = ˜τ/ +√ +2n with ˜τ > 0 and γ ∈ +� +1, +0.75 +1+τ/2 +� +. In addition, by +(4.2), the parameters τ > 0 and σ > 0 in (5.6) satisfies 2nτσ < +4 +3(1 + τ +2 ). In +our numerical results, we consider τ = ˜τ/ +√ +2n and σ = 1/(γ˜τ +√ +2n) with γ ∈ +�1, 0.751 +1+τ/2 +�. For a given n, we follow the way in [34] to randomly generate 20 +matrices C via C = rand(n); C = (C+C’)./2 and report the average performance. +The initial points are always chosen as X0 = +1 +neneT +n and y0 = 0. By (4.10), we +stop both algorithms when the relative KKT residual �Rk := max{dk, pk} ≤ 10−8 +with pk = τ −1∥Xk − Xk−1∥F and dk = +���� +� Xken − en +(Xk)Ten − en +�����. +For both algorithms, we test a series of ˜τ ∈ 10a with a = [0.2 : 0.01 : 0.6]. +The comparison results are depicted in Figure 2. In the figures (e)-(h), the “ratio” +is computed according to (5.3) with iter taken as the iteration number of PDHG +(5.6) with γ = 1, and in the figures (i)-(l), the “ratio” is computed according +to (5.3) with iter taken as the iteration number of eBALM (5.5) with γ = 1. +From these figures, we can draw the following observations. (i) Both PDHG and +eBALM benefit from choosing a larger stepsize, namely, a smaller γ. For a large +range of ˜τ, PDHG with γ = +0.751 +1+τ/2 is more than 30% faster than PDHG with +γ = 1 and eBALM with γ = +0.75 +1+τ/2 is more than 15% faster than eBALM with +γ = 1. (ii) eBALM with γ = +0.75 +1+τ/2 performs best among the four algorithms, and +it is even about more than 50% faster than the classical PDHG with γ = 1 for +˜τ ∈ 10[0.35,0.6]. +To further investigate the effect of ˜τ on the performance of different algorithms, +as done in Section 5.1, we present the performance of each algorithm with ˜τbest in +Table 2. From this table, we can see that even with the best possible parameter ˜τ, +both PDHG and BALM with a smaller γ (means the larger stepsize in updating +y) still have better performance than the corresponding algorithm with larger γ +for this problem. Besides, eBALM with γ = +0.75 +1+τ/2 has the best performance, +compared with the classical PDHG with γ = 1, it saves about 40% of iteration +numbers. + +Understanding PrePDHG: a view of indefinite proximal ADMM +31 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +iteration +(a) n = 200 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +2200 +iteration +(b) n = 400 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +500 +1000 +1500 +2000 +2500 +3000 +iteration +(c) n = 600 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +500 +1000 +1500 +2000 +2500 +3000 +3500 +iteration +(d) n = 800 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +0 +10 +20 +30 +40 +50 +60 +70 +80 +ratio +(e) n = 200 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +0 +10 +20 +30 +40 +50 +60 +70 +80 +ratio +(f) n = 400 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +0 +10 +20 +30 +40 +50 +60 +70 +80 +ratio +(g) n = 600 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +0 +10 +20 +30 +40 +50 +60 +70 +80 +ratio +(h) n = 800 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +ratio +(i) n = 200 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +ratio +(j) n = 400 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +0 +5 +10 +15 +20 +25 +30 +35 +40 +ratio +(k) n = 600 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +0 +5 +10 +15 +20 +25 +30 +35 +ratio +(l) n = 800 +Fig. 2: Comparison of eBALM (5.5) with γ = 1.0 and γ = +0.75 +1+τ/2 and PDHG (5.6) +with γ = 1.0 and γ = +0.751 +1+τ/2 for problem (5.4). +5.3 Earth Mover’s Distance +Given two discrete mass distributions ρ0 and ρ1 over the M × N grid, comput- +ing the earth mover’s distance between them can be formulated as the following +optimization problem (see [33] for instance): +min +m∈R2M×N ∥m∥1,2 +s.t. +div(m) + ρ1 − ρ0 = 0, +(5.7) +where m = +�m1 +m2 +� +is the sought flux vector on the M × N grid with m1, m2 ∈ +RM×N and m1 +M,j = 0 for j = 1, . . . , N and m2 +i,N = 0 for i = 1, . . . , M. Here, +∥m∥1,2 := �M +i=1 +�N +j=1 +� +(m1)2 +i,j + (m2)2 +i,j. The 2D discrete divergence operator +div(m) : R2M×N → RM×N is defined as +div(m)i,j = h +� +m1 +i,j − m1 +i−1,j + m2 +i,j − m2 +i,j−1 +� +, +where h is the grid stepsize, m1 +0,j = 0 for j = 1, . . . , N and m2 +i,0 = 0 for i = +1, . . . , M. Let x = +�vec(m1) +vec(m2) +� +∈ R2MN, then problem (5.7) is a form of (4.11) +with b = vec(ρ0 −ρ1) and the matrix K ∈ RMN×2MN satisfies Kx = vec(div(m)). + +32 +Y. Ma, X. Cai, B. Jiang & D. Han +1 +2 +3 +4 +5 +6 +7 +10-6 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +iteration +105 +(a) i-eBALM: itera- +tion versus τ +1 +2 +3 +4 +5 +6 +7 +10-6 +-5 +0 +5 +10 +15 +20 +25 +ratio +(b) +i-eBALM: +ratio +versus τ +1 +2 +3 +4 +5 +6 +7 +10-6 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +iteration +105 +(c) eBALM-sGS: iter- +ation versus τ +1 +2 +3 +4 +5 +6 +7 +10-6 +0 +5 +10 +15 +20 +25 +30 +ratio +(d) eBALM-sGS: ra- +tio versus τ +Fig. +3: +Comparison +on +iteration +and +ratio +of +i-eBALM +with +γ += +{1.00,0.90,0.85,0.77} and eBALM-sGS with γ = {1.00,0.90,0.85,0.75}. Note that +i-eBALM with γ = 1 is exactly iPrePDHG in [36]. +We consider two versions of PrePDHG, namely, eBALM (4.13) and eBALM- +sGS (4.14) to solve problem (5.7). For eBALM (4.13), due to the particular struc- +ture of KKT explored in [36, Section 4], we only performed two epochs of block co- +ordinate descent method as done in [36]. We name this implementation i-eBALM. +Moreover, we take θ = 0 in (4.13) since its performance is very similar to that +of very small θ. It should be mentioned that when γ = 1 in eBALM (4.13), +it becomes the iPrePDHG proposed in [36]. Note that [36] proved the conver- +gence of iPrePDHG under the strong convexity of the objective, which does not +hold for problem (5.7). Besides, the convergence of i-eBALM (4.13) remains un- +known, although it performs well. We consider four choices of γ. For eBALM +(4.13), we take γ ∈ {1.00,0.90,0.85,0.77} and for eBALM-sGS (4.14), we take +γ ∈ {1.00,0.90,0.85, 0.75}. Note that the lower bound of γ to guarantee the con- +vergence of eBALM-sGS (4.14) is 0.75, see Lemma 4.1. Actually, in our numerical +tests, eBALM-sGS (4.14) with γ = 0.749 always diverges. +For this problem, we have ∥b∥ ≈ 0.009. Therefore, we replace the term ∥Kxk+1− +b∥ in (4.10) by ∥Kxk+1 −b∥/∥b∥ and stop each algorithm when the iteration num- +ber exceeds 200,000 or the relative KKT residual +�Rk := max{dk, pk} ≤ tol := 5 × 10−5, +where pk = τ −1∥xk − xk−1∥ and dk = ∥Kxk − b∥/∥b∥. The initial x0 and y0 +are both taken as all-zero vectors. Besides, we adopt the same problem setting +in [33,36], namely, M = N = 256, h = (N − 1)/4. +The comparison results among different γ are reported in Figure 3. In this +figure, for each fixed τ ∈ {1, 1.1, . . . , 6.9, 7} × 10−6, the saved ratio in terms of +iteration number is defined as (5.3), where iter is taken as the corresponding +method with γ = 1. From the figures, we can see that both eBALM and eBALM- +sGS benefit from choosing small γ, which enlarges the stepsize in updating yk+1 +in some sense. In particular, for eBALM, when τ ≥ 4 × 10−6, the saved ratios of +taking γ = 0.77,0.85, 0.90 are about 20%, 15% and 10%, respectively. For eBALM- +sGS, when τ ≥ 3 × 10−6, the saved ratios of taking γ = 0.77,0.85,0.90 are about +25%, 15% and 10%, respectively. Besides, we also know that eBALM-sGS always +perform worse than i-eBALM, although the former has a convergence guarantee +while the latter does not. +As done in Section 5.1, we present the results corresponding to the best τ +in Figure 4 and Table 3. From them, we observe that choosing small γ can still + +Understanding PrePDHG: a view of indefinite proximal ADMM +33 +3 +3.5 +4 +4.5 +5 +5.5 +iterations +104 +10-5 +10-4 +10-3 +(a) i-eBALM: dk ver- +sus iterations +3 +3.5 +4 +4.5 +5 +5.5 +iterations +104 +10-5 +10-4 +10-3 +(b) i-eBALM: pk ver- +sus iterations +3 +3.5 +4 +4.5 +5 +5.5 +6 +6.5 +7 +7.5 +iterations +104 +10-5 +10-4 +10-3 +(c) +eBALM-sGS: +dk +versus iterations +3 +3.5 +4 +4.5 +5 +5.5 +6 +6.5 +7 +7.5 +iterations +104 +10-5 +10-4 +10-3 +10-2 +(d) +eBALM-sGS: +pk +versus iterations +Fig. 4: Comparison on pk and dk of i-eBALM with γ = {1.00,0.90,0.85,0.77} and +eBALM-sGS with γ = {1.00,0.90,0.85,0.75}. The parameter of each method is +taken as τbest in Table 3. Note that i-eBALM with γ = 1 is exactly iPrePDHG +in [36]. +Table 3: Results of i-eBALM and eBALM-sGS with best τ. +γ +τbest +time +iter +∥m∥1,2 +ratio +i-eBALM +1.00 +3.4e-6 +139.6 +52461 +0.671770 +0.00 +0.90 +3.6e-6 +131.2 +49715 +0.671770 +5.23 +0.85 +3.7e-6 +125.5 +48362 +0.671770 +7.81 +0.77 +3.9e-6 +119.2 +45990 +0.671770 +12.33 +eBALM-sGS +1.00 +2.4e-6 +166.1 +74024 +0.671770 +0.00 +0.90 +2.6e-6 +155.7 +70105 +0.671769 +5.29 +0.85 +2.6e-6 +153.4 +68241 +0.671770 +7.81 +0.75 +2.8e-6 +142.7 +63955 +0.671770 +13.60 +accelerate the corresponding method with γ = 1, and the saved ratio is always +more than 12% when we take γ = 0.77 in eBALM and γ = 0.75 in eBALM-sGS. +Again note that to achieve such improvement, we only need to change a parameter +in the original method without increasing any additional cost. We also know from +Table 3 that the saved ratios shown in this table are not as large as those in Figure +3. However, choosing the best τ from a portion of candidates is time-consuming +and impractical for both i-eBALM and eBALM-sGS. +Finally, in Figure 5, we show the solutions obtained by eBALM-sGS with +different tolerance and the ground truth obtained by running CVX in several +hours, see [36]. We can see that eBALM-sGS with tolerance tol = 5 × 10−5 can +return a solution with satisfactory precision. +5.4 CT Reconstruction +Let xtrue ∈ Rn with n = MN and M = N = 256 be a true image. Given a vector +of line-integration values b = Rxtrue ∈ Rm, where R ∈ Rm×n is a system matrix +for 2D fan-beam CT with a curved detector, the CT image reconstruction aims to +recover xtrue via solving the following optimization problem: +min +x∈Rn Φ(x) := 1 +2∥Rx − b∥2 + λ∥Dx∥1, +(5.8) + +34 +Y. Ma, X. Cai, B. Jiang & D. Han +(a) i-eBALM, tol = +5 × 10−1, mk +r = 3.6 × +10−1, iter = 842 +(b) i-eBALM, tol = +5 × 10−3, mk +r = 1.7 × +10−2, iter = 8404 +(c) i-eBALM, tol = +5 × 10−5, mk +r = 8.9 × +10−4, iter = 45990 +(d) groundtruth from +[36] +(e) eBALM-sGS, tol += 5 × 10−1, mk +r += +3.6 × 10−1, +iter += +1171 +(f) eBALM-sGS, tol += 5 × 10−3, mk +r += +1.7 × 10−2, +iter += +11729 +(g) eBALM-sGS, tol += 5 × 10−5, mk +r += +8.5 × 10−4, +iter += +63955 +(h) groundtruth from +[36] +Fig. 5: Mass distributions ρ0 and ρ1 are both with size 256×256. The white stand- +ing cat is ρ0 and the black crouching cat is ρ1. The red or blue curves are the flux +that moves the standing cat ρ0 into the crouching cat ρ1. The ground truth flux, +denoted by mcvx, is obtained by CVX after several hours. The earth mover’s dis- +tance between ρ0 and ρ1 is 0.671783. The term mk +r = ∥mk −mcvx∥/∥mcvx∥, where +mk is the flux obtained by each method. The data matrices ρ0, ρ1, and mcvx are +downloaded from https://github.com/xuyunbei/Inexact-preconditioning. +where D ∈ R2n×n is the 2D discrete gradient operator with h = 1 (see [36, Section +4] for instance) and λ = 1 is a regularization parameter. +To avoid solving the linear system related to the matrices R and D, as done +in [36] and [45], we understand problem (5.8) as a form of (P) with +f(x) = 0, +g(z) = 1 +2∥p − b∥2 + λ∥q∥1, +z = +�p +q +� +∈ Rm+2n, +K = +�R +D +� +. +We choose the variable metric matrices M1 and M2 via (4.7), wherein K1 and K2 +are R and D, respectively and σ = (τ∥R∥2)−1, P2 = θIn. The constant θ is taken +as 10−3 in our experiments. According to Remark 4.4, we have the parameter +γ ≥ 3/4. Note that the dual variable y can be decomposed as y = +�y1 +y2 +� +with +y1 ∈ Rm and y2 ∈ R2n. The main iteration scheme of PrePDHG (3.1) for solving + +Understanding PrePDHG: a view of indefinite proximal ADMM +35 +problem (5.8) is given as follows: + + + + + + + + + + + + + + + + + +xk+1 = xk − τ +2γ +� +RTyk +1 + DTyk +2 +� +, +(5.9a) +yk+1 +1 += τ∥R∥2yk +1 + R(2xk+1 − xk) − b +1 + τ∥R∥2 +, +(5.9b) +yk+1 +2 += argmin +∥y2∥∞≤λ +1 +2 +���y2 − yk +2 +��� +2 +τDDT+θI2n +− +� +y2, D(2xk+1 − xk) +� +. +(5.9c) +The y2-subproblem in (5.9c) does not take a closed-form solution. However, thanks +to the special structure of D, [36] developed an efficient block coordinate descent +(BCD) method to solve (5.9c). To guarantee the convergence of PrePDHG (5.9), +theoretically, we need to run many BCD epochs to solve (5.9c) almost exactly. +However, this may be time-consuming as observed in [36]. As suggested by [36], +we find that running two BCD steps is enough to make the PrePDHG (5.9) per- +form well. Hence, in our numerical experiments, we only apply two BCD steps +in solving the y2-subproblem. Considering that [36] has shown the superiority of +their proposed inexact preconditioned PDHG (iPrePDHG) over other variants of +PDHG, here we mainly compare our PrePDHG (5.9) with iPrePDHG therein. It +should be mentioned that iPrePDHG corresponds to our PrePDHG (5.9) with +γ = 1. For PrePDHG (5.9), we consider three versions with γ = 5/6, γ = 3/4 and +γ = 1/2, respectively. Although there is no convergence guarantee for the last one, +it performs very well in our numerical experiments. +Given a vector z = [0, ν, 2ν, . . . , 360 − ν]T containing the projection angles in +degrees, we generate a test problem by using the fancurvedtomo function from the +AIR Tools II package [19] with input N and z. In our numerical results, we consider +ν ∈ {6, 9, 12, 15, 18, 24, 30, 36}. The starting points of PrePDHG and iPrePDHG +are both taken as x0 = 0 and y0 = 0. We stop each algorithm at (xk, yk) when the +KKT residual R(xk, yk) ≤ 5 × 10−6, where R(xk, yk) is computed according to +R(xk, yk) += max +� +∥RTyk +1 + DTyk +2∥, ∥Rxk − yk +1 − b∥, dist +� +Dxk, ∂I∥y2∥∞≤λ(yk +2) +�� +. +For each fixed ν, we test a series of τ ∈ 10a with a = [−a1 : 0.02 : −a1 + 1]. +The parameter a1 is 3.5 for ν = 6, 3.2 for ν = 9, 12 or 15, 2.8 for ν = 18 or 24, and +2.3 for ν = 30 or 36. The results are presented in Figure 6. In these figures, the +term ratio is computed according to (5.3), wherein “iter” is the iteration number +corresponding to γ = 1, namely, iPrePDHG. From these figures, we can see that +taking smaller γ (meaning the larger stepsize in updating the primal variable x, +see (5.9a)) can always speed up the performance of PrePDHG. More specifically, +the saved ratios of taking γ = 3/4, the theoretical lower bound, are about 13% for +ν = 6, 9, 12 and about 25% for ν ≥ 15 for a large portion of τ. On the other hand, +although taking γ = 1/2 has no convergence guarantee since 1/2 is smaller than +the theoretical lower bound of γ, it always has the best performance for a large +portion of τ. The corresponding saved ratios are more than 25% for ν = 6, 9, 12, +more than 40% for ν = 15, and even 50% for ν ≥ 18 for a large portion of τ. +The numerical results corresponding to the best τ, denoted by τbest, for each +instance are reported in Table 4. From this table, we can see that, compared with +iPrePDHG, PrePDHG with smaller γ is always faster. For γ = 5/6, it can save + +36 +Y. Ma, X. Cai, B. Jiang & D. Han +-3.5 +-3 +-2.5 +5000 +5500 +6000 +6500 +7000 +7500 +8000 +8500 +9000 +9500 +10000 +iteration +(a) ν = 6 +-3.2 +-3 +-2.8 +-2.6 +-2.4 +-2.2 +5000 +6000 +7000 +8000 +9000 +10000 +11000 +12000 +13000 +iteration +(b) ν = 9 +-3.2 +-3 +-2.8 +-2.6 +-2.4 +-2.2 +4000 +4500 +5000 +5500 +6000 +6500 +7000 +7500 +8000 +iteration +(c) ν = 12 +-3.2 +-3 +-2.8 +-2.6 +-2.4 +-2.2 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +iteration +104 +(d) ν = 15 +-2.8 +-2.6 +-2.4 +-2.2 +-2 +-1.8 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +iteration +104 +(e) ν = 18 +-2.8 +-2.6 +-2.4 +-2.2 +-2 +-1.8 +4000 +6000 +8000 +10000 +12000 +14000 +iteration +(f) ν = 24 +-2.2 +-2 +-1.8 +-1.6 +-1.4 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +iteration +(g) ν = 30 +-2.3 +-2.2 +-2.1 +-2 +-1.9 +-1.8 +-1.7 +-1.6 +-1.5 +-1.4 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +iteration +104 +(h) ν = 36 +-3.5 +-3 +-2.5 +-10 +-5 +0 +5 +10 +15 +20 +25 +30 +35 +ratio +(i) ν = 6 +-3.2 +-3 +-2.8 +-2.6 +-2.4 +-2.2 +-5 +0 +5 +10 +15 +20 +25 +30 +35 +ratio +(j) ν = 9 +-3.2 +-3 +-2.8 +-2.6 +-2.4 +-2.2 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +ratio +(k) ν = 12 +-3.2 +-3 +-2.8 +-2.6 +-2.4 +-2.2 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +ratio +(l) ν = 15 +-2.8 +-2.6 +-2.4 +-2.2 +-2 +-1.8 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +ratio +(m) ν = 18 +-2.8 +-2.6 +-2.4 +-2.2 +-2 +-1.8 +0 +10 +20 +30 +40 +50 +ratio +(n) ν = 24 +-2.3 +-2.2 +-2.1 +-2 +-1.9 +-1.8 +-1.7 +-1.6 +-1.5 +-1.4 +0 +10 +20 +30 +40 +50 +ratio +(o) ν = 30 +-2.3 +-2.2 +-2.1 +-2 +-1.9 +-1.8 +-1.7 +-1.6 +-1.5 +-1.4 +-10 +0 +10 +20 +30 +40 +50 +ratio +(p) ν = 36 +Fig. 6: Comparison of PrePDHG with γ = {1, 5/6,3/4, 1/2} for CT reconstruction +problem (5.8). Note that PrePDHG with γ = 1 is exactly iPrePDHG in [36]. +about 8% of iteration number; for γ = 3/4, it can save about 13% of iteration +number. More interesting, PrePDHG with γ = 1/2 can save about 30% of itera- +tion number. This tells that reducing the parameter γ (in a reasonable range) in +PrePDHG for the CT reconstruction problem can still bring some benefits even +though the so-called best stepsize τ is chosen. However, it should be emphasized +again that selecting the best stepsize τ is very hard in practice. +6 Conclusions +In this paper, we investigate the PrePDHG algorithm from the iPADMM point +of view. We establish the equivalence between PrePDHG and iPADMM, based on +which we can obtain a tight convergence condition for PrePDHG. Some counter- +examples are given to show the tightness of the convergence condition we estab- + +Understanding PrePDHG: a view of indefinite proximal ADMM +37 +Table 4: Performance of PrePDHG and iPre-PDHG with best τ for CT recon- +struction problem (5.8). In the table, “a” stands for iPrePDHG, “b”, “c” and “d” +stands for PrePDHG with γ = 5/6, γ = 3/4 and γ = 1/2, respectively. +log10(τbest) +time +iter +ratio % +θ +a +b +c +d +a +b +c +d +a +b +c +d +b +c +d +6 +-2.74 -2.80 -2.84 -2.92 96.8 89.5 84.9 70.3 6441 5990 5674 4690 +7.0 11.9 27.2 +9 +-2.58 -2.62 -2.64 -2.72 76.0 69.4 66.1 53.6 6544 5932 5677 4613 +9.4 13.2 29.5 +12 -2.44 -2.50 -2.50 -2.60 54.5 49.2 47.4 37.6 5416 4866 4675 3725 10.2 13.7 31.2 +15 -2.36 -2.40 -2.42 -2.50 59.3 54.5 51.7 41.6 6456 5926 5635 4539 +8.2 12.7 29.7 +18 -2.08 -2.12 -2.14 -2.24 37.8 34.5 32.7 26.4 4393 4010 3800 3094 +8.7 13.5 29.6 +24 -2.24 -2.28 -2.30 -2.42 36.5 33.7 32.0 26.2 4673 4271 4054 3321 +8.6 13.2 28.9 +30 -1.64 -1.68 -1.70 -1.80 19.5 17.8 17.0 13.8 2655 2431 2307 1879 +8.4 13.1 29.2 +36 -1.54 -1.58 -1.62 -1.70 21.0 19.3 18.3 15.3 2954 2703 2571 2073 +8.5 13.0 29.8 +lished for PrePDHG. This result subsumes the latest convergence condition for the +original PDHG and derives an interesting by-product, namely, the dual stepsize of +the BALM can be extended to 4/3 other than 1. Besides, based on the equivalence +between PrePDHG and iPADMM, we also establish the global convergence and +the ergodic and non-ergodic sublinear convergence rate of PrePDHG. In order +to make PrePDHG practical, we also discuss the various choices of the proxi- +mal terms. A variety of numerical results on the matrix game, projection onto +the Birkhoff polytope, earth mover’s distance, and CT reconstruction show the +efficiency of PrePDHG with improved convergence conditions. Considering that +the subproblems in PrePDHG are still hard to solve in some cases, it would be +interesting to investigate the inexact version of PrePDHG in future work. +Data availability statements +The authors confirm that all data generated or analyzed during this study are in- +cluded in the paper. 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UCLA CAM Report 34, 8–34 (2008) + diff --git a/ftE1T4oBgHgl3EQfMAOS/content/tmp_files/load_file.txt b/ftE1T4oBgHgl3EQfMAOS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd72fca319d52de9a58df7f499fb6469a5a50ed7 --- /dev/null +++ b/ftE1T4oBgHgl3EQfMAOS/content/tmp_files/load_file.txt @@ -0,0 +1,2192 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf,len=2191 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='02984v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='OC] 8 Jan 2023 Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (will be inserted by the editor) Understanding the convergence of the preconditioned PDHG method: a view of indefinite proximal ADMM Yumin Ma · Xingju Cai · Bo Jiang · Deren Han Received: date / Accepted: date Abstract The primal-dual hybrid gradient (PDHG) algorithm is popular in solv- ing min-max problems which are being widely used in a variety of areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' To improve the applicability and efficiency of PDHG for different application scenarios, we fo- cus on the preconditioned PDHG (PrePDHG) algorithm, which is a framework covering PDHG, alternating direction method of multipliers (ADMM), and other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We give the optimal convergence condition of PrePDHG in the sense that the key parameters in the condition can not be further improved, which fills the theoretical gap in the-state-of-art convergence results of PrePDHG, and ob- tain the ergodic and non-ergodic sublinear convergence rates of PrePDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The theoretical analysis is achieved by establishing the equivalence between PrePDHG and indefinite proximal ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Besides, we discuss various choices of the proxi- mal matrices in PrePDHG and derive some interesting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For example, the convergence condition of diagonal PrePDHG is improved to be tight, the dual step- size of the balanced augmented Lagrangian method can be enlarged to 4/3 from Xingju Cai is supported by the NSFC grants 12131004 and 11871279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Bo Jiang is supported by the NSFC grant 11971239 and the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (21KJA110002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Deren Han is supported by the NSFC grants 2021YFA1003600, 12126603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Yumin Ma School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing, 210023, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' E-mail: mayumin@nufe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='cn Xingju Cai School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing 210023, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' E-mail: caixingju@njnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='cn Bo Jiang School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing 210023, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' E-mail: jiangbo@njnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='cn Deren Han (Corresponding author) LMIB, School of Mathematical Sciences, Beihang University, Beijing 100191, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' E-mail: handr@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='cn 2 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han 1, and a balanced augmented Lagrangian method with symmetric Gauss-Seidel iterations is also explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Numerical results on the matrix game, projection onto the Birkhoff polytope, earth mover’s distance, and CT reconstruction verify the effectiveness and superiority of PrePDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Keywords Preconditioned PDHG · Indefinite proximal ADMM · Tight convergence condition · Enhanced balanced ALM Mathematics Subject Classification (2020) 90C08 · 90C25 · 90C47 1 Introduction In this paper, we consider the convex-concave min-max problem: min x∈Rn max y∈Rm L(x, y) := f(x) + ⟨Kx, y⟩ − g∗(y), (PD) where K ∈ Rm×n, f : Rn → (−∞, +∞], and g : Rm → (−∞, +∞] are proper closed convex functions, g∗ is the convex conjugate of g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=', g∗(y) = supz∈Rm{⟨z, y⟩− g(z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Here ⟨·, ·⟩ denotes the standard inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The primal and dual formu- lations of problem (PD) are, respectively, given as min x∈Rn f(x) + g(Kx) (P) and min y∈Rm f ∗(−KTy) + g∗(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (D) Such problems have wide applications in matrix completion [4], image denoising [7,44], compressed sensing [20], earth mover’s distance [33], computer vision [41], CT reconstruction [45], magnetic resonance imaging [46], robust face recognition [47] and image restoration [50], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' An efficient method to solve (PD) is the primal-dual hybrid gradient (PDHG) algorithm which was originally proposed by Zhu and Chan [50] and further de- veloped by Chambolle and Pock [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The recursion of the PDHG for (PD) reads as: PDHG procedure for (PD): Let τ > 0 and σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For given (xk, yk), the new iterate (xk+1, yk+1) is generated by: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 xk+1 = argmin x∈Rn f(x) + � Kx, yk� + 1 2τ ���x − xk��� 2 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1a) yk+1 = argmin y∈Rm g∗(y) − � K(2xk+1 − xk), y � + 1 2σ ∥y − yk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1b) Here, ∥ · ∥ means the vector ℓ2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1), τ, σ > 0 are the primal and dual stepsize parameters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Chambolle and Pock [7] and He and Yuan [24] established the convergence of PDHG under the condition τσ∥K∥2 < 1, in which ∥K∥ is the spectral norm of the matrix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' This condition is improved to τσ∥K∥2 ≤ 1 by Condat [13] and further enhanced to τσ∥K∥2 < 4 3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) Understanding PrePDHG: a view of indefinite proximal ADMM 3 very recently by He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' [21] for a special case of (PD), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=', g∗(y) = ⟨b, y⟩ and b ∈ Rm, other than general g∗(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Under the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2), the convergence of PDHG (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) for (PD) with general g∗(·) is established in [30] and [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For more results about the convergence of PDHG, readers can refer to [6,23,28,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' As observed in [40] that for cases when ∥K∥ may not be estimated easily, or it might be very large, the practical convergence of the PDHG (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) significantly slows down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' To overcome this issue, we are concerned in this paper with a general algorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=', the preconditioned PDHG (PrePDHG), which is given as: PrePDHG procedure for (PD): Let M1 ∈ Rn×n and M2 ∈ Rm×m be sym- metric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For given (xk, yk), the new iterate (xk+1, yk+1) is generated by: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 xk+1 = argmin x∈Rn f(x) + � Kx, yk� + 1 2 ���x − xk��� 2 M1 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3a) yk+1 = argmin y∈Rm g∗(y) − � K(2xk+1 − xk), y � + 1 2∥y − yk∥2 M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3b) Here, ∥z∥2 M1 = ⟨z, M1z⟩ for a vector z ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Obviously, the PrePDHG (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) re- duces to the PDHG (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) by taking M1 = τ −1In and M2 = σ−1Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' More impor- tantly, by taking other specific forms of M1 and M2, the framework of PrePDHG can take several other algorithms as special cases, see Section 4 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The PrePDHG is first proposed by Pock and Chambolle [40]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' They established the convergence of the PrePDHG (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) under the condition � M1 −KT −K M2 � ≻ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) see [40, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For a symmetric positive matrix M, denote M − 1 2 as the square root of M −1, namely, M −1/2M −1/2 = M −1, then condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) is equivalent to (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) M1 ≻ 0, M2 ≻ 0, ���M − 1 2 2 KM − 1 2 1 ��� < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) Besides, [40] also proposed a family of diagonal preconditioners for M1 and M2, which make the subproblems easier to solve and guarantee the convergence of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' From the point view of an indefinite proximal point algorithm, Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' [27] showed that the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) can be improved to M1 + 1 2Σf ≻ 0, M2 + 1 2Σg∗ ≻ 0, ���� � M2 + 1 2Σg∗ �− 1 2 K � M1 + 1 2Σf �− 1 2 ���� < 1, where Σf and Σg∗ are symmetric semidefinite matrices related to f and g∗ (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Since min-max problems are equivalent to constrained or composite optimiza- tion problems under certain conditions, some literatures focus on understanding 1 The setting in [40] is for the general finite-dimensional vector space other than the Eu- clidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For simplicity of presentation, we focus on the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' However, our results in this paper can be easily extended to the general finite-dimensional vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 4 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han PDHG and PrePDHG from various perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For example, the equivalence be- tween PDHG and linearized alternating direction method of multipliers (ADMM) is discussed in [14, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Similarly, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' [36] established the equivalence be- tween PrePDHG for (PD) and positive semidefinite proximal ADMM (sPADMM) for an equivalent problem of (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Based on the equivalence and the convergence analysis of the first-order primal-dual algorithm in [8], Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' [36] established the ergodic convergence result (but without sequence convergence) of PrePDHG under the condition � M1 −KT −K M2 � ⪰ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) and also considered some inexact versions of PrePDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that a similar condi- tion of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) is extended for infinite dimensional Hilbert space in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Very recently, under condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6), Jiang and Vandenberghe [31] showed convergence of iterates for Bregman PDHG, of which PrePDHG is a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' As mentioned above, when M1 = τ −1In and M2 = σ−1Im, the PrePDHG (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) reduces to the original PDHG (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' However, the convergence condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) degrades into τσ∥K∥2 ≤ 1 other than (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' This raises a natural question: can we obtain a tighter convergence condition of PrePDHG to fill this gap?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Motivated by [36], we intend to investigate PrePDHG from the perspective of proximal ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' A known result is that indefinite proximal ADMM (iPADMM), with weaker convergence conditions, outperforms positive semidefinite proximal ADMM (sPADMM) [5, 10, 11, 17, 18, 22, 32, 37, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In this paper, we restudy the PrePDHG (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) from the point view of iPADMM other than sPADMM as done in [36] and give positive answers to the above question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The main contributions of this paper are as follows: Firstly, we establish the equivalence between PrePDHG for (PD) and iPADMM for an equivalent problem of (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Based on the equivalence, we improve the con- vergence condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) of the PrePDHG to � 4 3 � M1 + 1 2Σf � KT K M2 � ≻ 0, which can be rewritten as (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) M1 + 1 2Σf ≻ 0, M2 ≻ 0, ����M − 1 2 2 K � M1 + 1 2Σf �− 1 2 ���� 2 < 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) Note that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) is exactly (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) when PrePDHG reduces to the original PDHG and Σf is taken as a zero matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Some counter-examples are given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 to illustrate that condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) is tight in the sense that the constants 4/3 and 1/2 can not be replaced by any larger numbers, namely, the inequality sign “<” can not be replaced by “≤”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Secondly, we establish the ergodic and non-ergodic sublinear convergence rate results of the PrePDHG both in the sense of the KKT residual and the function value residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' To the best of our knowledge, the sublinear convergence rate based on the KKT residual is new for PDHG-like methods since the existing results mainly focus on the function value residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' And for the function value residual measurement, our sublinear rate result is the first non-ergodic result since the existing results are all ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The numerical experiments in Section 5 show that the KKT residual is more practical than the function value residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Understanding PrePDHG: a view of indefinite proximal ADMM 5 Thirdly, we discuss some practical choices of M1 and M2 and get some in- teresting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For example, condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) is tight for PDHG (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' the sharp range of parameters for diagonal PrePDHG is given, and the dual stepsize of the balanced ALM (BALM) [25] can be enlarged to 4/3 from 1, and we rename it an enhanced BALM (eBALM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Besides, we explore the eBALM with symmetric Gauss-Seidel iterations (eBALM-sGS), which can be understood as a special case of PrePDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Finally, we perform four groups of numerical experiments on solving the ma- trix game, projection onto the Birkhoff polytope, earth mover’s distance, and CT reconstruction problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We choose proper M1 and M2 and the numerical results verify the effectiveness of the choices of M1 and M2 and the superiority of the PrePDHG (with tighter convergence condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Some notations and preliminaries are pre- sented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In Section 3, we first establish the equivalence between PrePDHG and iPADMM and then develop the global convergence of PrePDHG from the iPADMM point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The existing convergence condition of PrePDHG is im- proved to be tight, as shown by counter-examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Then, the sublinear convergence rate of the PrePDHG is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We revisit the choices of M1 and M2 in Section 4 and get some new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In Section 5, we perform numerical experiments on four practical problems to verify the effectiveness of the PrePDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Some concluding remarks are made in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 2 Notations and Preliminaries We use ∥x∥1, ∥x∥, and ∥x∥∞ to denote the ℓ1, ℓ2 and ℓ∞ norm of the vector x respectively, and ∥A∥ to denote the spectral norm of the matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We use vec(A) to denote a vector formulated by stacking the columns of A one by one, from first to last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We slightly abuse the notation ∥x∥2 M := ⟨x, Mx⟩ as long as M is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' When M is symmetric positive semidefinite, we use M 1 2 to represent the square root of M, namely, M 1 2 M 1 2 = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For symmetric matrices A and B, A ⪰ (≻) B means that A − B is positive semidefinite (positive definite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For a symmetric matrix P ∈ Rn×n, we can always decompose it as P = P+ − P− with P+, P− ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We name this decomposition a DC decomposition of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that the DC decomposition of a symmetric matrix is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We adopt some standard notations in convex analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' see [43] for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The distance from a point x to a nonempty convex closed set S ⊆ Rn is denoted as dist(x, S) = miny∈S ∥y − x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For any proper closed convex function f : Rn → (−∞, +∞] and ¯x ∈ domf := {x ∈ Rn | f(x) < +∞}, the subdifferential at ¯x is defined as ∂f(¯x) := {ξ ∈ Rn | f(x) ≥ f(¯x) + ⟨ξ, x − ¯x⟩ , ∀x ∈ Rn}, in which any ξ is a subgradient at ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Moreover, there exists a symmetric positive semidefinite matrix Σf such that for all x1, x2 ∈ Rn and ξ1 ∈ ∂f(x1), ξ2 ∈ ∂f(x2), ⟨ξ1 − ξ2, x1 − x2⟩ ≥ ∥x1 − x2∥2 Σf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) For any proper closed convex function f, the convex conjugate of f is defined as f ∗(y) := supx∈Rn{⟨x, y⟩ − f(x)}, and we have y ∈ ∂f(x) ⇔ x ∈ ∂f ∗(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) 6 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han Given a symmetric matrix M with M + Σf ≻ 0, we define the generalized proximal operator as proxM f (x) := argmin z∈Rn f(z) + 1 2∥z − x∥2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) If M = τ −1In for some τ > 0, we simply denote proxτf (x) := proxM f (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Let ˜f(·) := f(·) − 1 2∥ · ∥2 Σf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Observing that f(z) + 1 2∥z − x∥2 M = ˜f(z) + 1 2 ���z − (M + Σf)−1Mx ��� 2 M+Σf + 1 2∥x∥2 M − 1 2∥(M + Σf)−1Mx∥2 M+Σf , we have an equivalent characterization of proxM f (x) as proxM f (x) = proxM+Σf ˜ f � (M + Σf)−1 Mx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) We now present a generalization of Moreau’s identity, see [12, Theorem 1 (ii)] or [2, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3], which is very useful in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 Let f : Rn → (−∞, +∞] be a proper closed convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Suppose M ≻ 0, then we have x = proxM f (x) + M −1proxM−1 f∗ (Mx), ∀x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In the following lemma, the equivalence between (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In [40], the authors proved that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) implies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Here we present a simple proof of the equivalence based on the well-known Schur complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 Let M1 ∈ Rn×n, M2 ∈ Rm×m be symmetric matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) is equivalent to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Proof By [48, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='12], we know that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) is equivalent to M1 ≻ 0, M2 ≻ 0, M1 − KTM −1 2 K ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Since M1 ≻ 0 and M2 ≻ 0, we have M1 − KTM −1 2 K ≻ 0 ⇐⇒ In − M − 1 2 1 KTM − 1 2 2 M − 1 2 2 KM − 1 2 1 ≻ 0 ⇐⇒ ∥M − 1 2 2 KM − 1 2 1 ∥ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' ⊓⊔ Throughout this paper, we assume that problem (PD) has a saddle point (x⋆, y⋆), which satisfies the optimality condition L(x⋆, y) ≤ L(x⋆, y⋆) ≤ L(x, y⋆), ∀x ∈ Rn, ∀y ∈ Rm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) and the KKT-type optimality condition 0 ∈ ∂f(x⋆) + KTy⋆, 0 ∈ ∂g∗(y⋆) − Kx⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) Understanding PrePDHG: a view of indefinite proximal ADMM 7 Such x⋆ and y⋆ are also optimal for (P) and (D), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Define the KKT residual mapping R : Rn × Rm → R as R(x, y) = max � dist(0, ∂f(x) + KTy), dist(0, ∂g∗(y) − Kx) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) Clearly, we have the following equivalent characterization of the optimality condi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 The KKT-type optimality condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) holds if and only if R(x⋆, y⋆) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Based on this, we define the ǫ-solution of problem (PD) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 Given ǫ ≥ 0, a pair (x, y) is called an ǫ-solution of problem (PD) if R(x, y) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that the KKT residual (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) may be difficult or expensive to calculate since it involves computing the distance of a point to a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' However, in some practical circumstances, the upper bound of R(x, y) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) could be easily obtained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' see the discussion in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In the rest of this section, we present the existing convergence and sublinear convergence rate results of iPADMM developed in [17], which are key to the con- vergence analysis of PrePDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that the algorithm in [17] is more general and takes iPADMM as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Here we display the corresponding results of iPADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Consider the convex minimization problem with linear constraints and a sep- arable objective function min x∈Rn1,y∈Rn2 θ1(x) + θ2(y) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ax + By = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8) where A ∈ Rm×n1, B ∈ Rm×n2, θ1 : Rn1 → (−∞, +∞], and θ2 : Rn2 → (−∞, +∞] are proper closed convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The augmented Lagrangian function of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8) is defined by: Lβ(x, y, λ) = θ1(x) + θ2(y) − ⟨λ, Ax + By⟩ + β 2 ∥Ax + By∥2, where λ is the corresponding Lagrange multiplier of the linear constraints and β > 0 is a penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The iPADMM for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8) in [17] is given as: iPADMM procedure for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8): Choose the symmetric indefinite matrices S and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For given (xk, yk, λk), the new iterate (xk+1, yk+1, λk+1) is generated by: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 xk+1 = argmin x∈X Lβ(x, yk, λk) + 1 2∥x − xk∥2 S, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9a) yk+1 = argmin y∈Y Lβ(xk+1, y, λk) + 1 2∥y − yk∥2 T , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9b) λk+1 = λk − β(Axk+1 + Byk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9c) Let Σ1 and Σ2 be the symmetric positive semidefinite matrices related to θ1 and θ2, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The sequence {(xk, yk, λk)} is denoted as {wk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Now we present the convergence results of iPADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 8 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 [17, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2] Let the sequence {wk} be generated by iPADMM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' If the proximal terms S and T are chosen such that S + 1 2Σ1 ⪰ 0, S + 1 2Σ1 + βATA ≻ 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='10) and T + Σ2 + βBTB ≻ 0, T + 1 2Σ2 + κ1(−2T− + Σ2) + κ2βBTB ≻ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='11) where κ1 = 1, κ2 ∈ (0, 3 4), and T− comes from one DC decomposition of T, then {wk} converges to an optimal solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 [17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1] Let the sequence {wk} be generated by iPADMM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' If the proximal terms S and T are chosen such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='11) hold, and S + 1 2Σ1 ⪰ c 2Σ1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='12) with c > 0, then we have min 1≤i≤k ∥wi − wi+1∥2 ˆ G = o(1/k), in which ˆG = \uf8eb \uf8ed S + Σ1 T + Σ2 + βBTB 1 β Im \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 [17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2] Let the sequence {wk} be generated by iPADMM (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' If the proximal terms S and T are chosen such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='11), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='12) hold, and T + 1 2Σ2 ≻ 0, then we have ∥wk − wk+1∥2 ˆ G = o(1/k), where ˆG is defined in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 3 The Preconditioned PDHG and its Convergence We first present the PrePDHG with practical stopping criterion for convex-concave min-max optimization (PD) in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We shall first establish an equivalence between PrePDHG and iPADMM, which is key to analyzing the algorithm, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 and deduce the global convergence of Algorithm 1 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 provides counter-examples to show the tightness of condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The sublinear convergence rate in both ergodic and non-ergodic sense is investigated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The PrePDHG is given in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that the stopping criterion R(xk+1, yk+1) ≤ ǫ can be replaced by R(xk+1, yk) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Understanding PrePDHG: a view of indefinite proximal ADMM 9 Algorithm 1: PrePDHG: Preconditioned PDHG for solving (PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 1 Initialization: Choose the initial points x0 ∈ Rn, y0 ∈ Rm, and set the tolerance ǫ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Choose the matrices M1 ∈ Rn×n and M2 ∈ Rm×m satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 2 for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , do 3 Update xk+1 and yk+1 as follows: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 xk+1 = argmin x∈Rn f(x) + � Kx, yk� + 1 2 ���x − xk��� 2 M1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1a) yk+1 = argmin y∈Rm g∗(y) − � K(2xk+1 − xk), y � + 1 2 ∥y − yk∥2 M2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1b) 4 if R(xk+1, yk+1) ≤ ǫ then 5 break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 6 end 7 end Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='24) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='26), we have R(xk+1, yk+1) ≤ ˆR(xk+1, yk+1) with ˆR(xk+1, yk+1) := max � ���KT(yk+1 − yk) − M1(xk+1 − xk) ��� , ���K(xk+1 − xk) − M2(yk+1 − yk) ��� � , which can be easily computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Therefore, if R(xk+1, yk+1) is difficult to com- pute, we can use the stopping criterion ˆR(xk+1, yk+1) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Similarly, by the first inequalities in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='30) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='31), we can also replace R(xk+1, yk) by its upper bound as ˆR(xk+1, yk) := max � ���M1(xk+1 − xk) ��� , ���K(xk − xk−1) + K(xk − xk+1) − M2(yk − yk−1) ��� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that for some special cases, such as g∗ is a linear function, a more compact upper bound of R(xk+1, yk+1) or R(xk+1, yk) can be obtained, see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 Equivalence of PrePDHG and iPADMM We first show that PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) can be understood as an iPADMM applied on the equivalent formulation of problem (P): min x∈Rn, u∈Rm g(u) + f(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' M − 1 2 2 (Kx − u) = 0, (P1) where M2 ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Let L1(u, x, λ) = g(u) + f(x) + � λ, M − 1 2 2 (Kx − u) � + 1 2∥Kx − u∥2 M−1 2 10 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han be the augmented Lagrangian function of problem (P1), where λ is the correspond- ing Lagrange multiplier of the linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Given the initial points x0 ∈ Rn and λ0 ∈ Rm, the main iterations of the iPADMM are given as \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 uk+1 = argmin u∈Rm L1(u, xk, λk), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2a) xk+1 = argmin x∈Rn L1(uk+1, x, λk) + 1 2∥x − xk∥2 M1−KTM−1 2 K, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2b) λk+1 = λk + M − 1 2 2 (Kxk+1 − uk+1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2c) where the proximal matrix M1 − KTM −1 2 K could be indefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2), there is only an additional proximal term in the second subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Using the notations of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4), we can equivalently formulate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) as \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 uk+1 = proxM−1 2 g � M 1 2 2 λk + Kxk� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3a) xk+1 = proxM1+Σf ˜ f � (M1 + Σf)−1M1xk − (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3b) (M1 + Σf)−1KTM −1 2 � M 1 2 2 λk + Kxk − uk+1�� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3c) λk+1 = λk + M − 1 2 2 (Kxk+1 − uk+1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3d) where ˜f(·) := f(·) − 1 2∥ · ∥2 Σf is defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Similarly, we can reformulate the iterations of PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) as \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 xk+1 = proxM1+Σf ˜ f � (M1 + Σf)−1M1xk − (M1 + Σ1)−1KTyk� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4a) yk+1 = proxM2 g∗ � yk + M −1 2 K(2xk+1 − xk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4b) We are now ready to deduce the equivalence between PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) and iPADMM (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) (or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4)) and iPADMM (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) (or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3)) are equiv- alent in the sense that the sequence generated by either algorithm can explicitly recover the sequence generated by the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Proof Let the sequence {(uk, xk, λk)} be generated by iPADMM (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) with initial points x0 ∈ Rn and λ0 ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Consider the transform yk = M −1 2 � M 1 2 2 λk + Kxk − uk+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) First, substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3c) yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1, we have from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3a) that M 1 2 2 λk + Kxk = uk+1 + M2proxM2 g∗ � M − 1 2 2 λk + M −1 2 Kxk� , which with the transform (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) implies yk = proxM2 g∗ � M − 1 2 2 λk + M −1 2 Kxk� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' This also tells yk+1 = proxM2 g∗ � M − 1 2 2 λk+1 + M −1 2 Kxk+1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) Understanding PrePDHG: a view of indefinite proximal ADMM 11 Besides, with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3d) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5), we have M − 1 2 2 λk+1 = M −1 2 K(xk+1 − xk) + yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Substituting this relation into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Now we can conclude that the sequence {(xk, yk)} is exactly the sequence generated by PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) with initial points x0 and y0 = M −1 2 (M 1 2 2 λ0 + Kx0 − u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' On the other hand, let the sequence {(xk, yk)} be generated by PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) with given initial points x0 ∈ Rn and y0 ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Consider the transforms λk+1 = M − 1 2 2 K(xk+1 − xk) + M 1 2 2 yk, uk+1 = M 1 2 2 λk + Kxk − M2yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Using a similar argument, we can show that {(uk, xk, λk)} is exactly the same sequence generated by iPADMM (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) and the initial points of x and λ are taken as x0 and M − 1 2 2 K(x1 − x0) + M 1 2 2 y0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We omit the details for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' ⊓⊔ ⊓⊔ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 If M1 = KTM −1 2 K, then iPADMM (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) reduces to the classical ADMM [15,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In this case, PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) is equivalent to the classical ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Based on the key observation that PrePDHG and iPADMM are equivalent, we next investigate the convergence of PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1), namely, Algorithm 1, via the well-established convergence results of iPADMM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' see [11, 17, 22, 32, 37, 49] for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Here, we mainly use the global and sublinear convergence rate results developed in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' It should be mentioned that Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' [36] also showed that PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) is equivalent to a proximal ADMM applied on the equivalent formulation of dual problem (D) as: min y∈Rm, v∈Rn g∗(y) + f ∗(v) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' M − 1 2 1 (KTy + v) = 0, where they require (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The recursion of the proximal ADMM therein is given as \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 yk+1 = argmin y∈Rm �L1(y, vk, λk) + 1 2∥y − yk∥M2−KM−1 1 KT, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7a) vk+1 = argmin v∈Rn �L1(yk+1, v, λk), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7b) ˜λk+1 = ˜λk + M − 1 2 1 (KTyk+1 + vk+1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7c) where �L1(y, v, ˜λ) = g∗(y) + f ∗(v) + � λ, M − 1 2 1 (KTy + v) � + 1 2∥KTy + v∥2 M−1 1 , in which ˜λ is the corresponding Lagrange multiplier of the linear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' A main difference between (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) lies in that the proximal term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) is in the second subproblem other than in the first subproblem as done by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' It is this key point that makes our condition on M1 and M2 weaker than that in [36] since the iPADMM can always allow more indefiniteness of the proximal term in the second subproblem other than that in the first subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 12 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 Global Convergence It is clear that condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) implies M1 − KTM −1 2 K ⪰ 0, which further means that the proximal matrix in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2b) is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' However, the well- explored convergence results of iPADMM tell that the proximal matrix M1 − KTM −1 2 K could be indefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Therefore, we could further improve the convergence condition of PrePDHG from the perspective of iPADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 Suppose condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) holds, that is, M1 + 1 2Σf ≻ 0, M2 ≻ 0, ����M − 1 2 2 K � M1 + 1 2Σf �− 1 2 ���� 2 < 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Then the sequence generated by iPADMM (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) converges to an optimal solution of (P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Proof Let the sequence {(uk, xk, λk)} be generated by iPADMM (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3, we take n1 := m, n2 := n, θ1 := g, θ2 := f, A := M −1/2 2 , B := −M −1/2 2 K, β = 1, S := 0, T := M1 − KTM −1 2 K, Σ1 := 0, Σ2 := Σf, and the parameter κ2 := 1 − ρ with ρ ∈ ( 1 4, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Then we immediately know that {(uk, xk, λk)} converges to an optimal solution of (P1) as long as there exists a DC decomposition of M1 − KTM −1 2 K and ρ ∈ (1/4,1) such that M1 +Σf ≻ 0, H := M1 + 3 2Σf −2 � M1 − KTM −1 2 K � − −ρKTM −1 2 K ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8) Now we only need to show the correctness of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8) under the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' If M1−KTM −1 2 K ⪰ 0, we can take its DC decomposition as (M1−KTM −1 2 K)+ = M1 − KTM −1 2 K and (M1 − KTM −1 2 K)− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Hence, for any ρ ∈ ( 1 4, 1), we have H = M1 + 3 2Σf − ρKTM −1 2 K = ρ(M1 − KTM −1 2 K) + (1 − ρ)M1 + 3 2Σf ≻ 0, where the last inequality is due to M1 + 1 2Σf ≻ 0 which comes from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Now, suppose M1 − KTM −1 2 K ̸⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) and the Schur complement theo- rem, we have 4 3 � M1 + 1 2Σf � ≻ KTM −1 2 K, namely, M1 − KTM −1 2 K + 1 3 (M1 + 2Σf) ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Let M := (M1 + 2Σf)− 1 2 � M1 − KTM −1 2 K � (M1 + 2Σf)− 1 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9) then obviously we have M + 1 3In ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Set M = UΣU T be the eigenvalue decompo- sition of M with U TU = UU T = In and the diagonal matrix Σ = diag(σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , σn) with σ1 ≥ · · · ≥ σp ≥ 0 > σp+1 ≥ · · · ≥ σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Then we have σn ∈ (− 1 3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Consider a DC decomposition of M as M+ = U max(0, Σ)U T and M− = U max(0, −Σ)U T, where the max-operator max(·, ·) takes the maximum of the two matrices entry-wisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' It is clear that M− ≺ |σn|In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9), we thus obtain a DC decomposition of M1 − KTM −1 2 K as (M1 − KTM −1 2 K)+ = (M1 + 2Σf) 1 2 M+ (M1 + 2Σf) 1 2 Understanding PrePDHG: a view of indefinite proximal ADMM 13 and (M1 − KTM −1 2 K)− = (M1 + 2Σf) 1 2 M− (M1 + 2Σf) 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='10) Choosing ρ = 1−2|σn| 1+|σn| ∈ ( 1 4, 1), with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='10) and M− ≺ |σn|In, we thus have (2 + ρ)(M1 − KTM −1 2 K)− ≺ (2 + ρ)|σn|(M1 + 2Σf) = (1 − ρ)(M1 + 2Σf) ⪯ (1 − ρ)M1 + 3 2Σf + ρ�M1 − KTM −1 2 K� +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Substituting � M1−KTM −1 2 K � + = M1−KTM −1 2 K + � M1−KTM −1 2 K � − into the above assertion, by some easy calculations, we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' ⊓⊔ Now we are ready to establish the convergence of PrePDHG (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 Let {(xk, yk)} be the sequence generated by Algorithm 1 with ǫ = 0 and M1, M2 satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Then {(xk, yk)} converges to a saddle point of (PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Proof Let the sequence {(uk, xk, λk)} be generated by iPADMM (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Since M1 and M2 satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7), we know from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 that {(uk, xk, λk)} converges to an optimal solution (u⋆, x⋆, λ⋆) of (P1), namely, 0 ∈ ∂f(x⋆) + KTM − 1 2 2 λ⋆, 0 ∈ ∂g(u⋆) − M − 1 2 2 λ⋆, Kx⋆ − u⋆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='11) Recalling the transform (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5), we know from the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 that {(xk, yk)} is exactly the sequence generated by PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) with x0 and y0 = M −1 2 (M 1 2 2 λ0+ Kx0−u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Since {(uk, xk, λk)} converges to (u⋆, x⋆, λ⋆), we know from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='11) that xk → x⋆ and yk → y⋆ := M − 1 2 2 λ⋆ and 0 ∈ ∂f(x⋆) + KTy⋆, 0 ∈ ∂g(Kx⋆) − y⋆, which with the fact that g is proper closed convex and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) shows 0 ∈ ∂f(x⋆) + KTy⋆, 0 ∈ ∂g∗(y⋆) − Kx⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' This means that (x⋆, y⋆) is a saddle point of (PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' ⊓⊔ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 Tightness of Condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) We first claim that condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) is tight in the sense that the constant “4/3” can not be replaced by any number larger than it, namely, the sign “<” can not be improved to “≤”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 Let {(xk, yk)} be the sequence generated by Algorithm 1 with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Suppose condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) is replaced by M1 + 1 2Σf ≻ 0, M2 ≻ 0, ����M − 1 2 2 K � M1 + 1 2Σf �− 1 2 ���� 2 ≤ ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='12) (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' If ρ1 ∈ (0, 4/3), then {(xk, yk)} converges to a saddle point of (PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' If ρ1 ≥ 4/3, then {(xk, yk)} is not necessarily convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 14 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han Proof The assertion of (a) comes from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 and the fact that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) is true if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='12) holds for any fixed ρ1 ∈ (0, 4/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' To prove (b), consider a simple instance of problem (PD) as min x∈R max y∈R xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13) Note that such an example is a special case of the one in [35, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2] by setting n = m = 1 and A = 1 therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' It is easy to see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13) has a unique saddle point (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For this problem, Σf = 0, K = 1, and M1, M2 take the form of M1 = 1/τ, M2 = 1/σ with τ, σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In this case, condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='12) becomes τ, σ > 0 and τσ ≤ ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We next show that if τ > 0, σ > 0, τσ = 4 3, �x0 y0 � ̸∈ S := � a �2 σ � : a ∈ R � , then the sequence generated by PrePDHG diverges, which is enough to finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Specifically, by some easy calculations, the PrePDHG recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) for prob- lem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13) reads as � xk+1 = xk − τyk, yk+1 = σxk + (1 − 2τσ) yk, which can be reformulated as �xk+1 yk+1 � = G �xk yk � with G := �1 −τ σ 1 − 2τσ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='14) Since τσ = 4/3, it is easy to verify that the two eigenvalues of G is −1 and 1/3 and G = V �1/3 0 0 −1 � V −1 with V = �2/σ τ/2 1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='15) We have from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='15) that �xk+1 yk+1 � = Gk+1 �x0 y0 � = V �3−k 0 0 (−1)k � V −1 �x0 y0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' It is obvious that ��xk+1 yk+1 �� is convergent ⇐⇒ V −1 �x0 y0 � = �a 0 � for some a ∈ R ⇐⇒ �x0 y0 � ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Hence, if �x0 y0 � ̸∈ S, then ��xk+1 yk+1 �� diverges and certainly will not converge to �0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' ⊓⊔ We next claim that condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) is tight in the sense that the constant “1/2” can not be replaced by any number larger than it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Understanding PrePDHG: a view of indefinite proximal ADMM 15 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 Let {(xk, yk)} be the sequence generated by Algorithm 1 with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Suppose condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) is replaced by M1 + ρ2Σf ≻ 0, M2 ≻ 0, ���M − 1 2 2 K (M1 + ρ2Σf)− 1 2 ��� 2 < 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='16) (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' If ρ2 ∈ (0, 1/2], then {(xk, yk)} converges to a saddle point of (PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' If ρ2 > 1/2, then {(xk, yk)} is not necessarily convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Proof The assertion of (a) comes from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 and the fact that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) is true if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='16) holds for any fixed ρ2 ∈ (0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' To prove (b), consider a simple instance of problem (PD) as min x∈R max y∈R 1 2x2 + xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='17) It is easy to see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='17) has a unique saddle point (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For this problem, Σf = 1, K = 1, and M1, M2 take the form of M1 = 1/τ,M2 = 1/σ with 1/τ + ρ2 > 0, σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In this case, condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='16) becomes 0 < σ < 4 3(1/τ + ρ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We only need to show that for any ρ2 ∈ (1/2,1] and ρ3 ∈ (1/2,ρ2) if 0 < σ = 4 3(1/τ + ρ3) < 4 3(1/τ + ρ2), then the sequence generated by PrePDHG is not necessarily convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' First, it is not hard to verify that the PrePDHG recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) for problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='17) reads as \uf8f1 \uf8f2 \uf8f3 xk+1 = (xk − τyk)/(1 + τ), yk+1 = � σ(1 − τ)xk + (1 + τ − 2τσ) yk� /(1 + τ), which can be reformulated as �xk+1 yk+1 � = �G �xk yk � with �G := 1 1 + τ � 1 −τ σ(1 − τ) 1 + τ − 2τσ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='18) The characteristic polynomial of �G is given as p(µ) = µ2 − 1 1 + τ � τ − 2(1 + 4ρ3τ) 3 � µ − 1 + 4ρ3τ 3(1 + τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Noting 1+τ τ ≥ 1 τ + ρ2 > 0, we have p(−1) = 2(1 − 2ρ3)τ 1 + τ < 0, which tells that at least one eigenvalue of �G is less than −1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=', ∥ �G∥ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' There- fore, the sequence {(xk, yk)} generated by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='18) is not necessarily convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' ⊓⊔ 16 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 Sublinear Convergence Rate We now investigate the sublinear convergence rate of PrePDHG (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 Let {(xk, yk)} be the sequence generated by Algorithm 1 with ǫ = 0 and M1, M2 satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Then we have min 1≤k≤t R(xk, yk) = o � 1 √ t � and min 1≤k≤t R(xk, yk−1) = o � 1 √ t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='19) Moreover, if condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) is replaced by M1 + 1 2Σf ≻ 0, M2 ≻ 0, ����M − 1 2 2 K � M1 + 1 2Σf �− 1 2 ���� < 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='20) then we have R(xt, yt) = o � 1 √ t � and R(xt, yt−1) = o � 1 √ t � , ∀t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='21) Proof First, let us bound the KKT residual Rk+1 := R(xk+1, yk+1) and Rk+1/2 := R(xk+1, yk) for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' From the optimality condition of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1a), we have − M1(xk+1 − xk) ∈ ∂f(xk+1) + KTyk, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='22) which implies KT(yk+1 − yk) − M1(xk+1 − xk) ∈ ∂f(xk+1) + KTyk+1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='23) and thus dist � 0, ∂f(xk+1) + KTyk+1� ≤ ���KT(yk+1 − yk) − M1(xk+1 − xk) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='24) Similarly, using the optimality condition of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1b), we have K(xk+1 − xk) − M2(yk+1 − yk) ∈ ∂g∗(yk+1) − Kxk+1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25) and thus dist � 0, ∂g∗(yk+1) − Kxk+1� ≤ ���K(xk+1 − xk) − M2(yk+1 − yk) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='26) Let ˆRk+1 = ���KT(yk+1 − yk) − M1(xk+1 − xk) ��� + ���K(xk+1 − xk) − M2(yk+1 − yk) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' By the Cauchy-Schwarz inequality, we have ˆRk+1 ≤ (∥K∥ + ∥M1∥) ∥xk+1 − xk∥ + ∥K∥ · ∥yk+1 − yk∥ + ∥M2(yk+1 − yk)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Since M2 ≻ 0, for any z ∈ Rm, we have ⟨z, M2z⟩ ≥ λmin(M2) ⟨z, z⟩ and ∥M2z∥2 = ⟨z, M2M2z⟩ ≤ ∥M2∥ ⟨z, M2z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Therefore, we have ∥yk+1−yk∥ ≤ 1 √ λmin(M2)∥yk+1− yk∥M2 and ∥M2(yk+1 −yk)∥ ≤ � ∥M2∥∥yk+1 −yk∥M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Then we immediately have ˆRk+1 ≤ c1∥xk+1 − xk∥ + c2∥yk+1 − yk∥M2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='27) Understanding PrePDHG: a view of indefinite proximal ADMM 17 where the constants c1 = ∥K∥ + ∥M1∥, c2 = ∥K∥ � λmin(M2) + � ∥M2∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='28) By the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) of R(x, y), it is not hard to obtain from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='24), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='26), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='27) that Rk+1 ≤ ˆRk+1 ≤ c1∥xk+1 − xk∥ + c2∥yk+1 − yk∥M2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='29) Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25) for k := k−1, we have K(xk−xk−1)−M2(yk−yk−1) ∈ ∂g∗(yk)−Kxk and thus K(xk − xk−1) + K(xk − xk+1) − M2(yk − yk−1) ∈ ∂g∗(yk) − Kxk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Hence, we have dist � 0, ∂g∗(yk) − Kxk+1� ≤ ���K(xk − xk−1) + K(xk − xk+1) − M2(yk − yk−1) ��� ≤ ∥K∥(∥xk − xk−1∥ + ∥xk − xk+1∥) + � ∥M2∥∥yk − yk−1∥M2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='30) where the second inequality uses ∥M2(yk − yk−1)∥ ≤ � ∥M2∥∥yk − yk−1∥M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' On the other hand, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='22) implies dist(0, ∂f(xk+1) + KTyk) ≤ ∥M1(xk+1 − xk)∥ ≤ ∥M1∥∥xk+1 − xk∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='31) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='30) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='31) together, and by the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) of R(x, y), we have Rk+1/2 ≤ c1∥xk+1 − xk∥ + ∥K∥ · ∥xk − xk−1∥ + � ∥M2∥∥yk − yk−1∥M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='32) Second, we estimate the upper bound of ∥yk+1 −yk∥M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2c) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5), we have M 1 2 2 yk = λk + M − 1 2 2 (Kxk − uk+1) = λk+1 + M − 1 2 2 K(xk − xk+1), which again with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) for k := k + 1 yields M 1 2 2 (yk+1 − yk) = M − 1 2 2 (Kxk+2 − uk+2) − M − 1 2 2 K(xk − xk+1) + M − 1 2 2 K(xk+1 − xk+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='33) Condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='20) tells KTM −1 2 K ≺ 4 3 � M1 + 1 2Σf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Thus, for any z ∈ Rn, we have ���M − 1 2 2 Kz ��� = ���M −1 2 Kz ��� M2 = ∥z∥KT M−1 2 K ≤ 2 √ 3∥z∥M1+ 1 2 Σf ≤ c3∥z∥, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='34) 18 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han where c3 = 2 � λmax(M1 + 1 2Σf)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Hence, noticing that ∥M 1/2 2 v∥ = ∥v∥M2 for any v ∈ Rm, we have from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='33) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='34) that ∥yk+1 − yk∥M2 ≤ ���M − 1 2 2 (Kxk+2 − uk+2) ��� + c3 ���xk+1 − xk�� + ��xk+1 − xk+2�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='29) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35), we obtain Rk+1 ≤ (c1 + c2c3) ��xk+1 − xk�� + c2c3 ��xk+1 − xk+2�� + c2 ���M − 1 2 2 (Kxk+2 − uk+2) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='36) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='32) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35) with k := k − 1, we have Rk+1/2 ≤ � c1 + � ∥M2∥c3 � ��xk+1 − xk�� + � ∥K∥ + � ∥M2∥c3 � ��xk − xk−1�� + � ∥M2∥ ��M − 1 2 2 (Kxk+1 − uk+1) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='37) Finally, similar to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1, if condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) holds, it is easy to see that the conditions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 for iPADMM (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Thus, by applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4, we have min 0≤k≤t � ∥xk − xk+1∥2 M1+Σf + ∥M − 1 2 2 (Kxk+1 − uk+1)∥2� = o(1/t), which with ∥xk − xk+1∥M1+Σf ≥ λmin(M1 + Σf)∥xk − xk+1∥, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='36) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='37) lead to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' If condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='20) holds, it is easy to see that the conditions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 for iPADMM (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Thus, by applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5, we have ∥xt − xt+1∥2 M1+Σf + ∥M − 1 2 2 (Kxt+1 − ut+1)∥2 = o(1/t), which with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='36) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='37) leads to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' ⊓⊔ It is immediate to establish the iteration complexity of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 If ǫ > 0, then Algorithm 1 stops in O(1/ǫ2) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Revisiting the optimality condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5), instead of using the KKT residual, we can also measure the quality of approximate solution (ˆx, ˆy) by giving an upper bound of the function value residual L(ˆx, y) − L(x, ˆy) for any x ∈ Rn and y ∈ Rm, see [7,8,28,29,36,42] and the references therein for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' However, the existing results for PDHG and PrePDHG under condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) are all ergodic, which always have the bound: L(¯xt, y) − L(x, ¯yt) ≤ ϕ1(x, x0) + ϕ2(y, y0) t , ∀x ∈ Rn, ∀y ∈ Rm, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='38) or L(¯xt, y⋆) − L(x⋆, ¯yt) ≤ ϕ1(x⋆, x0) + ϕ2(y⋆, y0) t , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='39) where ¯xt = 1 t �t i=1 xi, ¯yt = 1 t �t i=1 yi, and ϕ1(·, ·), ϕ2(·, ·) are some nonnegative functions and (x⋆, y⋆) is a saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Here, we aim to investigate some non-ergodic results with the help of our established bounds for the KKT residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Understanding PrePDHG: a view of indefinite proximal ADMM 19 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 Let {(xk, yk)} be the sequence generated by Algorithm 1 with ǫ = 0 and M1, M2 satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Let (x∞, y∞) be the limit point of {(xk, yk)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Define a constant ¯c = supk≥0{∥xk − x∞∥ + ∥yk − y∞∥} and denote k(t) := argmin 1≤k≤t � c1∥xk+1 − xk∥M1+ 1 2 Σf + c2∥yk+1 − yk∥M2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='40) where c1 and c2 are defined (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Then for t ≥ 1, L(xk(t), y) − L(x, yk(t)) ≤ o(1/ √ t) (¯c + ∥x∞ − x∥ + ∥y∞ − y∥) , ∀x ∈ Rn, ∀y ∈ Rm (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='41) and L(xk(t), y∞) − L(x∞, yk(t)) ≤ o(1/ √ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='42) Moreover, if condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) is replaced by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='20), then for any t ≥ 1 L(xt, y) − L(x, yt) ≤ o(1/ √ t) (¯c + ∥x∞ − x∥ + ∥y∞ − y∥) , ∀x ∈ Rn, ∀y ∈ Rm (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='43) and L(xt, y∞) − L(x∞, yt) ≤ o(1/ √ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='44) Proof By the convexity of f(x) + � x, KTyk+1� and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='23), we have f(xk+1) + � xk+1, KTyk+1� − f(x) − � x, KTyk+1� ≤ � xk+1 − x, KT(yk+1 − yk) − M1(xk+1 − xk) � , ∀x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45) Similarly, by the convexity of g∗(y) − � y, Kxk+1� and the optimality condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25), we have � g∗(yk+1) − � yk+1, Kxk+1�� − � g∗(y) − � y, Kxk+1�� ≤ � yk+1 − y, K(xk+1 − xk) − M2(yk+1 − yk) � , ∀y ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='46) Summing up (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='46), for any x ∈ Rn and y ∈ Rm, we have L(xk+1, y) − L(x, yk+1) ≤ � xk+1 − x, KT(yk+1 − yk) − M1(xk+1 − xk) � + � yk+1 − y, K(xk+1 − xk) − M2(yk+1 − yk) � ≤ � c1∥xk+1 − xk∥M1+ 1 2 Σf + c2∥yk+1 − yk∥M2 � � ∥xk+1 − x∥ + ∥yk+1 − y∥ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='47) where the second inequality uses the Cauchy-Schwarz inequality and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Suppose M1 and M2 satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' From the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2, we know that min 1≤k≤t � c1∥xk+1 − xk∥M1+ 1 2 Σf + c2∥yk+1 − yk∥M2 � = o � 1 √ t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='48) 20 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han Note that for any k, by the Cauchy-Schwarz inequality and the definition of ¯c, we have ∥xk+1 − x∥ + ∥yk+1 − y∥ ≤ ∥xk+1 − x∞∥ + ∥yk+1 − y∞∥ + ∥x∞ − x∥ + ∥y∞ − y∥ ≤ ¯c + ∥x∞ − x∥ + ∥y∞ − y∥, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='49) which together with the definition of k(t) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='40), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='47), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='48) implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Suppose M1 and M2 satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' From the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2, we know that c1∥xk+1 − xk∥M1+ 1 2 Σf + c2∥yk+1 − yk∥M2 = o � 1 √ k � , which with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='47) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='49) implies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1), we know that (x∞, y∞) is a saddle point of (PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Thus, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='42) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='44) follow directly from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='41) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='43), respectively, by setting x = x∞ and y = y∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' ⊓⊔ Some remarks on our results about the sublinear convergence rate of PrePDHG are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' First, to the best of our knowledge, the sublinear rate based on the KKT residual R(xk+1, yk+1) or R(xk+1, yk) is new for PDHG like methods since the existing results mainly focus on (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Compared with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='38), the upper bounds of the KKT residual R(xk+1, yk+1) or R(xk+1, yk) are always computable, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Our sublinear rate result for the KKT residual also tells that Algorithm 1 can return an ǫ-solution in O(1/ǫ2) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Second, for the function value residual measurement, our sublinear rate result is the first non-ergodic result since the existing results are all ergodic, see [7,8,28,29] for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' It should be clear that our non-ergodic results are o(1/ √ t) while the existing ergodic results are O(1/t) both under the condition that (x, y) is in a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' It remains unknown whether the non-ergodic result can be improved to O(1/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' To end this section, we briefly discuss a dual formulation of the PrePDHG recursion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) in the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 In Section 2, we assume that problem (PD) has a saddle point, which means that solving (PD) is equivalent to solving the following problem: min y∈Rm max x∈Rn g∗(y) − ⟨y, Kx⟩ − f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='50) Using PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) to solve (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='50) and based on the symmetry of the primal and dual variables between (PD) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='50) (the primal variable x in (PD) is the dual variable in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='50) and vice versa), we can obtain the other PrePDHG recursion, which can also be used to solve (PD): \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 xk+1 = argmin x∈Rn f(x) + � Kx, 2yk − yk−1� + 1 2 ���x − xk��� 2 Q1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='51a) yk+1 = argmin y∈Rm g∗(y) − � Kxk+1, y � + 1 2∥y − yk∥2 Q2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='51b) where the symmetric matrices Q1 ∈ Rn×n and Q2 ∈ Rm×m satisfy � Q1 −KT −K 4 3 �Q2 + 1 2Σg∗� � ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Understanding PrePDHG: a view of indefinite proximal ADMM 21 Consider an equivalent formulation of problem (D) (note that (D) is also the primal formulation of problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='50)) min z∈Rn, y∈Rm f ∗(z) + g∗(y) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Q − 1 2 1 (z + KTy) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (D1) The iPADMM recursion for (D1) is given as \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 zk+1 = argmin z∈Rn ¯L1(z, yk, λk), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='52a) yk+1 = argmin y∈Rm ¯L1(zk+1, y, λk) + 1 2∥y − yk∥2 Q2−KQ−1 1 KT, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='52b) λk+1 = λk + Q − 1 2 1 (zk+1 + KTyk+1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='52c) where ¯L1(z, y, λ) = f ∗(z) + g∗(y) + � λ, Q−1/2 1 (z + KTy) � + 1 2∥z + KTy∥2 Q−1 1 is the augmented Lagrangian function of (D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Using the same process in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1, we can show the equivalence between (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='51) and the iPADMM (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The convergence results of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='51) can thus be established similar to that in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We omit the details for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 4 Revisit on the Choices of M1 and M2 In this section, we revisit PrePDHG and discuss the choices of M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Specif- ically, with the choices in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2, PrePDHG gives improved versions of the original PDHG and PDHG with diagonal preconditioners, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3, we investigate the choice of M1 = τ −1In, M2 = γτKKT + P and its extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4, we consider a special case when g∗(y) = ⟨b, y⟩ and dis- cuss an enhanced BALM (eBALM) and an eBALM with symmetric Gauss-Seidel iterations (eBALM-sGS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 M1 = τ −1In, M2 = σ−1Im If the proximal operators of f and g∗ are both easy to compute, we can simply take M1 = τ −1In, M2 = σ−1Im with τ, σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In this case, PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) reduces to the original PDHG (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1), which can be reformulated as \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 xk+1 = proxτf � xk − τKTyk� , yk+1 = proxσg∗ � yk + σK(2xk+1 − xk) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) where proxτf(x) is defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Define a constant λf min := λmin(Σf) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For such choices of M1 and M2, we have ����M − 1 2 2 K � M1 + 1 2Σf �− 1 2 ���� 2 ≤ σ∥K∥2 1/τ + (1/2)λf min = τσ∥K∥2 1 + (τ/2)λf min .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 22 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han To make condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) hold, we obtain the convergence condition of the PDHG (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) as τ, σ > 0, τσ∥K∥2 < 4 3 � 1 + τλf min 2 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) which can imply (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Besides, we also know from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) is tight in the sense that the constant 4/3 could not be enlarged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 If λf min is not easy to estimate or f has no more property beyond convexity, we can set λf min as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Moreover, in the following part of this section, to make the discussion precise, we choose Σf = 0 and λf min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We refer to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 for one exception, wherein there holds that Σf = In2 and λf min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 Diagonal M1 and M2 If both f and g take the separable structures, namely, f(x) := �n j=1 fj(xj), g∗(y) = �m i=1 g∗ i (yi), and the proximal operators of fj and g∗ i are all easy to compute, we can consider the following choices of diagonal M1 and M2, which were first proposed in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 For any α ∈ [0, 2] and γ1, γ2 > 0, let M1 = γ1 diag(τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , τn) with τj = δ + m � i=1 |Kij|2−α, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , n, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) M2 = γ2 diag(σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , σm) with σi = δ + n � j=1 |Kij|α, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , m, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) where δ ≥ 0 is chosen such that τj, σi are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' If γ1γ2 > 3 4, then such M1 and M2 satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Proof By [40, Lemma 2], we know ∥(M2/γ2)−1/2K(M1/γ1)−1/2∥ ≤ 1, which im- plies that ∥M − 1 2 2 KM − 1 2 1 ∥2 ≤ 1 γ1γ2 < 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' ⊓⊔ With choices (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4), PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) becomes \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 xk+1 j = proxτjfj � xk j − τj(KTyk)j � , j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , n, yk+1 i = proxσig∗ i � yk i + σi(K(2xk+1 − xk))i � , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 Taking γ1 = γ2 = 1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) yields the diagonal precondi- tioners in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 tells that γ1γ2 > 3 4 in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 is tight in the sense that “>” can not be improved to “≥”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Understanding PrePDHG: a view of indefinite proximal ADMM 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 M1 = τ −1In, M2 = γτKKT + P and Extensions Another choice is M1 = τ −1In and M2 = τKKT + θIm with τ, θ > 0, which was proposed in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Here, we consider a relaxed version of such choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 Let P ∈ Rm×m be a nonzero symmetric positive semidefinite matrix such that KKT + P ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Choose M1 = τ −1In, M2 = γτKKT + P, τ > 0, γ ≥ 3 4, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Proof It is easy to see that M2 ≻ 0 from KKT +P ≻ 0 with KKT ⪰ 0 and P ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Hence, We have ���M − 1 2 2 KM − 1 2 1 ��� 2 = τλmax � KT � γτKKT + P �−1 K � = 1 γ λmax �� γτKKT + P �−1 (γτKKT) � < 1 γ λmax �� γτKKT + P �−1 � γτKKT + P �� ≤ 4 3, where the first inequality is due to P ⪰ 0 but P ̸= 0, and the second one relies on γ ≥ 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' ⊓⊔ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 Similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5), letting ˆP ∈ Rm×m be a nonzero symmetric positive semidefinite matrix such that KTK + ˆP ≻ 0, we can choose M1 = γσKTK + ˆP, M2 = σ−1In, σ > 0, γ ≥ 3 4 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) such that condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that very recently Bai [1] considered (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) with γ = 1 and symmetric positive definite P and ˆP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In some problem, such as CT reconstruction in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4, g∗(y) takes a separable structure as g∗(y) = g∗ 1(y1)+g∗ 2(y2) with y = �y1 y2 � , y1 ∈ Rm1, y2 ∈ Rm2, in which the proximal of g1 takes a closed form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In this case, motivated by [36], we can partition K as K = �K1 K2 � with K1 ∈ Rm1×n, K2 ∈ Rm2×n and choose M1 = 2γ τ In, M2 = �σ−1Im1 0 0 τK2KT 2 + P2 � with τ, σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 Let P2 ∈ Rm2×m2 be a nonzero symmetric positive semidefinite matrix such that K2KT 2 + P2 ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Let τ, σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' If (τσ∥K1∥2 + 1)/γ ≤ 8/3 and M1 and M2 are chosen according to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7), then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 24 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han Proof It is easy to see that M2 ≻ 0 from K2KT 2 + P2 ≻ 0 with K2KT 2 ⪰ 0 and P2 ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We thus have ���M − 1 2 2 KM − 1 2 1 ��� 2 = τ 2γ λmax � σK1KT 1 + � τK2KT 2 + P2 �−1/2 K2 � τK2KT 2 + P2 �−1/2� ≤ τ 2γ σ∥K1∥2 + 1 2γ λmax �� τK2KT 2 + P2 �−1 (τK2KT 2 ) � < τσ∥K1∥2 + 1 2γ ≤ 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' ⊓⊔ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 A particular choice in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) is τ > 0, σ > 0, and τσ∥K1∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In this case, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 yields γ ≥ 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 A Special Case g∗ = ⟨b, y⟩ and Beyond In this subsection, we mainly consider the case when g∗ is a linear function, for which with choice (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5), the y-subproblem in PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) can be efficiently solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Some more general cases of g∗ are also discussed at the end of this subsec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Given a vector b ∈ Rm, we consider g(y) = I{b} and g∗(y) = sup z∈Rm{⟨z, y⟩ − g(z)} = ⟨b, y⟩ , where I{b} is the indicator function of the singleton {b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Hence, problem (PD) becomes min x∈Rn max y∈Rm L(x, y) := f(x) + ⟨y, Kx⟩ − ⟨b, y⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8) The recursions of PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8) are given as \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 xk+1 = proxτf � xk − τKTyk� , yk+1 = yk + (γτKKT + P)−1 � K(2xk+1 − xk) − b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 For g∗(y) = ⟨b, y⟩, compared with the results in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1, we can obtain more compact upper bounds of R(xk+1, yk+1) and R(xk+1, yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='24) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7), we have R(xk+1, yk+1) ≤ max{∥KT(yk+1 − yk) − τ −1(xk+1 − xk)∥, ∥Kxk+1 − b∥}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Besides, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='22) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7), we have R(xk+1, yk) ≤ max{∥τ −1(xk+1 − xk)∥, ∥Kxk+1 − b∥}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='10) Understanding PrePDHG: a view of indefinite proximal ADMM 25 We next consider two choices of P, where the y-subproblem in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9) is easy to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The first one is to choose γ = 1 and P = θIm for some θ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9) reduces to the balanced ALM (BALM) [25] for solving the following convex optimization problem min x∈Rn f(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Kx = b, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='11) which corresponds to the primal formulation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8) (see Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 for two instances of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' BALM procedure: Let τ > 0 and θ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For given (xk, yk), the new iterate (xk+1, yk+1) is generated by: \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 xk+1 = proxτf � xk − τKTyk� , yk+1 = yk + (τKKT + θIm)−1 � K(2xk+1 − xk) − b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='12) In [25], He and Yuan proved the convergence of BALM (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='12) in an elegant way by using the framework of variational inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that the parameters τ and θ can be arbitrary positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' By applying the results in Section 3, we obtain an enhanced BALM (eBALM), with global convergence and sublinear convergence rate, as follows: eBALM procedure: Let τ > 0, θ > 0 and γ ≥ 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For given (xk, yk), the new iterate (xk+1, yk+1) is generated by: \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 xk+1 = proxτf � xk − τKTyk� , yk+1 = yk + γ−1(τKKT + θIm)−1 � K(2xk+1 − xk) − b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 Note that γ−1 is taken as 1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='12) and can be any number in (0, 4/3] in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Therefore, compared with BALM, the stepsize of y-subproblem in eBALM can be enlarged to 4/3 from 1 Moreover, 4/3 is a tight upper bound of γ−1 according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Next, we discuss the case when the inverse of the matrix in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13) does not take a closed form or solving the corresponding linear system is difficult;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' see the earth mover’s distance problem in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In this case, we can use the block Gauss-Seidel method or the conjugate gradient method to inexactly solve the corresponding linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' However, the convergence issues are beyond the scope of this paper, and we refer the readers to [28,29,36] and the reference therein for some discussion on the inexact PDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' As an alternative, we can adopt one block symmetric Gauss-Seidel (sGS) iteration to solve the linear system inexactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' By the sGS decomposition theorem developed by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' [34], this approach corresponds to taking P as a specific positive definite matrix in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' More specifically, let Q = γτKKT + θIm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Suppose that Q takes the block structure Q = \uf8eb \uf8ec \uf8ed Q1,1 · · · Q1,s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' QT 1,s · · · Qs,s \uf8f6 \uf8f7 \uf8f8 , 26 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han where Qi,j ∈ Rmi×nj for 1 ≤ i, j ≤ s and Qi,i is symmetric positive definite and its inverse is easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that if (γτKKT)i,i is positive definite, then θ can be chosen to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Let U = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 Q1,2 · · · Q1,s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Qs−1,s 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , D = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed Q1,1 Q2,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Qs,s \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Suppose U ̸= 0, otherwise, the y-subproblem in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13) takes closed form solution since the inverse of Qi,i is easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Taking ˜P = UD−1U T, by [34, Theorem 1], we have Q + ˜P = (D + U)D−1(D + U T) ≻ 0 and that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9) with P = θIm + UD−1U T is equivalent to the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' eBALM-sGS procedure: Let τ > 0, θ > 0, γ ≥ 3/4 and Q = γτKKT + θIm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For given (xk, yk), the new iterate (xk+1, yk+1) is generated by: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 xk+1 = proxτf � xk − τKTyk� , ¯bk+1 = K(2xk+1 − xk) − b, ¯yk+1 i = yk i + Q−1 i,i � ¯bk+1 i − i−1 � j=1 QT j,iyk j − s � j=i+1 Qi,j ¯yk+1 j � , i = s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , 2, yk+1 i = yk i + Q−1 i,i � ¯bk+1 i − i−1 � j=1 QT j,iyk+1 j − s � j=i+1 Qi,j¯yk+1 j � , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , s, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='14) where ¯yk+1 i , yk+1 i ∈ Rmi for 1 ≤ i ≤ s and yk+1 = � (yk+1 1 )T, · · · , (yk+1 s )T�T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We name (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='14) as enhanced BALM with symmetric Gauss-Seidel iterations (eBALM- sGS) for solving problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2, we have the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 Suppose U ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Let τ > 0, θ > 0, and γ ≥ 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Then the sequence {(xk, yk)} generated by eBALM-sGS (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='14) converges to an optimal solution of problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Moreover, for t ≥ 1, we have min 1≤k≤t dist(0, ∂f(xk) + KTyk) = o � 1 √ t � , min 1≤k≤t dist(0, ∂f(xk) + KTyk−1) = o � 1 √ t � , and min 1≤k≤t ∥Kxk − b∥ = o � 1 √ t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Understanding PrePDHG: a view of indefinite proximal ADMM 27 If γ ≥ 1, θ > 0, the sublinear rate results are refined as dist(0, ∂f(xt) + KTyt) = o � 1 √ t � , dist(0, ∂f(xt) + KTyt−1) = o � 1 √ t � and ∥Kxt − b∥ = o � 1 √ t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 If the i-th block (γτKKT)i,i is positive definite for any 1 ≤ i ≤ s, then θ > 0 in the above lemma becomes θ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' To end this subsection, we consider a more general scenario that g∗ takes the block separable structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=', y = � yT 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , yT s �T and g∗(y) = �s j=1 gj(yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In this case, the y-subproblem in PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) can be efficiently solved by cyclic proximal block coordinate descent method, see [36] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 5 Numerical Experiments In this section, we present plenty of numerical results on the matrix game, pro- jection onto the Birkhoff polytope, earth mover’s distance, and CT reconstruction problems to verify the superiority of the larger range of the corresponding param- eters in our PrePDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The codes are written in MATLAB (Release 2017b) and run in macOS 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 on a MacBook Pro with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9GHz Intel Core i7 processor with 16GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 Matrix Game Let ∆n = {x ∈ Rn | �n i=1 xi = 1, x ≥ 0} be the standard unit simplex in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Given a matrix K ∈ Rm×n, we consider the min-max matrix game min x∈∆n max y∈∆m ⟨Kx, y⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) This problem is a form of problem (PD) with f and g∗ chosen as the indicator functions of ∆n and ∆m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The main iterations of PDHG (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) are thus given as \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 xk+1 = Proj∆n � xk − τKTyk� , yk+1 = Proj∆m � yk + σK(2xk+1 − xk) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) where Proj∆n(·) is the projection operator onto the simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For this problem, λf min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2), the stepsizes σ > 0 and τ > 0 satisfy τσ∥K∥2 < 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In our numerical results, we consider τ = ˜τ/∥K∥ and σ = 1/(γ˜τ∥K∥) with γ ∈ {1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='751} (the requirement on γ is γ > 3/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that γ = 1 corresponds to the original PDHG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1, we stop the algorithm when the iterations exceed 106 or max{∥KT(yk+1 − yk) − τ −1(xk+1 − xk)∥, ∥K(xk+1 − xk) − σ−1(yk+1 − yk)∥} ≤ 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The starting points are always chosen as x0 = 1 n �1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , 1�T ∈ Rn and 28 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 iteration 104 (a) Test 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 7 iteration 104 (b) Test 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 iteration 105 (c) Test 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 iteration 104 (d) Test 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 15 10 5 0 5 10 15 20 25 30 35 ratio (e) Test 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 5 0 5 10 15 20 25 30 ratio (f) Test 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 10 5 0 5 10 15 20 25 30 35 40 ratio (g) Test 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0 5 10 15 20 25 ratio (h) Test 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 1: Comparison of PDHG (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) with different values of γ for matrix game problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' y0 = 1 m �1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , 1�T ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We follow the way in [9, 38] to generate the matrix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The corresponding Matlab commands are given as: i) m = 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' n = 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' A = rand(m,n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' ii) m = 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' n = 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' A = randn(m,n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' iii) m = 500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' n = 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' A = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' *randn(m,n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' iv) m = 1000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' n = 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' A = sprand(m,n,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For each case, we randomly generate the matrix K 20 times and report the average performance of each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We test a series of ˜τ ∈ 10a with a = [a1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='01 : a1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4] with a1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 for Test 1 and Test 2, and a1 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 for Test 3 and a1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 for Test 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The comparison results among different γ are reported in Figure 1, wherein the saved ratio in terms of iteration number is defined as ratio = �iter − iter iter × 100 � %, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) where the baseline iteration number “iter” is taken as the iteration number of PDHG with γ = 1 and “iter” means the iteration number of PDHG with a chosen γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' From these figures, we can see that PDHG with smaller γ always has better performance than the classical PDHG with γ = 1, and for a large range of ˜τ, the saved ratio is more than 20% for Tests 1-3 and is more than 15% for Test 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We also observe that the performance of PDHG with different γ might depend on the choice of ˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Therefore, to make a fair comparison, for PDHG with fixed γ, we take the best ˜τ (in terms of the lowest iteration number), denoted by ˜τbest, from the set 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The comparison results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' This table shows that PDHG with smaller γ is still better than PDHG with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='751, the saved ratio is always more than 22%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that such improvement only needs to change a parameter in the original PDHG without additional cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Understanding PrePDHG: a view of indefinite proximal ADMM 29 Table 1: Performance of PDHG (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) with best ˜τ for problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In the table, “a”, “b”, “c”, and “d” stands for PDHG (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2) with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='90, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85, and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='751, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 10 × log10(˜τbest) time iter ratio % Test a b c d a b c d a b c d b c d 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6e-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5e-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5e-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3e-1 17278 16350 15644 14500 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7e-1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4e-1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3e-1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0e-1 52040 49075 47458 44458 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9 -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9e1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6e1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5e1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3e1 202919 183475 173976 157250 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7e1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6e1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6e1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4e1 56122 52226 50205 45723 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 Projection onto the Birkhoff Polytope Given a matrix C ∈ Rn×n, computing its projection onto the Birkhoff polytope can be formulated as min X∈Bn 1 2∥X − C∥2 F, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) where Bn := {X ∈ Rn×n | Xen = en, XTen = en, X ≥ 0} with en ∈ Rn being the all-one vector is known as the Birkhoff polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) has wide applications in solving the optimization problems involving permutations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' see [26, 34] and the references therein for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Let x = vec(X), problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4) can be seen as a special instance of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='11) with f(x) = 1 2∥x − vec(C)∥2 + IX with X = Rn2 + and IX being the indicator function of the set X , and K = �eT n ⊗ In In ⊗ eT n � , b = e2n, where ⊗ is the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For such K, we have ∥K∥2 = 2n (see [21] for instance) and � KKT + θI2n �−1 = 1 n + θ I2n + 1 2nθ + θ2 � n n+θeneT n −eneT −eneT n n n+θ eneT n � , θ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We consider two particular choices of PrePDHG (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The first one is eBALM (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13), whose main iterations are given as: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 Xk+1 = 1 1 + τ Proj+ � Xk + τC − τ � yk 1eT n + en(yk 2)T�� , ak+1 = eT n(2Xk+1 − Xk)en + n + θ, yk+1 = yk + 1 γτ(n + θ) � (2Xk+1 − Xk)en (2Xk+1 − Xk)Ten � − ak+1 γτ(n + θ)(2n + θ)e2n, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) where Proj+(·) is the projection operator over Rn×n + and θ is taken as 10−4, yk 1 ∈ Rn is the vector formulated by the first n components of yk and yk 2 ∈ Rn is the vector formulated by the last n components of yk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The second one is PDHG (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) with main iterations given as: \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 Xk+1 = 1 1 + τ Proj+ � Xk + τC − τ � yk 1eT n + en(yk 2)T�� , yk+1 = yk + σ � (2Xk+1 − Xk)en (2Xk+1 − Xk)Ten � − σe2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) 30 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han Table 2: Performance of eBALM (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) and PDHG (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) with best ˜τ for problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In the table, “a” and “b” stands for PDHG (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='751 1+τ/2, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' “c” and “d” stands for eBALM (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75 1+τ/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' log10(˜τbest) time iter ratio % n a b c d a b c d a b c d b c d 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='44 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1e-1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6e-2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5e-2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2e-2 471 398 336 280 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6e-1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2e-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8e-1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9e-1 676 574 485 408 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5e-1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4e-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7e-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1e-1 835 714 598 506 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6e-1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2e-1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0e-1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9e-1 1068 913 769 652 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 Note that for problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4), we have Σf = In2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2, we know that the parameters τ > 0 and γ > 0 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) should satisfy γ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75 1+τ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In our numerical results, we consider τ = ˜τ/ √ 2n with ˜τ > 0 and γ ∈ � 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75 1+τ/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In addition, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2), the parameters τ > 0 and σ > 0 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) satisfies 2nτσ < 4 3(1 + τ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In our numerical results, we consider τ = ˜τ/ √ 2n and σ = 1/(γ˜τ √ 2n) with γ ∈ �1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='751 1+τ/2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For a given n, we follow the way in [34] to randomly generate 20 matrices C via C = rand(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' C = (C+C’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='/2 and report the average performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The initial points are always chosen as X0 = 1 neneT n and y0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='10), we stop both algorithms when the relative KKT residual �Rk := max{dk, pk} ≤ 10−8 with pk = τ −1∥Xk − Xk−1∥F and dk = ���� � Xken − en (Xk)Ten − en �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For both algorithms, we test a series of ˜τ ∈ 10a with a = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='01 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The comparison results are depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In the figures (e)-(h), the “ratio” is computed according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) with iter taken as the iteration number of PDHG (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) with γ = 1, and in the figures (i)-(l), the “ratio” is computed according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3) with iter taken as the iteration number of eBALM (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' From these figures, we can draw the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (i) Both PDHG and eBALM benefit from choosing a larger stepsize, namely, a smaller γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For a large range of ˜τ, PDHG with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='751 1+τ/2 is more than 30% faster than PDHG with γ = 1 and eBALM with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75 1+τ/2 is more than 15% faster than eBALM with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (ii) eBALM with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75 1+τ/2 performs best among the four algorithms, and it is even about more than 50% faster than the classical PDHG with γ = 1 for ˜τ ∈ 10[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' To further investigate the effect of ˜τ on the performance of different algorithms, as done in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1, we present the performance of each algorithm with ˜τbest in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' From this table, we can see that even with the best possible parameter ˜τ, both PDHG and BALM with a smaller γ (means the larger stepsize in updating y) still have better performance than the corresponding algorithm with larger γ for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Besides, eBALM with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75 1+τ/2 has the best performance, compared with the classical PDHG with γ = 1, it saves about 40% of iteration numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Understanding PrePDHG: a view of indefinite proximal ADMM 31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 200 400 600 800 1000 1200 1400 1600 iteration (a) n = 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 400 600 800 1000 1200 1400 1600 1800 2000 2200 iteration (b) n = 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 500 1000 1500 2000 2500 3000 iteration (c) n = 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 500 1000 1500 2000 2500 3000 3500 iteration (d) n = 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0 10 20 30 40 50 60 70 80 ratio (e) n = 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0 10 20 30 40 50 60 70 80 ratio (f) n = 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0 10 20 30 40 50 60 70 80 ratio (g) n = 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0 10 20 30 40 50 60 70 80 ratio (h) n = 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0 5 10 15 20 25 30 35 40 45 ratio (i) n = 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0 5 10 15 20 25 30 35 40 45 ratio (j) n = 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0 5 10 15 20 25 30 35 40 ratio (k) n = 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0 5 10 15 20 25 30 35 ratio (l) n = 800 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 2: Comparison of eBALM (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5) with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75 1+τ/2 and PDHG (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6) with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='751 1+τ/2 for problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 Earth Mover’s Distance Given two discrete mass distributions ρ0 and ρ1 over the M × N grid, comput- ing the earth mover’s distance between them can be formulated as the following optimization problem (see [33] for instance): min m∈R2M×N ∥m∥1,2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' div(m) + ρ1 − ρ0 = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) where m = �m1 m2 � is the sought flux vector on the M × N grid with m1, m2 ∈ RM×N and m1 M,j = 0 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , N and m2 i,N = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Here, ∥m∥1,2 := �M i=1 �N j=1 � (m1)2 i,j + (m2)2 i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The 2D discrete divergence operator div(m) : R2M×N → RM×N is defined as div(m)i,j = h � m1 i,j − m1 i−1,j + m2 i,j − m2 i,j−1 � , where h is the grid stepsize, m1 0,j = 0 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , N and m2 i,0 = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Let x = �vec(m1) vec(m2) � ∈ R2MN, then problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7) is a form of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='11) with b = vec(ρ0 −ρ1) and the matrix K ∈ RMN×2MN satisfies Kx = vec(div(m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 32 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han 1 2 3 4 5 6 7 10-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 iteration 105 (a) i-eBALM: itera- tion versus τ 1 2 3 4 5 6 7 10-6 5 0 5 10 15 20 25 ratio (b) i-eBALM: ratio versus τ 1 2 3 4 5 6 7 10-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2 iteration 105 (c) eBALM-sGS: iter- ation versus τ 1 2 3 4 5 6 7 10-6 0 5 10 15 20 25 30 ratio (d) eBALM-sGS: ra- tio versus τ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 3: Comparison on iteration and ratio of i-eBALM with γ = {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='90,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='77} and eBALM-sGS with γ = {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='90,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that i-eBALM with γ = 1 is exactly iPrePDHG in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We consider two versions of PrePDHG, namely, eBALM (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13) and eBALM- sGS (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='14) to solve problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For eBALM (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13), due to the particular struc- ture of KKT explored in [36, Section 4], we only performed two epochs of block co- ordinate descent method as done in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We name this implementation i-eBALM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Moreover, we take θ = 0 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13) since its performance is very similar to that of very small θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' It should be mentioned that when γ = 1 in eBALM (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13), it becomes the iPrePDHG proposed in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that [36] proved the conver- gence of iPrePDHG under the strong convexity of the objective, which does not hold for problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Besides, the convergence of i-eBALM (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13) remains un- known, although it performs well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We consider four choices of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For eBALM (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='13), we take γ ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='90,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='77} and for eBALM-sGS (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='14), we take γ ∈ {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='90,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that the lower bound of γ to guarantee the con- vergence of eBALM-sGS (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='14) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Actually, in our numerical tests, eBALM-sGS (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='14) with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='749 always diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For this problem, we have ∥b∥ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Therefore, we replace the term ∥Kxk+1− b∥ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='10) by ∥Kxk+1 −b∥/∥b∥ and stop each algorithm when the iteration num- ber exceeds 200,000 or the relative KKT residual �Rk := max{dk, pk} ≤ tol := 5 × 10−5, where pk = τ −1∥xk − xk−1∥ and dk = ∥Kxk − b∥/∥b∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The initial x0 and y0 are both taken as all-zero vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Besides, we adopt the same problem setting in [33,36], namely, M = N = 256, h = (N − 1)/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The comparison results among different γ are reported in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In this figure, for each fixed τ ∈ {1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9, 7} × 10−6, the saved ratio in terms of iteration number is defined as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3), where iter is taken as the corresponding method with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' From the figures, we can see that both eBALM and eBALM- sGS benefit from choosing small γ, which enlarges the stepsize in updating yk+1 in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In particular, for eBALM, when τ ≥ 4 × 10−6, the saved ratios of taking γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='77,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='90 are about 20%, 15% and 10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For eBALM- sGS, when τ ≥ 3 × 10−6, the saved ratios of taking γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='77,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='90 are about 25%, 15% and 10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Besides, we also know that eBALM-sGS always perform worse than i-eBALM, although the former has a convergence guarantee while the latter does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' As done in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1, we present the results corresponding to the best τ in Figure 4 and Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' From them, we observe that choosing small γ can still Understanding PrePDHG: a view of indefinite proximal ADMM 33 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 iterations 104 10-5 10-4 10-3 (a) i-eBALM: dk ver- sus iterations 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 iterations 104 10-5 10-4 10-3 (b) i-eBALM: pk ver- sus iterations 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 iterations 104 10-5 10-4 10-3 (c) eBALM-sGS: dk versus iterations 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 iterations 104 10-5 10-4 10-3 10-2 (d) eBALM-sGS: pk versus iterations Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 4: Comparison on pk and dk of i-eBALM with γ = {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='90,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='77} and eBALM-sGS with γ = {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='00,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='90,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The parameter of each method is taken as τbest in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that i-eBALM with γ = 1 is exactly iPrePDHG in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Table 3: Results of i-eBALM and eBALM-sGS with best τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' γ τbest time iter ∥m∥1,2 ratio i-eBALM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4e-6 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 52461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='671770 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6e-6 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 49715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='671770 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7e-6 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 48362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='671770 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='77 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9e-6 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 45990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='671770 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='33 eBALM-sGS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4e-6 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 74024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='671770 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='90 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6e-6 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 70105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='671769 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6e-6 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 68241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='671770 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8e-6 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 63955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='671770 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='60 accelerate the corresponding method with γ = 1, and the saved ratio is always more than 12% when we take γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='77 in eBALM and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='75 in eBALM-sGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Again note that to achieve such improvement, we only need to change a parameter in the original method without increasing any additional cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We also know from Table 3 that the saved ratios shown in this table are not as large as those in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' However, choosing the best τ from a portion of candidates is time-consuming and impractical for both i-eBALM and eBALM-sGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Finally, in Figure 5, we show the solutions obtained by eBALM-sGS with different tolerance and the ground truth obtained by running CVX in several hours, see [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We can see that eBALM-sGS with tolerance tol = 5 × 10−5 can return a solution with satisfactory precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 CT Reconstruction Let xtrue ∈ Rn with n = MN and M = N = 256 be a true image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Given a vector of line-integration values b = Rxtrue ∈ Rm, where R ∈ Rm×n is a system matrix for 2D fan-beam CT with a curved detector, the CT image reconstruction aims to recover xtrue via solving the following optimization problem: min x∈Rn Φ(x) := 1 2∥Rx − b∥2 + λ∥Dx∥1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8) 34 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han (a) i-eBALM, tol = 5 × 10−1, mk r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 × 10−1, iter = 842 (b) i-eBALM, tol = 5 × 10−3, mk r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 × 10−2, iter = 8404 (c) i-eBALM, tol = 5 × 10−5, mk r = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9 × 10−4, iter = 45990 (d) groundtruth from [36] (e) eBALM-sGS, tol = 5 × 10−1, mk r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 × 10−1, iter = 1171 (f) eBALM-sGS, tol = 5 × 10−3, mk r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 × 10−2, iter = 11729 (g) eBALM-sGS, tol = 5 × 10−5, mk r = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 × 10−4, iter = 63955 (h) groundtruth from [36] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 5: Mass distributions ρ0 and ρ1 are both with size 256×256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The white stand- ing cat is ρ0 and the black crouching cat is ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The red or blue curves are the flux that moves the standing cat ρ0 into the crouching cat ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The ground truth flux, denoted by mcvx, is obtained by CVX after several hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The earth mover’s dis- tance between ρ0 and ρ1 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='671783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The term mk r = ∥mk −mcvx∥/∥mcvx∥, where mk is the flux obtained by each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The data matrices ρ0, ρ1, and mcvx are downloaded from https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='com/xuyunbei/Inexact-preconditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' where D ∈ R2n×n is the 2D discrete gradient operator with h = 1 (see [36, Section 4] for instance) and λ = 1 is a regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' To avoid solving the linear system related to the matrices R and D, as done in [36] and [45], we understand problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8) as a form of (P) with f(x) = 0, g(z) = 1 2∥p − b∥2 + λ∥q∥1, z = �p q � ∈ Rm+2n, K = �R D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We choose the variable metric matrices M1 and M2 via (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7), wherein K1 and K2 are R and D, respectively and σ = (τ∥R∥2)−1, P2 = θIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The constant θ is taken as 10−3 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' According to Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4, we have the parameter γ ≥ 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that the dual variable y can be decomposed as y = �y1 y2 � with y1 ∈ Rm and y2 ∈ R2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The main iteration scheme of PrePDHG (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1) for solving Understanding PrePDHG: a view of indefinite proximal ADMM 35 problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8) is given as follows: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 xk+1 = xk − τ 2γ � RTyk 1 + DTyk 2 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9a) yk+1 1 = τ∥R∥2yk 1 + R(2xk+1 − xk) − b 1 + τ∥R∥2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9b) yk+1 2 = argmin ∥y2∥∞≤λ 1 2 ���y2 − yk 2 ��� 2 τDDT+θI2n − � y2, D(2xk+1 − xk) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9c) The y2-subproblem in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9c) does not take a closed-form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' However, thanks to the special structure of D, [36] developed an efficient block coordinate descent (BCD) method to solve (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' To guarantee the convergence of PrePDHG (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9), theoretically, we need to run many BCD epochs to solve (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9c) almost exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' However, this may be time-consuming as observed in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' As suggested by [36], we find that running two BCD steps is enough to make the PrePDHG (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9) per- form well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Hence, in our numerical experiments, we only apply two BCD steps in solving the y2-subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Considering that [36] has shown the superiority of their proposed inexact preconditioned PDHG (iPrePDHG) over other variants of PDHG, here we mainly compare our PrePDHG (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9) with iPrePDHG therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' It should be mentioned that iPrePDHG corresponds to our PrePDHG (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9) with γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For PrePDHG (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9), we consider three versions with γ = 5/6, γ = 3/4 and γ = 1/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Although there is no convergence guarantee for the last one, it performs very well in our numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Given a vector z = [0, ν, 2ν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' , 360 − ν]T containing the projection angles in degrees, we generate a test problem by using the fancurvedtomo function from the AIR Tools II package [19] with input N and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In our numerical results, we consider ν ∈ {6, 9, 12, 15, 18, 24, 30, 36}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The starting points of PrePDHG and iPrePDHG are both taken as x0 = 0 and y0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We stop each algorithm at (xk, yk) when the KKT residual R(xk, yk) ≤ 5 × 10−6, where R(xk, yk) is computed according to R(xk, yk) = max � ∥RTyk 1 + DTyk 2∥, ∥Rxk − yk 1 − b∥, dist � Dxk, ∂I∥y2∥∞≤λ(yk 2) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For each fixed ν, we test a series of τ ∈ 10a with a = [−a1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='02 : −a1 + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The parameter a1 is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 for ν = 6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 for ν = 9, 12 or 15, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 for ν = 18 or 24, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 for ν = 30 or 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The results are presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In these figures, the term ratio is computed according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3), wherein “iter” is the iteration number corresponding to γ = 1, namely, iPrePDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' From these figures, we can see that taking smaller γ (meaning the larger stepsize in updating the primal variable x, see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9a)) can always speed up the performance of PrePDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' More specifically, the saved ratios of taking γ = 3/4, the theoretical lower bound, are about 13% for ν = 6, 9, 12 and about 25% for ν ≥ 15 for a large portion of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' On the other hand, although taking γ = 1/2 has no convergence guarantee since 1/2 is smaller than the theoretical lower bound of γ, it always has the best performance for a large portion of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The corresponding saved ratios are more than 25% for ν = 6, 9, 12, more than 40% for ν = 15, and even 50% for ν ≥ 18 for a large portion of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The numerical results corresponding to the best τ, denoted by τbest, for each instance are reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' From this table, we can see that, compared with iPrePDHG, PrePDHG with smaller γ is always faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' For γ = 5/6, it can save 36 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Cai, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Jiang & D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Han 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10000 iteration (a) ν = 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 5000 6000 7000 8000 9000 10000 11000 12000 13000 iteration (b) ν = 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 4000 4500 5000 5500 6000 6500 7000 7500 8000 iteration (c) ν = 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 iteration 104 (d) ν = 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2 iteration 104 (e) ν = 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 4000 6000 8000 10000 12000 14000 iteration (f) ν = 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 2000 4000 6000 8000 10000 12000 14000 iteration (g) ν = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2 iteration 104 (h) ν = 36 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 10 5 0 5 10 15 20 25 30 35 ratio (i) ν = 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 5 0 5 10 15 20 25 30 35 ratio (j) ν = 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0 5 10 15 20 25 30 35 40 45 ratio (k) ν = 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 0 5 10 15 20 25 30 35 40 45 ratio (l) ν = 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 0 5 10 15 20 25 30 35 40 45 50 ratio (m) ν = 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 0 10 20 30 40 50 ratio (n) ν = 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 0 10 20 30 40 50 ratio (o) ν = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='4 10 0 10 20 30 40 50 ratio (p) ν = 36 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 6: Comparison of PrePDHG with γ = {1, 5/6,3/4, 1/2} for CT reconstruction problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Note that PrePDHG with γ = 1 is exactly iPrePDHG in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' about 8% of iteration number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' for γ = 3/4, it can save about 13% of iteration number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' More interesting, PrePDHG with γ = 1/2 can save about 30% of itera- tion number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' This tells that reducing the parameter γ (in a reasonable range) in PrePDHG for the CT reconstruction problem can still bring some benefits even though the so-called best stepsize τ is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' However, it should be emphasized again that selecting the best stepsize τ is very hard in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' 6 Conclusions In this paper, we investigate the PrePDHG algorithm from the iPADMM point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' We establish the equivalence between PrePDHG and iPADMM, based on which we can obtain a tight convergence condition for PrePDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Some counter- examples are given to show the tightness of the convergence condition we estab- Understanding PrePDHG: a view of indefinite proximal ADMM 37 Table 4: Performance of PrePDHG and iPre-PDHG with best τ for CT recon- struction problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In the table, “a” stands for iPrePDHG, “b”, “c” and “d” stands for PrePDHG with γ = 5/6, γ = 3/4 and γ = 1/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' log10(τbest) time iter ratio % θ a b c d a b c d a b c d b c d 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='74 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='80 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='84 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='92 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='2 36 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='54 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='58 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='62 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='70 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 2954 2703 2571 2073 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='0 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='8 lished for PrePDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' This result subsumes the latest convergence condition for the original PDHG and derives an interesting by-product, namely, the dual stepsize of the BALM can be extended to 4/3 other than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Besides, based on the equivalence between PrePDHG and iPADMM, we also establish the global convergence and the ergodic and non-ergodic sublinear convergence rate of PrePDHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' In order to make PrePDHG practical, we also discuss the various choices of the proxi- mal terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' A variety of numerical results on the matrix game, projection onto the Birkhoff polytope, earth mover’s distance, and CT reconstruction show the efficiency of PrePDHG with improved convergence conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Considering that the subproblems in PrePDHG are still hard to solve in some cases, it would be interesting to investigate the inexact version of PrePDHG in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' Data availability statements The authors confirm that all data generated or analyzed during this study are in- cluded in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content=' The data matrices ρ0, ρ1, and mcvx in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE1T4oBgHgl3EQfMAOS/content/2301.02984v1.pdf'} +page_content='3 are from [36] and downloaded at https://github.' 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Most of +these efforts have concentrated in energy efficient control of the +HVAC (Heating, Ventilation, and Air conditioning) system, which +is usually the core system in charge of controlling buildings’ +conditioning and ventilation. However, in practice, HVAC systems +alone cannot control every aspect of conditioning and comfort +that affects buildings’ occupants. Modern lighting, blind and +window systems, usually considered as independent systems, +when present, can significantly affect building energy use, and +perhaps more importantly, user comfort in terms of thermal, +air quality and illumination conditions. For example, it has been +shown that a blind system can provide 12%∼35% reduction in +cooling load in summer while also improving visual comfort. In +this paper, we take a holistic approach to deal with the trade- +offs between energy use and comfort in commercial buildings. +We developed a system called OCTOPUS, which employs a +novel deep reinforcement learning (DRL) framework that uses +a data-driven approach to find the optimal control sequences +of all building’s subsystems, including HVAC, lighting, blind +and window systems. The DRL architecture includes a novel +reward function that allows the framework to explore the trade- +offs between energy use and users’ comfort, while at the same +time enable the solution of the high-dimensional control problem +due to the interactions of four different building subsystems. In +order to cope with OCTOPUS’s data training requirements, we +argue that calibrated simulations that match the target building +operational points are the vehicle to generate enough data to be +able to train our DRL framework to find the control solution for +the target building. In our work, we trained OCTOPUS with 10- +year weather data and a building model that is implemented +in the EnergyPlus building simulator, which was calibrated +using data from a real production building. Through extensive +simulations we demonstrate that OCTOPUS can achieve 14.26% +and 8.1% energy savings compared with the state-of-the art rule- +based method in a LEED Gold Certified building and the latest +DRL-based method available in the literature respectively, while +maintaining human comfort within a desired range. +Index Terms—HVAC, Energy efficiency, Optimal control, Deep +reinforcement learning +I. INTRODUCTION +Energy saving in buildings is important to society, as +buildings consume 32% energy and 51% electricity demand +worldwide [2], [3]. Rule-based control (RBC) is widely used to +set the actuators (e.g., heating or cooling temperature, and fan +speed) in the HVAC (heating, ventilation, and air-conditioning) +system. The ”rules” in RBC are usually set as some static +thresholds or simple control loops based on the experience of +engineers and facility managers. The thresholds and simple +A preliminary version of this work was published in the Proceedings of +ACM BuildSys 2019 [1]. +control rules may not be optimal and have to be adapted to +new buildings at commissioning time. Many times these rules +are updated in an ad-hoc manner, based on experience and +feedback from occupants and/or trial and error performed by +HVAC engineers during the operational use of the building. As +a result, many model-based approaches have been developed +to model the thermal dynamics of a building and execute a +control algorithm on top of the model, such as Proportional +Integral Derivative (PID) [4] and Model Predictive Control +(MPC) [5]. However, the complexity of the thermal dynamics +and the various influencing factors are hard to be precisely +modeled, which is why the models tend to be simplified +in order deal with the parameter-fitting data requirements +and computational complexity when solving the optimization +problem [5]. +To tackle the limitations of the model-based methods, +some model-free approaches have been proposed based on +reinforcement learning (RL) for HVAC control, including Q- +learning [6] and Deep Reinforcement Learning (DRL) [7]. +With RL, an optimal control policy can be learned by the +trial-and-error interaction between a control agent and a +building, without explicitly modeling the system dynamics. By +adopting a deep neural network as the control agent, DRL- +based schemes can handle large state and action space in +building control [7]. Some recent work [7], [8] has shown +that DRL can provide real-time control for building energy +efficiency. However, all existing methods only consider a +single subsystem in buildings, e.g., the HVAC system [8] or the +heating system [7], ignoring some other subsystems that can +affect performance from the energy use and/or user comfort +point of view. +At present, more and more buildings are been equipped +with automatically-adjustable windows and blinds. For exam- +ple, motor-operated windows and blinds, like the intelligent +products from GEZE [9], have been installed using an effective +natural ventilation strategy [10]. In addition, researchers have +studied the potential of energy saving by jointly controlling +the HVAC system and another subsystem, like blind [11], +lighting [12], and window [13]. For example, the energy +consumed by HVAC can be reduced by 17%∼47% if window- +based natural ventilation is enabled [13]. +In this work, we argue that a holistic approach that considers +all available subsystems (HVAC, blinds, windows, lights) in +buildings, which have complex and non-trivial interactions +should be used in coordination to achieve a specific en- +ergy efficiency/comfort goal. Figure 1 shows a depiction of +a modern building that includes multiple subsystems (e.g., +HVAC, window, blind and lighting) that work together to +arXiv:2301.11510v1 [cs.LG] 27 Jan 2023 + +2 +Supply Fan +Return Fan +Thermostat +Illuminance +Temperature +CO2 Sensor +Room +Return Air +Supply Air +HVAC system +Lighting system +Window +system +VAV +Thermal comfort +Visual comfort +Indoor air quality +Blind +system +Fig. 1: Four Subsystems in a Typical Building. +guarantee human comfort goals, including thermal comfort, +visual comfort, and indoor air quality goals. For example, +indoor temperature can be influenced by three subsystems, +like setting the HVAC temperature (adjusting the discharge +temperature set points at the VAV level), and/or adjusting blind +slats (allowing external sunlight to heat indoor air) and/or the +window system (enabling exchange of indoor and outdoor air). +To achieve more efficient energy management in buildings, +we propose to study the joint control problem of four subsys- +tems of a building to meet three human comfort metrics as +depicted in Figure 2. The energy consumption of a building +is determined by four subsystems and their interaction. It is +challenging to control four subsystems jointly, since they may +have opposite outcomes on different human comfort metrics. +For example, opening the window can improve indoor air +quality and save the energy consumed by the HVAC system +for ventilation, but it may also reduce (in winter) or increase +(in summer) indoor temperature. To handle the temperature +variation caused by the open window, the HVAC system may +need to spend more energy rather than the energy saved by +natural ventilation. +This paper presents a customized DRL-based control sys- +tem, named OCTOPUS, which controls four subsystems of +a building to meet three human comfort requirements with +the best energy efficiency. It leverages all the advantages of +DRL-based control, including fast adaptation to new buildings, +real-time actuation and being able to handle a large state +space. However, to control four subsystems jointly in a unified +framework, we need to tackle three main challenges: +High-Dimension Control Actions. With a uniform DRL +framework, OCTOPUS needs to decide a control action +for four subsystems jointly and periodically, including the +heating/cooling air temperature of the HVAC system, the +brightness level of electric lights, the blind slat range and +the open proportion of the window. Each subsystem adds +one dimension in the action space. The goal of OCTOPUS +is to select the best action combination As from the set of +all possible combinations Aall that meet the requirement of +human comfort with the lowest energy consumption. Since +each subsystem can set its actuator to a large number of +discrete values, e.g., we have 66 possible values to set the +zone temperature by the HVAC system, the set of all possible +action combinations Aall is extremely large, i.e., 2,371,842 +actions in our case. +To solve this problem, we leverage a novel neural architec- +Indoor Air Quality +Thermal Comfort +Visual Comfort +HVAC System +Window System +Blind System +Lighting System +Fig. 2: Relationship between Four Subsystems and Three +Human Comfort Metrics +ture featuring a shared representation followed by four network +branches, one for each action dimension. In addition, from the +shared representation, a state value is obtained that links the +joint interrelations in the action space, and it is added to the +output of the four previous branches. This approach achieves a +linear increase in the number of network outputs by allowing +independence for each action dimension. +Reward Function. To explore the potential energy saving +energy across four subsystems while considering three human +comforts, we formulate this problem into an optimization +problem. We define a reward function in our DRL framework +to solve the optimization problem. The novel reward function +jointly combines energy consumption, thermal comfort, visual +comfort, and indoor air quality, offering better control and +more flexibility to meet the unique requirement of users. +Data Training Requirements. While model-free approaches +in general, and RL techniques in particular, are very powerful, +their main weakness is the amount of data required to train +them properly. The amount of training data should be in +proportion to the action space, which in our case it is +very large. This issue is very important since we cannot +expect building stakeholders to have years of building data +readily available so we can use OCTOPUS. Instead, we use +a calibrated building simulator combined with weather data +that is readily available, in order to generate as much training +data as we needed. We trained our OCTOPUS system with +10-year of weather data of two areas; one is Merced, CA, and +the other one in Chicago, IL, due to their distinct weather +characteristics. The critical point is that this method allows to +train OCTOPUS for any building under any weather profile, as +long as there is a repository of weather data for the location, +and a few months of building data to perform the calibration +of the simulator. +We highlight the main contributions of the paper as follows: +• +To the best of our knowledge, this is the first work that +leverages DRL to balance the tradeoff between energy use +and human comfort in a holistic manner. +• OCTOPUS adopts a special reward function and a new DRL +architecture to tackle the challenges imposed by the combined +joint control of four subsystems with a very large action space. +• We tackle the issue of data training requirement by adopting +a simulation strategy for data generation, and spending effort +in calibrating the simulations to make them as close as possible +to the target building. This allows our system to generate as +much data as needed within a finite amount of time. +II. RELATED WORK +Conventional control of the HVAC system. Model pre- +dictive control (MPC) models have been developed for HVAC + +3 +control. It is a planning-based method that solves an optimal +control problem iteratively over a receding time horizon. Some +of the advantages of MPC are that it takes into consideration +future disturbances and that it can handle multiple constraints +and objectives, e.g., energy and comfort [5]. +However, it can be argued that the main roadblock prevent- +ing widespread adoption of MPC is its reliance on a model +[14], [15]. By some estimates, modeling can account for up to +75% of the time and resources required for implementing MPC +in practice [16]. Because buildings are highly heterogeneous, +a custom model is required for each thermal zone or building +under control. +There are two paradigms for modeling building dynamics: +physics-based and statistics-based [14]. Physics-based models, +e.g., EnergyPlus, utilize physical knowledge and material +properties of a building to create detailed representation of the +building dynamics. A major shortcoming is that such models +are not control-oriented. Nonetheless, it is not impossible to +use such models for control [17]. For instance, exhaustive +search optimization is used to derive control policy for an +EnergyPlus model [18]. Furthermore, physics-based model +requires significant modeling effort, because they have a large +number of free parameters to be specified by engineers (e.g., +2,500 parameters for a medium-sized building [19]); and +information required for determining these parameters are +scattered in different design documents [20]. +Statistical models assume a parametric model form, which +may or may not have physical underpinnings, and identifies +model parameters directly from data. Dinh et al. [21] propose +a hybrid control that combines MPC and direct imitation +learning to reduce energy cost while maintaining a comfortable +indoor temperature. While this approach is potentially scal- +able, a practical problem is that the experimental conditions +required for accurate identification of building systems fall +outside of normal building operations [22]. +Conventional control of multiple subsystems. Blind sys- +tem should be considered as an integral part of fenestration +system design for commercial and office buildings, in order to +balance daylighting requirements versus the need to reduce +solar gains. The impact of glazing area, shading device +properties and shading control on building cooling and lighting +demand was calculated using a coupled lighting and thermal +simulation module [11]. The interactions between cooling and +lighting energy use in perimeter spaces were evaluated as a +function of window-to-wall ratio and shading parameters. +The impacts of window operation on building performance +was investigated [13] for different types of ventilation systems +including natural ventilation, mixed-mode ventilation, and +conventional VAV systems in a medium-size reference office +building. While the results highlighted the impacts of window +operation on energy use and comfort and identified HVAC +energy savings with mixed-mode ventilation during summer +for various climates, the control for window opening fraction +was estimated by experience and is not salable for different +kinds of buildings. +Kolokotsa et al. [23] develop an energy efficient fuzzy +controller based on a genetic algorithm to control four +subsystems (HVAC, lighting, window, and blind) and meet +the occupant requirements of human comfort. However, the +genetic algorithm requires a few minutes to hours to generate +one control action and thus is not practical to be used in real +building control. +RL-based control of the HVAC system. With the devel- +opment of deep learning [24], [25] and deep reinforcement +learning [26], [27], many works apply RL for HVAC control. +RL control can be a “model-free” control method, i.e., an RL +agent has no prior knowledge about the controlled process. +RL learns an optimal control strategy by “trial-and-error”. +Therefore, it can be an online learning method that learns an +optimal control strategy during actual building operations. Pe- +dro et al. [28] investigated the application of a reinforcement- +learning-based supervisory control approach, which actively +learns how to appropriately schedule thermostat temperature +setpoints. However, in HVAC control, online learning may +introduce unstable and poor control actions at the initial stage +of the learning. In addition, it may take a long time (e.g. over +50 days reported in [28]) for an RL agent to converge to a +stable control policy for some cases. Therefore, some studies +choose to use an HVAC simulator to the train the RL agent +offline [29]. +Unlike MPC, simulators with arbitrary high complexity can +be directly used to train RL agents because of its “model-free” +nature. Li et al. [6] adopt Q learning for HVAC control. Dala- +magkidis et al. [30] design a Linear Reinforcement Learning +Controller (LRLC) using linear function approximation of the +state-action value function to meet the thermal comfort with +minimal energy consumption. However, the tabular Q learning +approaches are not suitable for problems with a large state +space, like the state of four subsystems. Le et al. [31], [32] +propose a control method of air free-cooled data centers in +tropics via DRL. Vazquez-Canteli et al. [33] develop a multi- +agent RL implementation for load shaping of grid-interactive +connected buildings. Ding et al. [34] design a model-based +RL method for multi-zone building control. Zhang et al. [7], +[35] implement and deploy a DRL-based control method for +radiant heating systems in a real-life office building. Gao et al. +[36] propose a deep deterministic policy gradients (DDPGs)- +based approach for learning the thermal comfort control policy. +Although the above works can improve the performance of +HVAC control, they only focused on HVAC subsystem. +III. MOTIVATION +In this section, we perform a set of preliminary simulations +in EnergyPlus [37] in order to understand the relationships +between the different subsystems and their impact on human +comfort in a building as described in Figure 2. This is also +used to gain trust that the simulator is being run correctly, with +intuitive results that can be understood. +Our goal is to study the effect of different subsystems to +three human comfort metrics. A single-floor office building +of 100 m2 at Merced, California is modeled. The building is +equipped with a north-facing single-panel window of 2 m2 and +an interior blind. The simulations are conducted with weather +data for the month of October. This is a shoulder season, with +outdoor temperatures being a bit cold, but mostly sunny days, +i.e. high solar gain. + +4 +0 +5 +10 +15 +20 +Time (h) +0.5 +0.0 +0.5 +1.0 +1.5 +Predictive Mean Vote +Baseline +Blind Open +Window Open +HVAC Open +Fig. 3: Thermal Comfort, PMV +6 +8 +10 +12 +14 +16 +18 +Time (h) +0 +1 +2 +3 +Illuminance (x1000 lux) +Blind Open +Blind Close +Fig. 4: Visual Comfort, Illuminance +0 +5 +10 +15 +20 +Time (h) +15 +20 +25 +30 +35 +40 +Temperature ( C) +Outdoor Temperature +Blind Close +Blind Open +Fig. 5: Temperature Effect +Figure 3 shows the effect of three subsystems on thermal +comfort. Predictive Mean Vote (PMV) is used to evaluate +thermal comfort. A PMV value that is close to zero represents +the best thermal comfort, with higher positive values meaning +people are hot, and lower negative values meaning people are +cold. A detailed description of PMV values and ranges will be +provided in Section IV-D2. The baseline case (green-solid) in +Figure 3 shows the case when all three subsystems are closed. +This case acts like a “fishtank” model, where the only effect +in the room is due to the solar gain during the day, with no +other interactions through any system but the window. +When only the blind is open (blue-dashed), the PMV value +can be affected from 1.45 to 1.75, showing an increase in +the temperature due to the increase of solar gain. This is more +prominent in the middle of the day, when the sun is at its apex. +When the window is open (red-dashed-dot), the PMV value is +lowered due to the temperature effect, colder outside air enters +the room, producing a colder, more comfortable temperature. +The HVAC system (black-dot) can maintain the PMV value +to an acceptable range (between -0.5 and +0.5) by forcing +air to be at the correct temperature through the room vents. +From the results of Figure 3, we can conclude that all these +three subsystems have an obvious impact on thermal comfort. +Figure 4 shows the illuminance measured at a place close to +the window from 5 am to 7 pm when the blind is open (green- +solid) and the room has natural light. Illuminance values from +500-1000 lux or higher are acceptable in most environments. +We clearly see that with the blind open, the values are within +this range for most of the day. +Figure 5 shows the indoor temperature when the blind is +open (red-dashed) or closed (blue-solid). The outdoor temper- +ature (green-dash-dot) is lower than the indoor temperature, +due to the ”fish tank” effect and the lack of window open +or an HVAC system on during the day. Combining the results +from Figures 4 and 5 we see that the blind system can save the +energy consumed by the lighting system by reducing the need +of artificial light, but it may also increase the energy used by +the HVAC system in order to maintain the load. However, for +lower outdoor temperatures in winter, the sunlight through the +blind can increase the indoor temperature and save the energy +of the HVAC system. +The simulations are conducted to show some examples of +the non-trivial interactions between subsystems and human +comfort. It is challenging to quantify the complex relationships +among different subsystems and the three human comfort +metrics and serves as motivation for our work. +IV. DESIGN OF OCTOPUS +In this section, we describe in detail the design of OC- +TOPUS, including a system overview, DRL-based building +control, branching dueling Q-Network, and reward function +calculation. +A. OCTOPUS Overview +The design goal of OCTOPUS is to meet the requirement of +human comfort by energy efficient control of four subsystems +in a building. +Our goal is to minimize the energy E consumed by all +subsystems in the building, including the energy used in +heating/cooling coils to heat and cool the air, the electricity +used in the water pumps and flow fans in the HVAC system, +electricity used by the lights, and the electricity used by the +motors to adjust the blinds and windows. +The value of E is constantly being affected by the vector +As, which is an action combination for four subsystems, which +belongs to the vector Aall that is all the possible action +combinations. +In addition to the minimization of energy, we would like to +maintain the human comfort metrics within a particular range. +This can be expressed as Pmin ≤ PMV ≤ Pmax, Vmin ≤ +V ≤ Vmax, and Imin ≤ I ≤ Imax. +PMV is a parameter that measures thermal comfort; V +measures visual comfort; and I measures indoor air quality. +The consumed energy E and the human comfort metrics +(PMV , V , and I) are determined by the current state of +all four subsystems, the outdoor weather and the action we +are about to take. They can be measured in real buildings or +calculated in a building simulator, like EnergyPlus, after the +action is executed. +The achieved human comfort results should fall into an +acceptable range to meet the requirements of users. We use +[Pmin, Pmax], [Vmin, Vmax], [Imin, Imax] to present the +accepted range for thermal comfort, visual comfort and indoor +air quality. They can be set by individual users according to +their preference, or by facility managers based on building +standards. The details on calculation of the above parameters +(E, PMV , V and I), the definition of an action (As) and +the settings of the human comfort ranges (e.g., [Pmin, Pmax]) +will be introduced in Section IV-D. +Our goal is to find the best As from Aall for each action +interval (15 mins in our implementation). The best As should +maintain the three human comfort metrics in their acceptable +ranges for the entire control interval with the lowest energy + +5 +Environment +Meta Data +HVAC +system +lighting +system +blind +system +Customized DRL +Control Method +Visual +Comfort +Thermal +Comfort +Indoor Air +Condition +Reward +Actuation +Energy +Consumption +window +system +Calibrated Model +EnergyPlus +Model +Initialization +On-demand +Optimization +Demand +Formulation +Fig. 6: OCTOPUS Architecture with Four Subsystems (including HVAC, lighting, blind and window systems) +consumption (E). To achieve this goal, we implement a +DRL-based control system for buildings. Figure 6 shows +the overview of OCTOPUS as a building control system. It +consists of three layers, i.e., building layer, control layer, and +user demand layer. The building layer is composed of the real +building or a building simulation model, and the sensor data +management components. It provides sensor data to the control +layer and executes the control actions generated by the latter. +The user demand layer quantifies the user requirement of three +human comfort metrics. The range of each human comfort +metric is then passed to the control layer, which searches for +the optimal control to meet the human comfort ranges with +minimal energy consumption. +B. DRL-based Building Control +1) Basics for DRL and DQN: In a standard RL framework, +as shown in Figure 7, an agent learns an optimal control policy +by trying different control actions to the environment. In our +case, the environment is a building simulation model due to the +extensive data requirements to train the system. With DRL, the +agent is implemented as a deep neural network (DNN). The +agent-environment interactions of one step can be expressed +as a tuple (St, At, St+1, Rt+1), where St is the environment’s +state at time t, At is the control action performed by the agent +at time t, St+1 is the resulting environment’ s state after the +agent has taken the action, Rt+1 is the reward received by the +agent from the environment. The goal of DNN agent training is +to learn an optimal control policy to maximize the accumulated +returned reward by taking different control actions. +2) State in OCTOPUS: The state is what the DRL agent +takes as input for each control step. In this study, the state is +a stack of the current and historical observations, as shown +below: +S = {obt, obt−1, ..., obt−n} , +(1) +where t is the current time step, n is the number of the +historical time steps to be considered, and each ob consists of +the following 15 items: outdoor air temperature (◦C), outdoor +air relative humidity (%), indoor air temperature(◦C), indoor +𝒂𝒄𝒕𝒊𝒐𝒏: 𝒂𝒕 +𝒔𝒕)𝟏 +𝒓𝒕)𝟏 +𝒔𝒕𝒂𝒕𝒆: 𝒔𝒕 +𝒓𝒆𝒘𝒂𝒓𝒅: 𝒓𝒕 +Agent +Environment +Fig. 7: Reinforcement Learning Model. +air relative humidity (%), diffuse solar radiation (W/m2), +direct solar radiation (W/m2), solar incident angle (◦), wind +speed (m/s), wind direction (degree from north), average PMV +(%), heating setpoint of the HVAC system (◦C), cooling +setpoint of the HVAC system (◦C), the dimming level of +lights (%), the window open percentage (%), and the blind +open angle (◦). All the values we can be calculated by the +EnergyPlus simulation model. Min-max normalization is used +to convert each item to a value within 0-1. +3) Action in OCTOPUS: The action is how the DRL +agent controls the environment. Given the state, we want the +agent to find the most suitable action combinations among +HVAC, lighting, blind and window system to balance energy +consumption and three human comfort metrics. There are four +action dimensions when considering these four subsystems, +represented as +At = {Ht, Lt, Bt, Wt} , +(2) +where At is the action combination of four subsystems at time +t. Ht is the temperature set-point of the HVAC system, which +can be set to 66 values. Lt is the dimming level of electric +lights. Bt is the blind slat angle. The range of blind slat can +be adjusted from 0 ◦ ∼ 180 ◦. Wt is the open percentage +of the window. Each of the above three actuation parameters +can be set to 33 values in our current implementation to +achieve a proper balance between control granularity and +calculation complexity. According to Equation 2, the total +number of possible actions in the action space is 2,371,842 +(66 × 33 × 33 × 33). Existing DRL architectures, like Deep + +6 +Branching Dueling Q-Network +State +Advantages dimension 4 +(Window system) +Advantages dimension 2 +(Lighting system) +State value +512 +128 +128 +128 +128 +Q-values +Advantages dimension 1 +(HVAC system) +Q-values +n +Action Combination +argmax +argmax +argmax +earning (Arash Tavakoli, Fabio Pardo, Petar Kormushev) +Action Branching Architectures for Deep Reinforcement L +1 +Advantages dimension 3 +(Blind system) +128 +Q-values +Q-values +argmax +Shared representation +256 +Fig. 8: The Specific Action Branching Network Implemented for the Proposed BDQ Agent +Q-Network (DQN) in [8] and Asynchronous Advantage Actor- +Critic (A3C) in [7], cannot work efficiently in our problem, +because the large number of actions requires to be explicitly +represented in the agent DNN network and it will significantly +increase the number of DNN parameters to be learned and +consequently the training time [38]. To solve this problem, +we leverage a novel neural architecture featuring a shared +representation followed by four network branches, one for +each action dimension. +4) Reward Function in OCTOPUS: Reward illustrates the +immediate evaluation of the control effects for each action +under a certain state. Both human comfort and energy con- +sumption should be incorporated. To define the reward func- +tion, a common approach is to use the Lagrangian Multiplier +function [39] to first convert the constrained formulation into +an unconstrained one: +R = −[ρ1Norm(E) + ρ2Norm(Tc) ++ρ3Norm(Vc) + ρ4Norm(Ic)], +(3) +where ρ1, ρ2, ρ3 and ρ4 are the Lagrangian multipliers. E +is energy consumption, Tc is thermal comfort, V c is visual +comfort and Ic is Indoor air quality. Norm(x) is a normal- +ization process, i.e., Norm(x) = (x - xmin )/(xmax - xmin) +to transform energy and three human comfort to the same +scale. This reward function merges the objective (e.g. energy +consumption) and constraint satisfaction (e.g. human comfort). +The reward consists of four parts, namely, the penalty for +the energy consumption of the HVAC and lighting system, +the penalty for the occupants’ thermal discomfort, the penalty +for the occupants’ visual discomfort and the penalty for the +occupants’ indoor air condition discomfort. Specifically, the +reward should be less, if more energy is consumed by the +HVAC system or the occupants feel uncomfortable about +the building thermal, visual and indoor air condition. The +details about how to define and formulate energy consumption +E, thermal comfort Tc, visual comfort V c and indoor air +condition Ic are explained in Section IV-D. +C. Branching Dueling Q-Network +To solve the high-dimensional action problem described in +Section IV-B3, OCTOPUS adopts a Branching Dueling Q- +Network (BDQ), which is a branching variant of the dueling +Double Deep Q-Network (DDQN). BDQ is a new neural +architecture featuring a shared decision module followed by +several network branches, one for each action dimension. BDQ +can scale robustly to environments with high dimensional +action spaces and even outperform the Deep Deterministic +Policy Gradient (DDPG) algorithm in the most challenging +task [40]. In our current implementation, we use a simulated +building model developed in EnergyPlus as the environment +for training and validation. Our BDQ-based agent interacts +with the EnergyPlus model. At each control step, it processes +the state (building and weather parameters) and generates a +combined action set for four subsystems. +Figure 8 demonstrates the action branching network of BDQ +agent. When a state is inputted, the shared decision module +computes a latent representation that is then used for the +calculation of the state value and the output of the network +(Advantages dimension in Figure 8) for each dimension +branch. The state value and the factorized advantages are then +combined, via a special aggregation layer, to output the Q- +values for each action dimension. These Q-values are then +queried for the generation of a joint-action tuple. The weights +of the fully connected neural layers are denoted by the gray +trapezoids and the size of each layer (i.e. number of units) is +depicted in the figure. +Training Process: The training process of the BDQ-based +control agent is outlined in Algorithm 1. At the beginning, +we first initialize a neural network Q with random weight +θ. Another neural network Q− with the same architecture +is also created. The outer ”for” loop controls the number of +training episodes, and the inner ”for” loop performs control +at each control time step within one training episode. During +the training process, the recent transition tuples (St, At, St+1, +Rt+1) are stored in the replay memory Λ from which a mini- +batch of samples will be generated for neural network training. +The variable At stores the control action in the last step, and +St and St+1 represent the building state in the previous and +current control time steps, respectively. At the beginning of +each time slot t, we first update four actions and obtain the +current state St+1. In line 7, the immediate reward Rt+1 is +calculated by Equation 3. A training mini-batch can be built +by randomly drawing some transition tuples from the memory. +We calculate the target vector and update the weights of +the neural network Q by using an Adam optimizer for every +control step t. Formally, for an action dimension d ∈ 1, ...N +with n discrete actions, a branch’s Q-value at state s ∈ S and +with action ad ∈ Ad is expressed in terms of the common +state value V (s) (the result of the shared representation layer + +7 +Algorithm 1: The Training Process of Our BDQ-Based +Agent +Input: The range of human comfort metrics and +maximum acceptable energy consumption +Output: A trained DRL agent +1 Initialize BDQ’s prediction Q with random weights θ; +2 Initialize BDQ’s target Q− with weight θ− = θ ; +3 for episode =0,1,...,M do +4 +Obtain the initial state St and At randomly; +5 +for control time step t = 0,1,...,T do +6 +Update Ht, Lt, Bt, Wt by the control action, +At; +7 +Calculate reward Rt+1 by Equation 3; +8 +Obtain current state observation St+1; +9 +Store (St, At, St+1, Rt+1) in reply memory Λ; +10 +Draw mini-batch sample transitions from Λ; +11 +Calculate the target vector and update weights +in neural network Q ; +12 +Update target network Q− +d (s, ad) using +Equation 5; +13 +Perform greedy descent iteratively to tune BDQ +by Equation 6. +in Figure 8) and the corresponding (state-dependent) action +advantage Ad(s, ad) of each branch (the result of the each +advantage dimension in Figure 8) by: +Qd(s, ad) = V (s) + (Ad(s, ad) − 1 +n +� +a′ +d∈Ad Ad(s, a +′ +d)). +(4) +The target network Q− will be updated with the latest +weights of the network Q every c control time steps. c is set +to 50 in our current implementation. Q− is used for inferring +the target value for the next c control steps. We use yd to +represent the maximum accumulative reward we can obtain in +the next c steps. yd can be calculated by temporal-difference +(TD) targets in a recursive fashion: +yd = R + γ 1 +N +� +d Q− +d (s +′, arg max +a′ +d⊆Ad +Qd(s +′, a +′ +d)), +(5) +where Q− +d denoting the branch d of the target network Q−; +R is the reward function result; and γ is discount factor. +Finally, at the end of the inner ”for” loop, we calculate the +following loss function every c control steps: +L = E(s,a,r,s′) ∼ D +�� +d(yd − Qd(s, ad))2� +, +(6) +where D denotes a (prioritized) experience replay buffer +and a denotes the joint-action tuple (a1, a2, ..., aN). The loss +function L should decrease as more training episodes are +performed. +D. Reward Calculation +This section describes how we calculate the reward function +in Equation 3, including energy cost E, thermal comfort T, +visual comfort V and indoor air condition I. +TABLE I: PMV Constants +Parameter +Value +Units +Metabolic rate +70 +W/m2 +Clothing Level +0.5 +clo +1) Energy Consumption: The energy consumption of a +building includes heating coil power Ph and cooling coil +power Pc and fan power Pf from the HVAC system and +electric light power Pl from the lighting system. We calculate +the reward function for energy consumption E during a time +slot as +E = (Ph + Pc + Pf + Pl) +(7) +The heating and cooling coil are used to cool or heat the +air and the fan is used to distribute the heating air or cooling +air to the zone. The electric lights are used for normal work +in the zone. They are calculated by EnergyPlus simulator in +our training and evaluation. In our current implementation, +we ignore the power consumed by the water pumps and the +motors to adjust blinds and windows, because it is relatively +small compared with the power consumption of the HAVC +system or the lighting systems, and can be safely ignored (less +than 1% total). +2) Human Comfort: We define and explain the measure- +ment of the three human comfort metrics. +Thermal Comfort: It is determined by the index PMV (Pre- +dictive Mean Vote) that is calculated by Fanger’s equation [41]. +PMV predicts the mean thermal sensation vote on a standard +scale for a large group of persons. The American Society +of Heating Refrigerating and Air Conditioning Engineers +(ASHRAE) developed the thermal comfort index by using +coding -3 for cold, -2 for cool, -1 for slightly cool, 0 for +natural, +1 for slightly warm, +2 for warm, and +3 for hot. +PMV has been adopted by the ISO 7730 standard [42]. The +ISO recommends maintaining PMV at level 0 with a tolerance +of 0.5 as the best thermal comfort. We calculate the reward +function for thermal comfort Tc during a time slot as +Tc = +� +0, +PMV ≤ P +|PMV − P|, +PMV > |P| +(8) +The occupants can feel comfort when PMV value is within +an acceptable range. We denote the range as [−P, P], where P +is the threshold for PMV value. If the PMV value lies within +[−P, P], it will not incur a penalty. Otherwise, it will incur +a penalty for the occupants’ dissatisfaction with the building +thermal condition. There are six primary factors that directly +affect thermal comfort that can be grouped in two categories: +personal factors - because they are characteristics of the +occupants - and environmental factors - which are conditions +of the thermal environment. The former are metabolic rate +and clothing level, the latter are air temperature, mean radiant +temperature, air speed and humidity. The PMV personal +factors parameters are shown in Table I. The PMV personal +factors environmental factors are obtained in real time from +EnergyPlus. + +8 +Visual Comfort: The research on visual comfort is dom- +inated by studies analyzing the presence of an adequate +amount of light where discomfort can be caused by either +too low or too high level of light as glare. In this paper, the +major glare metric is illuminance range [43]. The illuminance +source includes daylight and electrical light. Thus, the main +subsystems that can have an impact on visual comfort are blind +system and lighting system. We calculate the reward function +for visual comfort Vc during a time slot as +Vc = +� +� +� +� +� +−F − ML, +F < ML +0, +ML ≤ F ≤ MH +F − MH, +F > MH +(9) +The occupants can feel comfort when illuminance value F +is within an acceptable range. We denote the range as [ML, +MH], where M is the threshold for illuminance value. If the +illuminance value lies within [ML, MH], it will not incur a +penalty. Otherwise, it will incur the penalty for the occupants’ +dissatisfaction with the building illuminance condition. +Indoor Air Quality: Carbon dioxide (CO2) concentration +in a building is used as a proxy for air quality [44]. The carbon +dioxide concentration comes from building’s users. There are +various other sources of pollution (NOx, Total Volatile Organic +Compounds (TVOC), respirable particles, etc.). Ventilation is +an important means for controlling indoor air quality (IAQ) +in buildings [45]. Ventilation in this work mainly comes from +the HVAC system and the window system. We calculate the +reward function for indoor air condition Ic during a time slot +as +Ic = +� +� +� +� +� +−C − AL, +C < AL +0, +AL ≤ C ≤ AH +C − AH, +C > AH +(10) +The occupants can feel comfort when carbon dioxide con- +centration value C is within an acceptable range. We denote +the range as [AL, AH], where A is the threshold for dioxide +concentration value. If the dioxide concentration value lies +within [AL, AH], it will not incur a penalty. Otherwise, it +will incur a penalty for the occupants’ dissatisfaction with the +building indoor air quality. +V. IMPLEMENTATION OF OCTOPUS +In this section, we illustrate in detail the implementation +of OCTOPUS including platform setup, HVAC modeling and +calibration, and OCTOPUS training. +A. Platform Setup +Figure. 9 shows a conceptual flow diagram of our build- +ing simulation and control platform. Our building model is +rendered using SketchUp [46]. It replicates a LEED Gold +Certified Building in our University Campus. Using Open- +Studio, the HVAC, lighting, blind and window system are +installed in the building/zones. The control scheme - OC- +TOPUS is implemented using Tensorflow, which is an open- +source machine learning library for Python. Using the Building +Control Virtual Test Bed (BCVTB), a Ptolemy II platform +that enables co-simulation across different models [47], we +Controller +(Matlab/Python) +Gateway +(BCVTB) +Building +(EnergyPlus) +Floor Plan +(SketchUp) +Thermal Zone +(OpenStudio) +Simulation Platform +Building Model +Design Process +Fig. 9: Workflow of Octopus +implement the control of each zone temperature set points, +blinds, lighting and window schedule during each action +time in EnergyPlus for our Building alongside weather data. +OCTOPUS is modeled using EnergyPlus version 8.6 [37]. We +train OCTOPUS based on 10-year weather data from two +different cities, Merced, CA and Chicago, IL due to their +distinct weather characteristics. The weather data for Merced +has intensive solar radiation and large variance in temperature, +while Chicago is classified as hot-summer humid continental +with four distinct seasons. To train our model, we define an +“episode” as one inner for loop of Algorithm 1. +B. Rule Based Method +We implement a rule-based method based on our current +campus building control policy. This policy was first set up +at commissioning time by a mechanical engineering company, +and then it was further optimized by two experienced HVAC +engineers when going over the LEED certification process. +First, we assign different zone temperature setpoints. Each +zone has a separate heating and cooling setpoint. The heating +setpoint is set to 70 ◦F, and the cooling setpoint to 74 ◦F during +the warm-up stage. The cooling setpoint is limited between +72◦F and 80◦F, and the heating setpoint is limited between +65◦F and 72◦F. Second, we set control restrictions and +actuator limits and control inputs are subject to the following +constraints: the heating setpoint should not exceed the cooling +setpoint minus 1 +◦F. The adjustment will move both the +existing heating and cooling setpoints upwards or downwards +by the same amount unless the limit has been reached. Third, +for the control Loops: two separate control loops operate to +maintain space temperature at setpoint, the Cooling Loop and +the Heating Loop. Both loops are continuously active. +C. HVAC System Description +The HVAC system we modeled is a single duct central +cooling HVAC with terminal reheat as shown in Figure 10. +The process begins at the supply fan in the air handler unit +(AHU), which supplies air for the zone. The supply fan’s air +first goes through a cooling coil, which cools the air to the +minimum required temperature needed for the zone. Before +air enters a zone, the air passes through a variable air volume + +9 +Supply Fan +Cooling Coil +Outside +Air +Exhaust +Air +Return Fan +AHU +Dampers +Heating Coil +VAV Damper +Supply Air +Return Air +Blind +Window +Light +room +Illuminance +Temperature +CO2 Sensor +Fig. 10: HVAC Single Duct VAV Terminal Reheat Layout. +Initial Proposed +Model +Weather Station +(https://darksky.net) +Interim model #1 +(weather data) +Interim model#2 +(occupancy +schedule) +Interim model #3 +(zone temperature +and HVAC energy) +Calibrated Model +Panasonic PIR and +Grid eye sensors +WebCTRL and +influx database +Actual +measured data +Calibrate Error +MBE, CVRMSE +Fig. 11: Building Model Calibration Process. +(VAV) unit that regulates the amount of air that flows into a +zone. Terminal reheat occurs when the heating coil increases +the temperature before discharging air into a zone. A discharge +setpoint temperature is selected for each zone and the VAV +ensures that the air is heated to this temperature for each zone. +The air supplied to the zone is mixed with the current zone air, +and some of the air is exhausted out of the zone to maintain +a constant static pressure. The return air from each zone is +mixed in the return duct, and then portions of it may enter the +economizer. +D. HVAC Modeling and Calibration +The purpose of the calibration is to ensure the building +energy model can generate energy use results close to the +measured values in the target building using actual inputs, +including weather, occupancy schedule, and the HVAC system +parameters and controls. +The building model calibration process is shown in Figure +11. The first step of the calibration is to collect the real +weather data from a public weather station for the period +to be tested. We use a Dark Sky’s API, a public weather +website, to collect real weather data for three months. The +second step is to replace the default occupancy schedules in +the simulator with the actual occupancy schedules collected +from the real target building using ThermoSense [48]. This +TABLE II: Model Calibration Parameters +Parameter +Range +Adoption +Infiltration Rate +0.01 m3 ∼ 0.5 m3 +0.05 m3 +Window Type +Single/Double Pane +Single +Window Area +1m2 ∼ 4m2 +2m2 +Window Thickness +3mm ∼ 6mm +3mm +Fan Efficiency +0.5 ∼ 0.8 +0.7 +Blind Type +Interior/Exterior Blind +Interior +Blind Thickness +1mm ∼ 6mm +1mm +TABLE III: Modeling Error after Calibration +MBE +CVRMSE +February (hourly temperature) +-1.48% +5.32% +March (hourly temperature) +-0.26% +4.95% +April (hourly temperature) +1.20% +5.06% +May (hourly temperature) +0.48% +4.38% +February - May(monthly energy) +-3.83% +12.33% +system was installed in the target building on our campus and +allows the collection of fine grain occupancy data at the zone +level in the building, allowing the evaluation using accurate +occupancy patterns. We used the hourly occupancy data from +3 months as the occupancy schedule in our simulated building +by EnergyPlus. The third step is to calibrate certain system and +control parameters to match those in the target building we +want to replicate. This involves multiple issues, including (a) +the selection of the parameters to be calibrated, (b) the range +of those parameters, and (c) the step used within the range. +In our work, we use an N-factorial design with 5 parameters +and ranges to be tested based on operational experience. We +tested different combinations of HVAC system parameters +(Infiltration rate) and control (mass flow rate, heating, and +cooling setpoints) and found the combination that minimized +the calibrated error (see below). The selected calibration +parameters are listed in the Table II with their calibration +ranges and value selected. The final step is to compare the +calibrated error between the calibrated model and the actual +measured zone temperature and energy consumption stored +in the operational building database. The whole calibration +process of modeling our building takes nearly one month. +ASHRAE Guideline 14-2002 [49] defines the evaluation +criteria to calibrate BEM models. According to the Guide- +line, monthly and hourly data can be used for calibration. +Mean Bias Error (MBE) and Coefficient of Variation of the +Root Mean Squared Error (CVRMSE) are used as evaluation +indices. The guideline states that the model should have an +MBE of 5% and a CVRMSE of 15% relative to monthly +calibration data. If hourly calibration data are used, these +requirements should be 10% and 30%, respectively. In our +case, hourly data is used to calculate the error metrics for +the average zone temperature. We choose monthly data to + +10 +TABLE IV: Human Comfort Statistical Results for Rule Based, DDQN-HVAC and OCTOPUS Schemes +Location +Method +Metric +PMV +Illuminance (lux) +CO2 Concentration +(ppm) +Energy Consumption +(kWh) +January +July +January +July +January +July +January +July +Merced +Rule Based +Method +Mean +0.03 +-0.25 +576.78 +646.45 +623.61 +668.03 +1990.99 +3583.03 +Std +0.11 +0.13 +152.54 +157.11 +120.64 +181.22 +Violation rate +0 +2% +0.94% +0 +0.3% +3.629% +DDQN-HVAC +[7] +Mean +-0.19 +0.28 +576.78 +646.45 +625.62 +648.01 +1859.10 +3335.58 +Std +0.21 +0.11 +152.54 +157.11 +122.62 +120.57 +Violation rate +2.99% +4.4% +0.94% +0 +0 +0.2% +OCTOPUS +Mean +-0.31 +0.27 +587.12 +569.88 +594.77 +612.33 +1756.24 +2941.46 +Std +0.2 +0.10 +382.27 +75.83 +111.59 +110.35 +Violation rate +5.7% +2.5% +0.26% +0.2% +1.31% +0.33% +Chicago +Rule Based +Method +Mean +-0.28 +-0.15 +583.27 +637.07 +610.26 +638.33 +3848.61 +3309.56 +Std +0.11 +0.02 +163.96 +151.37 +63.94 +151.37 +Violation rate +3.09% +0 +1.1% +0 +0 +0 +DDQN-HVAC +[7] +Mean +-0.32 +0.24 +583.27 +637.07 +612.74 +649.32 +3605.21 +3078.67 +Std +0.08 +0.07 +163.96 +151.37 +65.09 +90.16 +Violation rate +3.7% +2.9% +1.1 % +0 +0 +0 +OCTOPUS +Mean +-0.4 +0.29 +598.34 +544.09 +640.31 +633.71 +3496.54 +2722.03 +Std +0.1 +0.11 +259.88 +55.37 +99.85 +111.04 +Violation rate +4.2% +1.47% +1.6 % +0 +1% +1.31% +TABLE V: Parameter Settings in DRL Algorithms +△tc +15 m +β1 +0.9 +Minibatch Size +64 +β2 +0.999 +Learning Rate +10−4 +Action Dimension +35040 +γ +0.99 +Action Space +2.37 ∗ 107 +calculate energy error metrics because energy data can only be +obtained monthly. The calibration results for zone temperature +and energy consumption are shown in Table III. It is shown +that less than 2% NMBE and less than 6% CVRMSE for the +zone temperature can be achieved with the optimal parameter +setting. We found that both the CVRMSE for the monthly +heating and cooling energy demand is relatively large, but the +NMBE and CVRMSE are still within the acceptable range. +This means the model can achieve accurate calculation for the +monthly energy. +E. OCTOPUS Training +10-year weather data for training from the two locations +tested (Merced, CA and Chicago, IL) is randomly divided, +with eight years used for training and the remaining two +years used for testing. The parameter settings in our DRL +Algorithms are shown in Table V. In our implementation of +OCTOPUS, we use the Adam optimizer [50] for gradient- +based optimization with a learning rate of 10−4. We train +the agent with a minibatch size of 64 and a discount factor +γ = 0.99. The target network is updated every 103 time +steps. We use the rectified non-linearity (or ReLU) [51] for +all hidden layers and linear activation on the output layers. +The network has two hidden layers with 512 and 256 units in +the shared network module and one hidden layer per branch +with 128 units. The weights are initialized using the Xavier +initialization [52] and the biases were initialized to zero. +We used the prioritized replay with a buffer size of 106 +and linear annealing of β from β0 = 0.4 to 1 over 2 x +106 steps. While an ϵ−greedy policy is often used with Q- +learning, random exploration (with an exploration probability) +in physical, continuous-action domains can be inefficient. To +explore actions well in our building environment, we decided +to sample actions from a Gaussian distribution with its mean +at the greedy actions and with a small fixed standard deviation +throughout the training to encourage life-long exploration. We +used a fixed standard deviation of 0.2 during training and zero +during evaluation. This exploration strategy yielded a mildly +better performance as compared to using an ϵ−greedy policy +with a fixed or linearly annealed exploration probability. The +duration of each time (action) slot is 15 minutes. We achieved +convergence of our reward function after 1000 episodes as +explained in Section VI-F. +VI. EVALUATION +In this section, we compare the performance of OCTOPUS +with the rule-based method and the latest DRL-based method. + +11 +5 +10 +15 +20 +25 +30 +DAY +30 +40 +50 +60 +70 +80 +Daily Total Energy Consumption(kWh) +Rule Based Method +DDQN-HVAC +OCTOPUS +Fig. 12: Daily Energy Consumption of Control Methods. +July(M) +July(C) +Average +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +Total Energy Consumption (kWh) +Rule Based Method +OCTOPUS_HVAC +OCTOPUS_HVAC_L +OCTOPUS_HVAC_L_B +OCTOPUS_HVAC_L_B_W +January(M) +January(C) +Fig. 13: Performance Contribution of Each Subsystem. +A. Experiment Setting +The implementation of the rule-based HVAC control has +been introduced in Section V-B. The rule-based method only +controls the HVAC system. For the conventional DRL-based +method, we implement the dueling DQN architecture used +in [7], which controls the water-based heating system. We +name that work as DDQN-HVAC in our comparison. Since +these two benchmarks do not control the light system, for a +fair comparison, we initialize the lights on in all experiments. +OCTOPUS may dim the lights if the blind is open during the +day. In addition, the two benchmarks always leave the blind +and window system closed. +The three human comfort metrics are measured by PMV, +Illuminance, and carbon dioxide concentration. We set the +acceptable range of three human comfort metrics according to +building standards and previous experiences in related work. +The comfort range of PMV is set to -0.5 to 0.5 [53]. The +comfort range of illuminance is set to 500-1000 lux [43]. The +comfort range of carbon dioxide concentration is set to 400- +1000 ppm [45]. +We use three control methods to control the building we +modeled in Section V for two months (January and July) and +at two places with distinct weather patterns. Table IV shows +the human comfort results of three control methods and their +energy consumption. The violation rate is calculated as the +time when the value of a human comfort metric falls beyond +its acceptable range divided by the total simulated time. Other +quality of service metrics, including the amount by the which +the violation occurred, or combination of amount and time +will be explored in future work. +0 +200 +400 +600 +800 +1000 +Episode +3200 +3000 +2800 +2600 +2400 +2200 +2000 +1800 +Reward +Episode Reward +Average Reward +Fig. 14: The Convergence of OCTOPUS. +B. Human Comfort +From the results in Table IV, we see that all three methods +can maintain the PMV value in the desired range for most +of the time since the violation rate is low. The average PMV +violation rate of OCTOPUS and DDQN-HVAC is higher than +the rule-based method by 2.19% and 2.22% respectively. The +reason for this is that the DRL-based methods try to save more +energy by setting the PMV to a value close to the boundary of +the acceptable range. It can be observed in Table IV that the +average PMV value of OCTOPUS and DDQN-HVAC (-0.36 +and -0.26) is closer to the range boundary (-0.5), compared +with the rule-based method (-0.13). +For both visual comfort and indoor air quality, the three +control methods provide a very small violation rate. For illumi- +nance, the mean illuminance value of OCTOPUS and DDQN- +HAVC is 590.69 lux and 610.89 lux respectively. OCTOPUS +saves energy by utilizing natural light as much as possible. +For indoor air quality, the average of CO2 concentration of +OCTOPUS, DDQN-HVAC, and rule-based method is 620.28 +ppm, 633.92 ppm, and 635.06 ppm. OCTOPUS adjusts both +window system and HVAC system to maintain the CO2 +concentration level within the desired range. DDQN-HVAC +and the rule-based method only use the HVAC system. +C. Energy Efficiency +The results in Table IV reveal that OCTOPUS save 14.26% +and 8.1% energy on average, compared with the rule-based +control method and DDQN-HVAC. In both cities, OCTOPUS +achieves similar performance gain. OCTOPUS reduces the +energy consumption of HVAC by using the other subsystems. +Figure 12 shows a daily energy consumption of three control +methods in January at Merced. In most days, OCTOPUS +consumes less energy than the other two methods; however, +OCTOPUS is not always the best although we see clear gains +towards the second half of the month due to a change in +weather temperature. The average range of outdoor temper- +ature changes from 2 ◦C ∼ 13 ◦C in the first half of the +month to -1 ◦C ∼ 18 ◦C in the second half of the month. +OCTOPUS could use external air with the window open for +more natural ventilation. +In Table IV, compared to the rule-based method and DDQN- +HVAC, OCTOPUS saves more energy in July (17.6% and +11.7%) than in January (10.05% and 3.9%). In July, the +outdoor air temperature range at Merced and Chicago is 15◦C + +12 +TABLE VI: Different Parameters for Reward Function in Octopus +Parameter +(ρ1,ρ2, ρ3, ρ4) +PMV +Illuminance +(lux) +CO2 Concentration +(ppm) +Energy (kWh) +Mean +Std +Mean +Std +Mean +Std +1, 1, 1, 1 +-0.36 +0.15 +587.35 +94.52 +587.25 +101.14 +3250.55 +5, 1, 1, 1 +-0.33 +0.16 +611.71 +131 +608.48 +175.1 +3221.20 +10, 1, 1, 1 +-0.31 +0.16 +624.97 +189.04 +647.77 +150.33 +3150.62 +2, 3, 1, 1 +-0.383 +0.10 +569.88 +75.83 +636.5 +179.46 +2941.46 +2, 5, 1, 1 +-0.481 +0.13 +689.23 +146.66 +616.02 +177.32 +2900.44 +∼ 42◦C and 15◦C ∼ 40◦C respectively. The window can be +opened when the temperature is within the acceptable range, +in order to save the energy consumed by the HVAC system. +However, in January, due to the cold weather at both places, +the windows stay closed most of the time and cannot make +much contribution to energy savings. +D. Performance Decomposition +We implement four versions of OCTOPUS to study the +energy saving contribution of each subsystem, i.e., OCTO- +PUS just with the HVAC system (OCTOPUS +HVAC), OC- +TOPUS with HVAC and lighting (OCTOPUS +HVAC L), +OCTOPUS with HVAC, lighting and blind (OCTOPUS +HVAC L B) and OCTOPUS with all four subsystems (OC- +TOPUS +HVAC L B W). Figure 13 depicts the energy con- +sumption of these four versions in two different months and at +two different places (Merced and Chicago). Compared with the +rule-based method, OCTOPUS +HVAC can save 6.16% more +energy by only considering HVAC. When the lighting system +is added in OCTOPUS +HVAC L, 2.73% more energy can +be saved. If the blind system is further added in OCTOPUS +HVAC L B 1.93% more energy can be saved. Finally, when +the window system is added in OCTOPUS +HVAC L B W, +3.44% more energy can be saved. Four subsystems make +different contributions to energy saving in January and July. +In January, four subsystems (i.e., HVAC, lighting, blind and +window) make 6.16%, 2.73%, 1.93% and 0% contribution +of energy savings respectively. In July, the contribution of +these subsystems changes to 5.9 %, 3.31 %, 1.99%, and 6.4% +respectively. The most obvious difference between these two +months is made by the window system (6.4%). The reason +for this has been explained above. In January, the windows +are closed almost all the time. In July, the cold outdoor air is +used to cool down the building instead of using HVAC system. +E. Hyperparameters Setting +The hyperparameters in the reward function (Equation 3) +are tuned to balance between the energy consumption and +human comfort. Table VI shows the performance results of +the trained DRL agents in the selected experiments of the +hyperparameters tuning. The total energy consumption and +the mean and standard deviation of the PMV, Illuminance +and carbon dioxide concentration are used as the evaluation +metrics. It is interesting to find that the control performance +results of the different hyperparameters are not intuitive. +For example, we would expect the bigger ρ1 and smaller +ρ2, ρ3, ρ4 to lead to lower energy consumption and just meet +the requirements of thermal comfort, visual comfort and indoor +air condition. However, the results in Table VI shows that +when increasing the weight of energy, energy consumption +does not necessarily decrease. Such counter-intuitive results +are possibly caused by the delayed reward problem that the +DRL agents are stuck in some local optimal areas during the +training. Out of the five experiments in Table VI, the fourth +row saves 17.9% of the energy consumption with only slightly +worse three human comfort quality in the testing model, which +comparably achieves the best balance between the human +comfort and energy consumption. Therefore, the parameters +in the fourth row are used for the trained agent. +F. Convergence of OCTOPUS training +Figure 14 shows that the accumulated reward of OCTOPUS +in each episode during a training process. We calculate the +reward function every control time step (15 minutes), and +thus one episode (one month) contains 2880 time steps. The +accumulated reward of one episode (episode reward in Figure +14) is the sum of the rewards of 2880 time steps. From the +results in Figure 14, we see that the episode reward increases +and tends to be stable as the number of training episodes +increases. When the episode reward does not change much, it +means that we cannot do further to improve the learned control +policy and thus the training process converges. As indicated in +Figure 14, the training reward fluctuates between two adjacent +episodes, because the number of time steps is large in one +episode, i.e., 2880. The rewards calculated at some of these +2880 time steps may vary dynamically because we randomly +choose some time steps by an exploration rate (determined by +a Gaussian distribution with a standard deviation of 0.2). At +these time steps, we do not use the action generated by the +agent, but randomly choose an action to avoid local minimum +convergence. If we smooth the episode reward using a sliding +window of 10 episodes, the average reward in Figure 14 is +more stable during the training. +VII. DISCUSSION +Deploying in a Real Building. Although we have de- +veloped a calibrated simulation model of a real building +on our campus for training and evaluation, we have not +deployed OCTOPUS in the building, because we do not have + +13 +access to automatic blind and window system at the moment. +We are seeking financial support to work with our facility +team for a possible upgrade. OCTOPUS is designed for real +deployment in buildings. For a new building, we need to build +an EnergyPlus model for it and calibrate the model using real +building operation data. After training the OCTOPUS control +agent using the calibrated simulation model and real weather +data, we can deploy the trained agent in the building for real- +time control. For a certain action interval (e.g., every 10 mins), +the OCTOPUS control agent takes the state of the building as +input and generates the control actions of four subsystems. +OCTOPUS can provide real-time control, as one inference +only takes 22 ms. We plan to deploy OCTOPUS in a real +building in our future work. +Scalability of OCTOPUS. OCTOPUS can work in a one- +zone building with one HVAC system, lighting zone, blind and +window. However, a realistic building (or even a small home) +is usually equipped with many lighting zones, blinds and +windows which may take different actions in one subsystem. +OCTOPUS may solve this scalability problem by increasing +the number of BDQ branches, i.e., each branch corresponds +to one subsystem in each zone of a building. We will tackle +this scalability problem in our future work. +Building Model Calibration. A critical component of +our architecture is the use of a calibrated building model +that is close to the target building, allowing us to generate +sufficient data for our training needs. However, getting a +calibrated model ”right” is a tedious process of trial-and-error +over a large number of parameters. Out of the thousands of +parameters available in EnergyPlus, we use our experience +and consulted experts to determine both the most important +parameters and a sensible range of values to explore (it took +us four weeks to get it ”right”). However, there is no magic +bullet, and this may become a problem, especially for unusual +building architectures or specialized HVAC systems that may +not be trivial to replicate in a simulation environment. +Accepting Users’ Feedback. Some existing work [54] +allows users to send their feedback to the control server. The +feedback can represent a user’s personalized preference on +different human comfort metrics and will be considered in the +control decision process. OCTOPUS can easily accept users’ +feedback to train a better agent model by making a small +modification, i.e., changing the calculated comfort values in +the reward function by the users’ feedback. This can be used +for the initial training or for updated training (once deployed). +For example, the OCTOPUS control agent can be trained +incrementally with a certain time interval (e.g., one month). +The newly-trained agent will be used for real-time. +VIII. CONCLUSIONS +This paper proposes OCTOPUS, a DRL-based control +system for buildings that holistically controls many subsys- +tems in modern buildings (e.g., HVAC, light, blind, window) +and manages the trade-offs between energy use and human +comfort. As part of our architecture, we develop a system +that addresses the issues of large action state, a novel reward +function based on energy and comfort, and data requirements +for training using existing historical weather data together with +a calibrated simulator for the target building. We compare our +results with both the state-of-art rule-based control scheme +obtained from a LEED Gold certified building, a DRL scheme +used for optimized heating in the literature, and show that we +can get 14.26% and 8.1% energy savings while maintaining +(and sometime even improving) human comfort values for +temperature, air quality and lighting. +REFERENCES +[1] X. Ding, W. Du, and A. Cerpa, “Octopus: Deep reinforcement learning +for holistic smart building control,” in ACM BuildSys, 2019. +[2] O. Lucon, D. ¨Urge-Vorsatz, A. Z. Ahmed, H. Akbari, P. Bertoldi, L. F. +Cabeza, N. Eyre, A. Gadgil, L. Harvey, Y. Jiang et al., “Buildings,” +2014. +[3] H. Rajabi, Z. Hu, X. Ding, S. Pan, W. Du, and A. Cerpa, “Modes: M ulti- +sensor o ccupancy d ata-driven e stimation s ystem for smart buildings,” +in Proceedings of the Thirteenth ACM International Conference on +Future Energy Systems, 2022, pp. 228–239. +[4] W. W. Shein, Y. Tan, and A. O. 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Erickson, and A. E. Cerpa, “Thermosense: Occupancy +thermal based sensing for hvac control,” in ACM BuildSys Workshop, +2013. +[49] A. Guideline, “Guideline 14-2002, measurement of energy and demand +savings,” American Society of Heating, Ventilating, and Air Conditioning +Engineers, Atlanta, Georgia, 2002. +[50] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” +arXiv preprint arXiv:1412.6980, 2014. +[51] X. Glorot, A. Bordes, and Y. Bengio, “Deep sparse rectifier neural +networks,” in AISTATS, 2011. +[52] X. Glorot and Y. Bengio, “Understanding the difficulty of training deep +feedforward neural networks,” in AISTATS, 2010. +[53] R. American Society of Heating and A.-C. E. S. 55, “Thermal environ- +mental conditions for human occupancy,” 2017. +[54] D. A. Winkler, A. Beltran, N. P. Esfahani, P. P. Maglio, and A. E. Cerpa, +“Forces: feedback and control for occupants to refine comfort and energy +savings,” in ACM UbiComp, 2016. + diff --git a/gdFJT4oBgHgl3EQfUyyg/content/tmp_files/load_file.txt b/gdFJT4oBgHgl3EQfUyyg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..933948d29794f5bcac5328e2abfa72f88db0ce58 --- /dev/null +++ b/gdFJT4oBgHgl3EQfUyyg/content/tmp_files/load_file.txt @@ -0,0 +1,1065 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf,len=1064 +page_content='1 Exploring Deep Reinforcement Learning for Holistic Smart Building Control Xianzhong Ding, Alberto Cerpa, Member, IEEE, and Wan Du, Member, IEEE Abstract—Recently, significant efforts have been done to improve quality of comfort for commercial buildings’ users while also trying to reduce energy use and costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Most of these efforts have concentrated in energy efficient control of the HVAC (Heating, Ventilation, and Air conditioning) system, which is usually the core system in charge of controlling buildings’ conditioning and ventilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, in practice, HVAC systems alone cannot control every aspect of conditioning and comfort that affects buildings’ occupants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Modern lighting, blind and window systems, usually considered as independent systems, when present, can significantly affect building energy use, and perhaps more importantly, user comfort in terms of thermal, air quality and illumination conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For example, it has been shown that a blind system can provide 12%∼35% reduction in cooling load in summer while also improving visual comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In this paper, we take a holistic approach to deal with the trade- offs between energy use and comfort in commercial buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We developed a system called OCTOPUS, which employs a novel deep reinforcement learning (DRL) framework that uses a data-driven approach to find the optimal control sequences of all building’s subsystems, including HVAC, lighting, blind and window systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The DRL architecture includes a novel reward function that allows the framework to explore the trade- offs between energy use and users’ comfort, while at the same time enable the solution of the high-dimensional control problem due to the interactions of four different building subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In order to cope with OCTOPUS’s data training requirements, we argue that calibrated simulations that match the target building operational points are the vehicle to generate enough data to be able to train our DRL framework to find the control solution for the target building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In our work, we trained OCTOPUS with 10- year weather data and a building model that is implemented in the EnergyPlus building simulator, which was calibrated using data from a real production building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Through extensive simulations we demonstrate that OCTOPUS can achieve 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='26% and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='1% energy savings compared with the state-of-the art rule- based method in a LEED Gold Certified building and the latest DRL-based method available in the literature respectively, while maintaining human comfort within a desired range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Index Terms—HVAC, Energy efficiency, Optimal control, Deep reinforcement learning I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' INTRODUCTION Energy saving in buildings is important to society, as buildings consume 32% energy and 51% electricity demand worldwide [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Rule-based control (RBC) is widely used to set the actuators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', heating or cooling temperature, and fan speed) in the HVAC (heating, ventilation, and air-conditioning) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The ”rules” in RBC are usually set as some static thresholds or simple control loops based on the experience of engineers and facility managers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The thresholds and simple A preliminary version of this work was published in the Proceedings of ACM BuildSys 2019 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' control rules may not be optimal and have to be adapted to new buildings at commissioning time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Many times these rules are updated in an ad-hoc manner, based on experience and feedback from occupants and/or trial and error performed by HVAC engineers during the operational use of the building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' As a result, many model-based approaches have been developed to model the thermal dynamics of a building and execute a control algorithm on top of the model, such as Proportional Integral Derivative (PID) [4] and Model Predictive Control (MPC) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, the complexity of the thermal dynamics and the various influencing factors are hard to be precisely modeled, which is why the models tend to be simplified in order deal with the parameter-fitting data requirements and computational complexity when solving the optimization problem [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' To tackle the limitations of the model-based methods, some model-free approaches have been proposed based on reinforcement learning (RL) for HVAC control, including Q- learning [6] and Deep Reinforcement Learning (DRL) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' With RL, an optimal control policy can be learned by the trial-and-error interaction between a control agent and a building, without explicitly modeling the system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' By adopting a deep neural network as the control agent, DRL- based schemes can handle large state and action space in building control [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Some recent work [7], [8] has shown that DRL can provide real-time control for building energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, all existing methods only consider a single subsystem in buildings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', the HVAC system [8] or the heating system [7], ignoring some other subsystems that can affect performance from the energy use and/or user comfort point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' At present, more and more buildings are been equipped with automatically-adjustable windows and blinds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For exam- ple, motor-operated windows and blinds, like the intelligent products from GEZE [9], have been installed using an effective natural ventilation strategy [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In addition, researchers have studied the potential of energy saving by jointly controlling the HVAC system and another subsystem, like blind [11], lighting [12], and window [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For example, the energy consumed by HVAC can be reduced by 17%∼47% if window- based natural ventilation is enabled [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In this work, we argue that a holistic approach that considers all available subsystems (HVAC, blinds, windows, lights) in buildings, which have complex and non-trivial interactions should be used in coordination to achieve a specific en- ergy efficiency/comfort goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Figure 1 shows a depiction of a modern building that includes multiple subsystems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', HVAC, window, blind and lighting) that work together to arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='11510v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='LG] 27 Jan 2023 2 Supply Fan Return Fan Thermostat Illuminance Temperature CO2 Sensor Room Return Air Supply Air HVAC system Lighting system Window system VAV Thermal comfort Visual comfort Indoor air quality Blind system Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 1: Four Subsystems in a Typical Building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' guarantee human comfort goals, including thermal comfort, visual comfort, and indoor air quality goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For example, indoor temperature can be influenced by three subsystems, like setting the HVAC temperature (adjusting the discharge temperature set points at the VAV level), and/or adjusting blind slats (allowing external sunlight to heat indoor air) and/or the window system (enabling exchange of indoor and outdoor air).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' To achieve more efficient energy management in buildings, we propose to study the joint control problem of four subsys- tems of a building to meet three human comfort metrics as depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The energy consumption of a building is determined by four subsystems and their interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' It is challenging to control four subsystems jointly, since they may have opposite outcomes on different human comfort metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For example, opening the window can improve indoor air quality and save the energy consumed by the HVAC system for ventilation, but it may also reduce (in winter) or increase (in summer) indoor temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' To handle the temperature variation caused by the open window, the HVAC system may need to spend more energy rather than the energy saved by natural ventilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This paper presents a customized DRL-based control sys- tem, named OCTOPUS, which controls four subsystems of a building to meet three human comfort requirements with the best energy efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' It leverages all the advantages of DRL-based control, including fast adaptation to new buildings, real-time actuation and being able to handle a large state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, to control four subsystems jointly in a unified framework, we need to tackle three main challenges: High-Dimension Control Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' With a uniform DRL framework, OCTOPUS needs to decide a control action for four subsystems jointly and periodically, including the heating/cooling air temperature of the HVAC system, the brightness level of electric lights, the blind slat range and the open proportion of the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Each subsystem adds one dimension in the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The goal of OCTOPUS is to select the best action combination As from the set of all possible combinations Aall that meet the requirement of human comfort with the lowest energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Since each subsystem can set its actuator to a large number of discrete values, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', we have 66 possible values to set the zone temperature by the HVAC system, the set of all possible action combinations Aall is extremely large, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', 2,371,842 actions in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' To solve this problem, we leverage a novel neural architec- Indoor Air Quality Thermal Comfort Visual Comfort HVAC System Window System Blind System Lighting System Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 2: Relationship between Four Subsystems and Three Human Comfort Metrics ture featuring a shared representation followed by four network branches, one for each action dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In addition, from the shared representation, a state value is obtained that links the joint interrelations in the action space, and it is added to the output of the four previous branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This approach achieves a linear increase in the number of network outputs by allowing independence for each action dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Reward Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' To explore the potential energy saving energy across four subsystems while considering three human comforts, we formulate this problem into an optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We define a reward function in our DRL framework to solve the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The novel reward function jointly combines energy consumption, thermal comfort, visual comfort, and indoor air quality, offering better control and more flexibility to meet the unique requirement of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Data Training Requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' While model-free approaches in general, and RL techniques in particular, are very powerful, their main weakness is the amount of data required to train them properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The amount of training data should be in proportion to the action space, which in our case it is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This issue is very important since we cannot expect building stakeholders to have years of building data readily available so we can use OCTOPUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Instead, we use a calibrated building simulator combined with weather data that is readily available, in order to generate as much training data as we needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We trained our OCTOPUS system with 10-year of weather data of two areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' one is Merced, CA, and the other one in Chicago, IL, due to their distinct weather characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The critical point is that this method allows to train OCTOPUS for any building under any weather profile, as long as there is a repository of weather data for the location, and a few months of building data to perform the calibration of the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We highlight the main contributions of the paper as follows: To the best of our knowledge, this is the first work that leverages DRL to balance the tradeoff between energy use and human comfort in a holistic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS adopts a special reward function and a new DRL architecture to tackle the challenges imposed by the combined joint control of four subsystems with a very large action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We tackle the issue of data training requirement by adopting a simulation strategy for data generation, and spending effort in calibrating the simulations to make them as close as possible to the target building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This allows our system to generate as much data as needed within a finite amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' RELATED WORK Conventional control of the HVAC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Model pre- dictive control (MPC) models have been developed for HVAC 3 control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' It is a planning-based method that solves an optimal control problem iteratively over a receding time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Some of the advantages of MPC are that it takes into consideration future disturbances and that it can handle multiple constraints and objectives, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', energy and comfort [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, it can be argued that the main roadblock prevent- ing widespread adoption of MPC is its reliance on a model [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' By some estimates, modeling can account for up to 75% of the time and resources required for implementing MPC in practice [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Because buildings are highly heterogeneous, a custom model is required for each thermal zone or building under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' There are two paradigms for modeling building dynamics: physics-based and statistics-based [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Physics-based models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', EnergyPlus, utilize physical knowledge and material properties of a building to create detailed representation of the building dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' A major shortcoming is that such models are not control-oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Nonetheless, it is not impossible to use such models for control [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For instance, exhaustive search optimization is used to derive control policy for an EnergyPlus model [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Furthermore, physics-based model requires significant modeling effort, because they have a large number of free parameters to be specified by engineers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', 2,500 parameters for a medium-sized building [19]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' and information required for determining these parameters are scattered in different design documents [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Statistical models assume a parametric model form, which may or may not have physical underpinnings, and identifies model parameters directly from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Dinh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' [21] propose a hybrid control that combines MPC and direct imitation learning to reduce energy cost while maintaining a comfortable indoor temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' While this approach is potentially scal- able, a practical problem is that the experimental conditions required for accurate identification of building systems fall outside of normal building operations [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Conventional control of multiple subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Blind sys- tem should be considered as an integral part of fenestration system design for commercial and office buildings, in order to balance daylighting requirements versus the need to reduce solar gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The impact of glazing area, shading device properties and shading control on building cooling and lighting demand was calculated using a coupled lighting and thermal simulation module [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The interactions between cooling and lighting energy use in perimeter spaces were evaluated as a function of window-to-wall ratio and shading parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The impacts of window operation on building performance was investigated [13] for different types of ventilation systems including natural ventilation, mixed-mode ventilation, and conventional VAV systems in a medium-size reference office building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' While the results highlighted the impacts of window operation on energy use and comfort and identified HVAC energy savings with mixed-mode ventilation during summer for various climates, the control for window opening fraction was estimated by experience and is not salable for different kinds of buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Kolokotsa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' [23] develop an energy efficient fuzzy controller based on a genetic algorithm to control four subsystems (HVAC, lighting, window, and blind) and meet the occupant requirements of human comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, the genetic algorithm requires a few minutes to hours to generate one control action and thus is not practical to be used in real building control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' RL-based control of the HVAC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' With the devel- opment of deep learning [24], [25] and deep reinforcement learning [26], [27], many works apply RL for HVAC control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' RL control can be a “model-free” control method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', an RL agent has no prior knowledge about the controlled process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' RL learns an optimal control strategy by “trial-and-error”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Therefore, it can be an online learning method that learns an optimal control strategy during actual building operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Pe- dro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' [28] investigated the application of a reinforcement- learning-based supervisory control approach, which actively learns how to appropriately schedule thermostat temperature setpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, in HVAC control, online learning may introduce unstable and poor control actions at the initial stage of the learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In addition, it may take a long time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' over 50 days reported in [28]) for an RL agent to converge to a stable control policy for some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Therefore, some studies choose to use an HVAC simulator to the train the RL agent offline [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Unlike MPC, simulators with arbitrary high complexity can be directly used to train RL agents because of its “model-free” nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' [6] adopt Q learning for HVAC control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Dala- magkidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' [30] design a Linear Reinforcement Learning Controller (LRLC) using linear function approximation of the state-action value function to meet the thermal comfort with minimal energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, the tabular Q learning approaches are not suitable for problems with a large state space, like the state of four subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Le et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' [31], [32] propose a control method of air free-cooled data centers in tropics via DRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Vazquez-Canteli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' [33] develop a multi- agent RL implementation for load shaping of grid-interactive connected buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' [34] design a model-based RL method for multi-zone building control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' [7], [35] implement and deploy a DRL-based control method for radiant heating systems in a real-life office building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' [36] propose a deep deterministic policy gradients (DDPGs)- based approach for learning the thermal comfort control policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Although the above works can improve the performance of HVAC control, they only focused on HVAC subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' MOTIVATION In this section, we perform a set of preliminary simulations in EnergyPlus [37] in order to understand the relationships between the different subsystems and their impact on human comfort in a building as described in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This is also used to gain trust that the simulator is being run correctly, with intuitive results that can be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Our goal is to study the effect of different subsystems to three human comfort metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' A single-floor office building of 100 m2 at Merced, California is modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The building is equipped with a north-facing single-panel window of 2 m2 and an interior blind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The simulations are conducted with weather data for the month of October.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This is a shoulder season, with outdoor temperatures being a bit cold, but mostly sunny days, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' high solar gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 4 0 5 10 15 20 Time (h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5 Predictive Mean Vote Baseline Blind Open Window Open HVAC Open Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 3: Thermal Comfort, PMV 6 8 10 12 14 16 18 Time (h) 0 1 2 3 Illuminance (x1000 lux) Blind Open Blind Close Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 4: Visual Comfort, Illuminance 0 5 10 15 20 Time (h) 15 20 25 30 35 40 Temperature ( C) Outdoor Temperature Blind Close Blind Open Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 5: Temperature Effect Figure 3 shows the effect of three subsystems on thermal comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Predictive Mean Vote (PMV) is used to evaluate thermal comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' A PMV value that is close to zero represents the best thermal comfort, with higher positive values meaning people are hot, and lower negative values meaning people are cold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' A detailed description of PMV values and ranges will be provided in Section IV-D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The baseline case (green-solid) in Figure 3 shows the case when all three subsystems are closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This case acts like a “fishtank” model, where the only effect in the room is due to the solar gain during the day, with no other interactions through any system but the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' When only the blind is open (blue-dashed), the PMV value can be affected from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='45 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='75, showing an increase in the temperature due to the increase of solar gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This is more prominent in the middle of the day, when the sun is at its apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' When the window is open (red-dashed-dot), the PMV value is lowered due to the temperature effect, colder outside air enters the room, producing a colder, more comfortable temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The HVAC system (black-dot) can maintain the PMV value to an acceptable range (between -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5 and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5) by forcing air to be at the correct temperature through the room vents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' From the results of Figure 3, we can conclude that all these three subsystems have an obvious impact on thermal comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Figure 4 shows the illuminance measured at a place close to the window from 5 am to 7 pm when the blind is open (green- solid) and the room has natural light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Illuminance values from 500-1000 lux or higher are acceptable in most environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We clearly see that with the blind open, the values are within this range for most of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Figure 5 shows the indoor temperature when the blind is open (red-dashed) or closed (blue-solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The outdoor temper- ature (green-dash-dot) is lower than the indoor temperature, due to the ”fish tank” effect and the lack of window open or an HVAC system on during the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Combining the results from Figures 4 and 5 we see that the blind system can save the energy consumed by the lighting system by reducing the need of artificial light, but it may also increase the energy used by the HVAC system in order to maintain the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, for lower outdoor temperatures in winter, the sunlight through the blind can increase the indoor temperature and save the energy of the HVAC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The simulations are conducted to show some examples of the non-trivial interactions between subsystems and human comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' It is challenging to quantify the complex relationships among different subsystems and the three human comfort metrics and serves as motivation for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' DESIGN OF OCTOPUS In this section, we describe in detail the design of OC- TOPUS, including a system overview, DRL-based building control, branching dueling Q-Network, and reward function calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS Overview The design goal of OCTOPUS is to meet the requirement of human comfort by energy efficient control of four subsystems in a building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Our goal is to minimize the energy E consumed by all subsystems in the building, including the energy used in heating/cooling coils to heat and cool the air, the electricity used in the water pumps and flow fans in the HVAC system, electricity used by the lights, and the electricity used by the motors to adjust the blinds and windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The value of E is constantly being affected by the vector As, which is an action combination for four subsystems, which belongs to the vector Aall that is all the possible action combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In addition to the minimization of energy, we would like to maintain the human comfort metrics within a particular range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This can be expressed as Pmin ≤ PMV ≤ Pmax, Vmin ≤ V ≤ Vmax, and Imin ≤ I ≤ Imax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' PMV is a parameter that measures thermal comfort;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' V measures visual comfort;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' and I measures indoor air quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The consumed energy E and the human comfort metrics (PMV , V , and I) are determined by the current state of all four subsystems, the outdoor weather and the action we are about to take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' They can be measured in real buildings or calculated in a building simulator, like EnergyPlus, after the action is executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The achieved human comfort results should fall into an acceptable range to meet the requirements of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We use [Pmin, Pmax], [Vmin, Vmax], [Imin, Imax] to present the accepted range for thermal comfort, visual comfort and indoor air quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' They can be set by individual users according to their preference, or by facility managers based on building standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The details on calculation of the above parameters (E, PMV , V and I), the definition of an action (As) and the settings of the human comfort ranges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', [Pmin, Pmax]) will be introduced in Section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Our goal is to find the best As from Aall for each action interval (15 mins in our implementation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The best As should maintain the three human comfort metrics in their acceptable ranges for the entire control interval with the lowest energy 5 Environment Meta Data HVAC system lighting system blind system Customized DRL Control Method Visual Comfort Thermal Comfort Indoor Air Condition Reward Actuation Energy Consumption window system Calibrated Model EnergyPlus Model Initialization On-demand Optimization Demand Formulation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 6: OCTOPUS Architecture with Four Subsystems (including HVAC, lighting, blind and window systems) consumption (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' To achieve this goal, we implement a DRL-based control system for buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Figure 6 shows the overview of OCTOPUS as a building control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' It consists of three layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', building layer, control layer, and user demand layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The building layer is composed of the real building or a building simulation model, and the sensor data management components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' It provides sensor data to the control layer and executes the control actions generated by the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The user demand layer quantifies the user requirement of three human comfort metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The range of each human comfort metric is then passed to the control layer, which searches for the optimal control to meet the human comfort ranges with minimal energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' DRL-based Building Control 1) Basics for DRL and DQN: In a standard RL framework, as shown in Figure 7, an agent learns an optimal control policy by trying different control actions to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In our case, the environment is a building simulation model due to the extensive data requirements to train the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' With DRL, the agent is implemented as a deep neural network (DNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The agent-environment interactions of one step can be expressed as a tuple (St, At, St+1, Rt+1), where St is the environment’s state at time t, At is the control action performed by the agent at time t, St+1 is the resulting environment’ s state after the agent has taken the action, Rt+1 is the reward received by the agent from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The goal of DNN agent training is to learn an optimal control policy to maximize the accumulated returned reward by taking different control actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 2) State in OCTOPUS: The state is what the DRL agent takes as input for each control step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In this study, the state is a stack of the current and historical observations, as shown below: S = {obt, obt−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', obt−n} , (1) where t is the current time step, n is the number of the historical time steps to be considered, and each ob consists of the following 15 items: outdoor air temperature (◦C), outdoor air relative humidity (%), indoor air temperature(◦C), indoor 𝒂𝒄𝒕𝒊𝒐𝒏: 𝒂𝒕 𝒔𝒕)𝟏 𝒓𝒕)𝟏 𝒔𝒕𝒂𝒕𝒆: 𝒔𝒕 𝒓𝒆𝒘𝒂𝒓𝒅: 𝒓𝒕 Agent Environment Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 7: Reinforcement Learning Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' air relative humidity (%), diffuse solar radiation (W/m2), direct solar radiation (W/m2), solar incident angle (◦), wind speed (m/s), wind direction (degree from north), average PMV (%), heating setpoint of the HVAC system (◦C), cooling setpoint of the HVAC system (◦C), the dimming level of lights (%), the window open percentage (%), and the blind open angle (◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' All the values we can be calculated by the EnergyPlus simulation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Min-max normalization is used to convert each item to a value within 0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 3) Action in OCTOPUS: The action is how the DRL agent controls the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Given the state, we want the agent to find the most suitable action combinations among HVAC, lighting, blind and window system to balance energy consumption and three human comfort metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' There are four action dimensions when considering these four subsystems, represented as At = {Ht, Lt, Bt, Wt} , (2) where At is the action combination of four subsystems at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Ht is the temperature set-point of the HVAC system, which can be set to 66 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Lt is the dimming level of electric lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Bt is the blind slat angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The range of blind slat can be adjusted from 0 ◦ ∼ 180 ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Wt is the open percentage of the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Each of the above three actuation parameters can be set to 33 values in our current implementation to achieve a proper balance between control granularity and calculation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' According to Equation 2, the total number of possible actions in the action space is 2,371,842 (66 × 33 × 33 × 33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Existing DRL architectures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' like Deep 6 Branching Dueling Q-Network State Advantages dimension 4 (Window system) Advantages dimension 2 (Lighting system) State value 512 128 128 128 128 Q-values Advantages dimension 1 (HVAC system) Q-values n Action Combination argmax argmax argmax earning (Arash Tavakoli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Fabio Pardo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Petar Kormushev) Action Branching Architectures for Deep Reinforcement L 1 Advantages dimension 3 (Blind system) 128 Q-values Q-values argmax Shared representation 256 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 8: The Specific Action Branching Network Implemented for the Proposed BDQ Agent Q-Network (DQN) in [8] and Asynchronous Advantage Actor- Critic (A3C) in [7], cannot work efficiently in our problem, because the large number of actions requires to be explicitly represented in the agent DNN network and it will significantly increase the number of DNN parameters to be learned and consequently the training time [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' To solve this problem, we leverage a novel neural architecture featuring a shared representation followed by four network branches, one for each action dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 4) Reward Function in OCTOPUS: Reward illustrates the immediate evaluation of the control effects for each action under a certain state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Both human comfort and energy con- sumption should be incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' To define the reward func- tion, a common approach is to use the Lagrangian Multiplier function [39] to first convert the constrained formulation into an unconstrained one: R = −[ρ1Norm(E) + ρ2Norm(Tc) +ρ3Norm(Vc) + ρ4Norm(Ic)], (3) where ρ1, ρ2, ρ3 and ρ4 are the Lagrangian multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' E is energy consumption, Tc is thermal comfort, V c is visual comfort and Ic is Indoor air quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Norm(x) is a normal- ization process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', Norm(x) = (x - xmin )/(xmax - xmin) to transform energy and three human comfort to the same scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This reward function merges the objective (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' energy consumption) and constraint satisfaction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' human comfort).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The reward consists of four parts, namely, the penalty for the energy consumption of the HVAC and lighting system, the penalty for the occupants’ thermal discomfort, the penalty for the occupants’ visual discomfort and the penalty for the occupants’ indoor air condition discomfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Specifically, the reward should be less, if more energy is consumed by the HVAC system or the occupants feel uncomfortable about the building thermal, visual and indoor air condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The details about how to define and formulate energy consumption E, thermal comfort Tc, visual comfort V c and indoor air condition Ic are explained in Section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Branching Dueling Q-Network To solve the high-dimensional action problem described in Section IV-B3, OCTOPUS adopts a Branching Dueling Q- Network (BDQ), which is a branching variant of the dueling Double Deep Q-Network (DDQN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' BDQ is a new neural architecture featuring a shared decision module followed by several network branches, one for each action dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' BDQ can scale robustly to environments with high dimensional action spaces and even outperform the Deep Deterministic Policy Gradient (DDPG) algorithm in the most challenging task [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In our current implementation, we use a simulated building model developed in EnergyPlus as the environment for training and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Our BDQ-based agent interacts with the EnergyPlus model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' At each control step, it processes the state (building and weather parameters) and generates a combined action set for four subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Figure 8 demonstrates the action branching network of BDQ agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' When a state is inputted, the shared decision module computes a latent representation that is then used for the calculation of the state value and the output of the network (Advantages dimension in Figure 8) for each dimension branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The state value and the factorized advantages are then combined, via a special aggregation layer, to output the Q- values for each action dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' These Q-values are then queried for the generation of a joint-action tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The weights of the fully connected neural layers are denoted by the gray trapezoids and the size of each layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' number of units) is depicted in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Training Process: The training process of the BDQ-based control agent is outlined in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' At the beginning, we first initialize a neural network Q with random weight θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Another neural network Q− with the same architecture is also created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The outer ”for” loop controls the number of training episodes, and the inner ”for” loop performs control at each control time step within one training episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' During the training process, the recent transition tuples (St, At, St+1, Rt+1) are stored in the replay memory Λ from which a mini- batch of samples will be generated for neural network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The variable At stores the control action in the last step, and St and St+1 represent the building state in the previous and current control time steps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' At the beginning of each time slot t, we first update four actions and obtain the current state St+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In line 7, the immediate reward Rt+1 is calculated by Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' A training mini-batch can be built by randomly drawing some transition tuples from the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We calculate the target vector and update the weights of the neural network Q by using an Adam optimizer for every control step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Formally, for an action dimension d ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='N with n discrete actions, a branch’s Q-value at state s ∈ S and with action ad ∈ Ad is expressed in terms of the common state value V (s) (the result of the shared representation layer 7 Algorithm 1: The Training Process of Our BDQ-Based Agent Input: The range of human comfort metrics and maximum acceptable energy consumption Output: A trained DRL agent 1 Initialize BDQ’s prediction Q with random weights θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 2 Initialize BDQ’s target Q− with weight θ− = θ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 3 for episode =0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=',M do 4 Obtain the initial state St and At randomly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 5 for control time step t = 0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=',T do 6 Update Ht, Lt, Bt, Wt by the control action, At;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 7 Calculate reward Rt+1 by Equation 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 8 Obtain current state observation St+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 9 Store (St, At, St+1, Rt+1) in reply memory Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 10 Draw mini-batch sample transitions from Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 11 Calculate the target vector and update weights in neural network Q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 12 Update target network Q− d (s, ad) using Equation 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 13 Perform greedy descent iteratively to tune BDQ by Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' in Figure 8) and the corresponding (state-dependent) action advantage Ad(s, ad) of each branch (the result of the each advantage dimension in Figure 8) by: Qd(s, ad) = V (s) + (Ad(s, ad) − 1 n � a′ d∈Ad Ad(s, a ′ d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' (4) The target network Q− will be updated with the latest weights of the network Q every c control time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' c is set to 50 in our current implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Q− is used for inferring the target value for the next c control steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We use yd to represent the maximum accumulative reward we can obtain in the next c steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' yd can be calculated by temporal-difference (TD) targets in a recursive fashion: yd = R + γ 1 N � d Q− d (s ′, arg max a′ d⊆Ad Qd(s ′, a ′ d)), (5) where Q− d denoting the branch d of the target network Q−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' R is the reward function result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' and γ is discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Finally, at the end of the inner ”for” loop, we calculate the following loss function every c control steps: L = E(s,a,r,s′) ∼ D �� d(yd − Qd(s, ad))2� , (6) where D denotes a (prioritized) experience replay buffer and a denotes the joint-action tuple (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', aN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The loss function L should decrease as more training episodes are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Reward Calculation This section describes how we calculate the reward function in Equation 3, including energy cost E, thermal comfort T, visual comfort V and indoor air condition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' TABLE I: PMV Constants Parameter Value Units Metabolic rate 70 W/m2 Clothing Level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5 clo 1) Energy Consumption: The energy consumption of a building includes heating coil power Ph and cooling coil power Pc and fan power Pf from the HVAC system and electric light power Pl from the lighting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We calculate the reward function for energy consumption E during a time slot as E = (Ph + Pc + Pf + Pl) (7) The heating and cooling coil are used to cool or heat the air and the fan is used to distribute the heating air or cooling air to the zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The electric lights are used for normal work in the zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' They are calculated by EnergyPlus simulator in our training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In our current implementation, we ignore the power consumed by the water pumps and the motors to adjust blinds and windows, because it is relatively small compared with the power consumption of the HAVC system or the lighting systems, and can be safely ignored (less than 1% total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 2) Human Comfort: We define and explain the measure- ment of the three human comfort metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Thermal Comfort: It is determined by the index PMV (Pre- dictive Mean Vote) that is calculated by Fanger’s equation [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' PMV predicts the mean thermal sensation vote on a standard scale for a large group of persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The American Society of Heating Refrigerating and Air Conditioning Engineers (ASHRAE) developed the thermal comfort index by using coding -3 for cold, -2 for cool, -1 for slightly cool, 0 for natural, +1 for slightly warm, +2 for warm, and +3 for hot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' PMV has been adopted by the ISO 7730 standard [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The ISO recommends maintaining PMV at level 0 with a tolerance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5 as the best thermal comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We calculate the reward function for thermal comfort Tc during a time slot as Tc = � 0, PMV ≤ P |PMV − P|, PMV > |P| (8) The occupants can feel comfort when PMV value is within an acceptable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We denote the range as [−P, P], where P is the threshold for PMV value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' If the PMV value lies within [−P, P], it will not incur a penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Otherwise, it will incur a penalty for the occupants’ dissatisfaction with the building thermal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' There are six primary factors that directly affect thermal comfort that can be grouped in two categories: personal factors - because they are characteristics of the occupants - and environmental factors - which are conditions of the thermal environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The former are metabolic rate and clothing level, the latter are air temperature, mean radiant temperature, air speed and humidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The PMV personal factors parameters are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The PMV personal factors environmental factors are obtained in real time from EnergyPlus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 8 Visual Comfort: The research on visual comfort is dom- inated by studies analyzing the presence of an adequate amount of light where discomfort can be caused by either too low or too high level of light as glare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In this paper, the major glare metric is illuminance range [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The illuminance source includes daylight and electrical light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Thus, the main subsystems that can have an impact on visual comfort are blind system and lighting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We calculate the reward function for visual comfort Vc during a time slot as Vc = � � � � � −F − ML, F < ML 0, ML ≤ F ≤ MH F − MH, F > MH (9) The occupants can feel comfort when illuminance value F is within an acceptable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We denote the range as [ML, MH], where M is the threshold for illuminance value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' If the illuminance value lies within [ML, MH], it will not incur a penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Otherwise, it will incur the penalty for the occupants’ dissatisfaction with the building illuminance condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Indoor Air Quality: Carbon dioxide (CO2) concentration in a building is used as a proxy for air quality [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The carbon dioxide concentration comes from building’s users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' There are various other sources of pollution (NOx, Total Volatile Organic Compounds (TVOC), respirable particles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Ventilation is an important means for controlling indoor air quality (IAQ) in buildings [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Ventilation in this work mainly comes from the HVAC system and the window system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We calculate the reward function for indoor air condition Ic during a time slot as Ic = � � � � � −C − AL, C < AL 0, AL ≤ C ≤ AH C − AH, C > AH (10) The occupants can feel comfort when carbon dioxide con- centration value C is within an acceptable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We denote the range as [AL, AH], where A is the threshold for dioxide concentration value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' If the dioxide concentration value lies within [AL, AH], it will not incur a penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Otherwise, it will incur a penalty for the occupants’ dissatisfaction with the building indoor air quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' IMPLEMENTATION OF OCTOPUS In this section, we illustrate in detail the implementation of OCTOPUS including platform setup, HVAC modeling and calibration, and OCTOPUS training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Platform Setup Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 9 shows a conceptual flow diagram of our build- ing simulation and control platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Our building model is rendered using SketchUp [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' It replicates a LEED Gold Certified Building in our University Campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Using Open- Studio, the HVAC, lighting, blind and window system are installed in the building/zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The control scheme - OC- TOPUS is implemented using Tensorflow, which is an open- source machine learning library for Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Using the Building Control Virtual Test Bed (BCVTB), a Ptolemy II platform that enables co-simulation across different models [47], we Controller (Matlab/Python) Gateway (BCVTB) Building (EnergyPlus) Floor Plan (SketchUp) Thermal Zone (OpenStudio) Simulation Platform Building Model Design Process Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 9: Workflow of Octopus implement the control of each zone temperature set points, blinds, lighting and window schedule during each action time in EnergyPlus for our Building alongside weather data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS is modeled using EnergyPlus version 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='6 [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We train OCTOPUS based on 10-year weather data from two different cities, Merced, CA and Chicago, IL due to their distinct weather characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The weather data for Merced has intensive solar radiation and large variance in temperature, while Chicago is classified as hot-summer humid continental with four distinct seasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' To train our model, we define an “episode” as one inner for loop of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Rule Based Method We implement a rule-based method based on our current campus building control policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This policy was first set up at commissioning time by a mechanical engineering company, and then it was further optimized by two experienced HVAC engineers when going over the LEED certification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' First, we assign different zone temperature setpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Each zone has a separate heating and cooling setpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The heating setpoint is set to 70 ◦F, and the cooling setpoint to 74 ◦F during the warm-up stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The cooling setpoint is limited between 72◦F and 80◦F, and the heating setpoint is limited between 65◦F and 72◦F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Second, we set control restrictions and actuator limits and control inputs are subject to the following constraints: the heating setpoint should not exceed the cooling setpoint minus 1 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The adjustment will move both the existing heating and cooling setpoints upwards or downwards by the same amount unless the limit has been reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Third, for the control Loops: two separate control loops operate to maintain space temperature at setpoint, the Cooling Loop and the Heating Loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Both loops are continuously active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' HVAC System Description The HVAC system we modeled is a single duct central cooling HVAC with terminal reheat as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The process begins at the supply fan in the air handler unit (AHU), which supplies air for the zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The supply fan’s air first goes through a cooling coil, which cools the air to the minimum required temperature needed for the zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Before air enters a zone, the air passes through a variable air volume 9 Supply Fan Cooling Coil Outside Air Exhaust Air Return Fan AHU Dampers Heating Coil VAV Damper Supply Air Return Air Blind Window Light room Illuminance Temperature CO2 Sensor Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 10: HVAC Single Duct VAV Terminal Reheat Layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Initial Proposed Model Weather Station (https://darksky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='net) Interim model #1 (weather data) Interim model#2 (occupancy schedule) Interim model #3 (zone temperature and HVAC energy) Calibrated Model Panasonic PIR and Grid eye sensors WebCTRL and influx database Actual measured data Calibrate Error MBE, CVRMSE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 11: Building Model Calibration Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' (VAV) unit that regulates the amount of air that flows into a zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Terminal reheat occurs when the heating coil increases the temperature before discharging air into a zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' A discharge setpoint temperature is selected for each zone and the VAV ensures that the air is heated to this temperature for each zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The air supplied to the zone is mixed with the current zone air, and some of the air is exhausted out of the zone to maintain a constant static pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The return air from each zone is mixed in the return duct, and then portions of it may enter the economizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' HVAC Modeling and Calibration The purpose of the calibration is to ensure the building energy model can generate energy use results close to the measured values in the target building using actual inputs, including weather, occupancy schedule, and the HVAC system parameters and controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The building model calibration process is shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The first step of the calibration is to collect the real weather data from a public weather station for the period to be tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We use a Dark Sky’s API, a public weather website, to collect real weather data for three months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The second step is to replace the default occupancy schedules in the simulator with the actual occupancy schedules collected from the real target building using ThermoSense [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This TABLE II: Model Calibration Parameters Parameter Range Adoption Infiltration Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='01 m3 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5 m3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='05 m3 Window Type Single/Double Pane Single Window Area 1m2 ∼ 4m2 2m2 Window Thickness 3mm ∼ 6mm 3mm Fan Efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='7 Blind Type Interior/Exterior Blind Interior Blind Thickness 1mm ∼ 6mm 1mm TABLE III: Modeling Error after Calibration MBE CVRMSE February (hourly temperature) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='48% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='32% March (hourly temperature) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='26% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='95% April (hourly temperature) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='20% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='06% May (hourly temperature) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='48% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='38% February - May(monthly energy) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='83% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='33% system was installed in the target building on our campus and allows the collection of fine grain occupancy data at the zone level in the building, allowing the evaluation using accurate occupancy patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We used the hourly occupancy data from 3 months as the occupancy schedule in our simulated building by EnergyPlus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The third step is to calibrate certain system and control parameters to match those in the target building we want to replicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This involves multiple issues, including (a) the selection of the parameters to be calibrated, (b) the range of those parameters, and (c) the step used within the range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In our work, we use an N-factorial design with 5 parameters and ranges to be tested based on operational experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We tested different combinations of HVAC system parameters (Infiltration rate) and control (mass flow rate, heating, and cooling setpoints) and found the combination that minimized the calibrated error (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The selected calibration parameters are listed in the Table II with their calibration ranges and value selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The final step is to compare the calibrated error between the calibrated model and the actual measured zone temperature and energy consumption stored in the operational building database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The whole calibration process of modeling our building takes nearly one month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' ASHRAE Guideline 14-2002 [49] defines the evaluation criteria to calibrate BEM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' According to the Guide- line, monthly and hourly data can be used for calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Mean Bias Error (MBE) and Coefficient of Variation of the Root Mean Squared Error (CVRMSE) are used as evaluation indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The guideline states that the model should have an MBE of 5% and a CVRMSE of 15% relative to monthly calibration data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' If hourly calibration data are used, these requirements should be 10% and 30%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In our case, hourly data is used to calculate the error metrics for the average zone temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We choose monthly data to 10 TABLE IV: Human Comfort Statistical Results for Rule Based, DDQN-HVAC and OCTOPUS Schemes Location Method Metric PMV Illuminance (lux) CO2 Concentration (ppm) Energy Consumption (kWh) January July January July January July January July Merced Rule Based Method Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='25 576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='78 646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='45 623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='61 668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='03 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='99 3583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='03 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='13 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='54 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='11 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='64 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='22 Violation rate 0 2% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='94% 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='3% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='629% DDQN-HVAC [7] Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='28 576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='78 646.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='45 625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='62 648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='01 1859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='10 3335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='58 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='11 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='54 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='11 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='62 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='57 Violation rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='99% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='94% 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='2% OCTOPUS Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='27 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='12 569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='88 594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='77 612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='33 1756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='24 2941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='46 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='10 382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='27 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='83 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='59 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='35 Violation rate 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='7% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='26% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='2% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='31% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='33% Chicago Rule Based Method Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='32 3605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='21 3078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='67 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='07 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='96 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='37 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='09 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='16 Violation rate 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='7% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='9% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='1 % 0 0 0 OCTOPUS Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='29 598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='34 544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='09 640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='31 633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='71 3496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='54 2722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='03 Std 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='11 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='88 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='37 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='85 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='04 Violation rate 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='2% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='47% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='6 % 0 1% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='31% TABLE V: Parameter Settings in DRL Algorithms △tc 15 m β1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='9 Minibatch Size 64 β2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='999 Learning Rate 10−4 Action Dimension 35040 γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='99 Action Space 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='37 ∗ 107 calculate energy error metrics because energy data can only be obtained monthly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The calibration results for zone temperature and energy consumption are shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' It is shown that less than 2% NMBE and less than 6% CVRMSE for the zone temperature can be achieved with the optimal parameter setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We found that both the CVRMSE for the monthly heating and cooling energy demand is relatively large, but the NMBE and CVRMSE are still within the acceptable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This means the model can achieve accurate calculation for the monthly energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS Training 10-year weather data for training from the two locations tested (Merced, CA and Chicago, IL) is randomly divided, with eight years used for training and the remaining two years used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The parameter settings in our DRL Algorithms are shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In our implementation of OCTOPUS, we use the Adam optimizer [50] for gradient- based optimization with a learning rate of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We train the agent with a minibatch size of 64 and a discount factor γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The target network is updated every 103 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We use the rectified non-linearity (or ReLU) [51] for all hidden layers and linear activation on the output layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The network has two hidden layers with 512 and 256 units in the shared network module and one hidden layer per branch with 128 units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The weights are initialized using the Xavier initialization [52] and the biases were initialized to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We used the prioritized replay with a buffer size of 106 and linear annealing of β from β0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='4 to 1 over 2 x 106 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' While an ϵ−greedy policy is often used with Q- learning, random exploration (with an exploration probability) in physical, continuous-action domains can be inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' To explore actions well in our building environment, we decided to sample actions from a Gaussian distribution with its mean at the greedy actions and with a small fixed standard deviation throughout the training to encourage life-long exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We used a fixed standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='2 during training and zero during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This exploration strategy yielded a mildly better performance as compared to using an ϵ−greedy policy with a fixed or linearly annealed exploration probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The duration of each time (action) slot is 15 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We achieved convergence of our reward function after 1000 episodes as explained in Section VI-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' EVALUATION In this section, we compare the performance of OCTOPUS with the rule-based method and the latest DRL-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 11 5 10 15 20 25 30 DAY 30 40 50 60 70 80 Daily Total Energy Consumption(kWh) Rule Based Method DDQN-HVAC OCTOPUS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 12: Daily Energy Consumption of Control Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' July(M) July(C) Average 0 500 1000 1500 2000 2500 3000 3500 4000 Total Energy Consumption (kWh) Rule Based Method OCTOPUS_HVAC OCTOPUS_HVAC_L OCTOPUS_HVAC_L_B OCTOPUS_HVAC_L_B_W January(M) January(C) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 13: Performance Contribution of Each Subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Experiment Setting The implementation of the rule-based HVAC control has been introduced in Section V-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The rule-based method only controls the HVAC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For the conventional DRL-based method, we implement the dueling DQN architecture used in [7], which controls the water-based heating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We name that work as DDQN-HVAC in our comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Since these two benchmarks do not control the light system, for a fair comparison, we initialize the lights on in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS may dim the lights if the blind is open during the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In addition, the two benchmarks always leave the blind and window system closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The three human comfort metrics are measured by PMV, Illuminance, and carbon dioxide concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We set the acceptable range of three human comfort metrics according to building standards and previous experiences in related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The comfort range of PMV is set to -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5 [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The comfort range of illuminance is set to 500-1000 lux [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The comfort range of carbon dioxide concentration is set to 400- 1000 ppm [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We use three control methods to control the building we modeled in Section V for two months (January and July) and at two places with distinct weather patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Table IV shows the human comfort results of three control methods and their energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The violation rate is calculated as the time when the value of a human comfort metric falls beyond its acceptable range divided by the total simulated time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Other quality of service metrics, including the amount by the which the violation occurred, or combination of amount and time will be explored in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 0 200 400 600 800 1000 Episode 3200 3000 2800 2600 2400 2200 2000 1800 Reward Episode Reward Average Reward Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' 14: The Convergence of OCTOPUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Human Comfort From the results in Table IV, we see that all three methods can maintain the PMV value in the desired range for most of the time since the violation rate is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The average PMV violation rate of OCTOPUS and DDQN-HVAC is higher than the rule-based method by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='19% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='22% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The reason for this is that the DRL-based methods try to save more energy by setting the PMV to a value close to the boundary of the acceptable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' It can be observed in Table IV that the average PMV value of OCTOPUS and DDQN-HVAC (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='36 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='26) is closer to the range boundary (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5), compared with the rule-based method (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For both visual comfort and indoor air quality, the three control methods provide a very small violation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For illumi- nance, the mean illuminance value of OCTOPUS and DDQN- HAVC is 590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='69 lux and 610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='89 lux respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS saves energy by utilizing natural light as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For indoor air quality, the average of CO2 concentration of OCTOPUS, DDQN-HVAC, and rule-based method is 620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='28 ppm, 633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='92 ppm, and 635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='06 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS adjusts both window system and HVAC system to maintain the CO2 concentration level within the desired range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' DDQN-HVAC and the rule-based method only use the HVAC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Energy Efficiency The results in Table IV reveal that OCTOPUS save 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='26% and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='1% energy on average, compared with the rule-based control method and DDQN-HVAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In both cities, OCTOPUS achieves similar performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS reduces the energy consumption of HVAC by using the other subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Figure 12 shows a daily energy consumption of three control methods in January at Merced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In most days, OCTOPUS consumes less energy than the other two methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' however, OCTOPUS is not always the best although we see clear gains towards the second half of the month due to a change in weather temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The average range of outdoor temper- ature changes from 2 ◦C ∼ 13 ◦C in the first half of the month to -1 ◦C ∼ 18 ◦C in the second half of the month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS could use external air with the window open for more natural ventilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In Table IV, compared to the rule-based method and DDQN- HVAC, OCTOPUS saves more energy in July (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='6% and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='7%) than in January (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='05% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='9%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In July, the outdoor air temperature range at Merced and Chicago is 15◦C 12 TABLE VI: Different Parameters for Reward Function in Octopus Parameter (ρ1,ρ2, ρ3, ρ4) PMV Illuminance (lux) CO2 Concentration (ppm) Energy (kWh) Mean Std Mean Std Mean Std 1, 1, 1, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='15 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='35 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='52 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='25 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='14 3250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='55 5, 1, 1, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='16 611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='71 131 608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='48 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='1 3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='20 10, 1, 1, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='16 624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='97 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='04 647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='77 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='33 3150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='62 2, 3, 1, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='10 569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='88 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='83 636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='5 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='46 2941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='46 2, 5, 1, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='481 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='13 689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='23 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='66 616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='02 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='32 2900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='44 ∼ 42◦C and 15◦C ∼ 40◦C respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The window can be opened when the temperature is within the acceptable range, in order to save the energy consumed by the HVAC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, in January, due to the cold weather at both places, the windows stay closed most of the time and cannot make much contribution to energy savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Performance Decomposition We implement four versions of OCTOPUS to study the energy saving contribution of each subsystem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', OCTO- PUS just with the HVAC system (OCTOPUS HVAC), OC- TOPUS with HVAC and lighting (OCTOPUS HVAC L), OCTOPUS with HVAC, lighting and blind (OCTOPUS HVAC L B) and OCTOPUS with all four subsystems (OC- TOPUS HVAC L B W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Figure 13 depicts the energy con- sumption of these four versions in two different months and at two different places (Merced and Chicago).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Compared with the rule-based method, OCTOPUS HVAC can save 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='16% more energy by only considering HVAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' When the lighting system is added in OCTOPUS HVAC L, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='73% more energy can be saved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' If the blind system is further added in OCTOPUS HVAC L B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='93% more energy can be saved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Finally, when the window system is added in OCTOPUS HVAC L B W, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='44% more energy can be saved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Four subsystems make different contributions to energy saving in January and July.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In January, four subsystems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', HVAC, lighting, blind and window) make 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='16%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='73%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='93% and 0% contribution of energy savings respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In July, the contribution of these subsystems changes to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='9 %, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='31 %, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='99%, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='4% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The most obvious difference between these two months is made by the window system (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='4%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The reason for this has been explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In January, the windows are closed almost all the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' In July, the cold outdoor air is used to cool down the building instead of using HVAC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Hyperparameters Setting The hyperparameters in the reward function (Equation 3) are tuned to balance between the energy consumption and human comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Table VI shows the performance results of the trained DRL agents in the selected experiments of the hyperparameters tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The total energy consumption and the mean and standard deviation of the PMV, Illuminance and carbon dioxide concentration are used as the evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' It is interesting to find that the control performance results of the different hyperparameters are not intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For example, we would expect the bigger ρ1 and smaller ρ2, ρ3, ρ4 to lead to lower energy consumption and just meet the requirements of thermal comfort, visual comfort and indoor air condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, the results in Table VI shows that when increasing the weight of energy, energy consumption does not necessarily decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Such counter-intuitive results are possibly caused by the delayed reward problem that the DRL agents are stuck in some local optimal areas during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Out of the five experiments in Table VI, the fourth row saves 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='9% of the energy consumption with only slightly worse three human comfort quality in the testing model, which comparably achieves the best balance between the human comfort and energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Therefore, the parameters in the fourth row are used for the trained agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Convergence of OCTOPUS training Figure 14 shows that the accumulated reward of OCTOPUS in each episode during a training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We calculate the reward function every control time step (15 minutes), and thus one episode (one month) contains 2880 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The accumulated reward of one episode (episode reward in Figure 14) is the sum of the rewards of 2880 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' From the results in Figure 14, we see that the episode reward increases and tends to be stable as the number of training episodes increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' When the episode reward does not change much, it means that we cannot do further to improve the learned control policy and thus the training process converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' As indicated in Figure 14, the training reward fluctuates between two adjacent episodes, because the number of time steps is large in one episode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', 2880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The rewards calculated at some of these 2880 time steps may vary dynamically because we randomly choose some time steps by an exploration rate (determined by a Gaussian distribution with a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' At these time steps, we do not use the action generated by the agent, but randomly choose an action to avoid local minimum convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' If we smooth the episode reward using a sliding window of 10 episodes, the average reward in Figure 14 is more stable during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' DISCUSSION Deploying in a Real Building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Although we have de- veloped a calibrated simulation model of a real building on our campus for training and evaluation, we have not deployed OCTOPUS in the building, because we do not have 13 access to automatic blind and window system at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We are seeking financial support to work with our facility team for a possible upgrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS is designed for real deployment in buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For a new building, we need to build an EnergyPlus model for it and calibrate the model using real building operation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' After training the OCTOPUS control agent using the calibrated simulation model and real weather data, we can deploy the trained agent in the building for real- time control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For a certain action interval (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', every 10 mins), the OCTOPUS control agent takes the state of the building as input and generates the control actions of four subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS can provide real-time control, as one inference only takes 22 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We plan to deploy OCTOPUS in a real building in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Scalability of OCTOPUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS can work in a one- zone building with one HVAC system, lighting zone, blind and window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, a realistic building (or even a small home) is usually equipped with many lighting zones, blinds and windows which may take different actions in one subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS may solve this scalability problem by increasing the number of BDQ branches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', each branch corresponds to one subsystem in each zone of a building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We will tackle this scalability problem in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Building Model Calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' A critical component of our architecture is the use of a calibrated building model that is close to the target building, allowing us to generate sufficient data for our training needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, getting a calibrated model ”right” is a tedious process of trial-and-error over a large number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Out of the thousands of parameters available in EnergyPlus, we use our experience and consulted experts to determine both the most important parameters and a sensible range of values to explore (it took us four weeks to get it ”right”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' However, there is no magic bullet, and this may become a problem, especially for unusual building architectures or specialized HVAC systems that may not be trivial to replicate in a simulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Accepting Users’ Feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' Some existing work [54] allows users to send their feedback to the control server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The feedback can represent a user’s personalized preference on different human comfort metrics and will be considered in the control decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' OCTOPUS can easily accept users’ feedback to train a better agent model by making a small modification, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', changing the calculated comfort values in the reward function by the users’ feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' This can be used for the initial training or for updated training (once deployed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' For example, the OCTOPUS control agent can be trained incrementally with a certain time interval (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', one month).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' The newly-trained agent will be used for real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' CONCLUSIONS This paper proposes OCTOPUS, a DRL-based control system for buildings that holistically controls many subsys- tems in modern buildings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=', HVAC, light, blind, window) and manages the trade-offs between energy use and human comfort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' As part of our architecture, we develop a system that addresses the issues of large action state, a novel reward function based on energy and comfort, and data requirements for training using existing historical weather data together with a calibrated simulator for the target building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdFJT4oBgHgl3EQfUyyg/content/2301.11510v1.pdf'} +page_content=' We compare our results with both the state-of-art rule-based control scheme obtained from a LEED Gold 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b/k9E3T4oBgHgl3EQf5wuC/content/tmp_files/2301.04784v1.pdf.txt @@ -0,0 +1,1179 @@ +arXiv:2301.04784v1 [gr-qc] 12 Jan 2023 +Analytical approximate solutions for scalarized AdS +black holes +De-Cheng Zoua∗, Bo Menga†, Ming Zhangb‡, Sheng-Yuan Lia§, +Meng-Yun Laic¶ and Yun Soo Myungd‖ +aCenter for Gravitation and Cosmology and College of Physical Science and Technology, +Yangzhou University, Yangzhou 225009, China +bFaculty of Science, Xi’an Aeronautical University, Xi’an 710077, China +cCollege of Physics and Communication Electronics, Jiangxi Normal University, +Nanchang 330022, China +dInstitute of Basic Sciences and Department of Computer Simulation, +Inje University, Gimhae 50834, Korea +Abstract +The spontaneous scalarization of Schwarzscild-AdS is investigated in the Einstein- +scalar-Gauss–Bonnet (ESGB) theory. Firstly, we construct scalarized AdS black holes +numerically. Secondly, making use of the homotopy analysis method (HAM), we ob- +tain analytical approximate solutions for scalarized AdS black holes in the ESGB +theory. It is found that scalarized AdS black holes constructed numerically are con- +sistent with analytical approximate solutions in the whole space. +∗e-mail address: dczou@yzu.edu.cn +†mb20210111@163.com; +‡zhangming@xaau.edu.cn; +§lishengyuan314159@hotmail.com; +¶mengyunlai@jxnu.edu.cn; +‖ysmyung@inje.ac.kr; +1 + +1 +Introduction +In general relativity (GR), the “no-hair theorem” has always been a hot topic. It allows +that a GR black hole can be described by three observables of mass M, electric charge +Q, and rotation parameter a = J/M [1, 2], and rules out a black hole coupled to a scalar +field in asymptotically flat spacetimes, on account of the divergence of scalar field on the +horizon [3, 4, 5]. In the 1990s, Damour and Esposito-Farese [6, 7] have first found a new +mechanism of spontaneous scalarization in scalar-tensor theory in neutron stars. +This +phenomenon has received a lot of attention lately. Considering a scalar field function f(φ) +coupling to the Gauss–Bonnet curvature term R2 +GB such as f(φ)R2 +GB [8, 9, 10, 11], scalarized +black hole solutions were found in ESGB theory, where the coupling term causes instability +near the event horizon of a Schwarzschild black hole and induces scalarized black holes. +Then, the so-called “no-hair theorem” of GR [12] can be avoided in ESGB theory. It is +worth pointing out that, in ESGB theory, there is no a priori guidance for determining +the coupling function f(φ). The coupling function f(φ) has a decisive influence on the +properties of the scalarized black holes. +For instance, Ref. [8] adopted the exponential +coupling f(φ) ∼ exp(βφ2), while Ref. [9] focused on the quadratic coupling f(φ) ∼ βφ2 +instead. These theories possess black holes with scalar hair, whose properties have been +investigated in great detail [13, 14, 15, 16, 17]. In addition, Ref. [18] has noticed that, +under radial perturbations, the scalarized black holes are unstable for a quadratic coupling, +whereas it is stable for an exponential form in the ESGB theory. Motivated by current and +future gravitational wave observations from black hole mergers, the axial [19] and polar [20] +perturbations of scalarized black holes have been investigated to obtain the quasinormal +modes (QNMs) in the ESGB theory since QNMs could describe the ringdown after merging. +It is well-known that the anti-de Sitter/conformal field theory (AdS/CFT) correspon- +dence provides a powerful framework for studying quantum mechanical aspects of black +hoes [21, 22]. In some scenarios, holographic duality has allowed us to bring CFT knowl- +edge to bear on black hole physics in asymptotically AdS space-time. Moreover, a scalar +field in an asymptotically AdS space-time can cause an asymptotic instability only if its +mass-squared µ2 +eff is less than the BF bound µ2 +BF [23]. Then, the SAdS black hole may +evolve to a scalarized AdS black hole through tachyonic instability, and the “no-hair the- +orem” can usually be circumvented. +Bakopoulos et al. +[24] have firstly discussed the +emergence of novel, regular black hole solutions in ESGB theory. Recently, the scalariza- +2 + +tion of AdS black holes with applications to holographic phase transitions was studied in +Einstein-scalar-Ricci-Gauss–Bonnet gravity [25]. In addition, Guo et al. have discussed the +holographic realization of scalarization in the ESGB gravity with a negative cosmological +constant [26], and a horizon curvature has an effect on the scalarization [27]. +Nevertheless, the numerical black hole solutions were obtained at fixed values of parame- +ters. From these numerical solutions, it is usually hard to give a clear picture for dependence +of the metric on physical parameters of the system. Moreover, these numerical solutions are +displayed by some curves in figures, instead of expressions in explicit form. It causes these +solutions of scalarized black hole to usually need to be re-calculated by colleagues in some +relevant research work. Fortunately, the general methods for parametrization of the black +hole space-times (continued fractions method (CFM) [28] and homotopy analysis method +(HAM) [29, 30]) were developed. The CFM has recently been applied with success in a +variety of contexts [31]-[35]. We stress here that the HAM is also a very powerful method +for obtaining analytical approximate solutions to various nonlinear differential equations +(including systems of nonlinear equations and arising in many different areas of science +and engineering [36]-[42]. Despite its popularity in many areas of science and engineering +over the years, the application of the HAM has been very limited in the fields of general +relativity and gravitation. Recently, this HAM has been adopted to derive analytic ap- +proximate solutions of field equations in Einstein–Weyl gravity [43, 44] as well as analytic +expression of Regge–Wheeler equations under the metric perturbations on Schwarzschild +space-time [45]. In this work, firstly, we construct scalarized AdS black holes numerically. +Secondly, making use of the HAM, we wish to obtain analytical approximate solutions for +scalarized AdS black holes in the ESGB theory. +The plan of our work is as follows. In Section 2, we investigate the tachyonic instability of +Schwarzschild AdS (SAdS) black holes under the linearized scalar perturbation in the ESGB +theory. Then, we construct numerical solutions of scalarized AdS black holes in Section 3. +Section 4 is devoted to deriving analytical approximation solutions by introducing the HAM, +where two solutions are accurate in the whole space outside the event horizon. Finally, we +end the paper with a discussion and conclusions in Section 5. +3 + +2 +Instability of SAdS black hole +The action for ESGB theory with a negative cosmological constant Λ is given by +SESGBC = +1 +16π +� +d4x√−g +� +R − 2Λ − 2∂µφ∂µφ + λ2φ2 +2 +R2 +GB +� +, +(1) +where λ is the scalar coupling constant, R the Ricci scalar, φ a scalar field, and R2 +GB the +Gauss–Bonnet term +R2 +GB = R2 − 4RµνRµν + RµνρσRµνρσ +(2) +with Ricci tensor Rµν and Riemann tensor Rµνρσ. +Varying the action (1) with scalar φ and metric gµν, one obtains the scalar field equation +□φ + λ2 +4 R2 +GBφ = 0 +(3) +and Einstein equation +Gµν = Λgµν + 2∂µφ∂νφ − (∂φ)2gµν − 2λ2∇ρ∇σ(φ2)Pµρνσ, +(4) +where Gµν = Rµν − (R/2)gµν is the Einstein tensor, and Pµρνσ is given by +Pµρνσ += +Rµρνσ + gµσRνρ − gµνRρσ + gνρRµσ − gρσRµν + R +2 (gµνgρσ − gµσgνρ). +(5) +Topological black holes are found without scalar hair as +ds2 +SAdS = −fk(r)dt2 + +1 +fk(r)dr2 + r2 � +dθ2 + sin2 θdϕ2� +(6) +with +fk(r) = k − 2M +r +− Λr2 +3 , +(7) +where Λ = −3/L2 with L the curvature radius of AdS space-time. The cases of k = 0, −1 +were discussed in [27]. Here, afterwards, we choose the k = 1 case of +f(r) = 1 − 2M +r +− Λr2 +3 +(8) +which corresponds to the SAdS black hole. From f(rh) = 0, the outer horizon radius rh of +SAdS black hole is obtained as +rh = − +1 +� +3MΛ2 + +√ +9M2Λ4 − Λ3�1/3 − +� +3MΛ2 + +√ +9M2Λ4 − Λ3�1/3 +Λ +, +(9) +4 + +where the horizon radius rh > 0 is always satisfied on account of a positive mass M > 0 +of a black hole and a negative cosmological constant Λ < 0. Moreover, the mass of SAdS +black hole is determined as +M = 1 +6rh +� +3 − Λr2 +h +� +. +(10) +Now, we discuss the dynamical stability, Breitenlohner–Freedman (BF) bound, and +tachyonic instability of SAdS black hole in the ESGB theory. For this purpose, we need +to consider two linearized equations which describe the propagation of metric perturbation +hµν and scalar perturbation δφ +δRµν(h) = ¯gµν +2 δR + Λhµν, +(11) +¯□δφ − µ2 +effδφ = 0, +(12) +which are obtained by linearizing Equations (3) and (4). As was pointed out in Refs. [46, +47, 48], it is clear that the SAdS black hole is dynamically stable when making use of the +Regge–Wheeler prescription under metric perturbation. In an asymptotically AdS space- +time, a scalar field can cause an asymptotic instability only if its mass-squared µ2 +eff is less +than the BF bound µ2 +BF = − 9 +4L2 ≡ 3Λ +4 [23]. One always finds µ2 +eff > µ2 +BF for large enough r +and thus the SAdS black hole is stable asymptotically against the formation of the scalar +field. However, if µ2 +eff < µ2 +BF in the intermediate region, the SAdS black hole may evolve to +a scalarized AdS black hole through tachyonic instability. In our case, the effective mass +µ2 +eff is fixed as +µ2 +eff = −λ2 +4 +¯R2 +GB = −2λ2Λ2 +3 +− 12λ2M2 +r6 +(13) +and the condition for asymptotic instability is obtained as +µ2 +eff < µ2 +BF : +−2λ2Λ2 +3 +< 3Λ +4 → Λ > −9 +8λ2. +(14) +Now, we are in a position to perform the numerical analysis for the tachyonic instability +of SAdS black hole in the ESGB theory. Taking into account the separation of variables, +δφ(t, r, θ, ϕ) = ψ(r) +r +Ylm(θ, ϕ)e−iωt, +(15) +and introducing a tortoise coordinate dr∗ = dr/(1 − 2M/r − Λr2/3), the radial part of +Equation (12) is given by +d2ψ +dr2 +∗ ++ +� +ω2 − Veff(r) +� +ψ(r) = 0, +(16) +5 + +where the effective potential Veff(r) takes the form +Veff(r) = +� +1 − 2M +r +− Λr2 +3 +��2M +r3 + l(l + 1) +r2 +− 2Λ +3 +� +1 + λ2Λ +� +− 12λ2M2 +r6 +� +. +(17) +In the next sections, we only consider the case of l = 0. +To determine the threshold of tachyonic instability, one has to solve the second-order +differential equation numerically +d2ψ +dr2 +∗ +− +� +Ω2 + Veff(r) +� +ψ(r) = 0, +(18) +which allows an exponentially growing mode of eΩt (ω = iΩ, Ω > 0) as an unstable mode. +Considering Ω = 0, we may solve the static linearized equation +d2ψ +dr2 +∗ +− Veff(r)ψ(r) = 0, +(19) +to find out the threshold unstable mode propagating around the fixed SAdS black hole +background. To impose the boundary conditions, we first consider the near-horizon ex- +pansion, which is used to set data outside the horizon for a numerical integration to near +infinity +ψ(r) = +� +i≥0 +ψi(r − rh)i. +(20) +In the asymptotic far region, Equation (19) becomes approximately +ψ′′(r) + 2 +rψ′(r) − 2 + 2λ2Λ +r2 +ψ(r) ≈ 0. +(21) +Then, we can obtain the boundary condition of ψ(r) ∼ r− 1 +2± 1 +2 +√ +9+8λ2Λ at large r. Therefore, +the numerical solution to Equation (19) can be performed by using the shooting method in +the region between the black hole horizon and infinity, seeking for a value of the eigenvalue +λ. These solutions are labelled by an integer n ∈ N0: n = 0 is the fundamental mode, +whereas n > 1 are excited states (overtones). We focus on the fundamental mode since +the fundamental solutions is usually stable. Varying −Λ/3, a set of bifurcation points con- +stitutes the existence curve (threshold curve for tachyonic instability). Figure 1a includes +three threshold curves of rh = 1, 2, 4. If one chooses rh = 1, the unstable region is the +upper of threshold curve while the stable region is the lower of threshold curve; see Fig- +ure 1b. In case of −Λ/3 → 0, the value of coupling parameter λ matches the threshold +6 + +rh�1 +rh�2 +rh�4 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +� +� +3 +1 +2 +3 +4 +Λ +Λth +SAdS����3� +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +� +� +3 +0.8 +0.9 +1.0 +1.1 +1.2 +Λ +Figure 1: +(Left) The existence curve for scalarized AdS black holes (threshold curve +λSAdS +th +(−Λ/3) of tachyonic instability) in the (−Λ/3, λ) plane for three different horizon +radii rh = 1, 2, 4; (Right) the unstable region is plotted for the horizon radius rh = 1 of +SAdS black holes. +value (λS +th = 0.852, 1.704, 3.408) for the fundamental mode of the Schwarzschild black hole +in [9, 11]. This result naturally leads to the fact that the SAdS black hole is unstable in +the upper region and thus there exist scalarized AdS black holes in the ESGB theory. +3 +Numerical Solutions for Scalarized AdS Black Holes +We consider static and spherically symmetric space-times as well as static and spherically +symmetric scalar field configuration. The space-time metric and scalar are chosen to be +ds2 = −A(r)dt2 + +1 +B(r)dr2 + r2 � +dθ2 + sin2 θdϕ2� +, +φ = φ(r). +(22) +Now we try to find the numerical solutions for scalarized AdS black hole in the ESGB +theory. For this purpose, we first introduce a coordinate transformation of z = rh +r so that +the metric functions can be derived in the compact region of 0 ≤ z ≤ 1, and A(r) and +B(r) become A = A(z) and B = B(z). Therefore, z = 0 always corresponds to infinity +(r → ∞), and z = 1 naturally corresponds to the event horizon r = rh of the black hole. +To utilize the threshold values for an unstable region in Figure 1b, we will choose rh = 1 +for the horizon radius of the black hole in the following numerical calculation. +On the other hand, the metric functions A(r) and B(r) in Equation (22) approach r2 as +r → ∞. In other words, the new metric functions A(z) and B(z) with 1/z2 are divergent +at z = 0. Then, we can further define new metric functions +Az(z) → z2A(z), +Bz(z) → z2B(z) +(23) +7 + +so that the new functions Az(z) and Bz(z) are always regular in the whole region of 0 ≤ +z ≤ 1. Fortunately, the scalar field φ(z) is always regular in the whole region under the +coordinate transformation z = rh +r . Then, we set +φz(z) → φ(z). +(24) +Substituting the new metric functions Equation (23) and scalar field Equation (24) into +Equations (4) and (5), we have +eq1 += +zBzA′ +z +� +−r2 +h + 2z(z2 − r2 +hΛ − 3Bz)φzφ′ +z +� ++ Az +� +r2 +h(−z2 + r2 +hΛ − 2z2ΛφzB′ +zφ′ +z) +−Bz +� +r2 +h(−3 + z2(1 + 4Λ)φ′2 +z ) + 4zφz((z2 − 2r2 +hΛ)φ′ +z + r2 +hzΛφ′′ +z) +� ++12zB2 +zφzφ′ +z +� += 0, +(25) +eq2 += +2r2 +hz2ΛBzφzA′ +zφ′ +z − Az +� +− r2 +hz2 + r4 +hΛ − r2 +hzB′ +z + 2z4φzB′ +zφ′ +z − 2r2 +hz2ΛφzB′ +zφ′ +z +−4zB2 +z +� +zφ′2 +z + φz(−φ′ +z + zφ′′ +z) +� ++ Bz +� +3r2 +h + z2(r2 +h + 4z2 − 4r2 +hΛ)φ′2 +z ++2zφz((2z2 + 4r2 +hΛ − 3zB′ +zφz + 2z(z2 − r2 +hΛ)φ′′ +z) +�� += 0, +(26) +eq3 += +z2λ2(z2 − Bz)BzφzA′2 +z + zAz +� +− z3λ2φzA′ +zB′ +z + 2λ2B2 +zφz(−3A′ +z + zA′′ +z) + zBzr2 +hA′ +zφ′ +z ++λ2zBzφz +� +A′ +z(2z + 3B′ +z) − 2z2A′′ +z +� � ++ A2 +z +� +12λ2B2 +zφz + z2B′ +z(2zλ2φz + r2 +hφ′ +z) +−2zBz +� +λ2φz(2z + 3B′ +z) + r2 +h(2φ′ +z − zφ′′ +z) +� � += 0, +(27) +where primes denote derivatives with respect to z. +In order to obtain the asymptotic form of scalarized AdS black holes, we solve three +Equations (25)–(27) numerically via a shooting method. Spherically symmetric black holes +have an event horizon (z = 1), where the metric functions Az and Bz vanish, and the scalar +field φz tends to a constant: +Az(z ≈ 1) = A1(1 − z) + A2(1 − z)2 + · · · , +(28) +Bz(z ≈ 1) = B1(1 − z) + B2(1 − z)2 + · · · , +(29) +φz(z ≈ 1) = φ0 + φ1(1 − z) + · · · , +(30) +where φ0 denotes the scalar field at the horizon. It is worth pointing out that the regularity +of a scalar field, and its first and second derivatives on the horizon give an additional +8 + +0.0 +0.5 +1.0 +1.5 +2.0 +ln r +0 +10 +20 +30 +40 +A�r� +A�r��B�r� +A�r��f�r� +0.5 +1.0 +1.5 +ln r +0.9 +1.0 +1.1 +1.2 +0 +1 +2 +3 +4 +ln r +0.00 +0.05 +0.10 +Φ�r� +Figure 2: The scalarized AdS black hole with λ = 0.892 and −Λ/3 = 0.457 belonging to +the fundamental branch of λ > λb = 0.886 (bifurcation point). Here, f(r) represents the +metric function (8) for the SAdS black hole. +condition +r6 +h − 8r4 +hλ4� +3 + 2r2 +hΛ +� +r2 +hΛ − 2 +� � +φ2 +0 − 48r2 +hλ8Λ +� +r2 +hΛ − 2 +� +φ4 +0 > 0, +(31) +which reduces to that for the Schwarzschild black hole in the limit of Λ → 0 [8]. +On the other hand, the metric functions and scalar field at the infinity (z → 0) should +satisfy the following boundary conditions: +Az = Bz = −Λr2 +h +3 , +φz = 0, +when +z → 0 +(r → ∞). +(32) +We fix rh = 1 for the horizon radius of the black hole during the numerical calculation. By +tunneling the coupling parameter λ and choosing different values of cosmological constant +Λ, we can obtain a nontrivial solution of scalarized AdS black holes in the ESGB gravity. +The numerical solution for fundamental branch is obtained by taking λ = 0.892 and −Λ/3 = +0.457 (greater than 0.886 of bifurcation point) (see Figure 2). We plot all figures in terms +of ln r and thus the horizon is always located at ln rh = 0. Here, f(r) represents the metric +function for the SAdS black hole with φSAdS(r) = 0. Notice that the metric functions A(r) +and B(r) display different behaviors in comparison to those for the SAdS black hole and +these approach the SAdS metric function f(r) as ln r increases. Moreover, a scalar field +φ(r) is a decreasing function with starting with 0.107, and its asymptotic value is zero. +4 +Analytical approximate solutions +In general, it is a difficult task to find exact solutions of nonlinear differential equations. In +Refs. [30, 49], the HAM was developed to obtain analytical approximate solutions to nonlin- +9 + +ear differential equations. Here, we wish to derive analytical approximate solutions for met- +ric functions Az(z), Bz(z) and a scalar field φz(z) by solving nonlinear Equations (25)–(27) +by using the HAM. If we succeed to find them, it will confirm the numerical solutions in +the previous section. +We assume the nonlinear operators Ni, which are suitable for a system of n-nonlinear +differential equations +Ni[yi(t)] = 0, +i = 1, 2, ..., n, +(33) +with unknown function yi(t) and a variable t. Then, the zero-order deformation equation +can be written as [30, 49] +(1 − q)L[φi(t; q) − yi0(t)] = qhiHi(t)Ni[φi(t; q)] +(34) +where L is an auxiliary linear operator with the property L[0] = 0, q ∈ [0, 1] is an em- +bedding parameter in topology (called the homotopy parameter), φi(t; q) are the solutions +of Equation (34) for q ∈ [0, 1], yi0(t) is the initial guesses, and hi ̸= 0 is the so-called +“convergence-control parameters”. Considering the property L[0] = 0, the solutions φi(t; q) +of Equation (34) vary continuously from the initial guess yi0(t) to the actual solution yi(t) +of Equation (33) when the parameter q increases from 0 to 1. Here, we set the auxiliary +functions Hi(t) = 1 without any restrictions. +On the other hand, we can also expand φi(t; q) as the Maclaurin series with respect to +q +φi(t; q) = yi0(t) + +∞ +� +m=1 +yim(t)qm, +yim(t) = 1 +m! +∂mφi(t; q) +∂qm +. +(35) +The proper choice of the initial approximation yi0(t), linear operator L, and convergence +control parameter hi will make the series expansion (35) convergency at q = 1. Therefore, +we obtain +yi(t) = φi(t; 1) = yi0(t) + +∞ +� +m=1 +yim(t). +(36) +Here the function yim(t) could be obtained by solving the mth order deformation equation. +Differentiating Equation (34) m times with respect to the parameter q, setting q = 0, and +dividing by m!, we find the mth order deformation equation +L[yim(t) − χmyim−1(t)] = hiRim(yim−1), +(37) +10 + +where +Rim(yim−1) = +1 +(m − 1)! +∂m−1Ni[φi(t; q)] +∂qm−1 +|q=0, +(38) +and +χm = + + + +0 : m ≤ 1 +1 : m > 1. +(39) +We define the partial sum yM +i (t) by +yM +i (t) = yi0(t) + +M +� +m=1 +yim(t), +(40) +where yM +i (t) are the Mth order approximate solutions of the original Equation (33). +In order to solve Equations (25)–(27) by means of the HAM, we choose the initial +approximations +Az0(z) = Bz0(z) = +� +z2 − r2 +hΛ +3 ++ z3 +3 (−3 + r2 +hΛ) +� +(1 − αz), +(41) +φz0(z) = 107 +1000 +� 72 +100z3 + 28 +100z +� +(42) +with an undetermined constant α and corresponding auxiliary linear operators [50] +L[φz] = d2φz +dz2 , +L[Bz] = dBz +dz , +L[Az] = d2Az +dz2 . +(43) +One can find that the chosen approximations satisfy the initial and boundary conditions, +since Az0 and Bz0 vanish at the event horizon (z = 1), and they reduce to − +r2 +hΛ +3 +as z → 0. +Moreover, the scalar field φz0(z) disappears at infinity and equals 0.107 near the horizon +(z = 1) . +Then, we use the HAM to secure analytical approximations for Equations (25)–(27) by +using the boundary conditions +Az(0) = −r2 +hΛ +3 , +Az(1) = Bz(1) = 0, +φz(0) = 0.107, +φz(1) = 0, +(44) +where we reserve one boundary condition Bz(0) = − +r2 +hΛ +3 +for later computations. The Mth +11 + +order approximations of Az, Bz, φz are written as +Az(α, hi, z) ≈ Az0(α, z) + +M +� +k=1 +Azk(α, hi, z), +(45) +Bz(α, hi, z) ≈ Bz0(α, z) + +M +� +k=1 +Bzk(α, hi, z), +(46) +φz(α, hi, z) ≈ φz0(z) + +M +� +k=1 +φzk(α, hi, z), +(47) +which include the unknown parameter α and the convergence-control parameter hi. +Considering the boundary condition Bz(0) = − +r2 +hΛ +3 +with Mth order approximate ex- +pression (45), one obtains +ΓM(α, hi) ≡ Bz0(α, 0) + +M +� +k=1 +Bzk(α, hi, 0) + r2 +hΛ +3 += 0, +(48) +where ΓM represents an expanded form of the constrained boundary condition. As long as +hi is given, a solution to Equation (48) is easily obtained. We use the technique developed +by Xu et al. [36] to find out the optimal values of hi. In principle, the technique seeks for +minimizing averaged square residual error of Equations (25)–(27) at the mth order +Em(α, hi) += +EN1 +m + EN2 +m + EN3 +m += +1 +S + 1 +S +� +k=0 +�� +N1[ +m +� +n=0 +Azn(zk), +m +� +n=0 +Bzn(zk), +m +� +n=0 +φzn(zk)] +�2 ++ +� +N2[ +m +� +n=0 +Azn(zk), +m +� +n=0 +Bzn(zk), +m +� +n=0 +φzn(zk)] +�2 ++ +� +N3[ +m +� +n=0 +Azn(zk), +m +� +n=0 +Bzn(zk), +m +� +n=0 +φzn(zk)] +�2� +(49) +with +zk = k∆z = k +S , +k = 0, 1, 2, · · · , S. +(50) +We choose S = 40 used with the purpose of optimization for each function. For our problem, +the residual error depends on both α and hi. In fact, both Em(α, hi) and ΓM(α, hi) contain +undetermined parameters: α and hi. Therefore, the optimal convergence-control parame- +ters hi can be determined from the minimum of Em(α, hi), and it is subjected additionally +12 + +to the algebraic Equation (48) which needs to secure the constant α. Mathematically, this +doubly coupled optimization problem implies +(α∗, h∗ +i ) = min{Em(α, hi), ΓM(α, hi) = 0}. +(51) +Considering the 2nd order (M = 2) approximation, we obtain h1 = 1, h2 = −0.00044, +h3 = −22.08711 and α = −0.00402. Importantly, the corresponding 2nd order of analytical +approximate solutions are determined as +Az(z) = +0.4572667 + 0.01498763z + z2 − 1.451489z3 − 0.005843966z4 − 0.06132506z5 ++0.02351596z6 − 0.3336103z7 + 0.5323985z8 − 0.6967972z9 + 1.750668z10 +−1.817722z11 + 1.820144z12 − 2.660147z13 + 1.822905z14 − 197263z15 ++1.916911z16 − 2.772337z17 + 4.148017z18 − 5.315132z19 + 6.836533z20 +−8.412892z21 + 8.246583z22 − 8.138851z23 + 7.799650z24 − 5.610461z25 ++4.227755z26 − 3.541047z27 + 1.900061z28 − 0.9300943z29 + 0.6721495z30 +−0.2165210z31 − 0.007895318z32, +(52) +Bz(z) = +0.4572613 + 0.001838552z + 1.000001z2 − 1.453246z3 − 0.005859518z4 +−2.118835 × 10−5z8 + 4.558038 × 10−5z9 − 7.453177 × 10−5z10 ++1.929635 × 10−4z11 − 2.202587 × 10−4z12 + 3.020311 × 10−4z13 +−3.686329 × 10−4z14 + 2.159995 × 10−4z15 − 3.031449 × 10−5z16 +−2.347457 × 10−4z17 + 4.985287 × 10−4z18 − 7.088783 × 10−4z19 ++6.819188 × 10−4z20 − 2.785920 × 10−4z21 + 5.640031 × 10−5z22 +−9.993102 × 10−5z23 + 7.812429 × 10−5z24 − 7.631288 × 10−5z25 ++1.244101 × 10−4z26 − 1.403398 × 10−4z27 + 1.307204 × 10−4z28 +−9.834623 × 10−5z29 + 4.387319 × 10−5z30 − 1.459050 × 10−5z31 ++1.058479 × 10−5z32, +(53) +13 + +Bznum +Bzana +0.0 +0.2 +0.4 +0.6 +0.8 +z +0.2 +0.4 +0.6 +Bznum +Bzana +0.0 +0.2 +0.4 +0.6 +0.8 +z +0.2 +0.4 +0.6 +Φznum +Φzana +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +z +0.1 +0.2 +0.3 +Figure 3: Comparison figures of metric functions Az, Bz and scalar field φz in the numerical +(solid curve) and analytical approximate (dashed curve) solutions. Here, we choose horizon +radius parameter rh = 1, λ = 0.892 and −Λ/3 = 0.457. +φz(z) = +0.03217793z + 0.07647953z3 + 0.0001430922z4 − 0.0003676241z5 +−0.0002630361z6 − 0.00002948385z7 − 0.004393940z8 + 0.002189734z9 +−0.008413222z10 + 0.007363797z11 − 0.003525100z12 + 0.002154606z13 ++0.01131688z14 − 0.01319623z15 + 0.01786729z16 − 0.009802537z17 +−0.005691524z18 + 0.004912230z19 − 0.004087058z20 + 0.0006557677z21 ++0.004245723z22 − 0.008432530z23 + 0.01134451z24 − 0.008496555z25 ++0.003261786z26 + 0.004689052z27 − 0.01143095z28 + 0.01200959z29 +−0.008546128z30 + 0.002794749z31 + 6.702130 × 10−5z32. +(54) +Now, we can compare the analytic approximate solutions with the numerical solutions +appeared in the previous section. We plot the analytic approximate solutions (Aana +z +, Bana +z +and φana +z +) and numerical solutions (Anum +z +, Bnum +z +and φnum +z +) in Figure 3 for rh = 1, λ = 0.892, +and −Λ/3 = 0.457. They are apparently consistent with each other. +It is interesting to check the accuracies of these analytic approximate solutions and +numerical solutions based on the field Equations (25)–(27). Substituting these solutions +into Equations (25)–(27), the total absolute errors from three field equations are obtained +as +∆Err ≡ |∆eq1| + |∆eq2| + |∆eq3| . +(55) +Total absolute errors of these analytic approximate and numerical solutions are displayed +in Figure 4. One can find that the total absolute error for analytic approximate solutions +is smaller than that for the numerical solutions. In other words, the analytic approximate +solutions are more accurate than the numerical solutions when solving nonlinear Equations +(25)–(27). +14 + +�Errana +�Errnum +0.2 +0.4 +0.6 +0.8 +1.0 +z +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +Figure 4: Total absolute errors for analytic approximate solutions (∆Errana) and numerical +solutions (∆Errnum) from three field equations. +�Φz +�Bz +�Az +0.2 +0.4 +0.6 +0.8 +1.0 +z +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +∆Φz +∆Bz +∆Az +0.2 +0.4 +0.6 +0.8 +1.0 +z +0 +2 +4 +6 +8 +Figure 5: Absolute (Left) and relative (Right) errors between numerical and analytical +approximate solutions. +To comparing these solutions, we further calculate the absolute differences between +numerical and analytical approximate solutions by taking +∆Az = |Anum +z +− Aana +z +| , +∆Bz = |Bnum +z +− Bana +z +| , +∆φz = |φnum +z +− φana +z +| . +(56) +The relative errors can be also calculated in the form of +δAz = |Anum +z +− Aana +z +| +Anum +z +× 100%, +δBz = |Bnum +z +− Bana +z +| +Bnum +z +× 100%, +δφz = |φnum +z +− φana +z +| +φnum +z +× 100%. +(57) +We find that main differences between numerical solutions and analytic approximation +solutions occur close to the event horizon for metric functions A(z) and B(z) and region +far from the black hole for scalar field function φ(z), see Figure 5. +15 + +5 +Conclusions and discussions +In this work, we investigated the spontaneous scalarization for SAdS black holes thoroughly +in ESGB theory. The SAdS black holes become prone to tachyonic instability triggered by +the strong space-time curvature in some region of the parameter space. Then, scalarized +AdS black holes could emerge from SAdS black holes at bifurcating points. +Numerical +solutions for scalarized AdS black holes are obtained for λ = 0.892 and −Λ/3 = 0.457. +Later, we derive the analytical approximate solutions for metric functions A(z) and +B(z) and scalar field φ(z) by using the HAM. The region and rate of convergence of the +series solution for the HAM does not depend on the choice of the initial guess function, +auxiliary linear operator, and an auxiliary function, but it can be effectively controlled by +using a convergence control parameter. Since the approximation is significantly accurate in +the entire space-time outside the event horizon, it can be used for studying the properties +of this particular black hole and the various phenomena. 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Vajravelu, Communications in Nonlinear Science and Numerical +Simulation, 14, (2009), 4078-4089. +20 + diff --git a/k9E3T4oBgHgl3EQf5wuC/content/tmp_files/load_file.txt b/k9E3T4oBgHgl3EQf5wuC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1a73a867f69138ebe4c52e9b3ebd0db70d43d46 --- /dev/null +++ b/k9E3T4oBgHgl3EQf5wuC/content/tmp_files/load_file.txt @@ -0,0 +1,794 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf,len=793 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='04784v1 [gr-qc] 12 Jan 2023 Analytical approximate solutions for scalarized AdS black holes De-Cheng Zoua∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Bo Menga†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Ming Zhangb‡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Sheng-Yuan Lia§,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Meng-Yun Laic¶ and Yun Soo Myungd‖ aCenter for Gravitation and Cosmology and College of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Yangzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Yangzhou 225009,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' China bFaculty of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Xi’an Aeronautical University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Xi’an 710077,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' China cCollege of Physics and Communication Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Jiangxi Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Nanchang 330022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' China dInstitute of Basic Sciences and Department of Computer Simulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Inje University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Gimhae 50834,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Korea Abstract The spontaneous scalarization of Schwarzscild-AdS is investigated in the Einstein- scalar-Gauss–Bonnet (ESGB) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Firstly, we construct scalarized AdS black holes numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Secondly, making use of the homotopy analysis method (HAM), we ob- tain analytical approximate solutions for scalarized AdS black holes in the ESGB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' It is found that scalarized AdS black holes constructed numerically are con- sistent with analytical approximate solutions in the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' ∗e-mail address: dczou@yzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='cn †mb20210111@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' ‡zhangming@xaau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' §lishengyuan314159@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' ¶mengyunlai@jxnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' ‖ysmyung@inje.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='kr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' 1 1 Introduction In general relativity (GR), the “no-hair theorem” has always been a hot topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' It allows that a GR black hole can be described by three observables of mass M, electric charge Q, and rotation parameter a = J/M [1, 2], and rules out a black hole coupled to a scalar field in asymptotically flat spacetimes, on account of the divergence of scalar field on the horizon [3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In the 1990s, Damour and Esposito-Farese [6, 7] have first found a new mechanism of spontaneous scalarization in scalar-tensor theory in neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' This phenomenon has received a lot of attention lately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Considering a scalar field function f(φ) coupling to the Gauss–Bonnet curvature term R2 GB such as f(φ)R2 GB [8, 9, 10, 11], scalarized black hole solutions were found in ESGB theory, where the coupling term causes instability near the event horizon of a Schwarzschild black hole and induces scalarized black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Then, the so-called “no-hair theorem” of GR [12] can be avoided in ESGB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' It is worth pointing out that, in ESGB theory, there is no a priori guidance for determining the coupling function f(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' The coupling function f(φ) has a decisive influence on the properties of the scalarized black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' For instance, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' [8] adopted the exponential coupling f(φ) ∼ exp(βφ2), while Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' [9] focused on the quadratic coupling f(φ) ∼ βφ2 instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' These theories possess black holes with scalar hair, whose properties have been investigated in great detail [13, 14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In addition, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' [18] has noticed that, under radial perturbations, the scalarized black holes are unstable for a quadratic coupling, whereas it is stable for an exponential form in the ESGB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Motivated by current and future gravitational wave observations from black hole mergers, the axial [19] and polar [20] perturbations of scalarized black holes have been investigated to obtain the quasinormal modes (QNMs) in the ESGB theory since QNMs could describe the ringdown after merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' It is well-known that the anti-de Sitter/conformal field theory (AdS/CFT) correspon- dence provides a powerful framework for studying quantum mechanical aspects of black hoes [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In some scenarios, holographic duality has allowed us to bring CFT knowl- edge to bear on black hole physics in asymptotically AdS space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Moreover, a scalar field in an asymptotically AdS space-time can cause an asymptotic instability only if its mass-squared µ2 eff is less than the BF bound µ2 BF [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Then, the SAdS black hole may evolve to a scalarized AdS black hole through tachyonic instability, and the “no-hair the- orem” can usually be circumvented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Bakopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' [24] have firstly discussed the emergence of novel, regular black hole solutions in ESGB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Recently, the scalariza- 2 tion of AdS black holes with applications to holographic phase transitions was studied in Einstein-scalar-Ricci-Gauss–Bonnet gravity [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In addition, Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' have discussed the holographic realization of scalarization in the ESGB gravity with a negative cosmological constant [26], and a horizon curvature has an effect on the scalarization [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Nevertheless, the numerical black hole solutions were obtained at fixed values of parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' From these numerical solutions, it is usually hard to give a clear picture for dependence of the metric on physical parameters of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Moreover, these numerical solutions are displayed by some curves in figures, instead of expressions in explicit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' It causes these solutions of scalarized black hole to usually need to be re-calculated by colleagues in some relevant research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Fortunately, the general methods for parametrization of the black hole space-times (continued fractions method (CFM) [28] and homotopy analysis method (HAM) [29, 30]) were developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' The CFM has recently been applied with success in a variety of contexts [31]-[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' We stress here that the HAM is also a very powerful method for obtaining analytical approximate solutions to various nonlinear differential equations (including systems of nonlinear equations and arising in many different areas of science and engineering [36]-[42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Despite its popularity in many areas of science and engineering over the years, the application of the HAM has been very limited in the fields of general relativity and gravitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Recently, this HAM has been adopted to derive analytic ap- proximate solutions of field equations in Einstein–Weyl gravity [43, 44] as well as analytic expression of Regge–Wheeler equations under the metric perturbations on Schwarzschild space-time [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In this work, firstly, we construct scalarized AdS black holes numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Secondly, making use of the HAM, we wish to obtain analytical approximate solutions for scalarized AdS black holes in the ESGB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' The plan of our work is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In Section 2, we investigate the tachyonic instability of Schwarzschild AdS (SAdS) black holes under the linearized scalar perturbation in the ESGB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Then, we construct numerical solutions of scalarized AdS black holes in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Section 4 is devoted to deriving analytical approximation solutions by introducing the HAM, where two solutions are accurate in the whole space outside the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Finally, we end the paper with a discussion and conclusions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' 3 2 Instability of SAdS black hole The action for ESGB theory with a negative cosmological constant Λ is given by SESGBC = 1 16π � d4x√−g � R − 2Λ − 2∂µφ∂µφ + λ2φ2 2 R2 GB � , (1) where λ is the scalar coupling constant, R the Ricci scalar, φ a scalar field, and R2 GB the Gauss–Bonnet term R2 GB = R2 − 4RµνRµν + RµνρσRµνρσ (2) with Ricci tensor Rµν and Riemann tensor Rµνρσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Varying the action (1) with scalar φ and metric gµν, one obtains the scalar field equation □φ + λ2 4 R2 GBφ = 0 (3) and Einstein equation Gµν = Λgµν + 2∂µφ∂νφ − (∂φ)2gµν − 2λ2∇ρ∇σ(φ2)Pµρνσ, (4) where Gµν = Rµν − (R/2)gµν is the Einstein tensor, and Pµρνσ is given by Pµρνσ = Rµρνσ + gµσRνρ − gµνRρσ + gνρRµσ − gρσRµν + R 2 (gµνgρσ − gµσgνρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (5) Topological black holes are found without scalar hair as ds2 SAdS = −fk(r)dt2 + 1 fk(r)dr2 + r2 � dθ2 + sin2 θdϕ2� (6) with fk(r) = k − 2M r − Λr2 3 , (7) where Λ = −3/L2 with L the curvature radius of AdS space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' The cases of k = 0, −1 were discussed in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Here, afterwards, we choose the k = 1 case of f(r) = 1 − 2M r − Λr2 3 (8) which corresponds to the SAdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' From f(rh) = 0, the outer horizon radius rh of SAdS black hole is obtained as rh = − 1 � 3MΛ2 + √ 9M2Λ4 − Λ3�1/3 − � 3MΛ2 + √ 9M2Λ4 − Λ3�1/3 Λ , (9) 4 where the horizon radius rh > 0 is always satisfied on account of a positive mass M > 0 of a black hole and a negative cosmological constant Λ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Moreover, the mass of SAdS black hole is determined as M = 1 6rh � 3 − Λr2 h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (10) Now, we discuss the dynamical stability, Breitenlohner–Freedman (BF) bound, and tachyonic instability of SAdS black hole in the ESGB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' For this purpose, we need to consider two linearized equations which describe the propagation of metric perturbation hµν and scalar perturbation δφ δRµν(h) = ¯gµν 2 δR + Λhµν, (11) ¯□δφ − µ2 effδφ = 0, (12) which are obtained by linearizing Equations (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' As was pointed out in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' [46, 47, 48], it is clear that the SAdS black hole is dynamically stable when making use of the Regge–Wheeler prescription under metric perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In an asymptotically AdS space- time, a scalar field can cause an asymptotic instability only if its mass-squared µ2 eff is less than the BF bound µ2 BF = − 9 4L2 ≡ 3Λ 4 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' One always finds µ2 eff > µ2 BF for large enough r and thus the SAdS black hole is stable asymptotically against the formation of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' However, if µ2 eff < µ2 BF in the intermediate region, the SAdS black hole may evolve to a scalarized AdS black hole through tachyonic instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In our case, the effective mass µ2 eff is fixed as µ2 eff = −λ2 4 ¯R2 GB = −2λ2Λ2 3 − 12λ2M2 r6 (13) and the condition for asymptotic instability is obtained as µ2 eff < µ2 BF : −2λ2Λ2 3 < 3Λ 4 → Λ > −9 8λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (14) Now, we are in a position to perform the numerical analysis for the tachyonic instability of SAdS black hole in the ESGB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Taking into account the separation of variables, δφ(t, r, θ, ϕ) = ψ(r) r Ylm(θ, ϕ)e−iωt, (15) and introducing a tortoise coordinate dr∗ = dr/(1 − 2M/r − Λr2/3), the radial part of Equation (12) is given by d2ψ dr2 ∗ + � ω2 − Veff(r) � ψ(r) = 0, (16) 5 where the effective potential Veff(r) takes the form Veff(r) = � 1 − 2M r − Λr2 3 ��2M r3 + l(l + 1) r2 − 2Λ 3 � 1 + λ2Λ � − 12λ2M2 r6 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (17) In the next sections, we only consider the case of l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' To determine the threshold of tachyonic instability, one has to solve the second-order differential equation numerically d2ψ dr2 ∗ − � Ω2 + Veff(r) � ψ(r) = 0, (18) which allows an exponentially growing mode of eΩt (ω = iΩ, Ω > 0) as an unstable mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Considering Ω = 0, we may solve the static linearized equation d2ψ dr2 ∗ − Veff(r)ψ(r) = 0, (19) to find out the threshold unstable mode propagating around the fixed SAdS black hole background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' To impose the boundary conditions, we first consider the near-horizon ex- pansion, which is used to set data outside the horizon for a numerical integration to near infinity ψ(r) = � i≥0 ψi(r − rh)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (20) In the asymptotic far region, Equation (19) becomes approximately ψ′′(r) + 2 rψ′(r) − 2 + 2λ2Λ r2 ψ(r) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (21) Then, we can obtain the boundary condition of ψ(r) ∼ r− 1 2± 1 2 √ 9+8λ2Λ at large r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Therefore, the numerical solution to Equation (19) can be performed by using the shooting method in the region between the black hole horizon and infinity, seeking for a value of the eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' These solutions are labelled by an integer n ∈ N0: n = 0 is the fundamental mode, whereas n > 1 are excited states (overtones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' We focus on the fundamental mode since the fundamental solutions is usually stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Varying −Λ/3, a set of bifurcation points con- stitutes the existence curve (threshold curve for tachyonic instability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Figure 1a includes three threshold curves of rh = 1, 2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' If one chooses rh = 1, the unstable region is the upper of threshold curve while the stable region is the lower of threshold curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' see Fig- ure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In case of −Λ/3 → 0, the value of coupling parameter λ matches the threshold 6 rh�1 rh�2 rh�4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='6 � � 3 1 2 3 4 Λ Λth SAdS����3� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='5 � � 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 Λ Figure 1: (Left) The existence curve for scalarized AdS black holes (threshold curve λSAdS th (−Λ/3) of tachyonic instability) in the (−Λ/3, λ) plane for three different horizon radii rh = 1, 2, 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (Right) the unstable region is plotted for the horizon radius rh = 1 of SAdS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' value (λS th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='852, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='704, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='408) for the fundamental mode of the Schwarzschild black hole in [9, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' This result naturally leads to the fact that the SAdS black hole is unstable in the upper region and thus there exist scalarized AdS black holes in the ESGB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' 3 Numerical Solutions for Scalarized AdS Black Holes We consider static and spherically symmetric space-times as well as static and spherically symmetric scalar field configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' The space-time metric and scalar are chosen to be ds2 = −A(r)dt2 + 1 B(r)dr2 + r2 � dθ2 + sin2 θdϕ2� , φ = φ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (22) Now we try to find the numerical solutions for scalarized AdS black hole in the ESGB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' For this purpose, we first introduce a coordinate transformation of z = rh r so that the metric functions can be derived in the compact region of 0 ≤ z ≤ 1, and A(r) and B(r) become A = A(z) and B = B(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Therefore, z = 0 always corresponds to infinity (r → ∞), and z = 1 naturally corresponds to the event horizon r = rh of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' To utilize the threshold values for an unstable region in Figure 1b, we will choose rh = 1 for the horizon radius of the black hole in the following numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' On the other hand, the metric functions A(r) and B(r) in Equation (22) approach r2 as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In other words, the new metric functions A(z) and B(z) with 1/z2 are divergent at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Then, we can further define new metric functions Az(z) → z2A(z), Bz(z) → z2B(z) (23) 7 so that the new functions Az(z) and Bz(z) are always regular in the whole region of 0 ≤ z ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Fortunately, the scalar field φ(z) is always regular in the whole region under the coordinate transformation z = rh r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Then, we set φz(z) → φ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (24) Substituting the new metric functions Equation (23) and scalar field Equation (24) into Equations (4) and (5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' we have eq1 = zBzA′ z � −r2 h + 2z(z2 − r2 hΛ − 3Bz)φzφ′ z � + Az � r2 h(−z2 + r2 hΛ − 2z2ΛφzB′ zφ′ z) −Bz � r2 h(−3 + z2(1 + 4Λ)φ′2 z ) + 4zφz((z2 − 2r2 hΛ)φ′ z + r2 hzΛφ′′ z) � +12zB2 zφzφ′ z � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (25) eq2 = 2r2 hz2ΛBzφzA′ zφ′ z − Az � − r2 hz2 + r4 hΛ − r2 hzB′ z + 2z4φzB′ zφ′ z − 2r2 hz2ΛφzB′ zφ′ z −4zB2 z � zφ′2 z + φz(−φ′ z + zφ′′ z) � + Bz � 3r2 h + z2(r2 h + 4z2 − 4r2 hΛ)φ′2 z +2zφz((2z2 + 4r2 hΛ − 3zB′ zφz + 2z(z2 − r2 hΛ)φ′′ z) �� = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (26) eq3 = z2λ2(z2 − Bz)BzφzA′2 z + zAz � − z3λ2φzA′ zB′ z + 2λ2B2 zφz(−3A′ z + zA′′ z) + zBzr2 hA′ zφ′ z +λ2zBzφz � A′ z(2z + 3B′ z) − 2z2A′′ z � � + A2 z � 12λ2B2 zφz + z2B′ z(2zλ2φz + r2 hφ′ z) −2zBz � λ2φz(2z + 3B′ z) + r2 h(2φ′ z − zφ′′ z) � � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (27) where primes denote derivatives with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In order to obtain the asymptotic form of scalarized AdS black holes, we solve three Equations (25)–(27) numerically via a shooting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Spherically symmetric black holes have an event horizon (z = 1), where the metric functions Az and Bz vanish, and the scalar field φz tends to a constant: Az(z ≈ 1) = A1(1 − z) + A2(1 − z)2 + · · · , (28) Bz(z ≈ 1) = B1(1 − z) + B2(1 − z)2 + · · · , (29) φz(z ≈ 1) = φ0 + φ1(1 − z) + · · · , (30) where φ0 denotes the scalar field at the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' It is worth pointing out that the regularity of a scalar field, and its first and second derivatives on the horizon give an additional 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 ln r 0 10 20 30 40 A�r� A�r��B�r� A�r��f�r� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='5 ln r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 0 1 2 3 4 ln r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='10 Φ�r� Figure 2: The scalarized AdS black hole with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='892 and −Λ/3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='457 belonging to the fundamental branch of λ > λb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='886 (bifurcation point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Here, f(r) represents the metric function (8) for the SAdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' condition r6 h − 8r4 hλ4� 3 + 2r2 hΛ � r2 hΛ − 2 � � φ2 0 − 48r2 hλ8Λ � r2 hΛ − 2 � φ4 0 > 0, (31) which reduces to that for the Schwarzschild black hole in the limit of Λ → 0 [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' On the other hand, the metric functions and scalar field at the infinity (z → 0) should satisfy the following boundary conditions: Az = Bz = −Λr2 h 3 , φz = 0, when z → 0 (r → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (32) We fix rh = 1 for the horizon radius of the black hole during the numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' By tunneling the coupling parameter λ and choosing different values of cosmological constant Λ, we can obtain a nontrivial solution of scalarized AdS black holes in the ESGB gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' The numerical solution for fundamental branch is obtained by taking λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='892 and −Λ/3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='457 (greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='886 of bifurcation point) (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' We plot all figures in terms of ln r and thus the horizon is always located at ln rh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Here, f(r) represents the metric function for the SAdS black hole with φSAdS(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Notice that the metric functions A(r) and B(r) display different behaviors in comparison to those for the SAdS black hole and these approach the SAdS metric function f(r) as ln r increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Moreover, a scalar field φ(r) is a decreasing function with starting with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='107, and its asymptotic value is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' 4 Analytical approximate solutions In general, it is a difficult task to find exact solutions of nonlinear differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' [30, 49], the HAM was developed to obtain analytical approximate solutions to nonlin- 9 ear differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Here, we wish to derive analytical approximate solutions for met- ric functions Az(z), Bz(z) and a scalar field φz(z) by solving nonlinear Equations (25)–(27) by using the HAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' If we succeed to find them, it will confirm the numerical solutions in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' We assume the nonlinear operators Ni, which are suitable for a system of n-nonlinear differential equations Ni[yi(t)] = 0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=', n, (33) with unknown function yi(t) and a variable t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Then, the zero-order deformation equation can be written as [30, 49] (1 − q)L[φi(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' q) − yi0(t)] = qhiHi(t)Ni[φi(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' q)] (34) where L is an auxiliary linear operator with the property L[0] = 0, q ∈ [0, 1] is an em- bedding parameter in topology (called the homotopy parameter), φi(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' q) are the solutions of Equation (34) for q ∈ [0, 1], yi0(t) is the initial guesses, and hi ̸= 0 is the so-called “convergence-control parameters”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Considering the property L[0] = 0, the solutions φi(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' q) of Equation (34) vary continuously from the initial guess yi0(t) to the actual solution yi(t) of Equation (33) when the parameter q increases from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Here, we set the auxiliary functions Hi(t) = 1 without any restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' On the other hand, we can also expand φi(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' q) as the Maclaurin series with respect to q φi(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' q) = yi0(t) + ∞ � m=1 yim(t)qm, yim(t) = 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' ∂mφi(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' q) ∂qm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (35) The proper choice of the initial approximation yi0(t), linear operator L, and convergence control parameter hi will make the series expansion (35) convergency at q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Therefore, we obtain yi(t) = φi(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' 1) = yi0(t) + ∞ � m=1 yim(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (36) Here the function yim(t) could be obtained by solving the mth order deformation equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Differentiating Equation (34) m times with respect to the parameter q, setting q = 0, and dividing by m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=', we find the mth order deformation equation L[yim(t) − χmyim−1(t)] = hiRim(yim−1), (37) 10 where Rim(yim−1) = 1 (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' ∂m−1Ni[φi(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' q)] ∂qm−1 |q=0, (38) and χm = \uf8f1 \uf8f2 \uf8f3 0 : m ≤ 1 1 : m > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (39) We define the partial sum yM i (t) by yM i (t) = yi0(t) + M � m=1 yim(t), (40) where yM i (t) are the Mth order approximate solutions of the original Equation (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In order to solve Equations (25)–(27) by means of the HAM, we choose the initial approximations Az0(z) = Bz0(z) = � z2 − r2 hΛ 3 + z3 3 (−3 + r2 hΛ) � (1 − αz), (41) φz0(z) = 107 1000 � 72 100z3 + 28 100z � (42) with an undetermined constant α and corresponding auxiliary linear operators [50] L[φz] = d2φz dz2 , L[Bz] = dBz dz , L[Az] = d2Az dz2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (43) One can find that the chosen approximations satisfy the initial and boundary conditions, since Az0 and Bz0 vanish at the event horizon (z = 1), and they reduce to − r2 hΛ 3 as z → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Moreover, the scalar field φz0(z) disappears at infinity and equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='107 near the horizon (z = 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Then, we use the HAM to secure analytical approximations for Equations (25)–(27) by using the boundary conditions Az(0) = −r2 hΛ 3 , Az(1) = Bz(1) = 0, φz(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='107, φz(1) = 0, (44) where we reserve one boundary condition Bz(0) = − r2 hΛ 3 for later computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' The Mth 11 order approximations of Az, Bz, φz are written as Az(α, hi, z) ≈ Az0(α, z) + M � k=1 Azk(α, hi, z), (45) Bz(α, hi, z) ≈ Bz0(α, z) + M � k=1 Bzk(α, hi, z), (46) φz(α, hi, z) ≈ φz0(z) + M � k=1 φzk(α, hi, z), (47) which include the unknown parameter α and the convergence-control parameter hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Considering the boundary condition Bz(0) = − r2 hΛ 3 with Mth order approximate ex- pression (45), one obtains ΓM(α, hi) ≡ Bz0(α, 0) + M � k=1 Bzk(α, hi, 0) + r2 hΛ 3 = 0, (48) where ΓM represents an expanded form of the constrained boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' As long as hi is given, a solution to Equation (48) is easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' We use the technique developed by Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' [36] to find out the optimal values of hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In principle, the technique seeks for minimizing averaged square residual error of Equations (25)–(27) at the mth order Em(α, hi) = EN1 m + EN2 m + EN3 m = 1 S + 1 S � k=0 �� N1[ m � n=0 Azn(zk), m � n=0 Bzn(zk), m � n=0 φzn(zk)] �2 + � N2[ m � n=0 Azn(zk), m � n=0 Bzn(zk), m � n=0 φzn(zk)] �2 + � N3[ m � n=0 Azn(zk), m � n=0 Bzn(zk), m � n=0 φzn(zk)] �2� (49) with zk = k∆z = k S , k = 0, 1, 2, · · · , S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (50) We choose S = 40 used with the purpose of optimization for each function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' For our problem, the residual error depends on both α and hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In fact, both Em(α, hi) and ΓM(α, hi) contain undetermined parameters: α and hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Therefore, the optimal convergence-control parame- ters hi can be determined from the minimum of Em(α, hi), and it is subjected additionally 12 to the algebraic Equation (48) which needs to secure the constant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Mathematically, this doubly coupled optimization problem implies (α∗, h∗ i ) = min{Em(α, hi), ΓM(α, hi) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (51) Considering the 2nd order (M = 2) approximation, we obtain h1 = 1, h2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='00044, h3 = −22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='08711 and α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='00402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Importantly, the corresponding 2nd order of analytical approximate solutions are determined as Az(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='4572667 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='01498763z + z2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='451489z3 − 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='686329 × 10−4z14 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='159995 × 10−4z15 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='031449 × 10−5z16 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='347457 × 10−4z17 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='985287 × 10−4z18 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='088783 × 10−4z19 +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='819188 × 10−4z20 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='785920 × 10−4z21 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='640031 × 10−5z22 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='993102 × 10−5z23 + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='812429 × 10−5z24 − 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='631288 × 10−5z25 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='244101 × 10−4z26 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='403398 × 10−4z27 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='307204 × 10−4z28 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='834623 × 10−5z29 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='387319 × 10−5z30 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='459050 × 10−5z31 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='058479 × 10−5z32, (53) 13 Bznum Bzana 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='8 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='6 Bznum Bzana 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='8 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='6 Φznum Φzana 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='3 Figure 3: Comparison figures of metric functions Az, Bz and scalar field φz in the numerical (solid curve) and analytical approximate (dashed curve) solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Here, we choose horizon radius parameter rh = 1, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='892 and −Λ/3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' φz(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='03217793z + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='07647953z3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0001430922z4 − 0.' metadata={'source': 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+page_content='008546128z30 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='002794749z31 + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='702130 × 10−5z32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (54) Now, we can compare the analytic approximate solutions with the numerical solutions appeared in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' We plot the analytic approximate solutions (Aana z , Bana z and φana z ) and numerical solutions (Anum z , Bnum z and φnum z ) in Figure 3 for rh = 1, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='892, and −Λ/3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' They are apparently consistent with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' It is interesting to check the accuracies of these analytic approximate solutions and numerical solutions based on the field Equations (25)–(27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Substituting these solutions into Equations (25)–(27), the total absolute errors from three field equations are obtained as ∆Err ≡ |∆eq1| + |∆eq2| + |∆eq3| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (55) Total absolute errors of these analytic approximate and numerical solutions are displayed in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' One can find that the total absolute error for analytic approximate solutions is smaller than that for the numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' In other words, the analytic approximate solutions are more accurate than the numerical solutions when solving nonlinear Equations (25)–(27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' 14 �Errana �Errnum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='10 Figure 4: Total absolute errors for analytic approximate solutions (∆Errana) and numerical solutions (∆Errnum) from three field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' �Φz �Bz �Az 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='025 ∆Φz ∆Bz ∆Az 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='0 z 0 2 4 6 8 Figure 5: Absolute (Left) and relative (Right) errors between numerical and analytical approximate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' To comparing these solutions, we further calculate the absolute differences between numerical and analytical approximate solutions by taking ∆Az = |Anum z − Aana z | , ∆Bz = |Bnum z − Bana z | , ∆φz = |φnum z − φana z | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (56) The relative errors can be also calculated in the form of δAz = |Anum z − Aana z | Anum z × 100%, δBz = |Bnum z − Bana z | Bnum z × 100%, δφz = |φnum z − φana z | φnum z × 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' (57) We find that main differences between numerical solutions and analytic approximation solutions occur close to the event horizon for metric functions A(z) and B(z) and region far from the black hole for scalar field function φ(z), see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' 15 5 Conclusions and discussions In this work, we investigated the spontaneous scalarization for SAdS black holes thoroughly in ESGB theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' The SAdS black holes become prone to tachyonic instability triggered by the strong space-time curvature in some region of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Then, scalarized AdS black holes could emerge from SAdS black holes at bifurcating points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Numerical solutions for scalarized AdS black holes are obtained for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='892 and −Λ/3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content='457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Later, we derive the analytical approximate solutions for metric functions A(z) and B(z) and scalar field φ(z) by using the HAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' The region and rate of convergence of the series solution for the HAM does not depend on the choice of the initial guess function, auxiliary linear operator, and an auxiliary function, but it can be effectively controlled by using a convergence control parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Since the approximation is significantly accurate in the entire space-time outside the event horizon, it can be used for studying the properties of this particular black hole and the various phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' The present work is considered as an important work because we confirm that numerical solutions are consistent with an analytical approximate solution for the scalarized AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' As an avenue of a further research, one may propose the related properties of scalarized AdS black hole (thermodynamics, Hawking radiation, particle motion, shadow, stability and QNMs) and compare them with SAdS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Acknowledgments We appreciate Rui-Hong Yue for helpful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Z acknowledges financial support from Outstanding Young Teacher Programme from Yangzhou University, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' 137050368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' L acknowledges financial support from the Initial Research Foundation of Jiangxi Normal University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9E3T4oBgHgl3EQf5wuC/content/2301.04784v1.pdf'} +page_content=' References [1] B.' metadata={'source': 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b/ltE4T4oBgHgl3EQftg2f/content/tmp_files/2301.05225v1.pdf.txt @@ -0,0 +1,1730 @@ +Domain Expansion of Image Generators +Yotam Nitzan1,2 +Micha¨el Gharbi1 +Richard Zhang1 +Taesung Park1 +Jun-Yan Zhu3 +Daniel Cohen-Or2 +Eli Shechtman1 +1 Adobe Research +2 Tel-Aviv University +3 Carnegie Mellon University +Source Generator +Dormant +direction +Domain Expansion (Ours) +Zombie +direction +Domain Adaptation +Dormant +direction +Figure 1. (center) Traversing the latent space of generative models along some directions changes the image significantly while traversing +others has no perceptible effect. We call directions of the latter type dormant. (left) Domain adaptation methods, transform the entire +generator from a source domain to a target domain, indicated by the color blue. (right) We introduce an approach for a new task – domain +expansion. Instead of fully transforming the generator, we expand it to include new data domains. Our method learns to represent the new +domain in a disentangled manner by repurposing a single dormant direction. +Abstract +Can one inject new concepts into an already trained gen- +erative model, while respecting its existing structure and +knowledge? We propose a new task – domain expansion +– to address this. Given a pretrained generator and novel +(but related) domains, we expand the generator to jointly +model all domains, old and new, harmoniously. First, we +note the generator contains a meaningful, pretrained latent +space. Is it possible to minimally perturb this hard-earned +representation, while maximally representing the new do- +mains? Interestingly, we find that the latent space offers un- +used, “dormant” directions, which do not affect the output. +This provides an opportunity: By “repurposing” these di- +rections, we can represent new domains without perturbing +the original representation. In fact, we find that pretrained +generators have the capacity to add several – even hundreds +– of new domains! Using our expansion method, one “ex- +panded” model can supersede numerous domain-specific +models, without expanding the model size. Additionally, a +single expanded generator natively supports smooth transi- +tions between domains, as well as composition of domains. +Code and project page available here. +1. Introduction +Recent domain adaptation techniques piggyback on the +tremendous success of modern generative image models [3, +12, 32, 40], by adapting a pretrained generator so it can +generate images from a new target domain. Oftentimes, +the target domain is defined with respect to the source do- +main [5,21,22], e.g., changing the “stylization” from a pho- +torealistic image to a sketch. +When such a relationship +holds, domain adaptation typically seeks to preserve the fac- +tors of variations learned in the source domain, and transfer +them to the new one (e.g., making the human depicted in +a sketch smile based on the prior from a face generator). +With existing techniques, however, the adapted model loses +the ability to generate images from the original domain. +In this work, we introduce a novel task — domain ex- +pansion. Unlike domain adaptation, we aim to augment the +space of images a single model can generate, without over- +riding its original behavior (see Fig. 1). Rather than view- +ing similar image domains as disjoint data distributions, we +treat them as different modes in a joint distribution. As a +result, the domains share a semantic prior inherited from +the original data domain. For example, the inherent factors +1 +arXiv:2301.05225v1 [cs.CV] 12 Jan 2023 + +of variation for photorealistic faces, such as pose and face +shape, can equally apply to the domain of “zombies”. +To this end, we carefully structure the model train- +ing process for expansion, respecting the original data do- +main. It is well-known that modern generative models with +low-dimensional latent spaces offer an intriguing, emer- +gent property – through training, the latent spaces represent +the factors of variation, in a linear and interpretable man- +ner [3, 6, 10, 12, 28, 30, 39, 40]. We wish to extend this ad- +vantageous behavior and represent the new domains along +linear and disentangled directions. Interestingly, it was pre- +viously shown that many latent directions have insignificant +perceptible effect on generated images [6]. Taking advan- +tage of this finding, we repurpose such directions to repre- +sent the new domains. +In practice, we start from an orthogonal decomposition +of the latent space [36] and identify a set of low-magnitude +directions that have no perceptible effect on the generated +images, which we call dormant. To add a new domain, +we select a dormant direction to repurpose. Its orthogo- +nal subspace, which we call base subspace, is sufficient +to represent the original domain [6]. +We aim to repur- +pose the dormant direction such that traversing it would +now cause a transition between the original and the new +domain. Specifically, the transition should be disentangled +from the original domain’s factors of variation. +To this +end, we define a repurposed affine subspace by transport- +ing the base subspace along the chosen dormant direction, +as shown in Fig. 3. We capture the new domain by applying +a domain adaptation method, transformed to operate only +on latent codes sampled from the repurposed subspace. A +regularization loss is applied on the base subspace to en- +sure that the original domain is preserved. The original do- +main’s factors of variation are implicitly preserved due to +the subspaces being parallel and the latent space being dis- +entangled. For multiple new domains, we simply repeat this +procedure across multiple dormant directions. +We apply our method to the StyleGAN [13] architecture, +with multiple datasets, and expand the generator with hun- +dreds of new factors of variation. Crucially, we show our +expanded model simultaneously generates high-quality im- +ages from both original and new domains, comparable to +specialized, domain-specific generators. Thus, a single ex- +panded generator supersedes hundreds of adapted genera- +tors, facilitating the deployment of generative models for +real-world applications. We additionally demonstrate that +the new domains are learned as global and disentangled fac- +tors of variation, alongside existing ones. This enables fine- +grained control over the generative process and paves the +way to new applications and capabilities, e.g., compositing +multiple domains (See Fig. 2). Finally, we conduct a de- +tailed analysis of key aspects of our method, such as the ef- +fect of the number of newly introduced domains, thus shed- +Dog +Cute +Siberian Husky +Sketch +Expanded Domains +Boar +Happy +Pop Art +Source Domain +Domain Composition +Siberian Husky + +Cute + Sketch +Boar + Happy + +Pop Art +Figure 2. Example of a domain expansion result. Starting from +dogs as the source domain, we expand a single generator to model +new domains such as facial expressions, breeds of dogs and other +animals, and artistic styles. Finally, as the representations are dis- +entangled, the expanded generator is able to generalize and com- +pose the different domains, although they were never seen jointly +in training. +ding light on our method and, in the process, on the nature +of the latent space of generative models. +To summarize, our contributions are as follows: +• We introduce a new task – domain expansion of a pre- +trained generative model. +• We propose a novel latent space structure that is amenable +to representing new knowledge in a disentangled manner, +while maintaining existing knowledge intact. +• We present a simple paradigm transforming domain adap- +tation methods into domain expansion methods. +• We demonstrate successful domain expansion to hun- +dreds of new domains and illustrate its advantage over +domain adaptation methods. +2. Related Work +Fine-tuning generative models. +Starting from a genera- +tor pretrained on a source domain and training it for a target +domain, often called fine-tuning, is a common technique ap- +plied for various purposes and settings. +Some works wish to model only the target domain. In +which case, the pretrained model is leveraged simply as an +efficient initialization, shortening the training time, and im- +proving image quality [11, 16, 19, 47, 50]. Others, wish to +learn the target domain alongside the source domain, in a +setting called continuous learning, and propose methods to +ensure that the source domain is not forgotten [34,44]. Al- +though in a single generator, the domains are modeled sep- +arately, each as its own class. +A prominent line of works have sought to make the target +domain inherit knowledge from the source domain [1, 2, 5, +14,17,21–23,31,33,42,43,51]. This approach allows gener- +alization beyond the target domain per-se and is especially +useful when training data is scarce. +Our work similarly involves fine-tuning, but for a novel +2 + +purpose. Our perspective is that, since the target domain +is introduced with knowledge from the source domain – it +is in essence, an expansion of it. Therefore, in contrast to +the aforementioned works, we aim to model the domains +jointly. The proposed method does not replace previous +fine-tuning methods, but allows applying them jointly, in +a plug-and-play manner. +Latent directions in generative models. +Generative +models learn to represent the factors of variation of ob- +served data in their latent space. Disentangled represen- +tations are especially useful as they facilitate intuitive con- +trol over the generative process. With recent architectures, +disentanglement miraculously emerges without interven- +tion [12, 24, 28, 39]. In such models, disentanglement is +manifested through the existence of linear latent directions, +each ideally controlling a single factor of variation. +Due to the spontaneous emergence of such directions, +numerous works have been proposed to identify them after +the model has been trained [6, 25, 35–37, 41, 45] and used +them for downstream applications, most commonly seman- +tic image editing. At the same time, it has also been ob- +served that some latent directions have no perceptible effect +on the generated images [6, 46]. These directions, which +we call dormant, were not previously leveraged for any pur- +pose. +In this work, we rely on existing methods to factorize +the latent space into such linear directions. As we aim to +expand the pretrained generator to additional domains, we +decide to explicitly encode the “new knowledge” along the +dormant directions, while keeping other directions intact. +This design ensures that the original domain is preserved +and that the different domains are represented in a disentan- +gled fashion. +3. Method +We start with a pretrained generator Gsrc that maps from +latent codes z ∈ Z ⊆ RD to images in a source domain +Dsrc, and a set of N domain adaptation tasks, each defined +by a loss function Li, i ∈ {1, . . . , N}. In domain adapta- +tion, fine-tuning Gsrc to minimize Li yields a generator Gi +that generates images from the new domain Di. In contrast, +our goal is domain expansion, which aims at training a sin- +gle expanded generator G+ that can simultaneously model +all the new domains ∪N +i=1Di, along with the original do- +main Dsrc. We want to ensure that the new domains Di are +disentangled from each other and also share the factors of +variation from the source domain, which remain intact. +Our solution is to partition the latent space into disjoint +subspaces, one for each new domain, and to restrict the ef- +fect of each domain adaptation to the corresponding sub- +space. To this end, we endow the latent space with an ex- +plicit structure that supports domain expansion (Sec. 3.1), +Dormant +G +(a) Domain Adaptation Training +(b) Domain Expansion Training +Repurposed +G +Figure 3. Our method transforms a domain adaptation task to a +domain expansion task. (a) Generator G is optimized to satisfy +the loss Li for every latent code in space. The entire generator +and latent space now represent the new domain, indicated with the +color blue. (b) Generator G is optimized to satisfy the same loss, +Li, only on a subspace Zi, dedicated to the new domain. Simul- +taneously, G is optimized to satisfy a regularization term Lreg on +a parallel subspace, Zbase, ensuring the original knowledge is pre- +served there. The generator and latent space now represent both +domains, indicated by being colored both blue and orange. The +latent direction between the two spaces was originally dormant in +generator G, and now represents a transition between the domains. +and optimize each domain adaptation loss only using la- +tents from specific subspace reserved for the new domain +(Sec. 3.2). Our decomposition reserves a base subspace for +the original domain Dsrc, on which we impose a regulariza- +tion objective to maintain the behavior of the source gen- +erator (Sec. 3.3). Fig. 3 gives an overview of our domain +expansion algorithm. +3.1. Structuring the Latent Space for Expansion +Modern generative models conveniently learn to repre- +sent the factors of variation along linear latent directions, +in a completely unsupervised manner [12, 26, 28, 39]. We +decide to explicitly extend this model by structuring the la- +tent space such that the effect defined by an adaptation task +would be represented along a single linear direction. For- +mally, there should exist some scalar s and latent direction +vi, for which images generated from G+(z), G+(z + svi), +relate to each other as the corresponding images from the +source and adapted generators Gsrc(z), Gi(z) do. +Concretely, following SeFA [36], we obtain a semantic +and orthogonal basis V of the latent space from the right +singular vectors (produced by SVD) of the very first gener- +ator layer, which acts on the latent space Z. With a similar +factorization technique [6], it was observed that a relatively +small subset of the basis vector is sufficient to represent +most of the generators Gsrc’s variability. Other basis vectors +have barely any perceptible effect on the generated images. +We find this to be the case with SeFA as well. We refer to +vectors with no perceptible effect as dormant. +As the dormant directions do not affect the model’s gen- +eration capabilities, they are available to be repurposed with +new desired behavior. We thus choose to represent the do- +mains Dsrc and Di in regions that are separated by only a +dormant direction. +3 + +Z +aseL +22Zi\regFormally, for each of the N adaptation tasks, we dedi- +cate a single dormant direction, vi, that will be repurposed. +The remaining directions {vN+1, . . . , vD} will remain in- +tact. We finally define a subspace of Z, dubbed the base +subspace, as +Zbase = span(vN+1, . . . , vD) + z +(1) +where z is the mean of the distribution over the latents used +to train the generator. Then, for each repurposed direction, +vi, we define a repurposed subspace Zi that is the base +subspace transported along direction vi by a predetermined +scalar size s. +Zi = Zbase + svi. +(2) +The choice of direction vi and scalar s are discussed in Ap- +pendices C.2 and C.4. +Our domain expansion training procedure described +hereafter will ensure subspace Zi is the only part of the +latent space affected by the training objective Li, and is +reserved to generate images from domain Di. Intuitively, +shifting the base subspace along direction vi aims to achieve +two goals: 1. preserve the factors of variations inherited +from Zbase, and 2. restrict the new factor of variation (cor- +responding to Di) to a single latent direction, vi. +3.2. From Domain Adaptation to Expansion +Having defined disjoint affine subspaces Zi of the latent +space Z for our new domains Di, we now describe how we +constrain each domain adaptation objective Li to affect only +the corresponding subspace. +The domain adaptation objective is applied to images +generated from latent codes z ∈ Z, sampled from distribu- +tion p(z) defined on the entire space Z. Commonly the dis- +tribution is a Gaussian, or is derived from it [12] but some +exceptions exist [21, 43]. Our strategy is to transform this +sample distribution into one restricted to the affine subspace +Zi. We do so by projecting the samples from p(z) onto Zi, +using a standard orthogonal projection operator +projZi(z) = +D +� +j=N+1 +(v⊤ +j (z − z))vj + z + svi. +(3) +Denoting by pi the sampling distribution over Z for each +of the new domains we seek to adapt, the training loss over +all tasks is defined as +Lexpand = +N +� +i=1 +Ez∼pi(z) Li(G(projZi(z))). +(4) +3.3. Regularization +Optimizing Lexpand lets us learn to generate data from +the new domains Di within a single generator, but unfortu- +nately it leaves the base subspace Zbase under-constrained +Base Subspace +Subspace +Sketch of a Dog +Funny Dog +Subspace +w\o +w\o +Unregularized +Regularized +Figure 4. Regularization prevents leakage. Without regulariza- +tion (top row), new factors of variation “Sketch” and “Funny” are +leaking into the base subspace and the other repurposed subspace. +Note, for example, that the image from the base subspace is both +a sketch and depicts a smiling dog. Our regularization, described +in Eq. (6), solves the issue (bottom row). +and, therefore, does not guarantee it will remain unaltered +during training. In practice, we observe that the effect of Li +“leaks” outside Zi, causing catastrophic forgetting [18] in +subspace Zbase, and undesirably affecting other subspaces +Zj. We show an example of this leakage in Fig. 4. +To prevent this failure mode, we explicitly enforce the +preservation of Gsrc’s behavior over the base subspace Zbase +by regularization. We adopt two successful regularization +techniques. First, we keep optimizing the generator with +the loss it was originally trained on, Lsrc, which is known +to mitigate forgetting [15]. Second, we apply replay align- +ment [44], which is a reconstruction loss that compares the +output of a frozen copy of the source generator to that pro- +duced by our generator. We use a weighted combination of +an L2 pixel loss and LPIPS [49]: +Lrecon = λlpipsLlpips(Gsrc(z), G(z))+ +λL2∥Gsrc(z) − G(z)∥2, +(5) +where λlpips = λL2 = 10 are weighting hyperparameters. +Not only does replay alignment preserve the source domain +Dsrc, it also has the added benefit of aligning G+ to the +source generator Gsrc, in the sense that they will produce +similar outputs given the same latent code z. +Crucially, we only regularize the base subspace Zbase, +since the subspaces Zi should be allowed to change to learn +the new behaviors. To this end, we project the latent codes +to the base subspace Zbase, before calculating the regular- +ization terms. Our overall regularization objective is thus: +Lreg = Ez∼psrc(z) +� +λsrcLsrc(G(projZbase(z)))+ +Lrecon(G(projZbase(z))) +� +, +(6) +where λsrc = 1 balances the two terms and psrc(z) is the +latent distribution over Z used to train Gsrc. Our final, reg- +ularized domain expansion objective is, therefore: +Lfull = Lexpand + Lreg. +(7) +4 + +vregGsrc +c(z)Z122OP4. Experiments +We evaluate our method and analyze its key character- +istics. Sec. 4.1 first details the experimental setting. We +start by analyzing the knowledge encoded along repurposed +directions and compare it to domain adaptation methods +(Sec. 4.2). We then delve deeper and evaluate the effects +(Sec. 4.3) and opportunities (Sec. 4.4) presented by expand- +ing a generator to multiple domains simultaneously. Last, +we demonstrate that the quality of the source domain is +maintained in the base subspace (Sec. 4.5). +Further experiments, results, and details are provided in +the supplementary. +4.1. Experimental Setting +We adopt StyleGAN2 [13] as the source generator archi- +tecture, for its disentangled latent space and because it has +been the dominant test bed for generative domain adapta- +tion methods in recent years [1,5,21,22,43,51]. +Latent space and subspaces. +Several latent spaces have +been considered in the context of StyleGAN. We use the +intermediate latent space W in all our experiments but note +it as Z for consistency. We use SeFA [36] for the orthogonal +decomposition of Z. As SeFA performs SVD, there is a +native indication to how dormant is a given latent direction +– the corresponding singular value. As singular values are +commonly sorted in decreasing orders, the last basis vectors +are most dormant. When expanding with N new domains, +unless specified otherwise, we repurpose the last N basis +vectors. We use s = 20 in all experiments. These decisions +are evaluated in greater depth in Appendices C.2 and C.4. +Adaptation methods. +We demonstrate our expansion +method with two domain adaptation tasks - StyleGAN- +NADA [5] and MyStyle [21]. These two tasks were cho- +sen as they differ significantly in key aspects – source of +supervision, sampling distribution and loss. +StyleGAN-NADA is a zero-shot, text-guided, domain +adaptation method. It takes as input a pair of text prompts, +tsource and ttarget, describing the desired transformation +source → target to be applied on the domain of the pre- +trained generator, Dsrc. The loss function L is given by +∆T = ET (ttarget) − ET (tsource) , +∆I = EI (G (z)) − EI (Gsrc (z)) , +L = 1 − ∆I · ∆T +|∆I| |∆T| , +(8) +where EI and ET are CLIP’s [29] image and text encoders +respectively. +MyStyle is a few-shot, image-supervised, domain +adaptation method. +As input, it takes a set of images +! +−𝑠 3 +0 +𝑠 +! +𝑠 3 +! +2𝑠 3 +! +4𝑠 3 +Sketch +Polar +Elephant +Botox +Lips +𝛼 = +Figure 5. Continuous traversal along repurposed directions. As +can be seen, the traversal between the base subspace (α = 0) and +repurposed subspace (α = s) portrays a smooth transition between +the source and newly introduced domains. Advantageously, the +semantic meaning of the repurposed direction is preserved in the +extrapolation, representing the opposite relationship between the +domains (α < 0) or exaggerations of it (α > s). +{xm}M +m=1 of an individual (M ∼ 100), and adapts Gsrc to +form a personalized prior for that individual. The generator +is trained to better reconstruct xm from their original latent +space inversions zm ∈ Z [38]. Formally, the loss function +is given by +L = +M +� +m=1 +[Llpips(G(zm), xm) + ∥G(zm) − xm∥2]. +(9) +where Llpips is again the LPIPS loss [49]. +Datasets and models. +We demonstrate our method on +four datasets – FFHQ [12], AFHQ Dog [4], LSUN Church +[48] and SD-Elephant [20]. The FFHQ model is expanded +with 105 new domains, 100 introduced with the expanded +variant of StyleGAN-NADA and 5 from the expanded vari- +ant of MyStyle. The AFHQ Dog, LSUN Chruch and SD- +Elephant are expanded with 50, 20, and 20 new domains +correspondingly, all introduced from the expanded variant +of StyleGAN-NADA. +4.2. Evaluating Domains Individually +Traversing a repurposed direction. +We start by investi- +gating what knowledge, if any, is encoded along the repur- +posed latent directions. To this end, starting from a random +latent code z ∈ Zbase, we individually traverse different re- +purposed directions, vi, and inspect the generated images +G+(z +αvi). Sample results from our dogs, elephants, and +faces models are displayed in Fig. 5. We find that each in- +dividual repurposed direction now successfully encodes the +desired factor of variation, in a global and continuous way. +Our training paradigm is inherently discrete – encour- +aging the source behavior on the base subspace (α = 0) +and the newly introduced effect on the repurposed space +5 + +StyleGAN-NADA +Ours +Barbie +Barack Obama +MyStyle +Figure 6. A random set of images generated by our generator +from repurposed subspaces (bottom) and by corresponding do- +main adaptation methods (top). The images are similar and dif- +ferences are subtle. +Method +User % (↑) +ID (↑) +Diversity×10 (↑) +StyleGAN-NADA +41.2% +- +2.42 ± 0.13 +Ours w/ NADA +58.8% +- +2.42 ± 0.13 +MyStyle +- +0.80 ± 0.06 +3.08 ± 0.15 +Ours w/ MyStyle +- +0.76 ± 0.05 +3.14 ± 0.14 +Table 1. Quantitative comparison of images generated from our +repurposed subspaces to those generated by corresponding domain +adaptation methods - StyleGAN-NADA [5] and MyStyle [21]. We +follow each adaptation method’s quantitative evaluation protocol. +(α = s). Therefore, obtaining a smooth effect might seem +surprising at first glance. Nevertheless, this phenomenon +can be clearly traced to the well-established observation that +generators are smooth with respect to their latent space [12]. +Behavior on the repurposed subspace. +We have trans- +formed adaptation tasks into expansion tasks by limiting +the training effect to the repurposed subspaces only. But, +for latents in repurposed subspaces (α = s), the domain +adaptation could be considered to have been applied as-is. +We next directly compare the images generated by our +generator from the repurposed subspace to the correspond- +ing images generated by the domain-adapted generator. We +inherit and repeat the quantitative evaluation protocols per- +formed by each of the adaption tasks. To compare qual- +ity with StyleGAN-NADA [5] we perform a two-alternative +forced choice user study. Users were asked to pick the im- +age that has higher-quality and better aligns with the target +text used for training. We gathered 1440 responses from +32 unique users. To compare quality with MyStyle [21], +we evaluate preservation of identity in generated images, +as observed by a face recognition network [9]. For both +methods, the diversity is compared based on intra-cluster +LPIPS [49] distance, first suggested by Ojha et al. [22]. We +use 10 domains for comparison with StyleGAN-NADA and +Better +Alignment +Worse +Leakage +(a) +Sketch Subspace +Sumo Subspace +Sketch Only +Sketch + 4 +Sketch + 49 +Sketch + 4 +Sketch + 49 +(b) +Figure 7. Investigating the effect of introducing multiple domains +simultaneously. (a) Reports the CLIP error of generated images +with the text “a sketch”, as a function of training iterations. Images +are generated from the “Sketch” and “Sumo” subspaces of models +trained with a different number of domains. (b) Depicts generated +images from models that have similar CLIP errors. As can be +seen, the sketch domain does not “leak” into the sumo subspace. +Additionally, introducing additional domains delays, but does not +prevent, the introduction of sketch. +5 for comparison with MyStyle. Note that we use a single +generator G+, expanded with 105 domains, while compet- +ing methods use a dedicated model per domain, 15, overall. +The results are reported in Tab. 1, and a qualitative sample +is provided in Fig. 6. +As can be seen, on the repurposed subspaces, our method +produces comparable images to that generated by the ded- +icated, domain-adapted generator. +Perhaps surprisingly, +users somewhat prefer our results over StyleGAN-NADA’s. +We speculate this is due to the significantly greater difficulty +of choosing hyperparameters for their training. +4.3. Effect of Domains on Each Other +Previous evaluation of individual repurposed directions +already indicates disentanglement between different factors +of variation. For example, “Barbie” images in Fig. 6 show +no sign of being caricature, Barack Obama, or any of the +other hundred factors of variation introduced to that gen- +erator. In this section, we delve deeper into evaluating the +effects of expanding with multiple factors of variation. +To this end, we train three models to expand the FFHQ +parent model with either 1, 5 or 50 new domains, all in- +duced by StyleGAN-NADA [5]. All models are expanded +with “Sketch”, and the latter two with “Sumo” as well +as other factors of variation. +We quantify the strength +of introduced factor of variation using CLIP error, the 1- +complement of the score produced by CLIP [29]. We use +the top-performing version of the CLIP encoder available, +6 + +Dwayne Johnson - #509 (MyStyle) +Cubism Art - #503 (NADA) +Pixar - #484 (NADA) +Tongue Out - #463 (NADA) +Towers & Night - #7 (Original) +Desert - #496 (NADA) +Figure 8. Composing multiple effects by simple latent traversal. In each grid, we start from the latent code that generates the top-left image +and traverse along two latent directions, represented by advancement in rows and columns. For each direction, we note the associated +domain, its ordinal number in the latent space’s basis, and the training method used (“NADA” or “MyStyle”) to learn the domain. As can be +seen, G+ has learned a disentangled representation, allowing meaningful composition of concepts. Specifically, note the disentanglement +between directions, as traversing left-right does not affect the magnitude of the effect corresponding to up-down traversal, and vice versa. +ViT-L/14, which is not used during training. We note that +simply minimizing CLIP error is not the objective, as it +might lead to favoring mode-collapsed and adversarial ex- +amples [5]. Nevertheless, together with qualitative inspec- +tion, it is useful for comparing different models. +In Fig. 7a, we report the CLIP error of generated images +with the text “a sketch”, as a function of the training itera- +tions. Images are generated from the “Sketch”, and if exists, +“Sumo” subspace. First, we observe that CLIP error is de- +creasing for “Sketch” subspaces in all models, as expected. +Conversely, CLIP error in the “Sumo” subspace does not +significantly change, indicating it is not becoming any more +or less of a sketch. This result quantitatively supports our +previous finding, that factors of variations do not interfere +with each other, and demonstrates it is true regardless of +the number of other factors of variations learned simultane- +ously. Additionally, we observe that expanding with addi- +tional factors of variation delays, but does not prevent, G+ +the introduction of ”Sketch“ effectively. The observed delay +is expected, as expanding with more variations corresponds +to G+ optimizing and balancing additional loss functions. +Generated samples from the sketch and sumo subspaces are +provided in Fig. 7b. +4.4. Compositionality +While accidental “leakage” between latent directions +during training is undesired, intentionally composing vari- +ations at test time is advantageous. For generative mod- +els with a disentangled latent space, summing together la- +tent directions aggregates their semantic meaning, and ide- +ally should not affect the magnitude of their effects if ap- +plied separately. For example, if direction v1 controls head +pose and direction v2 controls an unrelated variation, im- +ages G(z + v1 + v2) and G(z + v1) should depict the same +head pose. +We find that the latent space of the expanded generator +G+ is disentangled, and variations can indeed be composed +effectively. Crucially, variations can be composed with each +other regardless of their originating training task, including +those on the base subspace, learned from the source domain. +Fig. 8 shows a sample of gradual composition results across +models and training tasks. +Comparison to existing techniques. +Several domain +adaptation methods [5, 14] have proposed techniques to +combine multiple variations. These methods still train a +separate generative model per variation, but combine their +effects in test-time. +Specifically, in the realm of CLIP- +supervised training, StyleGAN-NADA [5] interpolates the +generators’ weights, while DiffusionCLIP [14] interpolates +intermediate activations of the generators. Next, we com- +pare the disentanglement of composition in our generator to +that made possible using these techniques. +For each method, we start with a setting that was opti- +mized to generate images that align with one of two text +prompts. In our case, this setting is G+ with latent codes +in certain subspaces. For the baselines, these settings are +dedicated generators with any latent code. Then, for each +method, we gradually introduce the variation described by +the other text prompt and generate the corresponding im- +ages. Finally, we measure normalized CLIP error between +generated images and the two prompts. We normalize all +errors by the error of the initial setting, to make the metric +comparable across methods and text prompts. Fig. 9 reports +7 + +(a) +Munch +Painting +Samurai +Sumo +Neanderthal +Composition +Composition +Diffusion- +CLIP +StyleGAN- +NADA +Ours +(b) +Figure 9. Comparing compositionality in our generator to methods +of combining multiple domains proposed by StyleGAN-NADA +[5] and DiffusionCLIP [14]. Starting from a setting optimized for +either text prompt #1 or #2, we gradually introduce the variation +described by the other text. (a) Reports the CLIP error to both +prompts along the gradual introduction, normalized to the error +obtained for each text prompt in isolation. (b) Portrays a sample +of qualitative results, where the composition is such that assigns +equal strengths to both effects. As can be seen in, both quantita- +tively and qualitatively, NADA and DiffusionCLIP directly trade- +off one effect for the other – strengthening the effect of one prompt +directly lessens that of the other. In contrast, our generator allows +true composition of modalities. Specifically, note that our genera- +tor is able to compose effects that are somewhat dependent, such +as Neanderthal & Sumo. +the mean and standard deviation of the CLIP error, on 10 +pairs of prompts, and provides a sample of qualitative re- +sults. As can be seen, both baseline methods directly trade- +off one domain for the other, expressed by a linear-looking +trend. Conversely, our method obtains significantly lower +errors and allows for a true composition of concepts. +4.5. Preservation of the Source Domain +Finally, we evaluate the preservation of the source mode +in G+. To this end, using FID [7], we compare the quality +of images generated from the base subspace Zbase of G+ +to those generated by the source generator Gsrc. Since the +Model +FFHQ +AFHQ +LSUN +Church +SD +Elephant +Parent +2.77 +7.43 +3.92 +2.30 +Ours +2.80 +7.51 +3.76 +2.70 +Only Lsrc +2.75±0.08 +7.38±0.09 +3.31±0.22 +3.91±0.67 +Table 2. We generate images from the base subspace and report +FID [7] (↓) with respect to source domain dataset. We compare +our FID to that of the source generator Gsrc. For reference, we +also continue training the source generator for the same number +of iterations with its original loss - Lsrc, and report the mean and +standard deviations of FID along the training. As can be seen, +on the base subspace, our models have comparable FID scores to +their parents. Furthermore, similar magnitude of change in FID are +observed by simply continuing training, indicating that the change +in FID might be, at least in part, due to “random” fluctuations. +generator is being trained, some change in FID is expected. +Therefore, we also report the average and standard devia- +tion over FID scores for a generator that simply continues +training, i.e., using only the original loss Lsrc. Results are +reported in Tab. 2 and vary between datasets. For FFHQ and +AFHQ, we observe a slight increase in FID, but one that is +within 0.5σ and 1.5σ of the random fluctuations of the ref- +erence, respectively. For LSUN Church, our model obtains +a lower FID score than the source generator, but slightly +higher than that obtained by continuing training, while for +the SD-Elephants, the opposite occurs. We conclude that +the expansion method might have a slight impact on FID, +but it is negligible. +5. Conclusions +We present a new problem – domain expansion – and +propose an approach to solve it. The core of our method +is to carefully structure the latent space, such that it is +amenable to learning additional knowledge, while keeping +the existing knowledge intact. Our method takes advantage +of the existence of dormant latent directions, and the task +itself implicitly relies on the capacity of the model weights +to represent more knowledge. If one of these assumptions +does not hold, it might not be possible to apply domain ex- +pansion. +However, the popularity of methods squeezing +neural networks, such as Knowledge Distillation [8], and +current estimates of the intrinsic dimensionality of image +datasets [27], indicate that these assumptions commonly +hold. In our experiments, we were able to expand to hun- +dreds of directions. A plausible limitation is that the model +can be expanded to a certain point but ultimately limited by +factors such as the latent space or network capacity. Over- +coming this limitation, perhaps by considering more com- +plex latent space structures, is an avenue for future work. +8 + +1.0 +Method +NADA +Normalized CLIP Error #2 +0.8 +DiffusionCLiP +ours +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +NormalizedCLiPError#1Acknowledgment. +We are grateful to Rinon Gal, Yossi +Gandelsman and Sheng-Yu Wang for their suggestions in +the early stages of this research. We also thank Rinon Gal, +Yossi Gandelsman, Kfir Aberman, Alon Nitzan, Omer Bar- +Tal, Nupur Kumari and Gaurav Parmar for proofreading the +draft. We also thank Nupur Kumari for a technical advice. +We finally thank Yogev Nitzan for his help with the user +study and for coming up with hundreds of textual prompts. +This work was done while Yotam Nitzan was an intern at +Adobe. 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In Proceedings of the +IEEE conference on computer vision and pattern recogni- +tion, pages 586–595, 2018. 4, 5, 6, 13 +[50] Shengyu Zhao, Zhijian Liu, Ji Lin, Jun-Yan Zhu, and Song +Han. +Differentiable augmentation for data-efficient gan +training. arXiv preprint arXiv:2006.10738, 2020. 2 +[51] Peihao Zhu, Rameen Abdal, John Femiani, and Peter Wonka. +Mind the gap: +Domain gap control for single shot do- +main adaptation for generative adversarial networks. arXiv +preprint arXiv:2110.08398, 2021. 2, 5 +10 + +A. Overview +In Appendix B, we consider a baseline for domain ex- +pansion and demonstrate it is inferior to our proposed +method. Next follows the main part of the supplementary, +Appendix C, in which we perform additional analysis and +experimentation of our method. Finally, in Appendix D, we +provide additional details completing the paper. +B. Domain Expansion Baseline Using Class- +Conditioning +In this section, we experiment with an alternative, base- +line, method to perform domain expansion. +Generative +models capturing multiple domains commonly use a class- +conditioning mechanism [3]. Adopting this approach, we +attempt to perform domain expansion by modeling domains +with classes. We find that this method does not work as well +as our proposed method. +Method. +We start with an unconditional pretrained gen- +erator, specifically StyleGAN [13]. We then make the gen- +erator condition on a one-hot vector, using the architecture +proposed by Karras et al. [11]. This change involves adding +a single MLP layer, whose input is the one-hot vector. Its +output is concatenated to the random latent code and then +fed to the generator. +The class-conditioned generator is trained in a similar +protocol to our method. The source domain uses class c = +0, which is analogous to the base subspace. Whenever the +0th class is sampled, we apply the original loss Lsrc and the +memory replay regularization (See Sec. 3.3). Formally, the +loss describing this training is +Lreg = Ez∼psrc(z) +� +λsrcLsrc(G(z, c = 0))+ +Lrecon(G(z, c = 0)) +� +, +(10) +where Lrecon is the memory-replay loss defined in Eq. (5) +and λsrc = 1 is a hyperparameter weighting the losses. +Other classes, analogous to repurposed subspaces, are ded- +icated to the newly introduced domains. Whenever the ith +class is sampled (i > 0), we apply the loss of the domain +adaptation task Li. Applied over all new domains, the ex- +pansion loss is formally given by +Lexpand = +N +� +i=1 +Ez∼pi(z) Li(G(z, c = i)). +(11) +The final training objective still reads as Lfull = Lexpand + +Lreg. +Experiments. +We expand an FFHQ [12] generator with +two new domains, “Sketch” and “Tolkien Elf”, introduced +using StyleGAN-NADA [5]. We display the generated im- +ages using the same z latent codes for the different classes +Fig. 10a. +We qualitatively observe that the expanded, class- +conditioned generator preserves the source domain well, +also expressed by preserving the FID [7] score. However, +for new domains, we observe degraded performance from +two aspects. First, the class-conditioned generator “leaks” +knowledge between the classes. For example, in Fig. 10a, +faces generated from the class dedicated to sketches also +have long, elf-like, ears. +Second, the domains are not +“aligned”. Despite being generated from the same z latent +codes, the images differ beyond the differences between do- +mains. For example, corresponding images from the source +domain and elf domain often portray different head poses +and facial expression. Therefore, it is not clear how can one +obtain the elf “version” of a given face image, limiting the +applications of such a model. +For reference, we display comparable results from our +expansion method in Fig. 10b. As can be seen, our method +does not suffer from these issues. +C. Additional Experiments +C.1. Latent Directions Analysis +Our method explicitly relies on the existence of dormant +directions and their distinction from non-dormant direc- +tions. We wish to emphasize that the dichotomous distinc- +tion between “dormant” and “non-dormant” is a simplifica- +tion. In Fig. 11, we report the mean LPIPS distance induced +to images by a 3σ traversal along each direction. As can be +seen, the distance is never exactly 0 and there is also no +clear discontinuity. Nevertheless, it is clear that later direc- +tions, usually those beyond 100, cause significantly smaller +perceptual change in the generated image. This behavior +can also be qualitatively observed in Fig. 12. +As discussed in Sec. 4.1, this “almost” monotonous be- +havior is expected as our latent directions are right-singular +vectors, sorted in decreasing order according to their corre- +sponding singular values [36]. +C.2. Effect of Choice of Direction for Domain +Our method dedicates a single dormant direction for ev- +ery newly introduced domain. As mentioned in Sec. 4.1, +all previous experiments used the last dormant directions, +sorted in decreasing order according to their corresponding +singular values. One might wonder: Why should one use the +last directions? And among the last directions, how should +one match a direction to a domain? +We now demonstrate that the specific choice of a latent +direction has no significant impact on results, as long as it is +dormant. To this end, we perform multiple expansions, each +with 5 new domains introduced by StyleGAN-NADA [5], +11 + +use /Users/yotamnitzan/projects/stylegan2-ada-pytorch/results/cgan/cgan_new_div_0d1 +Source +Domain +Sketch +Elf +Source +Domain +Sketch +Elf +(a) Class-conditioned baseline +use /Users/yotamnitzan/projects/stylegan2-ada-pytorch/results/cgan/cgan_new_div_0d1 +Source +Domain +Sketch +Elf +Source +Domain +Sketch +Elf +(b) Our domain expansion method +Figure 10. Experimenting with a class-conditioned baseline for domain expansion. (a) Images generated from a class-conditioned expanded +model from the same z latent codes for the source, sketch, and elf domains. The source domain is preserved well in its dedicated class. +However, the newly introduced domains “leak” information, expressed in long, elf-like, ears in the sketch domain. Additionally, the +different domains are not well-aligned, as changing the domain also results in unrelated changes to head pose and facial expressions. (b) +Comparable results from our domain expansion method, provided for reference. As can be seen, using our method, the domains do not +interfere with each other and are well-aligned. +starting from a single generator pretrained on AFHQ [4]. +For 4 of the new domains – “Siberian Husky”, “Pixar”, +“Funny Dog”, “Boar” – we dedicate the same directions in +all experiments. Specifically, we use directions 507 − 510, +respectively. Directions numbers refer to their location in +the decreasingly sorted right-singular vector set. Recall that +the dimension of the latent space is 512, hence these direc- +tions are among the last ones. For the last domain, “Sketch”, +we vary the dedicated direction, using one of the directions +200, 300, 400, 500, 511. We run the expansion twice with +different random seeds. +We study how the choice of direction for the Sketch do- +main affects its performance. In Fig. 13 (top) we report the +CLIP error of images generated from the “Sketch” subspace +with the prompt “a sketch” as a function of training itera- +tions. We additionally display sample of generated images +from each model in Fig. 13 (bottom). As can be seen, sim- +ilar results are produced from different repurposed direc- +tions. Specifically, visual differences observed using differ- +ent directions, are similar to those observed using the same +directions but with different random seeds. This indicates +that the differences between directions are negligible and +might be entirely due to random chance. +Nevertheless, we do observe that certain directions min- +imize the CLIP error slightly more efficiently, across ran- +dom seeds. We therefore run additional 5 expansions, using +“Bear” instead of “Sketch”. We now observe a different or- +dering of directions. We therefore conclude, that even if +slight, imperceptible, differences exist between directions, +they are not consistent across domains. +In summary, the choice of dormant direction has little to +no effect. This result is arguably intuitive, as all dormant +directions might be considered equivalent, having insignif- +icant effect on generated images. Therefore, our choice of +using the last directions is almost arbitrary, only motivated +by the fact that they are the “most dormant”. Similarly, no +technique is required to match an direction to a domain, and +one can simply pick a dormant direction randomly. +C.3. Repurposing Non-Dormant Directions +Aiming at domain expansion, preserving the source do- +main is integral. Since the non-dormant directions span the +variations of the source domains, we explicitly kept them +intact, and repurposed only dormant directions. Neverthe- +less, the training method itself could be identically applied +to non-dormant directions. One simply needs to dedicate +a non-dormant direction to capture the new domain. We +next demonstrate that applying our method to non-dormant +directions is still effective and enables capabilities beyond +domain expansion. +Traversing the 1st latent direction in the generator pre- +trained on FFHQ [12], makes people in generated images +appear older and more masculine. Some users might decide +that they associate having a full beard with being older and +more masculine. To support such behavior, we fine-tune the +generator with a transformed StyleGAN-NADA [5], to cap- +ture “a person with a beard” along the 1st direction. We +display images generated along traversals of the 1st direc- +tion, before and after tuning, in Fig. 15a. As can be seen, +the generator now represents having a beard, along its 1st +latent direction, in addition to its previous behavior. +The capability to add new concepts in addition to exist- +ing ones does not depend on the close relationship between +the two in the last examples. To demonstrate this point, we +tune the generator to capture “Elf” along its 8th direction, +which originally encodes head pose (and a few other prop- +12 + +0 +100 +200 +300 +400 +500 +Direction +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +LPIPS Distance +(a) FFHQ +0 +100 +200 +300 +400 +500 +Direction +0.0 +0.2 +0.4 +0.6 +LPIPS Distance +(b) LSUN Church +Figure 11. Magnitude of perceptual effect caused by traversing +different directions. Directions are sorted in decreasing order ac- +cording to their corresponding singular values. For each direction, +we measure the LPIPS distance [49] between images from two la- +tent codes distanced by a 3σ traversal along the direction. As can +be seen, the effect caused by the traversal diminishes quickly and +the majority of directions are dormant. +erties). Results are displayed in Fig. 15b. +Previous results are clearly not solving domain expan- +sion, as they alter the original behavior of the source do- +main. Instead, one might say they adapt the domain mod- +eled by the generator. Nevertheless, there exists a profound +difference to existing domain adaptation methods. Our re- +sulting generator does not completely overriding the source +domain. Instead, in a precise and controllable manner, it +modifies individual factors of variation. Therefore, a user +can carefully rewrite [1,43] the semantic rules of a genera- +tive model, allowing greater control and freedom. +C.4. Distance to Repurposed Subspace +Repurposed subspaces are defined by transporting the +base subspace along a linear direction by a predetermined +scalar size s (See Eq. (3) in the main paper). All results +in the paper, across domains and variations used s = 20. +We next evaluate the effect the hyperparameter s has on re- +sults. To this end, we perform multiple expansions of an +FFHQ [12] generator with 100 new variations, while vary- +ing the value of s. +We measure CLIP errors (introduced in Sec. 4.3) of im- +ages generated from repurposed subspaces and the corre- +sponding target text used for training, as a function of train- +ing iterations. In Fig. 16a we report the results for two varia- +tions - “Marge Simpson” and “Tolkein Elf”. As can be seen, +for all s > 0, CLIP error decreases as training progresses, +and it decreases “faster” for greater values of the parameter +s. Even with ×10 more iterations, the model trained with +s = 5 does not reach the CLIP error of the model trained +with s = 20. +Images generated from the repurposed subspaces are dis- +played in Fig. 16b. For each value of s, we use the check- +point that resulted in the closest CLIP error to that obtained +by a favored s = 20 checkpoint. As can be seen, not only +training time is affected by parameter s, but the visual ef- +fects captured by training vary significantly. +We observe that models trained with greater values of +parameter s depict a more significant change with respect +to the source domain. When parameter s is too small (e.g., +s ≤ 5), the model captures only few, simple characteristics +of the new domain. On the other hand, when parameter s +is too large (e.g., s = 50), the model commonly generates +images that are blurry, have color artifacts or even do not +capture the target text well. For example, with the target text +“Marge Simpson”, the model learns to generate images with +blue skin rather than blue hair. We note that these undesired +artifacts cannot be mitigated by training with a large value +of parameter s originally, and use a smaller one in test-time, +as demonstrated in Fig. 17. +Following these results, we conclude that the parameter +s has a regularizing effect. Placing the domains “closer” +in the latent space causes them to be more similar in im- +age space as well. Conversely, placing the domains further +apart allows the new domain to capture more drastic, out- +of-domain effects. +Eventually, choosing a value for parameter s is subject +to user preference. In our experiments, we have found that +values in the range of [10, 30] offer satisfying results, across +different source and expanded domains. +We last note that the regularization effect of parameter s +could be explained by the existence of a globally consistent +“pace of change” of the generator with respect to the latent +space. With StyleGAN, such behavior is explicitly encour- +aged using a Perceptual Path Length (PPL) regularization +term [13]. Nevertheless, we observe identical results when +omitting this regularization during our expansion. +C.5. How Many Domains Can Fit? +So far, the largest number of new domains used for ex- +pansion was 105. The results from Appendix C.1 indicated +that there might be up to 400 dormant directions. Could +13 + +(a) FFHQ +(b) LSUN Church +Figure 12. Visualization of ±3σ traversal along latent directions in the FFHQ [12] and LSUN Church [48] models, obtained using +SeFA [36]. Directions shown are sorted from least (v0, top) to most (v511, bottom) dormant. As can be seen, later directions are dormant – +not affecting the generated image. We over-sample early directions for clarity. In practice, over 80% of directions are dormant. +14 + +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +Training Iterations +0.78 +0.79 +0.80 +0.81 +0.82 +0.83 +CLIP Error ( +) +Domain: Sketch +Repurposed Dimension +200 +300 +400 +500 +511 +200 +511 +300 +400 +500 +Seed 0 +(a) Seed 0 +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +Training Iterations +0.78 +0.79 +0.80 +0.81 +0.82 +0.83 +CLIP Error ( +) +Domain: Sketch +Repurposed Dimension +200 +300 +400 +500 +511 +200 +511 +300 +400 +500 +Seed 1 +(b) Seed 1 +Figure 13. We expand a generator pretrained on AFHQ [4] with 5 domains, varying the dormant direction dedicated to the “sketch” domain. +We repeat the expansion twice, with different random seeds. Top - reporting CLIP error of images generated from the sketch domain with +the text “a sketch”. Bottom - a sample of generated images from checkpoints obtaining CLIP error closest to the horizontal black line. As +can be seen, images generated using different repurposed dimensions differ only slightly. Specifically, changing the random seed induces +similar difference. +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +Training Iterations +0.76 +0.77 +0.78 +0.79 +0.80 +0.81 +CLIP Error ( +) +Domain: Bear +Repurposed Dimension +200 +300 +400 +500 +511 +Figure 14. Similar to Fig. 13, using a “bear” domain instead of +“sketch”. As can be seen, dimensions are ordered differently in +terms of minimizing CLIP error, as compared to their order for +sketch. +they all be repurposed? +We apply our method to expand a generator pretrained +on FFHQ with 400 new domains, repurposing the last (and +perhaps all) dormant directions. Incredibly, the expansion +succeeds. We find that the expansion follows the same find- +ings discussed in Sec. 4.3 – training is slower, yet quality is +uncompromised. Specifically, the FID score from the base +subspace is 2.83 compared to 2.80 in our model expanded +with 105 domains. We display images generated from this +model in the accompanying video and in Figs. 18 to 20. +C.6. Additional Compositions Results +In Figs. 21 and 22 we provide additional qualitative re- +sults displaying compositionality in expanded generators. +D. Additional Details +D.1. Training Time and Iterations +When expanding the generator with a single new do- +main, our training requires roughly twice the number of it- +erations to obtain comparable effects. The difference is a +direct result of our additional regularization terms. With ad- +ditional domains, we observe a roughly linear relationship +between the number of domains and the required training it- +erations. For example, the FFHQ model expanded with 105 +iterations was trained for 40K iterations, while the model +with 400 iterations was trained for 150K iterations. +Note that different training objective might require a dif- +ferent number of iterations. StyleGAN-NADA [5] specifi- +cally heavily relies on early-stopping. An ideal domain ex- +pansion method could consider this issue, and sample train- +ing objectives to apply non-uniformly. In practice, we did +not observe this to be an issue, probably due to our method +optimizing numerous objectives simultaneously. +15 + +200 +300 +400 +500 +511Before +After +1st direction +Before +After +(a) +Before +After +8th direction +Before +After +(b) +Figure 15. Using our training method with non-dormant direc- +tion rewrites existing semantic rules and adds new concepts on top +of existing ones. (a) Traversing the 1st direction originally made +people older and more masculine. After fine-tuning, it also adds +a beard. (b) Traversing the 8th direction originally turned people +heads. After fine-tuning it also turns them to elves. +D.2. Transformation of Loss Function +As explained in Sec. 3.2, transforming a given domain +adaptation task to perform domain expansion requires lim- +iting the samples latent codes. +The loss function itself, +in principal, is left unchanged. +This is exactly the case +for MyStyle [21]. For StyleGAN-NADA [5], however, we +made a subtle modification to the loss function. +StyleGAN-NADA computes its loss with respect to a +frozen copy of the source generator (See Sec. 4.1). This is +done in order to maintain access to the source domain, de- +spite it vanishing from the adapted generator during train- +ing. Conversely, using our method, the source domain is +preserved along the base subspace. We take advantage of +this fact and modify the loss only slightly. Instead of us- +ing a frozen generator to generate images from the source +domain, we simply use our expanded generator and latent +codes from the base subspace. +16 + +0 +50000 +100000 +150000 +200000 +250000 +300000 +350000 +400000 +Training Iterations +0.72 +0.74 +0.76 +0.78 +0.80 +0.82 +CLIP Error ( +) +Domain: Marge Simpson +s +0 +1 +5 +10 +20 +50 +0 +50000 +100000 +150000 +200000 +250000 +300000 +350000 +400000 +Training Iterations +0.70 +0.72 +0.74 +0.76 +0.78 +CLIP Error ( +) +Domain: Tolkein elf +s +0 +1 +5 +10 +20 +50 +(a) +s = 0 +s = 1 +s = 5 +s = 10 +s = 20 +s = 50 +Marge Simpson +Tolkien Elf +(b) +Figure 16. Evaluating the effect of the distance between the base and repurposed subspace, s. (a) We compare CLIP error as a function +of training iterations, between models trained with different values of parameter s. (b) Generated images from models having CLIP error +as close as possible to the black horizontal line. As can be seen, increasing s corresponds to faster minimization of CLIP error. However, +even with comparable CLIP errors, visual effect might vary significantly. Large values of parameter s are often associated with undesired +artifacts. We find that values between [10, 30] are usually preferable. +𝛼 = 0 +𝛼 = 10 +𝛼 = 40 +𝛼 = 20 +𝛼 = 30 +𝛼 = 50 +Base +subspace +Repurposed +subspace +Figure 17. Interpolation between the base subspace and the repur- +posed subspace where s = 50. As can be seen, undesired behavior +occurring at repurposed subspace (e.g. blue skin Marge Simpson) +cannot be mitigated by traversing shorter distances in test time. +The choice of parameter s is crucial in training time. +17 + +Gsrc +c(z)Figure 18. Subset 1/3 of generated images from a model expanded with 400 domains. +18 + +odigliani painting +leftnostri +Pho +lisplayed on a compute +Mar +turbar +OsoRr culpture +poctor Who +son eatingFigure 19. Subset 2/3 of generated images from a model expanded with 400 domains. +19 + +h no eyebrows +Crying Person +Viking +erson with tattoos +alvin klein +targarFigure 20. Subset 3/3 of generated images from a model expanded with 400 domains. +20 + + Photo taken in the Night +Ches +Koala +Shrek +Harry Potter +Kermit the Frog +Person with curly blond hair +Person with +eneyebrows +Digital Art +sketch +mustache +two horns +eyebrows +Satar +Statue of Liberty +Finger +ames Bond +BaldPerson +Jeag izzue +erson with +awide nose +lia Roberts + squirrel +ng person +Person with +curly purple hair +princes +MCDonaland +panda bear +Pixel Art +Roman emperor +bleeding nose +reencheeks +with no mouth +Dragon +Caricature +ersonwith +isplaye +irtphon +igerhain +nonkey +President +alePerson +opeye +person with curly ginger hairFigure 21. Composition of factors of variation introduced to a generator pretrained on FFHQ [12]. Following the format of Fig. 8 +. +21 + +Figure 22. Composition of factors of variation introduced to generators pretrained on LSUN Church [48] and SD-Elephant [20]. Following +the format of Fig. 8 +. +22 + diff --git a/ltE4T4oBgHgl3EQftg2f/content/tmp_files/load_file.txt b/ltE4T4oBgHgl3EQftg2f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..49313f072e2868bce51dd98780d21c22d29559a6 --- /dev/null +++ b/ltE4T4oBgHgl3EQftg2f/content/tmp_files/load_file.txt @@ -0,0 +1,950 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf,len=949 +page_content='Domain Expansion of Image Generators Yotam Nitzan1,2 Micha¨el Gharbi1 Richard Zhang1 Taesung Park1 Jun-Yan Zhu3 Daniel Cohen-Or2 Eli Shechtman1 1 Adobe Research 2 Tel-Aviv University 3 Carnegie Mellon University Source Generator Dormant direction Domain Expansion (Ours) Zombie direction Domain Adaptation Dormant direction Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (center) Traversing the latent space of generative models along some directions changes the image significantly while traversing others has no perceptible effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We call directions of the latter type dormant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (left) Domain adaptation methods, transform the entire generator from a source domain to a target domain, indicated by the color blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (right) We introduce an approach for a new task – domain expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Instead of fully transforming the generator, we expand it to include new data domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our method learns to represent the new domain in a disentangled manner by repurposing a single dormant direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Abstract Can one inject new concepts into an already trained gen- erative model, while respecting its existing structure and knowledge?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We propose a new task – domain expansion – to address this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Given a pretrained generator and novel (but related) domains, we expand the generator to jointly model all domains, old and new, harmoniously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' First, we note the generator contains a meaningful, pretrained latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Is it possible to minimally perturb this hard-earned representation, while maximally representing the new do- mains?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Interestingly, we find that the latent space offers un- used, “dormant” directions, which do not affect the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This provides an opportunity: By “repurposing” these di- rections, we can represent new domains without perturbing the original representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In fact, we find that pretrained generators have the capacity to add several – even hundreds – of new domains!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Using our expansion method, one “ex- panded” model can supersede numerous domain-specific models, without expanding the model size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Additionally, a single expanded generator natively supports smooth transi- tions between domains, as well as composition of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Code and project page available here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Introduction Recent domain adaptation techniques piggyback on the tremendous success of modern generative image models [3, 12, 32, 40], by adapting a pretrained generator so it can generate images from a new target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Oftentimes, the target domain is defined with respect to the source do- main [5,21,22], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=', changing the “stylization” from a pho- torealistic image to a sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' When such a relationship holds, domain adaptation typically seeks to preserve the fac- tors of variations learned in the source domain, and transfer them to the new one (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=', making the human depicted in a sketch smile based on the prior from a face generator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' With existing techniques, however, the adapted model loses the ability to generate images from the original domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In this work, we introduce a novel task — domain ex- pansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Unlike domain adaptation, we aim to augment the space of images a single model can generate, without over- riding its original behavior (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Rather than view- ing similar image domains as disjoint data distributions, we treat them as different modes in a joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As a result, the domains share a semantic prior inherited from the original data domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For example, the inherent factors 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='05225v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='CV] 12 Jan 2023 of variation for photorealistic faces, such as pose and face shape, can equally apply to the domain of “zombies”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To this end, we carefully structure the model train- ing process for expansion, respecting the original data do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' It is well-known that modern generative models with low-dimensional latent spaces offer an intriguing, emer- gent property – through training, the latent spaces represent the factors of variation, in a linear and interpretable man- ner [3, 6, 10, 12, 28, 30, 39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We wish to extend this ad- vantageous behavior and represent the new domains along linear and disentangled directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Interestingly, it was pre- viously shown that many latent directions have insignificant perceptible effect on generated images [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Taking advan- tage of this finding, we repurpose such directions to repre- sent the new domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In practice, we start from an orthogonal decomposition of the latent space [36] and identify a set of low-magnitude directions that have no perceptible effect on the generated images, which we call dormant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To add a new domain, we select a dormant direction to repurpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Its orthogo- nal subspace, which we call base subspace, is sufficient to represent the original domain [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We aim to repur- pose the dormant direction such that traversing it would now cause a transition between the original and the new domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Specifically, the transition should be disentangled from the original domain’s factors of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To this end, we define a repurposed affine subspace by transport- ing the base subspace along the chosen dormant direction, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We capture the new domain by applying a domain adaptation method, transformed to operate only on latent codes sampled from the repurposed subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' A regularization loss is applied on the base subspace to en- sure that the original domain is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The original do- main’s factors of variation are implicitly preserved due to the subspaces being parallel and the latent space being dis- entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For multiple new domains, we simply repeat this procedure across multiple dormant directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We apply our method to the StyleGAN [13] architecture, with multiple datasets, and expand the generator with hun- dreds of new factors of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Crucially, we show our expanded model simultaneously generates high-quality im- ages from both original and new domains, comparable to specialized, domain-specific generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Thus, a single ex- panded generator supersedes hundreds of adapted genera- tors, facilitating the deployment of generative models for real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We additionally demonstrate that the new domains are learned as global and disentangled fac- tors of variation, alongside existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This enables fine- grained control over the generative process and paves the way to new applications and capabilities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=', compositing multiple domains (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Finally, we conduct a de- tailed analysis of key aspects of our method, such as the ef- fect of the number of newly introduced domains, thus shed- Dog Cute Siberian Husky Sketch Expanded Domains Boar Happy Pop Art Source Domain Domain Composition Siberian Husky + Cute + Sketch Boar + Happy + Pop Art Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Example of a domain expansion result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Starting from dogs as the source domain, we expand a single generator to model new domains such as facial expressions, breeds of dogs and other animals, and artistic styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Finally, as the representations are dis- entangled, the expanded generator is able to generalize and com- pose the different domains, although they were never seen jointly in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' ding light on our method and, in the process, on the nature of the latent space of generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To summarize, our contributions are as follows: We introduce a new task – domain expansion of a pre- trained generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We propose a novel latent space structure that is amenable to representing new knowledge in a disentangled manner, while maintaining existing knowledge intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We present a simple paradigm transforming domain adap- tation methods into domain expansion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We demonstrate successful domain expansion to hun- dreds of new domains and illustrate its advantage over domain adaptation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Related Work Fine-tuning generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Starting from a genera- tor pretrained on a source domain and training it for a target domain, often called fine-tuning, is a common technique ap- plied for various purposes and settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Some works wish to model only the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In which case, the pretrained model is leveraged simply as an efficient initialization, shortening the training time, and im- proving image quality [11, 16, 19, 47, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Others, wish to learn the target domain alongside the source domain, in a setting called continuous learning, and propose methods to ensure that the source domain is not forgotten [34,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Al- though in a single generator, the domains are modeled sep- arately, each as its own class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' A prominent line of works have sought to make the target domain inherit knowledge from the source domain [1, 2, 5, 14,17,21–23,31,33,42,43,51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This approach allows gener- alization beyond the target domain per-se and is especially useful when training data is scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our work similarly involves fine-tuning, but for a novel 2 purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our perspective is that, since the target domain is introduced with knowledge from the source domain – it is in essence, an expansion of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Therefore, in contrast to the aforementioned works, we aim to model the domains jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The proposed method does not replace previous fine-tuning methods, but allows applying them jointly, in a plug-and-play manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Latent directions in generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Generative models learn to represent the factors of variation of ob- served data in their latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Disentangled represen- tations are especially useful as they facilitate intuitive con- trol over the generative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' With recent architectures, disentanglement miraculously emerges without interven- tion [12, 24, 28, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In such models, disentanglement is manifested through the existence of linear latent directions, each ideally controlling a single factor of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Due to the spontaneous emergence of such directions, numerous works have been proposed to identify them after the model has been trained [6, 25, 35–37, 41, 45] and used them for downstream applications, most commonly seman- tic image editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' At the same time, it has also been ob- served that some latent directions have no perceptible effect on the generated images [6, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' These directions, which we call dormant, were not previously leveraged for any pur- pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In this work, we rely on existing methods to factorize the latent space into such linear directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As we aim to expand the pretrained generator to additional domains, we decide to explicitly encode the “new knowledge” along the dormant directions, while keeping other directions intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This design ensures that the original domain is preserved and that the different domains are represented in a disentan- gled fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Method We start with a pretrained generator Gsrc that maps from latent codes z ∈ Z ⊆ RD to images in a source domain Dsrc, and a set of N domain adaptation tasks, each defined by a loss function Li, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In domain adapta- tion, fine-tuning Gsrc to minimize Li yields a generator Gi that generates images from the new domain Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In contrast, our goal is domain expansion, which aims at training a sin- gle expanded generator G+ that can simultaneously model all the new domains ∪N i=1Di, along with the original do- main Dsrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We want to ensure that the new domains Di are disentangled from each other and also share the factors of variation from the source domain, which remain intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our solution is to partition the latent space into disjoint subspaces, one for each new domain, and to restrict the ef- fect of each domain adaptation to the corresponding sub- space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To this end, we endow the latent space with an ex- plicit structure that supports domain expansion (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='1), Dormant G (a) Domain Adaptation Training (b) Domain Expansion Training Repurposed G Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our method transforms a domain adaptation task to a domain expansion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (a) Generator G is optimized to satisfy the loss Li for every latent code in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The entire generator and latent space now represent the new domain, indicated with the color blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (b) Generator G is optimized to satisfy the same loss, Li, only on a subspace Zi, dedicated to the new domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Simul- taneously, G is optimized to satisfy a regularization term Lreg on a parallel subspace, Zbase, ensuring the original knowledge is pre- served there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The generator and latent space now represent both domains, indicated by being colored both blue and orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The latent direction between the two spaces was originally dormant in generator G, and now represents a transition between the domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' and optimize each domain adaptation loss only using la- tents from specific subspace reserved for the new domain (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our decomposition reserves a base subspace for the original domain Dsrc, on which we impose a regulariza- tion objective to maintain the behavior of the source gen- erator (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 3 gives an overview of our domain expansion algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Structuring the Latent Space for Expansion Modern generative models conveniently learn to repre- sent the factors of variation along linear latent directions, in a completely unsupervised manner [12, 26, 28, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We decide to explicitly extend this model by structuring the la- tent space such that the effect defined by an adaptation task would be represented along a single linear direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For- mally, there should exist some scalar s and latent direction vi, for which images generated from G+(z), G+(z + svi), relate to each other as the corresponding images from the source and adapted generators Gsrc(z), Gi(z) do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Concretely, following SeFA [36], we obtain a semantic and orthogonal basis V of the latent space from the right singular vectors (produced by SVD) of the very first gener- ator layer, which acts on the latent space Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' With a similar factorization technique [6], it was observed that a relatively small subset of the basis vector is sufficient to represent most of the generators Gsrc’s variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Other basis vectors have barely any perceptible effect on the generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We find this to be the case with SeFA as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We refer to vectors with no perceptible effect as dormant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As the dormant directions do not affect the model’s gen- eration capabilities, they are available to be repurposed with new desired behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We thus choose to represent the do- mains Dsrc and Di in regions that are separated by only a dormant direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 3 Z aseL 22Zi\\regFormally, for each of the N adaptation tasks, we dedi- cate a single dormant direction, vi, that will be repurposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The remaining directions {vN+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' , vD} will remain in- tact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We finally define a subspace of Z, dubbed the base subspace, as Zbase = span(vN+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' , vD) + z (1) where z is the mean of the distribution over the latents used to train the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Then, for each repurposed direction, vi, we define a repurposed subspace Zi that is the base subspace transported along direction vi by a predetermined scalar size s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Zi = Zbase + svi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (2) The choice of direction vi and scalar s are discussed in Ap- pendices C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our domain expansion training procedure described hereafter will ensure subspace Zi is the only part of the latent space affected by the training objective Li, and is reserved to generate images from domain Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Intuitively, shifting the base subspace along direction vi aims to achieve two goals: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' preserve the factors of variations inherited from Zbase, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' restrict the new factor of variation (cor- responding to Di) to a single latent direction, vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' From Domain Adaptation to Expansion Having defined disjoint affine subspaces Zi of the latent space Z for our new domains Di, we now describe how we constrain each domain adaptation objective Li to affect only the corresponding subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The domain adaptation objective is applied to images generated from latent codes z ∈ Z, sampled from distribu- tion p(z) defined on the entire space Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Commonly the dis- tribution is a Gaussian, or is derived from it [12] but some exceptions exist [21, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our strategy is to transform this sample distribution into one restricted to the affine subspace Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We do so by projecting the samples from p(z) onto Zi, using a standard orthogonal projection operator projZi(z) = D � j=N+1 (v⊤ j (z − z))vj + z + svi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (3) Denoting by pi the sampling distribution over Z for each of the new domains we seek to adapt, the training loss over all tasks is defined as Lexpand = N � i=1 Ez∼pi(z) Li(G(projZi(z))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (4) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Regularization Optimizing Lexpand lets us learn to generate data from the new domains Di within a single generator, but unfortu- nately it leaves the base subspace Zbase under-constrained Base Subspace Subspace Sketch of a Dog Funny Dog Subspace w\\o w\\o Unregularized Regularized Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Regularization prevents leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Without regulariza- tion (top row), new factors of variation “Sketch” and “Funny” are leaking into the base subspace and the other repurposed subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Note, for example, that the image from the base subspace is both a sketch and depicts a smiling dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our regularization, described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (6), solves the issue (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' and, therefore, does not guarantee it will remain unaltered during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In practice, we observe that the effect of Li “leaks” outside Zi, causing catastrophic forgetting [18] in subspace Zbase, and undesirably affecting other subspaces Zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We show an example of this leakage in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To prevent this failure mode, we explicitly enforce the preservation of Gsrc’s behavior over the base subspace Zbase by regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We adopt two successful regularization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' First, we keep optimizing the generator with the loss it was originally trained on, Lsrc, which is known to mitigate forgetting [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Second, we apply replay align- ment [44], which is a reconstruction loss that compares the output of a frozen copy of the source generator to that pro- duced by our generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We use a weighted combination of an L2 pixel loss and LPIPS [49]: Lrecon = λlpipsLlpips(Gsrc(z), G(z))+ λL2∥Gsrc(z) − G(z)∥2, (5) where λlpips = λL2 = 10 are weighting hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Not only does replay alignment preserve the source domain Dsrc, it also has the added benefit of aligning G+ to the source generator Gsrc, in the sense that they will produce similar outputs given the same latent code z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Crucially, we only regularize the base subspace Zbase, since the subspaces Zi should be allowed to change to learn the new behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To this end, we project the latent codes to the base subspace Zbase, before calculating the regular- ization terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our overall regularization objective is thus: Lreg = Ez∼psrc(z) � λsrcLsrc(G(projZbase(z)))+ Lrecon(G(projZbase(z))) � , (6) where λsrc = 1 balances the two terms and psrc(z) is the latent distribution over Z used to train Gsrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our final, reg- ularized domain expansion objective is, therefore: Lfull = Lexpand + Lreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (7) 4 vregGsrc c(z)Z122OP4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Experiments We evaluate our method and analyze its key character- istics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='1 first details the experimental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We start by analyzing the knowledge encoded along repurposed directions and compare it to domain adaptation methods (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We then delve deeper and evaluate the effects (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='3) and opportunities (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='4) presented by expand- ing a generator to multiple domains simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Last, we demonstrate that the quality of the source domain is maintained in the base subspace (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Further experiments, results, and details are provided in the supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Experimental Setting We adopt StyleGAN2 [13] as the source generator archi- tecture, for its disentangled latent space and because it has been the dominant test bed for generative domain adapta- tion methods in recent years [1,5,21,22,43,51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Latent space and subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Several latent spaces have been considered in the context of StyleGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We use the intermediate latent space W in all our experiments but note it as Z for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We use SeFA [36] for the orthogonal decomposition of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As SeFA performs SVD, there is a native indication to how dormant is a given latent direction – the corresponding singular value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As singular values are commonly sorted in decreasing orders, the last basis vectors are most dormant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' When expanding with N new domains, unless specified otherwise, we repurpose the last N basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We use s = 20 in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' These decisions are evaluated in greater depth in Appendices C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Adaptation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We demonstrate our expansion method with two domain adaptation tasks - StyleGAN- NADA [5] and MyStyle [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' These two tasks were cho- sen as they differ significantly in key aspects – source of supervision, sampling distribution and loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' StyleGAN-NADA is a zero-shot, text-guided, domain adaptation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' It takes as input a pair of text prompts, tsource and ttarget, describing the desired transformation source → target to be applied on the domain of the pre- trained generator, Dsrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The loss function L is given by ∆T = ET (ttarget) − ET (tsource) , ∆I = EI (G (z)) − EI (Gsrc (z)) , L = 1 − ∆I · ∆T |∆I| |∆T| , (8) where EI and ET are CLIP’s [29] image and text encoders respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' MyStyle is a few-shot, image-supervised, domain adaptation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As input, it takes a set of images !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' −𝑠 3 0 𝑠 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 𝑠 3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 2𝑠 3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4𝑠 3 Sketch Polar Elephant Botox Lips 𝛼 = Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Continuous traversal along repurposed directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, the traversal between the base subspace (α = 0) and repurposed subspace (α = s) portrays a smooth transition between the source and newly introduced domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Advantageously, the semantic meaning of the repurposed direction is preserved in the extrapolation, representing the opposite relationship between the domains (α < 0) or exaggerations of it (α > s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' {xm}M m=1 of an individual (M ∼ 100), and adapts Gsrc to form a personalized prior for that individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The generator is trained to better reconstruct xm from their original latent space inversions zm ∈ Z [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Formally, the loss function is given by L = M � m=1 [Llpips(G(zm), xm) + ∥G(zm) − xm∥2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (9) where Llpips is again the LPIPS loss [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Datasets and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We demonstrate our method on four datasets – FFHQ [12], AFHQ Dog [4], LSUN Church [48] and SD-Elephant [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The FFHQ model is expanded with 105 new domains, 100 introduced with the expanded variant of StyleGAN-NADA and 5 from the expanded vari- ant of MyStyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The AFHQ Dog, LSUN Chruch and SD- Elephant are expanded with 50, 20, and 20 new domains correspondingly, all introduced from the expanded variant of StyleGAN-NADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Evaluating Domains Individually Traversing a repurposed direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We start by investi- gating what knowledge, if any, is encoded along the repur- posed latent directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To this end, starting from a random latent code z ∈ Zbase, we individually traverse different re- purposed directions, vi, and inspect the generated images G+(z +αvi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Sample results from our dogs, elephants, and faces models are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We find that each in- dividual repurposed direction now successfully encodes the desired factor of variation, in a global and continuous way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our training paradigm is inherently discrete – encour- aging the source behavior on the base subspace (α = 0) and the newly introduced effect on the repurposed space 5 StyleGAN-NADA Ours Barbie Barack Obama MyStyle Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' A random set of images generated by our generator from repurposed subspaces (bottom) and by corresponding do- main adaptation methods (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The images are similar and dif- ferences are subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Method User % (↑) ID (↑) Diversity×10 (↑) StyleGAN-NADA 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='13 Ours w/ NADA 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='8% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='13 MyStyle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='15 Ours w/ MyStyle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='14 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Quantitative comparison of images generated from our repurposed subspaces to those generated by corresponding domain adaptation methods - StyleGAN-NADA [5] and MyStyle [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We follow each adaptation method’s quantitative evaluation protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (α = s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Therefore, obtaining a smooth effect might seem surprising at first glance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Nevertheless, this phenomenon can be clearly traced to the well-established observation that generators are smooth with respect to their latent space [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Behavior on the repurposed subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We have trans- formed adaptation tasks into expansion tasks by limiting the training effect to the repurposed subspaces only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' But, for latents in repurposed subspaces (α = s), the domain adaptation could be considered to have been applied as-is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We next directly compare the images generated by our generator from the repurposed subspace to the correspond- ing images generated by the domain-adapted generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We inherit and repeat the quantitative evaluation protocols per- formed by each of the adaption tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To compare qual- ity with StyleGAN-NADA [5] we perform a two-alternative forced choice user study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Users were asked to pick the im- age that has higher-quality and better aligns with the target text used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We gathered 1440 responses from 32 unique users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To compare quality with MyStyle [21], we evaluate preservation of identity in generated images, as observed by a face recognition network [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For both methods, the diversity is compared based on intra-cluster LPIPS [49] distance, first suggested by Ojha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We use 10 domains for comparison with StyleGAN-NADA and Better Alignment Worse Leakage (a) Sketch Subspace Sumo Subspace Sketch Only Sketch + 4 Sketch + 49 Sketch + 4 Sketch + 49 (b) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Investigating the effect of introducing multiple domains simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (a) Reports the CLIP error of generated images with the text “a sketch”, as a function of training iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Images are generated from the “Sketch” and “Sumo” subspaces of models trained with a different number of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (b) Depicts generated images from models that have similar CLIP errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, the sketch domain does not “leak” into the sumo subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Additionally, introducing additional domains delays, but does not prevent, the introduction of sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 5 for comparison with MyStyle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Note that we use a single generator G+, expanded with 105 domains, while compet- ing methods use a dedicated model per domain, 15, overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The results are reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 1, and a qualitative sample is provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, on the repurposed subspaces, our method produces comparable images to that generated by the ded- icated, domain-adapted generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Perhaps surprisingly, users somewhat prefer our results over StyleGAN-NADA’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We speculate this is due to the significantly greater difficulty of choosing hyperparameters for their training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Effect of Domains on Each Other Previous evaluation of individual repurposed directions already indicates disentanglement between different factors of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For example, “Barbie” images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 6 show no sign of being caricature, Barack Obama, or any of the other hundred factors of variation introduced to that gen- erator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In this section, we delve deeper into evaluating the effects of expanding with multiple factors of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To this end, we train three models to expand the FFHQ parent model with either 1, 5 or 50 new domains, all in- duced by StyleGAN-NADA [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' All models are expanded with “Sketch”, and the latter two with “Sumo” as well as other factors of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We quantify the strength of introduced factor of variation using CLIP error, the 1- complement of the score produced by CLIP [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We use the top-performing version of the CLIP encoder available, 6 Dwayne Johnson - #509 (MyStyle) Cubism Art - #503 (NADA) Pixar - #484 (NADA) Tongue Out - #463 (NADA) Towers & Night - #7 (Original) Desert - #496 (NADA) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Composing multiple effects by simple latent traversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In each grid, we start from the latent code that generates the top-left image and traverse along two latent directions, represented by advancement in rows and columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For each direction, we note the associated domain, its ordinal number in the latent space’s basis, and the training method used (“NADA” or “MyStyle”) to learn the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, G+ has learned a disentangled representation, allowing meaningful composition of concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Specifically, note the disentanglement between directions, as traversing left-right does not affect the magnitude of the effect corresponding to up-down traversal, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' ViT-L/14, which is not used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We note that simply minimizing CLIP error is not the objective, as it might lead to favoring mode-collapsed and adversarial ex- amples [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Nevertheless, together with qualitative inspec- tion, it is useful for comparing different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 7a, we report the CLIP error of generated images with the text “a sketch”, as a function of the training itera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Images are generated from the “Sketch”, and if exists, “Sumo” subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' First, we observe that CLIP error is de- creasing for “Sketch” subspaces in all models, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Conversely, CLIP error in the “Sumo” subspace does not significantly change, indicating it is not becoming any more or less of a sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This result quantitatively supports our previous finding, that factors of variations do not interfere with each other, and demonstrates it is true regardless of the number of other factors of variations learned simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Additionally, we observe that expanding with addi- tional factors of variation delays, but does not prevent, G+ the introduction of ”Sketch“ effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The observed delay is expected, as expanding with more variations corresponds to G+ optimizing and balancing additional loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Generated samples from the sketch and sumo subspaces are provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Compositionality While accidental “leakage” between latent directions during training is undesired, intentionally composing vari- ations at test time is advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For generative mod- els with a disentangled latent space, summing together la- tent directions aggregates their semantic meaning, and ide- ally should not affect the magnitude of their effects if ap- plied separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For example, if direction v1 controls head pose and direction v2 controls an unrelated variation, im- ages G(z + v1 + v2) and G(z + v1) should depict the same head pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We find that the latent space of the expanded generator G+ is disentangled, and variations can indeed be composed effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Crucially, variations can be composed with each other regardless of their originating training task, including those on the base subspace, learned from the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 8 shows a sample of gradual composition results across models and training tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Comparison to existing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Several domain adaptation methods [5, 14] have proposed techniques to combine multiple variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' These methods still train a separate generative model per variation, but combine their effects in test-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Specifically, in the realm of CLIP- supervised training, StyleGAN-NADA [5] interpolates the generators’ weights, while DiffusionCLIP [14] interpolates intermediate activations of the generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Next, we com- pare the disentanglement of composition in our generator to that made possible using these techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For each method, we start with a setting that was opti- mized to generate images that align with one of two text prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In our case, this setting is G+ with latent codes in certain subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For the baselines, these settings are dedicated generators with any latent code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Then, for each method, we gradually introduce the variation described by the other text prompt and generate the corresponding im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Finally, we measure normalized CLIP error between generated images and the two prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We normalize all errors by the error of the initial setting, to make the metric comparable across methods and text prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 9 reports 7 (a) Munch Painting Samurai Sumo Neanderthal Composition Composition Diffusion- CLIP StyleGAN- NADA Ours (b) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Comparing compositionality in our generator to methods of combining multiple domains proposed by StyleGAN-NADA [5] and DiffusionCLIP [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Starting from a setting optimized for either text prompt #1 or #2, we gradually introduce the variation described by the other text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (a) Reports the CLIP error to both prompts along the gradual introduction, normalized to the error obtained for each text prompt in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (b) Portrays a sample of qualitative results, where the composition is such that assigns equal strengths to both effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen in, both quantita- tively and qualitatively, NADA and DiffusionCLIP directly trade- off one effect for the other – strengthening the effect of one prompt directly lessens that of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In contrast, our generator allows true composition of modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Specifically, note that our genera- tor is able to compose effects that are somewhat dependent, such as Neanderthal & Sumo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' the mean and standard deviation of the CLIP error, on 10 pairs of prompts, and provides a sample of qualitative re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, both baseline methods directly trade- off one domain for the other, expressed by a linear-looking trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Conversely, our method obtains significantly lower errors and allows for a true composition of concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Preservation of the Source Domain Finally, we evaluate the preservation of the source mode in G+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To this end, using FID [7], we compare the quality of images generated from the base subspace Zbase of G+ to those generated by the source generator Gsrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Since the Model FFHQ AFHQ LSUN Church SD Elephant Parent 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='77 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='30 Ours 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='80 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='70 Only Lsrc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='08 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='91±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='67 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We generate images from the base subspace and report FID [7] (↓) with respect to source domain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We compare our FID to that of the source generator Gsrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For reference, we also continue training the source generator for the same number of iterations with its original loss - Lsrc, and report the mean and standard deviations of FID along the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, on the base subspace, our models have comparable FID scores to their parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Furthermore, similar magnitude of change in FID are observed by simply continuing training, indicating that the change in FID might be, at least in part, due to “random” fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' generator is being trained, some change in FID is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Therefore, we also report the average and standard devia- tion over FID scores for a generator that simply continues training, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=', using only the original loss Lsrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Results are reported in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 2 and vary between datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For FFHQ and AFHQ, we observe a slight increase in FID, but one that is within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='5σ and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='5σ of the random fluctuations of the ref- erence, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For LSUN Church, our model obtains a lower FID score than the source generator, but slightly higher than that obtained by continuing training, while for the SD-Elephants, the opposite occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We conclude that the expansion method might have a slight impact on FID, but it is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Conclusions We present a new problem – domain expansion – and propose an approach to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The core of our method is to carefully structure the latent space, such that it is amenable to learning additional knowledge, while keeping the existing knowledge intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our method takes advantage of the existence of dormant latent directions, and the task itself implicitly relies on the capacity of the model weights to represent more knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' If one of these assumptions does not hold, it might not be possible to apply domain ex- pansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' However, the popularity of methods squeezing neural networks, such as Knowledge Distillation [8], and current estimates of the intrinsic dimensionality of image datasets [27], indicate that these assumptions commonly hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In our experiments, we were able to expand to hun- dreds of directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' A plausible limitation is that the model can be expanded to a certain point but ultimately limited by factors such as the latent space or network capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Over- coming this limitation, perhaps by considering more com- plex latent space structures, is an avenue for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='0 Method NADA Normalized CLIP Error #2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='8 DiffusionCLiP ours 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='0 NormalizedCLiPError#1Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We are grateful to Rinon Gal, Yossi Gandelsman and Sheng-Yu Wang for their suggestions in the early stages of this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We also thank Rinon Gal, Yossi Gandelsman, Kfir Aberman, Alon Nitzan, Omer Bar- Tal, Nupur Kumari and Gaurav Parmar for proofreading the draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We also thank Nupur Kumari for a technical advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We finally thank Yogev Nitzan for his help with the user study and for coming up with hundreds of textual prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This work was done while Yotam Nitzan was an intern at Adobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This research was supported in part by the Israel Science Foundation (grants no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 2492/20 and 3441/21), Len Blavatnik and the Blavatnik family foundation, and The Tel Aviv University Innovation Laboratories (TILabs).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='03365, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 5, 14, 22 [49] Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shecht- man, and Oliver Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The unreasonable effectiveness of deep features as a perceptual metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recogni- tion, pages 586–595, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4, 5, 6, 13 [50] Shengyu Zhao, Zhijian Liu, Ji Lin, Jun-Yan Zhu, and Song Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Differentiable augmentation for data-efficient gan training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='10738, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 2 [51] Peihao Zhu, Rameen Abdal, John Femiani, and Peter Wonka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Mind the gap: Domain gap control for single shot do- main adaptation for generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='08398, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 2, 5 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Overview In Appendix B, we consider a baseline for domain ex- pansion and demonstrate it is inferior to our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Next follows the main part of the supplementary, Appendix C, in which we perform additional analysis and experimentation of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Finally, in Appendix D, we provide additional details completing the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Domain Expansion Baseline Using Class- Conditioning In this section, we experiment with an alternative, base- line, method to perform domain expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Generative models capturing multiple domains commonly use a class- conditioning mechanism [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Adopting this approach, we attempt to perform domain expansion by modeling domains with classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We find that this method does not work as well as our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We start with an unconditional pretrained gen- erator, specifically StyleGAN [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We then make the gen- erator condition on a one-hot vector, using the architecture proposed by Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This change involves adding a single MLP layer, whose input is the one-hot vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Its output is concatenated to the random latent code and then fed to the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The class-conditioned generator is trained in a similar protocol to our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The source domain uses class c = 0, which is analogous to the base subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Whenever the 0th class is sampled, we apply the original loss Lsrc and the memory replay regularization (See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Formally, the loss describing this training is Lreg = Ez∼psrc(z) � λsrcLsrc(G(z, c = 0))+ Lrecon(G(z, c = 0)) � , (10) where Lrecon is the memory-replay loss defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (5) and λsrc = 1 is a hyperparameter weighting the losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Other classes, analogous to repurposed subspaces, are ded- icated to the newly introduced domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Whenever the ith class is sampled (i > 0), we apply the loss of the domain adaptation task Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Applied over all new domains, the ex- pansion loss is formally given by Lexpand = N � i=1 Ez∼pi(z) Li(G(z, c = i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (11) The final training objective still reads as Lfull = Lexpand + Lreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We expand an FFHQ [12] generator with two new domains, “Sketch” and “Tolkien Elf”, introduced using StyleGAN-NADA [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We display the generated im- ages using the same z latent codes for the different classes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We qualitatively observe that the expanded, class- conditioned generator preserves the source domain well, also expressed by preserving the FID [7] score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' However, for new domains, we observe degraded performance from two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' First, the class-conditioned generator “leaks” knowledge between the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 10a, faces generated from the class dedicated to sketches also have long, elf-like, ears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Second, the domains are not “aligned”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Despite being generated from the same z latent codes, the images differ beyond the differences between do- mains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For example, corresponding images from the source domain and elf domain often portray different head poses and facial expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Therefore, it is not clear how can one obtain the elf “version” of a given face image, limiting the applications of such a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For reference, we display comparable results from our expansion method in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 10b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, our method does not suffer from these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Additional Experiments C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Latent Directions Analysis Our method explicitly relies on the existence of dormant directions and their distinction from non-dormant direc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We wish to emphasize that the dichotomous distinc- tion between “dormant” and “non-dormant” is a simplifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 11, we report the mean LPIPS distance induced to images by a 3σ traversal along each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, the distance is never exactly 0 and there is also no clear discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Nevertheless, it is clear that later direc- tions, usually those beyond 100, cause significantly smaller perceptual change in the generated image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This behavior can also be qualitatively observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='1, this “almost” monotonous be- havior is expected as our latent directions are right-singular vectors, sorted in decreasing order according to their corre- sponding singular values [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Effect of Choice of Direction for Domain Our method dedicates a single dormant direction for ev- ery newly introduced domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='1, all previous experiments used the last dormant directions, sorted in decreasing order according to their corresponding singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' One might wonder: Why should one use the last directions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' And among the last directions, how should one match a direction to a domain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We now demonstrate that the specific choice of a latent direction has no significant impact on results, as long as it is dormant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To this end, we perform multiple expansions, each with 5 new domains introduced by StyleGAN-NADA [5], 11 use /Users/yotamnitzan/projects/stylegan2-ada-pytorch/results/cgan/cgan_new_div_0d1 Source Domain Sketch Elf Source Domain Sketch Elf (a) Class-conditioned baseline use /Users/yotamnitzan/projects/stylegan2-ada-pytorch/results/cgan/cgan_new_div_0d1 Source Domain Sketch Elf Source Domain Sketch Elf (b) Our domain expansion method Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Experimenting with a class-conditioned baseline for domain expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (a) Images generated from a class-conditioned expanded model from the same z latent codes for the source, sketch, and elf domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The source domain is preserved well in its dedicated class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' However, the newly introduced domains “leak” information, expressed in long, elf-like, ears in the sketch domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Additionally, the different domains are not well-aligned, as changing the domain also results in unrelated changes to head pose and facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (b) Comparable results from our domain expansion method, provided for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, using our method, the domains do not interfere with each other and are well-aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' starting from a single generator pretrained on AFHQ [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For 4 of the new domains – “Siberian Husky”, “Pixar”, “Funny Dog”, “Boar” – we dedicate the same directions in all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Specifically, we use directions 507 − 510, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Directions numbers refer to their location in the decreasingly sorted right-singular vector set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Recall that the dimension of the latent space is 512, hence these direc- tions are among the last ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For the last domain, “Sketch”, we vary the dedicated direction, using one of the directions 200, 300, 400, 500, 511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We run the expansion twice with different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We study how the choice of direction for the Sketch do- main affects its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 13 (top) we report the CLIP error of images generated from the “Sketch” subspace with the prompt “a sketch” as a function of training itera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We additionally display sample of generated images from each model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 13 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, sim- ilar results are produced from different repurposed direc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Specifically, visual differences observed using differ- ent directions, are similar to those observed using the same directions but with different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This indicates that the differences between directions are negligible and might be entirely due to random chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Nevertheless, we do observe that certain directions min- imize the CLIP error slightly more efficiently, across ran- dom seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We therefore run additional 5 expansions, using “Bear” instead of “Sketch”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We now observe a different or- dering of directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We therefore conclude, that even if slight, imperceptible, differences exist between directions, they are not consistent across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In summary, the choice of dormant direction has little to no effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This result is arguably intuitive, as all dormant directions might be considered equivalent, having insignif- icant effect on generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Therefore, our choice of using the last directions is almost arbitrary, only motivated by the fact that they are the “most dormant”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Similarly, no technique is required to match an direction to a domain, and one can simply pick a dormant direction randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Repurposing Non-Dormant Directions Aiming at domain expansion, preserving the source do- main is integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Since the non-dormant directions span the variations of the source domains, we explicitly kept them intact, and repurposed only dormant directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Neverthe- less, the training method itself could be identically applied to non-dormant directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' One simply needs to dedicate a non-dormant direction to capture the new domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We next demonstrate that applying our method to non-dormant directions is still effective and enables capabilities beyond domain expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Traversing the 1st latent direction in the generator pre- trained on FFHQ [12], makes people in generated images appear older and more masculine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Some users might decide that they associate having a full beard with being older and more masculine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To support such behavior, we fine-tune the generator with a transformed StyleGAN-NADA [5], to cap- ture “a person with a beard” along the 1st direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We display images generated along traversals of the 1st direc- tion, before and after tuning, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 15a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, the generator now represents having a beard, along its 1st latent direction, in addition to its previous behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The capability to add new concepts in addition to exist- ing ones does not depend on the close relationship between the two in the last examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To demonstrate this point, we tune the generator to capture “Elf” along its 8th direction, which originally encodes head pose (and a few other prop- 12 0 100 200 300 400 500 Direction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='6 LPIPS Distance (a) FFHQ 0 100 200 300 400 500 Direction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='6 LPIPS Distance (b) LSUN Church Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Magnitude of perceptual effect caused by traversing different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Directions are sorted in decreasing order ac- cording to their corresponding singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For each direction, we measure the LPIPS distance [49] between images from two la- tent codes distanced by a 3σ traversal along the direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, the effect caused by the traversal diminishes quickly and the majority of directions are dormant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' erties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 15b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Previous results are clearly not solving domain expan- sion, as they alter the original behavior of the source do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Instead, one might say they adapt the domain mod- eled by the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Nevertheless, there exists a profound difference to existing domain adaptation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Our re- sulting generator does not completely overriding the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Instead, in a precise and controllable manner, it modifies individual factors of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Therefore, a user can carefully rewrite [1,43] the semantic rules of a genera- tive model, allowing greater control and freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Distance to Repurposed Subspace Repurposed subspaces are defined by transporting the base subspace along a linear direction by a predetermined scalar size s (See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (3) in the main paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' All results in the paper, across domains and variations used s = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We next evaluate the effect the hyperparameter s has on re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' To this end, we perform multiple expansions of an FFHQ [12] generator with 100 new variations, while vary- ing the value of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We measure CLIP errors (introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='3) of im- ages generated from repurposed subspaces and the corre- sponding target text used for training, as a function of train- ing iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 16a we report the results for two varia- tions - “Marge Simpson” and “Tolkein Elf”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, for all s > 0, CLIP error decreases as training progresses, and it decreases “faster” for greater values of the parameter s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Even with ×10 more iterations, the model trained with s = 5 does not reach the CLIP error of the model trained with s = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Images generated from the repurposed subspaces are dis- played in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 16b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For each value of s, we use the check- point that resulted in the closest CLIP error to that obtained by a favored s = 20 checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, not only training time is affected by parameter s, but the visual ef- fects captured by training vary significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We observe that models trained with greater values of parameter s depict a more significant change with respect to the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' When parameter s is too small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=', s ≤ 5), the model captures only few, simple characteristics of the new domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' On the other hand, when parameter s is too large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=', s = 50), the model commonly generates images that are blurry, have color artifacts or even do not capture the target text well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For example, with the target text “Marge Simpson”, the model learns to generate images with blue skin rather than blue hair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We note that these undesired artifacts cannot be mitigated by training with a large value of parameter s originally, and use a smaller one in test-time, as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Following these results, we conclude that the parameter s has a regularizing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Placing the domains “closer” in the latent space causes them to be more similar in im- age space as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Conversely, placing the domains further apart allows the new domain to capture more drastic, out- of-domain effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Eventually, choosing a value for parameter s is subject to user preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In our experiments, we have found that values in the range of [10, 30] offer satisfying results, across different source and expanded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We last note that the regularization effect of parameter s could be explained by the existence of a globally consistent “pace of change” of the generator with respect to the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' With StyleGAN, such behavior is explicitly encour- aged using a Perceptual Path Length (PPL) regularization term [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Nevertheless, we observe identical results when omitting this regularization during our expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' How Many Domains Can Fit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' So far, the largest number of new domains used for ex- pansion was 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The results from Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='1 indicated that there might be up to 400 dormant directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Could 13 (a) FFHQ (b) LSUN Church Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Visualization of ±3σ traversal along latent directions in the FFHQ [12] and LSUN Church [48] models, obtained using SeFA [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Directions shown are sorted from least (v0, top) to most (v511, bottom) dormant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, later directions are dormant – not affecting the generated image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We over-sample early directions for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In practice, over 80% of directions are dormant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 14 0 500 1000 1500 2000 2500 3000 3500 4000 Training Iterations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='83 CLIP Error ( ) Domain: Sketch Repurposed Dimension 200 300 400 500 511 200 511 300 400 500 Seed 0 (a) Seed 0 0 500 1000 1500 2000 2500 3000 3500 4000 Training Iterations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='83 CLIP Error ( ) Domain: Sketch Repurposed Dimension 200 300 400 500 511 200 511 300 400 500 Seed 1 (b) Seed 1 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We expand a generator pretrained on AFHQ [4] with 5 domains, varying the dormant direction dedicated to the “sketch” domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We repeat the expansion twice, with different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Top - reporting CLIP error of images generated from the sketch domain with the text “a sketch”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Bottom - a sample of generated images from checkpoints obtaining CLIP error closest to the horizontal black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, images generated using different repurposed dimensions differ only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Specifically, changing the random seed induces similar difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 0 500 1000 1500 2000 2500 3000 3500 4000 Training Iterations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='81 CLIP Error ( ) Domain: Bear Repurposed Dimension 200 300 400 500 511 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 13, using a “bear” domain instead of “sketch”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, dimensions are ordered differently in terms of minimizing CLIP error, as compared to their order for sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' they all be repurposed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We apply our method to expand a generator pretrained on FFHQ with 400 new domains, repurposing the last (and perhaps all) dormant directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Incredibly, the expansion succeeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We find that the expansion follows the same find- ings discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='3 – training is slower, yet quality is uncompromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Specifically, the FID score from the base subspace is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='83 compared to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='80 in our model expanded with 105 domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We display images generated from this model in the accompanying video and in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 18 to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Additional Compositions Results In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 21 and 22 we provide additional qualitative re- sults displaying compositionality in expanded generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Additional Details D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Training Time and Iterations When expanding the generator with a single new do- main, our training requires roughly twice the number of it- erations to obtain comparable effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The difference is a direct result of our additional regularization terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' With ad- ditional domains, we observe a roughly linear relationship between the number of domains and the required training it- erations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For example, the FFHQ model expanded with 105 iterations was trained for 40K iterations, while the model with 400 iterations was trained for 150K iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Note that different training objective might require a dif- ferent number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' StyleGAN-NADA [5] specifi- cally heavily relies on early-stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' An ideal domain ex- pansion method could consider this issue, and sample train- ing objectives to apply non-uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' In practice, we did not observe this to be an issue, probably due to our method optimizing numerous objectives simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 15 200 300 400 500 511Before After 1st direction Before After (a) Before After 8th direction Before After (b) Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Using our training method with non-dormant direc- tion rewrites existing semantic rules and adds new concepts on top of existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (a) Traversing the 1st direction originally made people older and more masculine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' After fine-tuning, it also adds a beard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (b) Traversing the 8th direction originally turned people heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' After fine-tuning it also turns them to elves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Transformation of Loss Function As explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='2, transforming a given domain adaptation task to perform domain expansion requires lim- iting the samples latent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The loss function itself, in principal, is left unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This is exactly the case for MyStyle [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' For StyleGAN-NADA [5], however, we made a subtle modification to the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' StyleGAN-NADA computes its loss with respect to a frozen copy of the source generator (See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' This is done in order to maintain access to the source domain, de- spite it vanishing from the adapted generator during train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Conversely, using our method, the source domain is preserved along the base subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We take advantage of this fact and modify the loss only slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Instead of us- ing a frozen generator to generate images from the source domain, we simply use our expanded generator and latent codes from the base subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 16 0 50000 100000 150000 200000 250000 300000 350000 400000 Training Iterations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='82 CLIP Error ( ) Domain: Marge Simpson s 0 1 5 10 20 50 0 50000 100000 150000 200000 250000 300000 350000 400000 Training Iterations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='78 CLIP Error ( ) Domain: Tolkein elf s 0 1 5 10 20 50 (a) s = 0 s = 1 s = 5 s = 10 s = 20 s = 50 Marge Simpson Tolkien Elf (b) Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Evaluating the effect of the distance between the base and repurposed subspace, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (a) We compare CLIP error as a function of training iterations, between models trained with different values of parameter s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' (b) Generated images from models having CLIP error as close as possible to the black horizontal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, increasing s corresponds to faster minimization of CLIP error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' However, even with comparable CLIP errors, visual effect might vary significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Large values of parameter s are often associated with undesired artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' We find that values between [10, 30] are usually preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 𝛼 = 0 𝛼 = 10 𝛼 = 40 𝛼 = 20 𝛼 = 30 𝛼 = 50 Base subspace Repurposed subspace Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Interpolation between the base subspace and the repur- posed subspace where s = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' As can be seen, undesired behavior occurring at repurposed subspace (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' blue skin Marge Simpson) cannot be mitigated by traversing shorter distances in test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' The choice of parameter s is crucial in training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 17 Gsrc c(z)Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Subset 1/3 of generated images from a model expanded with 400 domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 18 odigliani painting leftnostri Pho lisplayed on a compute Mar turbar OsoRr culpture poctor Who son eatingFigure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Subset 2/3 of generated images from a model expanded with 400 domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 19 h no eyebrows Crying Person Viking erson with tattoos alvin klein targarFigure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Subset 3/3 of generated images from a model expanded with 400 domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='Photo taken in the Night ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='President ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='alePerson ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='opeye ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content='person with curly ginger hairFigure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Composition of factors of variation introduced to a generator pretrained on FFHQ [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Following the format of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 21 Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Composition of factors of variation introduced to generators pretrained on LSUN Church [48] and SD-Elephant [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' Following the format of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE4T4oBgHgl3EQftg2f/content/2301.05225v1.pdf'} diff --git a/m9E0T4oBgHgl3EQfZQCv/content/tmp_files/2301.02319v1.pdf.txt b/m9E0T4oBgHgl3EQfZQCv/content/tmp_files/2301.02319v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..82da9bb53b85ff700c525b1887c742d145271b07 --- /dev/null +++ b/m9E0T4oBgHgl3EQfZQCv/content/tmp_files/2301.02319v1.pdf.txt @@ -0,0 +1,1893 @@ +arXiv:2301.02319v1 [cond-mat.mes-hall] 5 Jan 2023 +Localized nonlinear excitations of a columnar chain of coronene molecules +Alexander V. Savin1, 2, ∗ and Sergey V. Dmitriev3, 4, † +1Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow 119991, Russia +2Plekhanov Russian University of Economics, Moscow 117997, Russia +3Institute of Molecule and Crystal Physics, Ufa Federal Research Centre of +Russian Academy of Sciences, Oktyabrya Ave. +151, 450075 Ufa, Russia +4Institute of Mathematics with Computing Centre, +Ufa Federal Research Centre of Russian Academy of Sciences, Ufa 450008, Russia +The nonlinear dynamics of a one-dimensional molecular crystal in the form of a chain of planar +coronene molecules is analyzed. Using molecular dynamics, it is shown that a chain of coronene +molecules supports acoustic solitons, rotobreathers, and discrete breathers. An increase in the size +of planar molecules in a chain leads to an increase in the number of internal degrees of freedom. +This results in an increase in the rate of emission of phonons from spatially localized nonlinear +excitations and a decrease in their lifetime. Presented results contribute to the understanding of +the effect of the rotational and internal vibrational modes of molecules on the nonlinear dynamics +of molecular crystals. +I. +INTRODUCTION +Molecular crystals can have a quasi-one-dimensional +morphology, for example, fullerene nanowhiskers con- +sisting of fullerene molecules [1], a columnar structure +of carbon nanotori [2, 3], B42 molecules [4], n-coronene +molecules [5–8], columnar discotic liquid crystals [9–11] +and many others. Finite-size particles of molecular crys- +tals have rotational degrees of freedom that can give rize +to such cointerintuitive effects as negative thermal expan- +sion [12–16] and auxeticity (negative Poisson’s ratio) [17– +21]. +Quasi-one-dimensional crystals can support various +spatially localized nonlinear excitations, their study is +important and is often considered in connection with the +transfer of energy, mass and information. If the molecules +that make up quasi-one-dimensional crystals, in addition +to translational, also have rotational and internal vibra- +tional degrees of freedom, then the variety of localized +excitations supported by them increases. +Let us note the most intensively studied spatially lo- +calized excitations in nonlinear lattices and crystals. +Compressive acoustic solitons are typically excited in +solids or metamaterials under shock loading [22–25]. +Acoustic solitons propagating at a speed exceeding the +speed of longitudinal sound were described in carbon +nanotube bundles [26], black phosphorene [27], graphene +and boron nitride [28]. It is shown that the attenuation +of compressive waves in black phosphorene occurs faster +than in graphene and boron nitride due to the greater +number of degrees of freedom in the translational cell of +phosphorene, which provides more channels for energy +emission [27]. +Rotobreathers are dynamical modes with a single rotat- +ing particle while neighboring particles oscillate with the +∗ asavin@chph.ras.ru +† dmitriev.sergey.v@gmail.com +amplitude decreasing exponentially with distance from +the rotating particle [29–32]. The works [33, 34] are de- +voted to the analysis of the stability of rotobreathers. +The effect of rotobreathers on heat capacity [29], ther- +mal conductivity [35, 36], and slow relaxation [37] was +analyzed within the framework of one-dimensional rota- +tor lattices. Rotobreathers were considered in a damped +driven rotator lattice [38] and in the lattices with geo- +metrical nonlinearities [39, 40]. The method of molecu- +lar dynamics [41] was used to describe the precession of a +rotating fullerene inside a fullerite crystal. The work [42] +shows the effect of C60 fullerite crystal deformation on the +rotational dynamics and shift of the center of mass of a +single C60 molecule. In the works [43–45] rotobreathers +in the form of carbon nanotubes rotating around their +axis in a carbon nanotube bundle were studied. +The +dynamics of a fullerene molecule rotating in a fullerite +crystal was studied in [46]. +Discrete breathers or intrinsic localized modes are the +large-amplitude, spatially localized vibrational modes in +defect-free nonlinear lattices [47–49]. Discrete breathers +are ubiquitous in nonlinear lattices and are investi- +gated in models described by the discrete nonlinear +Schr¨odinger equation [50], in Josephson superconducting +junctions [51, 52], in granular crystals [53], in a mass- +spring chain [54], and in magnetic systems [55–57]. In- +teratomic interactions are non-linear, so different crystals +support discrete breathers [58–61]. In real discrete sys- +tems, e.g. in crystals, one deals with quasi-breathers that +are not exactly periodic single-frequency modes [62]. A +discrete breather in the form of a single fullerene molecule +oscillating with a large amplitude in a fullerite crystal [46] +and a single oscillating carbon nanotube in a nanotube +bundle [45] were studied by the method of molecular dy- +namics. +Most popular approaches to the study of nonlinear ex- +citations in molecular crystals are the use of molecular +dynamics [2, 3] and coarse-grained models [5, 7, 63, 64]. +The aim of this study is to analyze the effect of in- +ternal vibrational degrees of freedom on the robustness + +2 +FIG. 1. Vertical chain of 10 n-coronene molecules C6n2H6n: +(a) n = 2 (coronene C24H12); (b) n = 3 (circumcoronene +C54H18); (c) n = 4 (dicircumcoronene C96H24). +Carbon +atoms (gray) form planar disk molecules, and hydrogen atoms +are located at the edges of the disks (shown in light gray). The +vertical axis of the chain is parallel to the z axis, the planar +molecules are parallel to the xy plane. The positions of neigh- +boring molecules in the chain differ by the shift along the z +axis and the relative rotation of the molecules in the xy plane +(shift ∆z and twist ∆φ steps of the chain). +of various spatially localized nonlinear excitations in +a quasi-one-dimensional chain of n-coronene molecules +with n = 2, 3, and 4 [5–7]. As the index n increases, the +size of the molecules and, consequently, the number of +internal degrees of freedom also increase. +In Sec. II, the structure of the n-coronene and the +molecular dynamics model used in this study are de- +scribed. +The spectrum of small-amplitude vibrations +of the n-coronene is analyzed in Sec. III. Sections from +IV to VI present the results of studying spatially local- +ized nonlinear excitations in the chains of n-coronene +molecules, namely, acoustic solitons, rotobreathers, and +discrete breathers, respectively. Our conclusions are for- +mulated in Sec. VII. +II. +MODEL +The n-coronene molecule C6n2H6n can be considered +as a graphene flake. Therefore, to describe the dynamics +of a coronene molecular crystal, one can use the force +field previously used for graphene nanoribbons. +To simplify the modeling, valence-bonded CH groups +of atoms at the edges of disk molecules will be considered +as a single carbon atom of mass 13mp, while all other +inner carbon atoms have the mass 12mp, where mp = +1.6601 × 10−27 kg is the proton mass. +The Hamiltonian of one molecule can be written as +H0 = +N0 +� +i=1 +�1 +2Mi( ˙ui, ˙ui) + Pi +� +, +(1) +where i is the number of an atom, N0 = 6n2 is the +(a) +n +m +(b) +m +k +n +(c) +m +k +n +l +(d) +m +k +n +l +(e) +m +k +n +l +FIG. 2. (Color online) Different types of interactions between +neighboring atoms belonging to the sets Ωj, j = 1, 2, 3, 4, 5. +(a) Valence interactions j = 1, (b) valence angles j = 2, (c-e) +different dihedral angles j = 3, 4, and 5, respectively. +number of atoms in the molecule, Mi is the mass of +the ith atom (there are 6n2 − 6n inner carbon atoms +of mass 12mp and 6n edge carbon atoms of mass 13mp), +ui = (xi(t), yi(t), zi(t)) is the three-dimensional vector +describing the position of ith atom at the time t. The +term Pi describes the interaction of the carbon atom with +the index i with the neighboring atoms. We emphasize +that the inner and edge atoms differ only in their masses, +and their interaction with each other is described by the +same potential. The potential depends on variations in +bond length, bond angles, and dihedral angles between +the planes formed by three neighboring carbon atoms and +it can be written in the form +P = +� +Ω1 +U1 + +� +Ω2 +U2 + +� +Ω3 +U3 + +� +Ω4 +U4 + +� +Ω5 +U5, +(2) +where Ωj, with j = 1, 2, 3, 4, 5, are the sets of configu- +rations describing different types of interactions between +neighbors. Members of these sets are shown in Fig. 2, and +all their rotated and mirrored versions should be taken +into account. +Potential U1(un, um) describes the energy due to +change in the length of a valence bond between atoms +with the indexes n and m, as shown in Fig. 2(a). The +potential U2(un, um, uk) describes the deformation en- +ergy of the angle between the valence bonds unum, and +umuk, see Fig. 2(b). Potentials Uj(un, um, uk, ul), j = 3, +4, and 5, describe the deformation energy associated with +a change in the angle between the planes unumuk and +ulukum, as shown in Figs. 2(c-e), respectively. +We use the potentials employed in the modeling of the +dynamics of large polymer macromolecules [65, 66] for +the valence bond coupling, +U1(u1, u2)=ǫ1{exp[−α0(ρ−ρ0)]−1}2, ρ=|u2−u1|, (3) +where ǫ1 is the energy of the valence bond and ρ0 is +the equilibrium length of the bond; the potential of the +valence angle is +U2(u1, u2, u3) = ǫ2(cos ϕ − cos ϕ0)2, (4) +cos ϕ = (u3 − u2, u1 − u2)/(|u3 − u2| · |u2 − u1|), +where the equilibrium value of the angle is cos ϕ0 = +cos(2π/3) = −1/2; the potential of the dihedral angle +is +Uj(u1, u2, u3, u4) = ǫj(1 + zj cos φ), +(5) +cos φ = (v1, v2)/(|v1| · |v2|), + +(c) +(b) +(a) +Z +x3 +v1 = (u2 − u1) × (u3 − u2), +v2 = (u3 − u2) × (u3 − u4), +where the sign zj = 1 for j = 3, 4 (the equilibrium value +of the torsional angle φ is φ0 = π) and zj = −1 for j = 5 +(φ0 = 0). +The values of the potential parameters are ǫ1 += +4.9632 eV, ρ0 = 1.418 ˚A, α0 = 1.7889 ˚A−1, ǫ2 = +1.3143 eV, and ǫ3 = 0.499 eV. They are found from the +frequency spectrum of small-amplitude oscillations of a +graphene sheet [67]. According to previous study [68], +the energy ǫ4 is close to the energy ǫ3, whereas ǫ5 ≪ ǫ4 +(|ǫ5/ǫ4| < 1/20). Therefore, we set ǫ4 = ǫ3 = 0.499 eV +and assume ǫ5 = 0, the latter means that we omit the +last term in the sum Eq. (2). More detailed discussion +and motivation of our choice of the interaction potentials +Eqs. (3-5) can be found in earlier publication [69]. +The interaction of two coronene molecules is described +by the potential +W(X1, X2) = +N0 +� +i=1 +N0 +� +j=1 +V (rij), +(6) +where the 3N0-dimensional vector Xk += +{uk,i}N0 +i=1 +(k = 1, 2) defines the coordinates of atoms of the k-th +molecules (vector uk,i specifies the coordinates of the i- +th atom of the k-th molecule), rij = |u2,j − u1,i| is the +distance between atoms. Nonvalence interactions of the +carbon atoms are described by the (6,12) Lennard-Jones +potential +V (r) = ǫc{[(rc/r)6 − 1]2 − 1}, +(7) +where ǫc = 0.002757 eV, rc = 3.807 ˚A [70]. +Hamiltonian of a chain of N molecules (see Fig. 1) can +be presented in the form +H = +N +� +n=1 +�1 +2(M ˙Xn, ˙Xn) + P(Xn) +� ++ +N−1 +� +n=1 +W(Xn, Xn+1) + +N−2 +� +n=1 +W(Xn, Xn+2), +(8) +where the first sum includes the kinetic and potential +energies of n-th molecule. The second and the third sums +describe the interaction between nearest and next-nearest +molecules, respectively. Here the vector Xn = {un,i}N0 +i=1 +specifies the coordinates of the atoms of n-th molecule, +M is the diagonal matrix of atom masses, P(Xn) is the +energy of n-th molecule, W(Xn, Xk) is the interaction +energy of n-th and k-th molecules. +III. +THE DISPERSION CURVES OF +SMALL-AMPLITUDE OSCILLATIONS +Let us consider the structure of a symmetric (spiral) +stack of planar n-coronene molecules with the symmetry +TABLE I. Values of shift ∆z and twist ∆φ parameters, max- +imum frequencies of out-of-plane ωop and in-plane ωip vibra- +tions, velocities of torsion vt and longitudinal vl sound for a +spiral stack of n-coronene C6n2H6n molecules. +n ∆z (˚A) ∆φ (◦) ωop (cm−1) ωip (cm−1) vt (m/s) vl (m/s) +2 +3.445 +30.0 +841.6 +1549.3 +217 +3170 +3 +3.411 +18.6 +883.7 +1580.4 +195 +3449 +4 +3.396 +12.6 +894.0 +1591.3 +250 +3591 +axis parallel to the z axis – see Fig. 1. In the ground state +of such a chain, the atomic coordinates of each successive +molecule are obtained from the coordinates of the previ- +ous molecule by translation along the z axis by a shift +∆z and rotation around the same axis by an angle ∆φ. +These are the shift and twist parameters: +xn+1,j = xn,j cos(∆φ) + yn,j sin(∆φ), +yn+1,j = −xn,j sin(∆φ) + yn,j cos(∆φ), +(9) +zn+1,j = zn,j + ∆z, +i = 1, ..., N0, n = 0, ±1, ±2, ... +Thus, the energy of the ground state is a function of +3N0 coordinates of N0 atoms of the first molecule X1 = +{u1,j}N0 +j=1, and the two geometry parameters, ∆z and +∆φ, where u1,j = (x1,j, y1,j, z1,j) is the vector position +of jth atom of the first molecule. +Finding the ground state reduces to the following min- +imization problem: +E = P(X1) + W(X1, X2) + W(X1, X3) +→ min : {u1,j}N0 +j=1, ∆φ, ∆z. +(10) +The problem (10) was solved numerically by the conju- +gate gradient method. The values of the shift ∆z and +the twist ∆φ steps of the chain of n-coronene molecules +are presented in Table I. +A vertical chain of molecules is a multistable system. +Numerical analysis shows that for n-coronene molecules +with n ≤ 4, the spiral structure defined by Eq. (9) is the +most energy-favorable ground state. +For analysis of small-amplitude oscillations of spiral +chain it is convenient to use local cylindrical coordinates +vn,j = (vn,j,1, vn,j,2, vn,j,3), given by the following ex- +pressions: +xn,j = x0 +n,j + vn,j,1 cos(φn,j) + vn,j,2 sin(φn,j), +yn,j = y0 +n,j − vn,j,1 sin(φn,j) + vn,j,2 cos(φn,j), (11) +zn,j = z0 +n,j + vn,j,3, +with u0 +n,j = (x0 +n,j, y0 +n,j, z0 +n,j), (n = 0, ±1, ±2, ...; j = +1, ..., N0) being coordinates of the atoms in the helix +ground state, and φn,j being angular coordinate of the +atom (n, j). With these new coordinates the Hamilto- +nian of the molecular chain Eq. (8) has the following + +4 + 0 + π/3 +2π/3 + +0 +500 +1000 +1500 +ω (cm−1) +q +(a) + 0 + π/3 +2π/3 + π + + + + +q +(b) +FIG. 3. Structure of 72 dispersion curves of a spiral chain +of coronene molecules C24H12 for (a) out-of-plane and (b) in- +plane vibrations. Black dots denote modes leading to the for- +mation of discrete breathers – localized nonlinear oscillations +of one molecule in the chain. +form +H = +� +n +�1 +2(M ˙vn, ˙vn) + P(vn, vn+1, vn+2) +� +, +(12) +where +vn += +{(vn,j,1, vn,j,2, vn,j,3)}N0 +j=1 +is +a +3N0- +dimensional vector, M is 3N0-dimensional diagonal mass +matrix. +From the Hamiltonian Eq. (12) the following system of +equations of motion can be derived: +−M¨vn = P1(vn, vn+1, vn+2) ++P2(vn−1, vn, vn+1) + P3(vn−2, vn−1, vn), +(13) +where Pi(v1, v2, v3) = ∂P/∂vi, i = 1, 2, 3. Within the +linear approximation, the system Eq. (13) obtains the +form +−M¨vn = B1vn +B2vn+1 +B∗ +2vn−1 +B3vn+2 +B∗ +3vn−2, +(14) +where the matrix elements are given as +B1 = P11 + P22 + P33, +B2 = P12 + P23, +B3 = P13, +and the partial derivative matrix is given as +Pij = +∂2P +∂vi∂vj +(0, 0, 0), +i, j = 1, 2, 3. +The solution to the system of linear equations Eq. (14) +can be found in the standard form +vn = Aw exp[i(qn − ωt)], +(15) + 0 + π/3 +2π/3 + +0 +500 +1000 +1500 +ω (cm−1) +q +(a) + 0 + π/3 +2π/3 + π + + + + +q +(b) +FIG. 4. Structure of 162 dispersion curves for a spiral chain +of circumcoronene molecules C54H18 for (a) out-of-plane and +(b) in-plane vibrations. Black dots indicate modes that lead +to the formation of discrete breathers – localized nonlinear +vibrations of one molecule in the chain. +where A is the linear mode amplitude, w is the eigen- +vector, ω is the phonon frequency with the dimension- +less wave number q ∈ [0, π]. Substituting Eq. (15) into +the system Eq. (14), we arrive at the following 3N0- +dimensional eigenvalue problem: +ω2Mw = C(q)w, +(16) +where Hermitian matrix +C(q) = B1 + B2 exp(iq) + B∗ +2 exp(−iq) ++B3 exp(2iq) + B∗ +3 exp(−2iq). +Using the substitution w = M−1/2e, problem Eq. (16) +can be rewritten in the form +ω2e = M−1/2C(q)M−1/2e +(17) +where e is the normalized eigenvector, (e, e) = 1. +Thus, to obtain the dispersion curves ωj(q), it is neces- +sary to find the eigenvalues and eigenvectors of the Her- +mitian matrix Eq. (17) of size 3N0 × 3N0 for each fixed +wavenumber 0 ≤ q ≤ π. +As a result, we obtain 3N0 +branches of the dispersion relation {ωj(q)}3N0 +j=1. +The planar structure of molecules in a spiral chain +leads to the division of its small-amplitude vibrations +into two-classes: +out-of-plane vibrations, when atoms +vibrate orthogonally to the molecular plane (all atoms +move along the z axis) and in-plane vibrations (all atoms +move in the xy plane). Two thirds of the branches corre- +spond to in-plane vibrations, while only one-third corre- +sponds to out-of-plane vibrations. The dispersion curves +are shown in Figs. 3 to 5. + +5 + 0 + π/3 +2π/3 + +0 +100 +200 +(a) +ω (cm−1) +q + 0 + π/3 +2π/3 + π + + + +(b) +q +FIG. 5. Dispersion curves in the low-frequency region for a +spiral chain of coronene molecules C24H12 for (a) out-of-plane +and (b) in-plane vibrations (three gray bands show the fre- +quency spectrum of the rotobreathers). The dashed straight +lines define the tangents to the dispersion curves emerging +from the zero point, corresponding to the velocities of the +longitudinal vl and torsion vt sound. +For the spiral chain of coronene molecules C24H12, the +dispersion curves of out-of-plane vibrations, see Fig. 3(a) +and Fig. 5(a), lie in the frequency range 0 ≤ ω ≤ ωop, +with the maximum frequency ωop = 842 cm−1. One dis- +persion curve ωl(q) starts from the origin (q = 0, ω = 0), +it describes the displacement of planar molecules along +the chain axis without internal deformations (longitudi- +nal acoustic vibrations of the chain). The tangent of this +dispersion curve at the origin gives the velocity of longi- +tudinal sound waves +vl = ∆z lim +q→0 +ωl(q) +q +. +The dispersion curves of in-plane oscillations, see +Fig. 3(b) and Fig. 5(b), lie in the frequency range 0 ≤ +ω ≤ ωip with the maximum frequency ωip = 1549 cm−1. +One dispersion curve ωt(q) starts from the origin and de- +scribes torsional acoustic oscillations (rotation of planar +molecules around the chain axis). The speed of long-wave +torsional vibrations (speed of torsional sound) is +vt = ∆z lim +q→0 +ωt(q) +q +. +In addition, one dispersion curve approaches the q axis +tangentially. +This curve describes the optical bending +vibrations of the chain. The frequency spectrum of in- +plane oscillations is characterized by the presence of a +0 +100 +200 +300 +400 +500 +0 +20 +40 +0 +0.1 +0.2 +(a) +t (ps) +n +En (eV) +0 +100 +200 +300 +400 +500 +0 +20 +40 +0 +0.3 +0.6 +(b) +t (ps) +n +En (eV) +0 +100 +200 +300 +400 +500 +0 +20 +40 +0 +0.4 +0.8 +(c) +t (ps) +n +En (eV) +FIG. 6. Formation of a supersonic acoustic soliton in a spiral +chain of (a) coronene, (b) circumcoronene, and (c) dicircum- +coronene molecules produced by longitudinal local compres- +sion at the end of the chain with amplitude az = 0.4 ˚A. The +distribution of energy in the chain En(t) at different times is +shown. The number of molecules in the chain is N = 500. +The dotted lines show the trajectory of motion with the ve- +locity of longitudinal sound vl to demonstrate the supersonic +motion of solitons. +gap in the low-frequency region. For a chain of coronene +molecules, the gap is from 10 to 203 cm−1 [see Fig. 5(b)], +and for a chain of circumcoronene molecules, from 9 to +141 cm−1 [see Fig. 4(b)]. +The values of the maximum frequencies ωop, ωip and +the speeds of sound vl, vt are given in Table I. As can be +seen from the table, the speed of longitudinal sound is 15 +times greater than the speed of torsional sound. + +6 + + + + + + +−0.2 +−0.1 +0 +(a) +ρn (A) +° + + + + + + +−0.1 +−0.05 +0 +(b) +ρn (A) +° +0 +100 +200 +300 +400 +500 +−0.08 +−0.04 +0 +(c) +n +ρn (A) +° +FIG. 7. Distribution of longitudinal compression during the +motion of an acoustic soliton along a chain of N = 500 +molecules of (a) coronene, (b) circumcoronene, (c) dicircum- +coronene. The distribution of relative longitudinal displace- +ments ρn of chain molecules at time t = 40 ps is shown for +the amplitude of the initial local compression of the chain end +az = 0.4 ˚A. The vertical dotted lines show the position of the +front of the acoustic phonon wave packet propagating with +the velocity vl. +IV. +ACOUSTIC SOLITONS +The interaction of neighboring planar molecules is de- +termined by the sum of interactions of all pairs of their +atoms Eq. (6), which are described by the Lennard-Jones +potential Eq. (7). The Lennard-Jones potential at small +interatomic distances is characterized by the hard-type +anharmonicity. Therefore, one can expect the possibil- +ity of propagation of compressive longitudinal acoustic +solitons moving at a speed exceeding the velocity of lon- +gitudinal sound vl. +To test the existence of supersonic acoustic solitons, +we simulate the propagation of initial local longitudinal +compression along a chain of molecules. Consider a spiral +chain of N = 500 molecules. Let us take the ground state +of the chain and at t = 0 shift the first two molecules +along the z axis by az. As a result, local longitudinal +compression occurs at the end of the chain. Having fixed +the position of these two molecules in the shifted state, +let us consider the propagation of local compression along +the chain. +1 +1.1 +1.2 +1.3 +1.4 +1.5 +0 +0.5 +1 +(a) +E (eV) +1 +1.1 +1.2 +1.3 +1.4 +1.5 +0 +0.3 +0.6 +(b) +s +Az (A) +° +FIG. 8. Dependence of (a) energy E of an acoustic soliton +and (b) longitudinal compression of the chain Az produced +by an acoustic soliton propagating in a chain of coronene +molecules on its dimensionless velocity s = v/vl. +Markers +show numerical values, solid curves show approximations ob- +tained by the least squares method E(s) = 3.36(s − 1)1.7 eV +and Az(s) = 0.93(s − 1)0,5 ˚A. +To simulate the dynamics of a chain with fixed ends, we +numerically integrate the system of equations of motion +corresponding to the Hamiltonian of the chain Eq. (8) +M ¨Xn = − ∂H +∂Xn +, n = 3, 4, ..., N − 2, +(18) +˙Xn ≡ 0, +n = 1, 2, N − 1, N, +with the initial conditions +Xn(0) = X0 +n + azez, +n = 1, 2 +Xn(0) = X0 +n, +n = 3, 4, ..., N, +(19) +˙Xn(0) = 0, +n = 1, 2, ...., N, +where +the +3N0-dimensional +vector +Xn += +{(xn,j, yn,j, zn,j)}N0 +j=1 +defines the +coordinates of the +atoms of n-th molecule, vectors {X0 +n}N +n=1 defines ground +state of molecular chain, ez is a unit vector directed +along the z axis, az > 0 is the amplitude of the initial +compression of the chain end. +Numerical integration of the system of equations of +motion (18) showed that the initial longitudinal compres- +sion of the chain edge with an amplitude az ≤ 0.6 ˚A for + +7 + 0 + π/3 +2π/3 + π +4π/3 +5π/3 + 2π +0 +0.02 +0.04 +0.06 +1 +2 +3 +ϕ +E (eV) +FIG. 9. Change in the energy of the chain E as the function +of the rotation angle ϕ of one molecule rotating around the z +axis in the chain of coronene, circumcoronene, and dicircum- +coronene (curves 1, 2, and 3, respectively). Only one molecule +rotates quasi-statically while the rest of the molecules remain +in their equilibrium positions. +coronene molecules always leads to the formation of a +supersonic acoustic soliton and a subsonic wave packet +of long-wavelength longitudinal acoustic phonons – see +Fig. 6 (a) and 7 (a). +A local area of compression is +formed in the chain, which moves along it with a con- +stant supersonic speed v > vl, keeping its shape. When +moving, the soliton breaks away from the wave packet +of phonons. This allows us to find its energy E and the +longitudinal compression of the chain Az: +E = +� +n +En, Az = +� +n +ρn, ρn = 1 +N0 +N0 +� +j=1 +(zn+1,j−zn,j−∆z), +where the summation is carried out only over the soliton +localization region. +Dependencies of the soliton energy E and chain com- +pression Az produced by the soliton on its dimensionless +velocity s = v/vl are shown in Fig. 8. As can be seen from +the figure, with increasing velocity, the soliton energy in- +creases as (s − 1)1.7, and the compression as (s − 1)1/2. +In chains of circumcoronene and dicircumcoronene +molecules, local longitudinal compression of the chain +end also leads to the formation of a supersonic localized +compression region. But the motion of this region is ac- +companied by the emission of phonons. As a result, the +energy and velocity of the soliton decrease monotonically, +see Figs. 6(b,c) and 7(b,c). The larger the molecule, the +more noticeable the emission of phonons. Therefore, it +can be concluded that a chain of n-coronene molecules +admits the existence of an exact acoustic soliton of longi- +tudinal compression only for n = 2, while for n > 2 there +is only a soliton-like excitation with a finite lifetime. +V. +ROTOBREATHERS +The structure of planar molecules allows their rotation +in chains around the z axis. The n-coronene molecule has +the shape of a regular hexagon, a rotation of one molecule +by 60◦ will transfer the chain to an equivalent state. If +we fix the positions of all molecules and rotate only one +molecule as a rigid body, then the rotation potential E(ϕ) +(dependence of the chain energy on the angle of rota- +tion of one molecule ϕ) can be obtained. This potential +is a periodic function with period π/3, see Fig. 9. +In +the approximation of absolutely rigid valence bonds, free +rotation requires overcoming energy barriers of height +0.26, 0.34 and 0.66 eV for the chain of coronene, circum- +coronene, and dicircumcoronene molecules, respectively. +These barriers are overcome at molecular rotation fre- +quencies above ω0 = 2.19, 1.11 and 0.87 cm−1. Thus, +the topology of the chain allows the existence of roto- +breathers (localized rotations of molecules). +In the approximation of absolutely rigid molecules, +their chains allow the existence of a rotobreathers with +an infinite frequency spectrum lying above frequency ω0. +The n-coronene molecule is not an absolutely rigid body, +it has 3N0−6 vibrational modes. The presence of internal +vibrations in a rotator (in our case, a planar n-coronene +molecule) leads to the appearance of band gaps (lacu- +nae) in the frequency spectrum of the rotobreather [44]. +At frequencies within these band gaps, the rotation leads +to resonance with the natural oscillations of the rotators +and the emission of phonons. Therefore, the presence of +internal vibrational modes in molecules should lead to a +significant narrowing of the frequency spectrum of the +rotobreather. +To find the rotobreather, we simulate the rotation of +one molecule at different initial frequencies in a chain of +N = 100 molecules. A viscous friction at the ends of +the chain is introduced, which ensures the absorption of +phonons emitted by the rotator. To do this, we numeri- +cally integrate the system of equations of motion +M ¨Xn = − ∂H +∂Xn +, +n = Nt + 1, ..., N − Nt, +(20) +M ¨Xn = − ∂H +∂Xn +− γM ˙Xn, +n ≤ Nt, n > N − Nt +with the friction coefficient γ = 1/tr, tr = 10 ps, Nt = 30. +Let us take the ground state of the chain and excite +the rotation of the central molecule nc = N/2 with the +frequency ω, i.e. take the initial conditions in the form +{Xn(0) = X0 +n}N +n=1, +˙Xn(0) = 0, +n ̸= nc +(21) +{ ˙xnc,j = −ωy0 +nc,j, ˙ync,j = ωx0 +nc,j, ˙znc,j = 0}N0 +j=1. +Thus, we set the rotation of one rotator in the chain with +the initial energy +E = 1 +2ω2 +N0 +� +j=1 +Mj(x0 +nc,j +2 + y0 +nc,j +2). + +8 +0 +5 +10 +15 +0 +10 +20 +30 +40 +50 +60 +70 + E (eV) + t (ns) +FIG. 10. Change in time of the energy of one rotator in the +chain of coronene molecules for different values of the initial +rotation frequency of central molecule, varying in the range +from ω = 3 to 22 cm−1 with a step of 0.25 cm−1. +Friction at the ends of the chain will ensure the ab- +sorption of phonons emitted by the rotator. Therefore, +depending on the value of the frequency ω, the rotator +either stops, having lost all the energy for phonon emis- +sion, or reaches a stationary rotation mode with a con- +stant frequency without phonon emission (rotobreather +mode). The change in the rotator energy E for various +initial values of the frequency ω in the chain of coronene +and circumcoronene molecules is shown in Figs. 10 and +11, respectively. +As can be seen from Fig. 10, for a chain of coronene +molecules, there are only three frequency ranges at which +a rotation at constant frequency of one molecule can +occur without emitting phonons: +[3.96, 4.54], [8.28, +9.09], and [16.33, 16.71] cm−1. +Thus, in the chain of +coronene molecules, the rotobreather has a frequency +spectrum consisting of only three narrow intervals, see +also Fig. 5(b), where the frequency spectrum of the ro- +tobreather is shown by gray bands. Rotation with other +frequencies leads to the emission of phonons. +Simulation of the dynamics of a rotator in a chain of +circumcoronene molecules showed that rotobreathers do +not exist in this chain. Here, at all values of the rota- +tion frequency, the rotator emits phonons and completely +loses energy, see Fig. 11. +There is only one frequency +ω = 22.6 cm−1 at which the radiation becomes less in- +tense, but does not completely disappear. +In a chain +0 +5 +10 +15 +0 +20 +40 +60 +80 +100 +120 +140 + E (eV) + t (ns) +FIG. 11. Change in time of the energy of one rotator in the +chain of circumcoronene molecules for different values of the +initial rotation frequency of central molecule, varying in the +range from ω = 2 to 13.75 cm−1 with a step of 0.25 cm−1. +The dashed line shows the energy corresponding to the rota- +tion frequency ω = 22.6 cm−1, at which the weakest phonon +emission occurs. +of dicircumcoronene molecules, the rotation of the ro- +tator at all frequencies leads to an even stronger emis- +sion of phonons and no rotobreather is formed. The ab- +sence of a rotobreather in the chains of circumcoronene +and dicircumcoronene molecules is explained by a denser +frequency spectrum of natural vibrations of molecules. +Here, in contrast to the coronene molecules, the rotation +of the rotator at all frequencies resonates with the natural +vibrations of the molecules. +VI. +DISCRETE BREATHERS +An isolated n-coronene molecule consists of N0 = 6n2 +atoms. It has 3N0 − 6 natural oscillations with non-zero +frequencies, {ωj}3N0 +j=7. +The first six eigenmodes have a +zero frequency ω1 = ... = ω6 = 0, they correspond to the +motion of a molecule as a rigid body (three translational +and three rotational degrees of freedom). +Eigenmodes +with non-zero frequencies are of two types: N0−2 out-of- +plane vibrations, when atoms move orthogonally to the +molecular plane, and 2N0 − 4 in-plane vibrations, when +atoms move in the molecular plane. +The coronene molecule has 22 out-of-plane vibra- +tions with frequencies 64.6, 117.7,..., 839.2 cm−1 and + +9 +0 +5 +10 +15 +20 +0 +0.02 +0.04 +0.06 +1 +2 +3 +4 +5 + E (eV) + t (ns) +FIG. 12. Dependence of the energy of vibrations of the central +molecule of a chain of coronene molecules on time at the initial +excitation of the j-th natural vibration: (curve 1) j = 17, +ωj = 236.3; (curve 2) j = 21, ωj = 278.8; (curve 3) j = 23, +ωj = 329.5; (curve 4) j = 33, ωj = 435.2; (curve 5) j = 47, +ωj = 839.2 cm−1. The initial atomic velocity used to excite +the vibrational mode in the central molecule is A = 10 ˚A/ps. +44 in-plane vibrations with frequencies 203.1, 236.3,..., +1546.2 cm−1. +The circumcoronene molecule has 52 +out-of-plane vibrations with frequencies 32.3, 60.9,..., +881.0 cm−1 and 104 in-plane vibrations with frequen- +cies 140.5, 162.0,..., 1576.3 cm−1. Let us check whether +the excitation of a high-amplitude natural oscillation of +one molecule can lead to the appearance of a discrete +breather in the chain – a nonlinear oscillation localized +on one molecule. +To find discrete breathers, we simulate high-amplitude +natural vibrations of one central molecule in a chain of +N = 100 molecules. At the ends of the chain, viscous +friction is introduced, which ensures the absorption of +phonons emitted by vibrations of the central molecule. +The system of equations of motion Eq. (20) is integrated +numerically with the initial conditions +Xn(0) = X0 +n, +˙Xn(0) = Aejδn,nc, n = 1, ..., N, +(22) +where A defines the magnitude of the initial velocity of +atoms of the central molecule, ej is the unit eigenvector of +the jth eigenmode of an isolated molecule (j = 7,...,3N0), +nc = N/2. The value of A determines the vibrational +energy of the molecule and it is chosen sufficiently large +to enter the regime of anharmonicity. +The dependencies of the vibrational energy of the cen- +tral molecule on time t are shown in Fig. 12. Numerical +integration of the system of equations of motion Eq. (20) +with the initial conditions Eq. (22) showed that three dy- +namics scenarios are possible: very fast damping of oscil- +lations (see Fig. 12, curve 3), slow damping (curves 1 and +4) and the formation of undamped oscillations (curves 2 +and 5). +The first two scenarios are typical for out-of- +plane vibrations, the last one – for in-plane vibrations. +The frequencies of the resulting discrete breathers are +shown by black dots in Figs. 3 and 4. Of all the out- +of-plane eigenmodes, only the oscillation with the max- +838 +840 +842 +0 +0.1 +0.2 +0.3 +(a) + E (eV) +1474 1476 1478 + + + + +(b) +ω (cm−1) +1494 +1496 +1498 + + + + +(c) +FIG. 13. Dependence of the energy E on the frequency ω for +a discrete breather based on the j eigenmode of the coronene +molecule: (a) j = 47, ωj = 839.2; (b) j = 67, ωj = 1470.0 (c) +j = 69, ωj = 1491.3 cm−1. +imum frequency can lead to the formation of a discrete +breather. For a chain of coronene molecules out of 44 in- +plane vibrations 24 can lead to the formation of a discrete +breather, and for a chain of circumcoronene molecules out +of 104 in-plane vibrations 31 produce discrete breathers. +The undamped vibrations are localized strictly on one +molecule. +The oscillations are anharmonic, their fre- +quency depends on the amplitude. A characteristic fea- +ture of localized oscillations (discrete breathers) is a lin- +ear decrease in their frequency with increasing energy, +see Fig. 13. As the oscillation amplitude increases, the +energy of the breather increases and the frequency de- +creases. Thus, n-coronene chains support gap discrete +breathers with a soft type of anharmonicity. The energy +of a discrete breather in a chain of coronene molecules can +reach 0.37 eV, and the width of the frequency spectrum +can reach 6 cm−1. +VII. +CONCLUSIONS +The linear phonon spectrum and nonlinear spatially +localized excitations, such as acoustic solitons, roto- +breathers, and discrete breathers in chains of n-coronene +molecules, are studied by the method of molecular dy- +namics. +Three members of the n-coronene were con- +sidered, namely coronene, circumcoronene and dicircum- +coronene (n = 2, 3 and 4 respectively). These molecules +include respectively N0 = 24, 54, and 96 carbon atoms +and have 3N0 − 6 vibrational degrees of freedom. +The size of molecules plays an important role in chain +dynamics. +The spectra of low-amplitude vibrations of +chains of coronene and circumcoronene molecules are +shown in Figs 3 and 4, respectively. +It can be seen +that the maximum frequencies of out-of-plane and in- +plane vibrations are approximately the same for chains +of coronene and circumcoronene molecules, but the spec- +trum of the latter is denser, since the number of degrees + +10 +of freedom is greater. The spectrum of a chain of dicir- +cumcoronene molecules is even denser. +It was found that a chain of coronene molecules sup- +ports the propagation of acoustic compressive solitons, +which practically do not emit energy when moving at +supersonic speed, see Fig. 6(a) and Fig. 7(a). +Similar +excitations in chains of circumcoronene and dicircum- +coronene molecules constantly lose energy, emitting low- +amplitude phonons, see Fig. 6(b,c) and Fig. 7(b,c). This +is because spiral chains have lower symmetry in the stack- +ing of larger molecules and more channels to radiate en- +ergy due to the greater number of vibrational degrees of +freedom. +A similar picture was observed for rotobreathers. Only +in a chain of coronene molecules a single molecule can +rotate with frequencies in certain ranges [shown in gray +in Fig. 5(b)], radiating no energy. In chains of circum- +coronene and dicircumcoronene molecules, a molecule ro- +tating at any frequency excites low-amplitude phonons, +constantly loses its energy, and eventually stops rotating. +The explanation lies in more resonances with a denser +phonon spectrum in chains with larger molecules. +As for discrete breathers, they are supported by all +three considered molecular chains. Discrete breathers are +in the form of single molecule vibrating at large ampli- +tude and radiating no energy. The frequencies of discrete +breathers are marked with black dots in Figs. 3 and 4 for +chains of coronene and circumcoronene molecules, respec- +tively. A discrete breather with out-of-plane oscillations, +see panels (a), is created only by the highest-frequency +out-of-plane mode. +On the other hand, a number of +in-plane vibrational modes create discrete breathers, see +panels (b). The frequency of discrete breathers decreases +with an increase in their energy, i.e. +soft-type anhar- +monicity is realized, see Fig. 13. +The results presented in this study illustrate the role +of the internal degrees of freedom of particles in the non- +linear dynamics of molecular chains. +ACKNOWLEDGMENTS +Computational facilities were provided by the Inter- +departmental Supercomputer Center of the Russian +Academy of Sciences. +The work of A.V.S. (statement +of the problem, numerical simulations, and writing the +manuscript) was supported by the Russian Science Foun- +dation, Grant No. 21-12-00229. S.V.D. thanks the finan- +cial support provided by the Grants Council of the Pres- +ident of the Russian Federation grant NSh-4320.2022.1.2 +(discussion of the results, writing the manuscript). +[1] A. +Kausar, +Fullerene +nanowhisker +nanocomposite- +current +stance +and +high-tech +opportunities, +Polym.-Plast. Technol. Mater. 61, 1908 (2022). +[2] A. V. Lun-Fu, A. M. 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B 82, 195422 (2010). +[70] R. +Setton, +Carbon +nanotubes +- +II. +Cohesion +and +formation +energy +of +cylindrical +nanotubes, +Carbon 34, 69 (1996). + diff --git a/m9E0T4oBgHgl3EQfZQCv/content/tmp_files/load_file.txt b/m9E0T4oBgHgl3EQfZQCv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..71b4bad1704b6c6b9c4e3279af4d96e9e19cd3b6 --- /dev/null +++ b/m9E0T4oBgHgl3EQfZQCv/content/tmp_files/load_file.txt @@ -0,0 +1,1099 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf,len=1098 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='02319v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='mes-hall] 5 Jan 2023 Localized nonlinear excitations of a columnar chain of coronene molecules Alexander V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Savin1, 2, ∗ and Sergey V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Dmitriev3, 4, † 1Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow 119991, Russia 2Plekhanov Russian University of Economics, Moscow 117997, Russia 3Institute of Molecule and Crystal Physics, Ufa Federal Research Centre of Russian Academy of Sciences, Oktyabrya Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 151, 450075 Ufa, Russia 4Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of Russian Academy of Sciences, Ufa 450008, Russia The nonlinear dynamics of a one-dimensional molecular crystal in the form of a chain of planar coronene molecules is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Using molecular dynamics, it is shown that a chain of coronene molecules supports acoustic solitons, rotobreathers, and discrete breathers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' An increase in the size of planar molecules in a chain leads to an increase in the number of internal degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' This results in an increase in the rate of emission of phonons from spatially localized nonlinear excitations and a decrease in their lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Presented results contribute to the understanding of the effect of the rotational and internal vibrational modes of molecules on the nonlinear dynamics of molecular crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' INTRODUCTION Molecular crystals can have a quasi-one-dimensional morphology, for example, fullerene nanowhiskers con- sisting of fullerene molecules [1], a columnar structure of carbon nanotori [2, 3], B42 molecules [4], n-coronene molecules [5–8], columnar discotic liquid crystals [9–11] and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Finite-size particles of molecular crys- tals have rotational degrees of freedom that can give rize to such cointerintuitive effects as negative thermal expan- sion [12–16] and auxeticity (negative Poisson’s ratio) [17– 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Quasi-one-dimensional crystals can support various spatially localized nonlinear excitations, their study is important and is often considered in connection with the transfer of energy, mass and information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' If the molecules that make up quasi-one-dimensional crystals, in addition to translational, also have rotational and internal vibra- tional degrees of freedom, then the variety of localized excitations supported by them increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Let us note the most intensively studied spatially lo- calized excitations in nonlinear lattices and crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Compressive acoustic solitons are typically excited in solids or metamaterials under shock loading [22–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Acoustic solitons propagating at a speed exceeding the speed of longitudinal sound were described in carbon nanotube bundles [26], black phosphorene [27], graphene and boron nitride [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' It is shown that the attenuation of compressive waves in black phosphorene occurs faster than in graphene and boron nitride due to the greater number of degrees of freedom in the translational cell of phosphorene, which provides more channels for energy emission [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Rotobreathers are dynamical modes with a single rotat- ing particle while neighboring particles oscillate with the ∗ asavin@chph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='ras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='ru † dmitriev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='sergey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='v@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='com amplitude decreasing exponentially with distance from the rotating particle [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The works [33, 34] are de- voted to the analysis of the stability of rotobreathers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The effect of rotobreathers on heat capacity [29], ther- mal conductivity [35, 36], and slow relaxation [37] was analyzed within the framework of one-dimensional rota- tor lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Rotobreathers were considered in a damped driven rotator lattice [38] and in the lattices with geo- metrical nonlinearities [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The method of molecu- lar dynamics [41] was used to describe the precession of a rotating fullerene inside a fullerite crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The work [42] shows the effect of C60 fullerite crystal deformation on the rotational dynamics and shift of the center of mass of a single C60 molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' In the works [43–45] rotobreathers in the form of carbon nanotubes rotating around their axis in a carbon nanotube bundle were studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The dynamics of a fullerene molecule rotating in a fullerite crystal was studied in [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Discrete breathers or intrinsic localized modes are the large-amplitude, spatially localized vibrational modes in defect-free nonlinear lattices [47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Discrete breathers are ubiquitous in nonlinear lattices and are investi- gated in models described by the discrete nonlinear Schr¨odinger equation [50], in Josephson superconducting junctions [51, 52], in granular crystals [53], in a mass- spring chain [54], and in magnetic systems [55–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' In- teratomic interactions are non-linear, so different crystals support discrete breathers [58–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' In real discrete sys- tems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' in crystals, one deals with quasi-breathers that are not exactly periodic single-frequency modes [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' A discrete breather in the form of a single fullerene molecule oscillating with a large amplitude in a fullerite crystal [46] and a single oscillating carbon nanotube in a nanotube bundle [45] were studied by the method of molecular dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Most popular approaches to the study of nonlinear ex- citations in molecular crystals are the use of molecular dynamics [2, 3] and coarse-grained models [5, 7, 63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The aim of this study is to analyze the effect of in- ternal vibrational degrees of freedom on the robustness 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Vertical chain of 10 n-coronene molecules C6n2H6n: (a) n = 2 (coronene C24H12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (b) n = 3 (circumcoronene C54H18);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (c) n = 4 (dicircumcoronene C96H24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Carbon atoms (gray) form planar disk molecules, and hydrogen atoms are located at the edges of the disks (shown in light gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The vertical axis of the chain is parallel to the z axis, the planar molecules are parallel to the xy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The positions of neigh- boring molecules in the chain differ by the shift along the z axis and the relative rotation of the molecules in the xy plane (shift ∆z and twist ∆φ steps of the chain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' of various spatially localized nonlinear excitations in a quasi-one-dimensional chain of n-coronene molecules with n = 2, 3, and 4 [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' As the index n increases, the size of the molecules and, consequently, the number of internal degrees of freedom also increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' II, the structure of the n-coronene and the molecular dynamics model used in this study are de- scribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The spectrum of small-amplitude vibrations of the n-coronene is analyzed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Sections from IV to VI present the results of studying spatially local- ized nonlinear excitations in the chains of n-coronene molecules, namely, acoustic solitons, rotobreathers, and discrete breathers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Our conclusions are for- mulated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' MODEL The n-coronene molecule C6n2H6n can be considered as a graphene flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Therefore, to describe the dynamics of a coronene molecular crystal, one can use the force field previously used for graphene nanoribbons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' To simplify the modeling, valence-bonded CH groups of atoms at the edges of disk molecules will be considered as a single carbon atom of mass 13mp, while all other inner carbon atoms have the mass 12mp, where mp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='6601 × 10−27 kg is the proton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The Hamiltonian of one molecule can be written as H0 = N0 � i=1 �1 2Mi( ˙ui, ˙ui) + Pi � , (1) where i is the number of an atom, N0 = 6n2 is the (a) n m (b) m k n (c) m k n l (d) m k n l (e) m k n l FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (Color online) Different types of interactions between neighboring atoms belonging to the sets Ωj, j = 1, 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (a) Valence interactions j = 1, (b) valence angles j = 2, (c-e) different dihedral angles j = 3, 4, and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' number of atoms in the molecule, Mi is the mass of the ith atom (there are 6n2 − 6n inner carbon atoms of mass 12mp and 6n edge carbon atoms of mass 13mp), ui = (xi(t), yi(t), zi(t)) is the three-dimensional vector describing the position of ith atom at the time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The term Pi describes the interaction of the carbon atom with the index i with the neighboring atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' We emphasize that the inner and edge atoms differ only in their masses, and their interaction with each other is described by the same potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The potential depends on variations in bond length, bond angles, and dihedral angles between the planes formed by three neighboring carbon atoms and it can be written in the form P = � Ω1 U1 + � Ω2 U2 + � Ω3 U3 + � Ω4 U4 + � Ω5 U5, (2) where Ωj, with j = 1, 2, 3, 4, 5, are the sets of configu- rations describing different types of interactions between neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Members of these sets are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 2, and all their rotated and mirrored versions should be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Potential U1(un, um) describes the energy due to change in the length of a valence bond between atoms with the indexes n and m, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The potential U2(un, um, uk) describes the deformation en- ergy of the angle between the valence bonds unum, and umuk, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Potentials Uj(un, um, uk, ul), j = 3, 4, and 5, describe the deformation energy associated with a change in the angle between the planes unumuk and ulukum, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 2(c-e), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' We use the potentials employed in the modeling of the dynamics of large polymer macromolecules [65, 66] for the valence bond coupling, U1(u1, u2)=ǫ1{exp[−α0(ρ−ρ0)]−1}2, ρ=|u2−u1|, (3) where ǫ1 is the energy of the valence bond and ρ0 is the equilibrium length of the bond;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' the potential of the valence angle is U2(u1, u2, u3) = ǫ2(cos ϕ − cos ϕ0)2, (4) cos ϕ = (u3 − u2, u1 − u2)/(|u3 − u2| · |u2 − u1|), where the equilibrium value of the angle is cos ϕ0 = cos(2π/3) = −1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' the potential of the dihedral angle is Uj(u1, u2, u3, u4) = ǫj(1 + zj cos φ), (5) cos φ = (v1, v2)/(|v1| · |v2|), (c) (b) (a) Z x3 v1 = (u2 − u1) × (u3 − u2), v2 = (u3 − u2) × (u3 − u4), where the sign zj = 1 for j = 3, 4 (the equilibrium value of the torsional angle φ is φ0 = π) and zj = −1 for j = 5 (φ0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The values of the potential parameters are ǫ1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='9632 eV, ρ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='418 ˚A, α0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='7889 ˚A−1, ǫ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3143 eV, and ǫ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='499 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' They are found from the frequency spectrum of small-amplitude oscillations of a graphene sheet [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' According to previous study [68], the energy ǫ4 is close to the energy ǫ3, whereas ǫ5 ≪ ǫ4 (|ǫ5/ǫ4| < 1/20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Therefore, we set ǫ4 = ǫ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='499 eV and assume ǫ5 = 0, the latter means that we omit the last term in the sum Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' More detailed discussion and motivation of our choice of the interaction potentials Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (3-5) can be found in earlier publication [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The interaction of two coronene molecules is described by the potential W(X1, X2) = N0 � i=1 N0 � j=1 V (rij), (6) where the 3N0-dimensional vector Xk = {uk,i}N0 i=1 (k = 1, 2) defines the coordinates of atoms of the k-th molecules (vector uk,i specifies the coordinates of the i- th atom of the k-th molecule), rij = |u2,j − u1,i| is the distance between atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Nonvalence interactions of the carbon atoms are described by the (6,12) Lennard-Jones potential V (r) = ǫc{[(rc/r)6 − 1]2 − 1}, (7) where ǫc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='002757 eV, rc = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='807 ˚A [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Hamiltonian of a chain of N molecules (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 1) can be presented in the form H = N � n=1 �1 2(M ˙Xn, ˙Xn) + P(Xn) � + N−1 � n=1 W(Xn, Xn+1) + N−2 � n=1 W(Xn, Xn+2), (8) where the first sum includes the kinetic and potential energies of n-th molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The second and the third sums describe the interaction between nearest and next-nearest molecules, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Here the vector Xn = {un,i}N0 i=1 specifies the coordinates of the atoms of n-th molecule, M is the diagonal matrix of atom masses, P(Xn) is the energy of n-th molecule, W(Xn, Xk) is the interaction energy of n-th and k-th molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' THE DISPERSION CURVES OF SMALL-AMPLITUDE OSCILLATIONS Let us consider the structure of a symmetric (spiral) stack of planar n-coronene molecules with the symmetry TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Values of shift ∆z and twist ∆φ parameters, max- imum frequencies of out-of-plane ωop and in-plane ωip vibra- tions, velocities of torsion vt and longitudinal vl sound for a spiral stack of n-coronene C6n2H6n molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' n ∆z (˚A) ∆φ (◦) ωop (cm−1) ωip (cm−1) vt (m/s) vl (m/s) 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='445 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='0 841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='6 1549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3 217 3170 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='411 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='6 883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='7 1580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='4 195 3449 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='396 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='6 894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='0 1591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3 250 3591 axis parallel to the z axis – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' In the ground state of such a chain, the atomic coordinates of each successive molecule are obtained from the coordinates of the previ- ous molecule by translation along the z axis by a shift ∆z and rotation around the same axis by an angle ∆φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' These are the shift and twist parameters: xn+1,j = xn,j cos(∆φ) + yn,j sin(∆φ), yn+1,j = −xn,j sin(∆φ) + yn,j cos(∆φ), (9) zn+1,j = zn,j + ∆z, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=', N0, n = 0, ±1, ±2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Thus, the energy of the ground state is a function of 3N0 coordinates of N0 atoms of the first molecule X1 = {u1,j}N0 j=1, and the two geometry parameters, ∆z and ∆φ, where u1,j = (x1,j, y1,j, z1,j) is the vector position of jth atom of the first molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Finding the ground state reduces to the following min- imization problem: E = P(X1) + W(X1, X2) + W(X1, X3) → min : {u1,j}N0 j=1, ∆φ, ∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (10) The problem (10) was solved numerically by the conju- gate gradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The values of the shift ∆z and the twist ∆φ steps of the chain of n-coronene molecules are presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' A vertical chain of molecules is a multistable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Numerical analysis shows that for n-coronene molecules with n ≤ 4, the spiral structure defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (9) is the most energy-favorable ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' For analysis of small-amplitude oscillations of spiral chain it is convenient to use local cylindrical coordinates vn,j = (vn,j,1, vn,j,2, vn,j,3), given by the following ex- pressions: xn,j = x0 n,j + vn,j,1 cos(φn,j) + vn,j,2 sin(φn,j), yn,j = y0 n,j − vn,j,1 sin(φn,j) + vn,j,2 cos(φn,j), (11) zn,j = z0 n,j + vn,j,3, with u0 n,j = (x0 n,j, y0 n,j, z0 n,j), (n = 0, ±1, ±2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=', N0) being coordinates of the atoms in the helix ground state, and φn,j being angular coordinate of the atom (n, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' With these new coordinates the Hamilto- nian of the molecular chain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (8) has the following 4 0 π/3 2π/3 0 500 1000 1500 ω (cm−1) q (a) 0 π/3 2π/3 π q (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Structure of 72 dispersion curves of a spiral chain of coronene molecules C24H12 for (a) out-of-plane and (b) in- plane vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Black dots denote modes leading to the for- mation of discrete breathers – localized nonlinear oscillations of one molecule in the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' form H = � n �1 2(M ˙vn, ˙vn) + P(vn, vn+1, vn+2) � , (12) where vn = {(vn,j,1, vn,j,2, vn,j,3)}N0 j=1 is a 3N0- dimensional vector, M is 3N0-dimensional diagonal mass matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' From the Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (12) the following system of equations of motion can be derived: −M¨vn = P1(vn, vn+1, vn+2) +P2(vn−1, vn, vn+1) + P3(vn−2, vn−1, vn), (13) where Pi(v1, v2, v3) = ∂P/∂vi, i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Within the linear approximation, the system Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (13) obtains the form −M¨vn = B1vn +B2vn+1 +B∗ 2vn−1 +B3vn+2 +B∗ 3vn−2, (14) where the matrix elements are given as B1 = P11 + P22 + P33, B2 = P12 + P23, B3 = P13, and the partial derivative matrix is given as Pij = ∂2P ∂vi∂vj (0, 0, 0), i, j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The solution to the system of linear equations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (14) can be found in the standard form vn = Aw exp[i(qn − ωt)], (15) 0 π/3 2π/3 0 500 1000 1500 ω (cm−1) q (a) 0 π/3 2π/3 π q (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Structure of 162 dispersion curves for a spiral chain of circumcoronene molecules C54H18 for (a) out-of-plane and (b) in-plane vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Black dots indicate modes that lead to the formation of discrete breathers – localized nonlinear vibrations of one molecule in the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' where A is the linear mode amplitude, w is the eigen- vector, ω is the phonon frequency with the dimension- less wave number q ∈ [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (15) into the system Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (14), we arrive at the following 3N0- dimensional eigenvalue problem: ω2Mw = C(q)w, (16) where Hermitian matrix C(q) = B1 + B2 exp(iq) + B∗ 2 exp(−iq) +B3 exp(2iq) + B∗ 3 exp(−2iq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Using the substitution w = M−1/2e, problem Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (16) can be rewritten in the form ω2e = M−1/2C(q)M−1/2e (17) where e is the normalized eigenvector, (e, e) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Thus, to obtain the dispersion curves ωj(q), it is neces- sary to find the eigenvalues and eigenvectors of the Her- mitian matrix Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (17) of size 3N0 × 3N0 for each fixed wavenumber 0 ≤ q ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' As a result, we obtain 3N0 branches of the dispersion relation {ωj(q)}3N0 j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The planar structure of molecules in a spiral chain leads to the division of its small-amplitude vibrations into two-classes: out-of-plane vibrations, when atoms vibrate orthogonally to the molecular plane (all atoms move along the z axis) and in-plane vibrations (all atoms move in the xy plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Two thirds of the branches corre- spond to in-plane vibrations, while only one-third corre- sponds to out-of-plane vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The dispersion curves are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 3 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 5 0 π/3 2π/3 0 100 200 (a) ω (cm−1) q 0 π/3 2π/3 π (b) q FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Dispersion curves in the low-frequency region for a spiral chain of coronene molecules C24H12 for (a) out-of-plane and (b) in-plane vibrations (three gray bands show the fre- quency spectrum of the rotobreathers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The dashed straight lines define the tangents to the dispersion curves emerging from the zero point, corresponding to the velocities of the longitudinal vl and torsion vt sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' For the spiral chain of coronene molecules C24H12, the dispersion curves of out-of-plane vibrations, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 3(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 5(a), lie in the frequency range 0 ≤ ω ≤ ωop, with the maximum frequency ωop = 842 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' One dis- persion curve ωl(q) starts from the origin (q = 0, ω = 0), it describes the displacement of planar molecules along the chain axis without internal deformations (longitudi- nal acoustic vibrations of the chain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The tangent of this dispersion curve at the origin gives the velocity of longi- tudinal sound waves vl = ∆z lim q→0 ωl(q) q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The dispersion curves of in-plane oscillations, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 3(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 5(b), lie in the frequency range 0 ≤ ω ≤ ωip with the maximum frequency ωip = 1549 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' One dispersion curve ωt(q) starts from the origin and de- scribes torsional acoustic oscillations (rotation of planar molecules around the chain axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The speed of long-wave torsional vibrations (speed of torsional sound) is vt = ∆z lim q→0 ωt(q) q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' In addition, one dispersion curve approaches the q axis tangentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' This curve describes the optical bending vibrations of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The frequency spectrum of in- plane oscillations is characterized by the presence of a 0 100 200 300 400 500 0 20 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='2 (a) t (ps) n En (eV) 0 100 200 300 400 500 0 20 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='6 (b) t (ps) n En (eV) 0 100 200 300 400 500 0 20 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='8 (c) t (ps) n En (eV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Formation of a supersonic acoustic soliton in a spiral chain of (a) coronene, (b) circumcoronene, and (c) dicircum- coronene molecules produced by longitudinal local compres- sion at the end of the chain with amplitude az = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='4 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The distribution of energy in the chain En(t) at different times is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The number of molecules in the chain is N = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The dotted lines show the trajectory of motion with the ve- locity of longitudinal sound vl to demonstrate the supersonic motion of solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' gap in the low-frequency region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' For a chain of coronene molecules, the gap is from 10 to 203 cm−1 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 5(b)], and for a chain of circumcoronene molecules, from 9 to 141 cm−1 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 4(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The values of the maximum frequencies ωop, ωip and the speeds of sound vl, vt are given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' As can be seen from the table, the speed of longitudinal sound is 15 times greater than the speed of torsional sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='1 0 (a) ρn (A) ° −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='05 0 (b) ρn (A) ° 0 100 200 300 400 500 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='04 0 (c) n ρn (A) ° FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Distribution of longitudinal compression during the motion of an acoustic soliton along a chain of N = 500 molecules of (a) coronene, (b) circumcoronene, (c) dicircum- coronene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The distribution of relative longitudinal displace- ments ρn of chain molecules at time t = 40 ps is shown for the amplitude of the initial local compression of the chain end az = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='4 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The vertical dotted lines show the position of the front of the acoustic phonon wave packet propagating with the velocity vl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' ACOUSTIC SOLITONS The interaction of neighboring planar molecules is de- termined by the sum of interactions of all pairs of their atoms Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (6), which are described by the Lennard-Jones potential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The Lennard-Jones potential at small interatomic distances is characterized by the hard-type anharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Therefore, one can expect the possibil- ity of propagation of compressive longitudinal acoustic solitons moving at a speed exceeding the velocity of lon- gitudinal sound vl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' To test the existence of supersonic acoustic solitons, we simulate the propagation of initial local longitudinal compression along a chain of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Consider a spiral chain of N = 500 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Let us take the ground state of the chain and at t = 0 shift the first two molecules along the z axis by az.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' As a result, local longitudinal compression occurs at the end of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Having fixed the position of these two molecules in the shifted state, let us consider the propagation of local compression along the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='5 1 (a) E (eV) 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='6 (b) s Az (A) ° FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Dependence of (a) energy E of an acoustic soliton and (b) longitudinal compression of the chain Az produced by an acoustic soliton propagating in a chain of coronene molecules on its dimensionless velocity s = v/vl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Markers show numerical values, solid curves show approximations ob- tained by the least squares method E(s) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='36(s − 1)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='7 eV and Az(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='93(s − 1)0,5 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' To simulate the dynamics of a chain with fixed ends, we numerically integrate the system of equations of motion corresponding to the Hamiltonian of the chain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (8) M ¨Xn = − ∂H ∂Xn , n = 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=', N − 2, (18) ˙Xn ≡ 0, n = 1, 2, N − 1, N, with the initial conditions Xn(0) = X0 n + azez, n = 1, 2 Xn(0) = X0 n, n = 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=', N, (19) ˙Xn(0) = 0, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='., N, where the 3N0-dimensional vector Xn = {(xn,j, yn,j, zn,j)}N0 j=1 defines the coordinates of the atoms of n-th molecule, vectors {X0 n}N n=1 defines ground state of molecular chain, ez is a unit vector directed along the z axis, az > 0 is the amplitude of the initial compression of the chain end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Numerical integration of the system of equations of motion (18) showed that the initial longitudinal compres- sion of the chain edge with an amplitude az ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='6 ˚A for 7 0 π/3 2π/3 π 4π/3 5π/3 2π 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='06 1 2 3 ϕ E (eV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Change in the energy of the chain E as the function of the rotation angle ϕ of one molecule rotating around the z axis in the chain of coronene, circumcoronene, and dicircum- coronene (curves 1, 2, and 3, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Only one molecule rotates quasi-statically while the rest of the molecules remain in their equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' coronene molecules always leads to the formation of a supersonic acoustic soliton and a subsonic wave packet of long-wavelength longitudinal acoustic phonons – see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 6 (a) and 7 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' A local area of compression is formed in the chain, which moves along it with a con- stant supersonic speed v > vl, keeping its shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' When moving, the soliton breaks away from the wave packet of phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' This allows us to find its energy E and the longitudinal compression of the chain Az: E = � n En, Az = � n ρn, ρn = 1 N0 N0 � j=1 (zn+1,j−zn,j−∆z), where the summation is carried out only over the soliton localization region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Dependencies of the soliton energy E and chain com- pression Az produced by the soliton on its dimensionless velocity s = v/vl are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' As can be seen from the figure, with increasing velocity, the soliton energy in- creases as (s − 1)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='7, and the compression as (s − 1)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' In chains of circumcoronene and dicircumcoronene molecules, local longitudinal compression of the chain end also leads to the formation of a supersonic localized compression region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' But the motion of this region is ac- companied by the emission of phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' As a result, the energy and velocity of the soliton decrease monotonically, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 6(b,c) and 7(b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The larger the molecule, the more noticeable the emission of phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Therefore, it can be concluded that a chain of n-coronene molecules admits the existence of an exact acoustic soliton of longi- tudinal compression only for n = 2, while for n > 2 there is only a soliton-like excitation with a finite lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' ROTOBREATHERS The structure of planar molecules allows their rotation in chains around the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The n-coronene molecule has the shape of a regular hexagon, a rotation of one molecule by 60◦ will transfer the chain to an equivalent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' If we fix the positions of all molecules and rotate only one molecule as a rigid body, then the rotation potential E(ϕ) (dependence of the chain energy on the angle of rota- tion of one molecule ϕ) can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' This potential is a periodic function with period π/3, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' In the approximation of absolutely rigid valence bonds, free rotation requires overcoming energy barriers of height 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='26, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='34 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='66 eV for the chain of coronene, circum- coronene, and dicircumcoronene molecules, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' These barriers are overcome at molecular rotation fre- quencies above ω0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='19, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='11 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='87 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Thus, the topology of the chain allows the existence of roto- breathers (localized rotations of molecules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' In the approximation of absolutely rigid molecules, their chains allow the existence of a rotobreathers with an infinite frequency spectrum lying above frequency ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The n-coronene molecule is not an absolutely rigid body, it has 3N0−6 vibrational modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The presence of internal vibrations in a rotator (in our case, a planar n-coronene molecule) leads to the appearance of band gaps (lacu- nae) in the frequency spectrum of the rotobreather [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' At frequencies within these band gaps, the rotation leads to resonance with the natural oscillations of the rotators and the emission of phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Therefore, the presence of internal vibrational modes in molecules should lead to a significant narrowing of the frequency spectrum of the rotobreather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' To find the rotobreather, we simulate the rotation of one molecule at different initial frequencies in a chain of N = 100 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' A viscous friction at the ends of the chain is introduced, which ensures the absorption of phonons emitted by the rotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' To do this, we numeri- cally integrate the system of equations of motion M ¨Xn = − ∂H ∂Xn , n = Nt + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=', N − Nt, (20) M ¨Xn = − ∂H ∂Xn − γM ˙Xn, n ≤ Nt, n > N − Nt with the friction coefficient γ = 1/tr, tr = 10 ps, Nt = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Let us take the ground state of the chain and excite the rotation of the central molecule nc = N/2 with the frequency ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' take the initial conditions in the form {Xn(0) = X0 n}N n=1, ˙Xn(0) = 0, n ̸= nc (21) { ˙xnc,j = −ωy0 nc,j, ˙ync,j = ωx0 nc,j, ˙znc,j = 0}N0 j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Thus, we set the rotation of one rotator in the chain with the initial energy E = 1 2ω2 N0 � j=1 Mj(x0 nc,j 2 + y0 nc,j 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 8 0 5 10 15 0 10 20 30 40 50 60 70 E (eV) t (ns) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Change in time of the energy of one rotator in the chain of coronene molecules for different values of the initial rotation frequency of central molecule, varying in the range from ω = 3 to 22 cm−1 with a step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='25 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Friction at the ends of the chain will ensure the ab- sorption of phonons emitted by the rotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Therefore, depending on the value of the frequency ω, the rotator either stops, having lost all the energy for phonon emis- sion, or reaches a stationary rotation mode with a con- stant frequency without phonon emission (rotobreather mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The change in the rotator energy E for various initial values of the frequency ω in the chain of coronene and circumcoronene molecules is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 10 and 11, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 10, for a chain of coronene molecules, there are only three frequency ranges at which a rotation at constant frequency of one molecule can occur without emitting phonons: [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='96, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='54], [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='28, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='09], and [16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='33, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='71] cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Thus, in the chain of coronene molecules, the rotobreather has a frequency spectrum consisting of only three narrow intervals, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 5(b), where the frequency spectrum of the ro- tobreather is shown by gray bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Rotation with other frequencies leads to the emission of phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Simulation of the dynamics of a rotator in a chain of circumcoronene molecules showed that rotobreathers do not exist in this chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Here, at all values of the rota- tion frequency, the rotator emits phonons and completely loses energy, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' There is only one frequency ω = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='6 cm−1 at which the radiation becomes less in- tense, but does not completely disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' In a chain 0 5 10 15 0 20 40 60 80 100 120 140 E (eV) t (ns) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Change in time of the energy of one rotator in the chain of circumcoronene molecules for different values of the initial rotation frequency of central molecule, varying in the range from ω = 2 to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='75 cm−1 with a step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='25 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The dashed line shows the energy corresponding to the rota- tion frequency ω = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='6 cm−1, at which the weakest phonon emission occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' of dicircumcoronene molecules, the rotation of the ro- tator at all frequencies leads to an even stronger emis- sion of phonons and no rotobreather is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The ab- sence of a rotobreather in the chains of circumcoronene and dicircumcoronene molecules is explained by a denser frequency spectrum of natural vibrations of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Here, in contrast to the coronene molecules, the rotation of the rotator at all frequencies resonates with the natural vibrations of the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' DISCRETE BREATHERS An isolated n-coronene molecule consists of N0 = 6n2 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' It has 3N0 − 6 natural oscillations with non-zero frequencies, {ωj}3N0 j=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The first six eigenmodes have a zero frequency ω1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' = ω6 = 0, they correspond to the motion of a molecule as a rigid body (three translational and three rotational degrees of freedom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Eigenmodes with non-zero frequencies are of two types: N0−2 out-of- plane vibrations, when atoms move orthogonally to the molecular plane, and 2N0 − 4 in-plane vibrations, when atoms move in the molecular plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The coronene molecule has 22 out-of-plane vibra- tions with frequencies 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='6, 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='7,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=', 839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='2 cm−1 and 9 0 5 10 15 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='06 1 2 3 4 5 E (eV) t (ns) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Dependence of the energy of vibrations of the central molecule of a chain of coronene molecules on time at the initial excitation of the j-th natural vibration: (curve 1) j = 17, ωj = 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (curve 2) j = 21, ωj = 278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (curve 3) j = 23, ωj = 329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (curve 4) j = 33, ωj = 435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (curve 5) j = 47, ωj = 839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='2 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The initial atomic velocity used to excite the vibrational mode in the central molecule is A = 10 ˚A/ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 44 in-plane vibrations with frequencies 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='1, 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=', 1546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='2 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The circumcoronene molecule has 52 out-of-plane vibrations with frequencies 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='9,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=', 881.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='0 cm−1 and 104 in-plane vibrations with frequen- cies 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='5, 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=', 1576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Let us check whether the excitation of a high-amplitude natural oscillation of one molecule can lead to the appearance of a discrete breather in the chain – a nonlinear oscillation localized on one molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' To find discrete breathers, we simulate high-amplitude natural vibrations of one central molecule in a chain of N = 100 molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' At the ends of the chain, viscous friction is introduced, which ensures the absorption of phonons emitted by vibrations of the central molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The system of equations of motion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (20) is integrated numerically with the initial conditions Xn(0) = X0 n, ˙Xn(0) = Aejδn,nc, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=', N, (22) where A defines the magnitude of the initial velocity of atoms of the central molecule, ej is the unit eigenvector of the jth eigenmode of an isolated molecule (j = 7,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=',3N0), nc = N/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The value of A determines the vibrational energy of the molecule and it is chosen sufficiently large to enter the regime of anharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The dependencies of the vibrational energy of the cen- tral molecule on time t are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Numerical integration of the system of equations of motion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (20) with the initial conditions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (22) showed that three dy- namics scenarios are possible: very fast damping of oscil- lations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 12, curve 3), slow damping (curves 1 and 4) and the formation of undamped oscillations (curves 2 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The first two scenarios are typical for out-of- plane vibrations, the last one – for in-plane vibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The frequencies of the resulting discrete breathers are shown by black dots in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Of all the out- of-plane eigenmodes, only the oscillation with the max- 838 840 842 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3 (a) E (eV) 1474 1476 1478 (b) ω (cm−1) 1494 1496 1498 (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Dependence of the energy E on the frequency ω for a discrete breather based on the j eigenmode of the coronene molecule: (a) j = 47, ωj = 839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (b) j = 67, ωj = 1470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='0 (c) j = 69, ωj = 1491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='3 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' imum frequency can lead to the formation of a discrete breather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' For a chain of coronene molecules out of 44 in- plane vibrations 24 can lead to the formation of a discrete breather, and for a chain of circumcoronene molecules out of 104 in-plane vibrations 31 produce discrete breathers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The undamped vibrations are localized strictly on one molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The oscillations are anharmonic, their fre- quency depends on the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' A characteristic fea- ture of localized oscillations (discrete breathers) is a lin- ear decrease in their frequency with increasing energy, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' As the oscillation amplitude increases, the energy of the breather increases and the frequency de- creases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Thus, n-coronene chains support gap discrete breathers with a soft type of anharmonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The energy of a discrete breather in a chain of coronene molecules can reach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='37 eV, and the width of the frequency spectrum can reach 6 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' CONCLUSIONS The linear phonon spectrum and nonlinear spatially localized excitations, such as acoustic solitons, roto- breathers, and discrete breathers in chains of n-coronene molecules, are studied by the method of molecular dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Three members of the n-coronene were con- sidered, namely coronene, circumcoronene and dicircum- coronene (n = 2, 3 and 4 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' These molecules include respectively N0 = 24, 54, and 96 carbon atoms and have 3N0 − 6 vibrational degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The size of molecules plays an important role in chain dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The spectra of low-amplitude vibrations of chains of coronene and circumcoronene molecules are shown in Figs 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' It can be seen that the maximum frequencies of out-of-plane and in- plane vibrations are approximately the same for chains of coronene and circumcoronene molecules, but the spec- trum of the latter is denser, since the number of degrees 10 of freedom is greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The spectrum of a chain of dicir- cumcoronene molecules is even denser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' It was found that a chain of coronene molecules sup- ports the propagation of acoustic compressive solitons, which practically do not emit energy when moving at supersonic speed, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 6(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Similar excitations in chains of circumcoronene and dicircum- coronene molecules constantly lose energy, emitting low- amplitude phonons, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 6(b,c) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 7(b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' This is because spiral chains have lower symmetry in the stack- ing of larger molecules and more channels to radiate en- ergy due to the greater number of vibrational degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' A similar picture was observed for rotobreathers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Only in a chain of coronene molecules a single molecule can rotate with frequencies in certain ranges [shown in gray in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 5(b)], radiating no energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' In chains of circum- coronene and dicircumcoronene molecules, a molecule ro- tating at any frequency excites low-amplitude phonons, constantly loses its energy, and eventually stops rotating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The explanation lies in more resonances with a denser phonon spectrum in chains with larger molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' As for discrete breathers, they are supported by all three considered molecular chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' Discrete breathers are in the form of single molecule vibrating at large ampli- tude and radiating no energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The frequencies of discrete breathers are marked with black dots in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 3 and 4 for chains of coronene and circumcoronene molecules, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' A discrete breather with out-of-plane oscillations, see panels (a), is created only by the highest-frequency out-of-plane mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' On the other hand, a number of in-plane vibrational modes create discrete breathers, see panels (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The frequency of discrete breathers decreases with an increase in their energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' soft-type anhar- monicity is realized, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The results presented in this study illustrate the role of the internal degrees of freedom of particles in the non- linear dynamics of molecular chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' ACKNOWLEDGMENTS Computational facilities were provided by the Inter- departmental Supercomputer Center of the Russian Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' The work of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' (statement of the problem, numerical simulations, and writing the manuscript) was supported by the Russian Science Foun- dation, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' 21-12-00229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E0T4oBgHgl3EQfZQCv/content/2301.02319v1.pdf'} +page_content='D.' 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b/mtFST4oBgHgl3EQfKjji/content/tmp_files/2301.13737v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..69464dc0289feba80ae2ca87480692cd10d689a8 --- /dev/null +++ b/mtFST4oBgHgl3EQfKjji/content/tmp_files/2301.13737v1.pdf.txt @@ -0,0 +1,1943 @@ +Under peer review. +SELF-CONSISTENT VELOCITY MATCHING OF PROBA- +BILITY FLOWS +Lingxiao Li +MIT CSAIL +lingxiao@mit.edu +Samuel Hurault +Univ. Bordeaux, Bordeaux INP, CNRS, IMB +samuel.hurault@math.u-bordeaux.fr +Justin Solomon +MIT CSAIL +jsolomon@mit.edu +ABSTRACT +We present a discretization-free scalable framework for solving a large class +of mass-conserving partial differential equations (PDEs), including the time- +dependent Fokker-Planck equation and the Wasserstein gradient flow. The main +observation is that the time-varying velocity field of the PDE solution needs to be +self-consistent: it must satisfy a fixed-point equation involving the flow character- +ized by the same velocity field. By parameterizing the flow as a time-dependent +neural network, we propose an end-to-end iterative optimization framework called +self-consistent velocity matching to solve this class of PDEs. Compared to existing +approaches, our method does not suffer from temporal or spatial discretization, +covers a wide range of PDEs, and scales to high dimensions. Experimentally, +our method recovers analytical solutions accurately when they are available and +achieves comparable or better performance in high dimensions with less train- +ing time compared to recent large-scale JKO-based methods that are designed for +solving a more restrictive family of PDEs. +1 +INTRODUCTION +Mass conservation is a ubiquitous phenomenon in dynamical systems arising from fluid dynamics, +electromagnetism, thermodynamics, and stochastic processes. Mathematically, mass conservation +is formulated as the continuity equation: +∂tpt(x) = −∇ · (vtpt), ∀x, t ∈ [0, T] +(1) +where pt : Rd → R is a scalar quantity such that the total mass +� +pt(x) is conserved with respect +to t, vt : Rd → Rd is a velocity field, and T > 0 is total time. We will assume, for all t ∈ [0, T], +pt ≥ 0 and +� +pt(x) dx = 1, i.e., pt is a probability density function. We use µt to denote the +probability measure with density pt. Once a pair (pt, vt) satisfies (1), the density pt is coupled with +vt in the sense that the evolution of pt in time is characterized by vt (Section 3.1). +We consider the subclass of mass-conserving PDEs that can be written in a single equation of the +form +∂tpt(x) = −∇ · (ft(x; µt)pt), ∀x, t ∈ [0, T] +(2) +where ft(·; µt) : Rd → Rd is a given function depending on µt, with initial condition µ0 = µ∗ +0 for +a given initial probability measure µ∗ +0 with density p∗ +0. +Different choices of ft lead to a large class of mass-conserving PDEs. For instance, given a func- +tional F : P2(Rd) → R on the space of probability distributions with finite second moments, if we +take +ft(x; µt) := −∇W2F(µt)(x), +(3) +where ∇W2F(µ) : Rd → Rd is the Wasserstein gradient of F, then the solution to (2) is the +Wasserstein gradient flow of F (Santambrogio, 2015, Chapter 8). Thus, solving (2) efficiently +1 +arXiv:2301.13737v1 [cs.LG] 31 Jan 2023 + +Under peer review. +allows us to optimize in the probability measure space. If we take +ft(x; µt) := bt(x) − Dt(x)∇ log pt(x), +(4) +where bt is a velocity field and Dt(x) is a positive-semidefinite matrix, then we obtain the time- +dependent Fokker-Planck equation Risken & Risken (1996), which describes the time evolution of +the probability flow undergoing drift bt and diffusion with coefficient Dt. +The predominant strategy to solve (2) is to use an Eulerian representation of the density field pt +on a discretized mesh or as a neural network (Raissi et al., 2019). However, these approaches do +not fully exploit the mass-conservation principle and are difficult to scale to high dimensions. Shen +et al. (2022) recently introduced the notion of self-consistency for the Fokker-Planck equation, a +Lagrangian formulation of (2) involving the velocity field of the flow. In this work, we extend their +notion of self-consistency to a more general class of mass-conserving PDEs of the form (2). To this +end, we develop an iterative optimization scheme called self-consistent velocity matching. With the +probability flow parameterized as a neural network, at each iteration, we refine the velocity field vt +of the current flow to match an estimate of ft evaluated using the network weights from the previous +iteration. This iterative formulation allows us to rewrite the velocity-matching objectives for certain +PDEs to get rid of the computationally expensive quantities such as ∇ log pt in the Fokker-Planck +equation. Moreover, our method is agnostic to the probability flow parameterization: we have empir- +ically found that the two popular ways of parameterizing the flow—as a time-varying pushforward +map (Biloˇs et al., 2021) and as a time-varying velocity field (Chen et al., 2018)—both have merits +in different scenarios. +Our method tackles mass-conserving PDEs of the form (2) in a unified manner without tempo- +ral or spatial discretization. Experimentally, it can recover true solutions faithfully for PDEs with +analytically-known solutions. Only recent neural JKO-based methods (Mokrov et al., 2021; Fan +et al., 2021; Alvarez-Melis et al., 2021) are capable of solving PDEs of the form (2) in high dimen- +sions, and these methods are specialized to Wasserstein gradient flows (3). Our algorithm achieves +comparable or better performance in our test cases compared to these JKO methods while using a +lower computational budget and without discretizing time. We further demonstrate the flexibility +of our method on a series of qualitative experiments for modeling flocks of birds, flows splashing +against obstacles, and computing smooth interpolation of measures, all without discretization. +2 +RELATED WORKS +Classical PDE solvers for mass-conserving PDEs such as the Fokker-Planck equation and the +Wasserstein gradient flow either use an Eulerian representation of the density and discretize space as +a grid or mesh Burger et al. (2010); Carrillo et al. (2015); Peyr´e (2015) or use a Lagrangian represen- +tation, which discretizes the flow as a collection of interacting particles simulated forward in time +Crisan & Lyons (1999); Westdickenberg & Wilkening (2010). Due to spatial discretization, these +methods struggle with high-dimensional problems. Hence, the rest of the section focuses solely on +recent neural network-based methods. +Physics-informed neural networks. +Physics-informed neural networks (PINNs) are prominent +methods that solve PDEs using deep learning (Raissi et al., 2019; Karniadakis et al., 2021). The +main idea is to minimize the residual of the PDE along with loss terms to enforce the boundary +conditions and to match observed data. Our notion of self-consistency is a Lagrangian analog of the +residual in PINN. Our velocity matching only occurs along the flow of the current solution where +interesting dynamics happen, while in PINNs the residual is evaluated on collocation points that +occupy the entire domain. Hence our method is particularly suitable for high-dimensional problems +where the dynamics have a low-dimensional structure. +Neural JKO methods. +Recent works (Mokrov et al., 2021; Alvarez-Melis et al., 2021; Fan et al., +2021) apply deep learning to the time-discretized JKO scheme (Jordan et al., 1998) to solve Wasser- +stein gradient flow (3). By pushing a reference measure through a chain of neural networks, usually +parameterized as input-convex neural networks (ICNNs) (Amos et al., 2017), these methods avoid +discretizing the space and are thus capable of solving high-dimensional problems. Mokrov et al. +(2021) optimize one ICNN to minimize Kullback-Leibler (KL) divergence plus a Wasserstein-2 dis- +tance term at each JKO step. This method is extended to other functionals by Alvarez-Melis et al. +2 + +Under peer review. +(2021). Fan et al. (2021) use the variational formulation of f-divergence to obtain a faster primal- +dual approach. +An often overlooked problem of neural JKO methods is that the total training time scales quadrati- +cally with the number of JKO steps: to draw samples for the current step, initial samples from the +reference measure must be passed through a long chain of neural networks, along with expensive +quantities like densities. However, using too few JKO steps results in large temporal discretization +errors. Moreover, the optimization at each step might not have fully converged before the next step +begins, resulting in an unpredictable accumulation of errors. In contrast, our method does not suf- +fer from temporal discretization and can be trained end-to-end. It outperforms these neural JKO +methods with less training time in most experiments we considered. +Velocity matching. +A few recent papers employ the idea of velocity matching to construct a flow +that follows a learned velocity field. di Langosco et al. (2021) simulate the Wasserstein gradient flow +of the KL divergence by learning a velocity field that drives a set of particles forward in time for +Bayesian posterior inference. The velocity field is refined on the fly based on the current positions +of the particles. Boffi & Vanden-Eijnden (2022) propose a similar method that applies to a more +general class of time-dependent Fokker-Planck equations. These two methods can only approximate +probability measures using finite particles and can have large temporal discretization errors similar +to JKO methods. Two recent methods (Liu et al., 2022; Lipman et al., 2022) use flow matching +for generative modeling by learning a velocity field that generates a probability path connecting a +reference distribution to the data distribution. Yet these two methods are not designed for solving +PDEs. +Most relevant to our work, Shen et al. (2022) propose the concept of self-consistency for the Fokker- +Planck equation, that the velocity field recovering the velocity field of the flow solution to the +Fokker-Planck equation must satisfy a fixed-point equation. They theoretically show that, under +certain regularity conditions, the Wasserstein-2 distance between the current solution and the true +solution is bounded by a term measuring the violation of the fixed-point equation (including up +to second-order spatial derivatives). Their algorithm minimizes such violation using neural ODE +parameterization (Chen et al., 2018) and the adjoint method. Our work extends the concept of self- +consistency to a wider class of PDEs in the form of (2). Unlike Shen et al. (2022), our method does +not optimize a fixed objective but instead carries out infinite-dimensional fixed-point iterations on +the self-consistency condition. While the experiments of Shen et al. (2022) are limited to a simple +2D example, presumably due to the computational cost of the higher-order spatial derivatives in their +objective, our method excels at solving a variety of large-scale problems. +3 +SELF-CONSISTENT VELOCITY MATCHING +3.1 +PROBABILITY FLOW OF THE CONTINUITY EQUATION +A key property of the continuity equation (1) is that any solution (pt, vt)t∈[0,T ] (provided pt is con- +tinuous with respect to t and vt is bounded) corresponds to a unique flow map {Φt(·) : Rd → +Rd}t∈[0,T ] that solves the ordinary differential equations (ODEs) (Ambrosio et al., 2005, Proposi- +tion 8.1.8) +Φ0(x) = x, d +dtΦt(x) = vt(Φt(x)), ∀x, t ∈ [0, T], +(5) +and the flow map satisfies µt = (Φt)#µ0 for all t ∈ [0, T], where (Φt)#µ0 to denote the push- +forward measure of µ0 by Φt. Moreover, the converse is true: any solution (Φt, vt) of (5) with +Lipschitz continuous and bounded vt is a solution of (1) with µt = (Φt)#µ0 (Ambrosio et al., 2005, +Lemma 8.1.6). Thus the Eulerian viewpoint of (1) is equivalent to the Lagrangian viewpoint of (5). +We next exploit this equivalence by modeling the probability flow using the Lagrangian viewpoint +so that it automatically satisfies the continuity equation (1). +3.2 +PARAMETRIZING PROBABILITY FLOWS +Our algorithm will be agnostic to the exact parameterization used to represent the probability flow. +As such, we need a way to parameterize the flow to access the following quantities for all t ∈ [0, T]: +3 + +Under peer review. +• Φt : Rd → Rd, the flow map at time t. Φt(x0) is the location of a particle at time t if it is +at x0 at time 0. We assume Φt is invertible; +• vt : Rd → Rd, the velocity field of the flow at time t. +• µt ∈ P(Rd), the probability measure at time t from which we can access samples and its +density pt. +We will assume all these quantities are sufficiently continuous and bounded to ensure the Eulerian +and Lagrangian viewpoints in Section 3.1 are equivalent. This can be achieved by using continuously +differentiable activation functions in the network architectures and assuming the network weights +are finite similar to the uniqueness arguments given in (Chen et al., 2018). We will use the following +two ways to parameterize the flow, modeling either the flow map Φt or the velocity field vt as a +neural network. +Time-dependent Invertible Push Forward (TIPF). We first parameterize a probability flow by +modeling Φt : Rd → Rd as an invertible network for every t. The network architecture is chosen so +that Φt has an analytical inverse with a tractable Jacobian determinant, similar to (Biloˇs et al., 2021). +We augment RealNVP (Dinh et al., 2016) so that the network for predicting scale and translation +takes t as an additional input. To enforce the initial condition, we need Φ0 to be the identity map. +This condition can be baked into the network architecture (Biloˇs et al., 2021) or enforced by adding +an additional loss term EX∼µ∗ +0∥Φ0(X) − X∥2. For brevity, we will from now on omit in the text this +additional loss term. The velocity field can be recovered via vt(x) = ∂tΦt(Φ−1 +t (x)). To recover the +density pt of µt = (Φt)#µ0, we use the change-of-variable formula log pt(x) = log p∗ +0(Φ−1 +t (x)) + +log det +��JΦ−1 +t (x) +��. +Neural ODE (NODE). We also parameterize a flow by modeling vt : Rd → Rd as a neural net- +work; this is used in Neural ODE (Chen et al., 2018). The network only needs to satisfy the minimum +requirement of being continuous. The flow map and the density can be recovered via numerical in- +tegration: Φt(x) = x + +� t +0 vs(Φs(x)) ds and log pt(Φt(x)) = log p∗ +0(x) − +� t +0 ∇ · vs(Φs(x)) ds, a +direct consequence of (1) also known as the instantaneous change-of-variable formula (Chen et al., +2018). To obtain the inverse of the flow map, we integrate along −vt. With NODE, the initial +condition µ0 = µ∗ +0 is obtained for free. +While the use of invertible coupling layers in TIPF allows efficient access to samples and densities, +TIPF becomes less effective in higher dimensions as many couple layers are needed to retain good +expressive power. In contrast, NODE puts little constraints on the network architecture, but nu- +merical integration can be slow and have errors. Handling the initial condition is trivial for NODE +while an additional loss term or special architecture is needed for TIPF. As we will show in the +experiments, both strategies have merits. +3.3 +FORMULATION +We now describe our algorithm for solving mass-conserving PDEs (2). A PDE of this form is +determined by ft(·; µt) : Rd → Rd plus the initial condition µ∗ +0. If a probability flow µt with flow +map Φt and velocity field vt satisfies the following self-consistency condition, +vt(x) = ft(x; µt), ∀x in the support of µt, +(6) +then the continuity equation of this flow implies the corresponding PDE (2) is solved. Conversely, +the velocity field of any solution of (2) will satisfy (6). Shen et al. (2022) develop this concept for +the Fokker-Planck equation, and here we generalize it to a wider class of PDEs of the form (2). +Hence, instead of solving (2) which is a condition on the density pt that might be hard to access, we +can solve (6) which is a more tractable condition on the velocity field vt that is readily accessible +using TIPF or NODE. +Let θ be the network weights that parameterize the probability flow using TIPF or NODE. The flow’s +measure, velocity field, and flow map at time t are denoted as µθ +t , vθ +t , Φθ +t respectively. One option +to solve (6) would be to minimize +min +θ +� T +0 +EX∼µθ +t +���vθ +t (X) − ft(X; µθ +t ) +��2� +dt. +(7) +This formulation is reminiscent of PINNs (Raissi et al., 2019) where a residual of the original PDE is +minimized. Direct optimization of (7) is challenging: while the integration over [0, T] and µθ +t can be +4 + +Under peer review. +approximated using Monte Carlo, to apply stochastic gradient descent, we must differentiate through +the µθ +t and ft: this can be either expensive or intractable depending on the network parameterization. +The algorithm by Shen et al. (2022) uses the adjoint method specialized to Fokker-Planck equations; +extending their approach to more general PDEs requires a closed-form formula for the time evolution +of the quantities within ft, which can only be obtained on a case-by-case basis. +Instead, we propose the following iterative optimization algorithm to solve (7). Let θk denote the +network weights at iteration k. We define iterates +θk+1 := arg min +θ +F(θ, θk). +(8) +where +F(θ, θk):= +� T +0 +EX∼µ +θk +t +����vθ +t (X) − ft(X; µθk +t ) +��� +2� +. +(9) +Effectively, in (9), we only match the velocity field vθ +t to what it should be according to ft based on +the network weights θk from the previous iteration. This scheme is an infinite-dimensional analog +to fixed-point iterations as vt is a continuous vector field. Since θk is fixed, minimizing (9) over +θ is a lot easier than directly minimizing (7), as vθ +t only needs to match a constant velocity field +ft(·; µθk +t ); we found a few steps of stochastic gradient descent sufficient for the optimization in (8) +(see a comparison in Figure 13). We call this iterative algorithm self-consistent velocity matching. +If ft depends on the density of µt only through the score ∇ log pt (corresponding to a diffusion term +in the PDE), then we can apply an integration-by-parts trick (Hyv¨arinen & Dayan, 2005) to get rid +of this density dependency by adding a divergence term of the velocity field. Suppose ft is from the +Fokker-Planck equation (4). Then the cross term in (9) after expanding the squared norm has the +following alternative expression. +Proposition 3.1. For every t ∈ [0, T], for ft defined in (4), assume vθ +t , Dt are bounded and continu- +ously differentiable, and µθ′ +t is a measure with a continuously differentiable density pθ′ +t that vanishes +in infinity and not at finite points, then we have +EX∼µθ′ +t +� +vθ +t (X)⊤ft(X; µθ′ +t ) +� += +EX∼µθ′ +t +� +vθ +t (X)⊤bt(X) + ∇ · +� +D⊤ +t (x)vθ +t (X) +�� +. +(10) +We provide the derivation in Appendix A. Minimizing (9) is then equivalent to minimizing the ex- +pectation of the squared norm of vθ +t plus the cross term (10), and access to pt is no longer needed. +This is useful for NODE parameterization since obtaining the score would otherwise require addi- +tional numerical integration. +3.4 +PRACTICAL ALGORITHM +We apply stochastic gradient descent to solve (9) using the Adam optimizer (Kingma & Ba, 2014). +Our algorithm is summarized in Algorithm 1. For sampling time steps t1, . . . , tL in [0, T], we use +stratified sampling where tl is uniformly sampled from [(l−1)T/L, lT/L]; such a sampling strategy +results in more stable training in our experiments. We retain the optimizer state of Adam from +iteration k to iteration k + 1. +We implemented our method using JAX (Bradbury et al., 2018) and FLAX (Heek et al., 2020). See +Appendix B for the implementation details. +4 +EXPERIMENTS +We show the efficiency and accuracy of our method on several PDEs of the form (2). We start with +three Wasserstein gradient flow experiments (Section 4.1, Section 4.2, Section 4.3) and compare +against JKO methods by Mokrov et al. (2021) and Fan et al. (2021). We will not compare against +Alvarez-Melis et al. (2021) since it is the same as JKO-ICNN except with a log det approximation; +we will not use such approximation to ensure accurate results. Next, we consider the time-dependent +5 + +Under peer review. +Algorithm 1 Self-consistent velocity matching +Input: ft(·, ·), µ∗ +0, T, Ntrain, Ninner, B, L. +Initialize network weights θ. +for k = 1, . . . , Ntrain do +θ′ ← θ. +for j = 1, . . . , Ninner do +Sample x1, . . . , xB ∼ µ∗ +0, t1, . . . , tL ∼ [0, T]. +yb,l ← Φθ′ +tl (xb), ∀b = 1, . . . , B, l = 1, . . . , L. +Minimize +1 +BL +� +b,l +���vθ +t (yb,l) − ft(yb,l; (Φθ′ +tl )#µ∗ +0) +��� +2 +over θ for one gradient step. +end for +end for +Output: optimized θ. +Fokker-Planck equation in Section 4.4 and compare it against the Euler-Maruyama method for sim- +ulating stochastic differential equations (Higham, 2001). Finally, in Section 4.5 we show that our +framework is capable of generating complicated dynamics in dimension 2. We will use SCVM- +TIPF and SCVM-NODE to denote our method with TIPF and NODE parameterization respectively. +We use JKO-ICNN to denote the method by Mokrov et al. (2021) and JKO-ICNN-PD to denote the +method by Fan et al. (2021) (PD for “primal-dual”). We use SDE-EM to denote the Euler-Maruyama +method. We implemented all competing methods in JAX—see more details in Appendix B. For JKO +methods, we always use 40 JKO steps. +Evaluation metrics. +For quantitative evaluation, we use the following metrics. To compare mea- +sures with density access, following Mokrov et al. (2021), we use the symmetric Kullback-Leibler +(symmetric KL) divergence, defined as SymKL(ρ1, ρ2) := KL(ρ1 ∥ ρ2) + KL(ρ2 ∥ ρ1), where +KL(ρ1 ∥ ρ2) := EX∼ρ1[log dρ1(X)/dρ2(X)]. When estimating symmetric KL divergence using sam- +ples, due to the finite sample size and the numerical error in estimating the log density, the estimated +divergence can be negative when it is close to zero—when this occurs we take absolute values. We +also consider an alternative f-divergence Df(ρ1 ∥ ρ2) := EX∼ρ2[(log ρ1(X)−log ρ2(X))2/2]. Com- +pared to KL divergence, sample estimates of Df are always positive. We similarly define the sym- +metric f-divergence SymDf(ρ1, ρ2) := Df(ρ1 ∥ ρ2) + Df(ρ2 ∥ ρ1). To compare measures with +only sample access, we consider the energy distance (Sz´ekely & Rizzo, 2013) and the Wasserstein-2 +distance (Bonneel et al., 2011). More details on the metric calculations are given in Appendix B.4. +4.1 +SAMPLING FROM MIXTURES OF GAUSSIANS +We consider computing the Wasserstein gradient flow of the KL divergence F(µ) = KL(µ ∥ µ∗) +where we have density access to the target measure µ∗. To fit into our framework, we set ft(x; µt) = +∇ log p∗(x)−∇ log pt(x) which matches (4) with bt(x) = ∇ log p∗(x) and Dt(x) = Id. Following +the experimental setup in Mokrov et al. (2021) and Fan et al. (2021), we take µ∗ to be a mixture of 10 +Gaussians with identity covariance and means sampled uniformed in [−5, 5]d. The initial measure +is µ∗ +0 = N(0, 16Id). We solve the corresponding Fokker-Planck PDE for a total time of T = 5 and +for d = 10, . . . , 60. As TIPF parameterization does not scale to high dimensions, we only consider +SCVM-NODE in this experiment. +Figure 1 shows the samples produced by SCVM-NODE align well with those from the target mea- +sure in dimension 60 at t = T. In Figure 14, we visualize µt produced by our method at irregular +time steps. +We quantitatively compare our solutions with those from Mokrov et al. (2021) and Fan et al. (2021). +In Figure 2, we plot various metrics for all methods at t = 5 (compared against the target distribu- +tion) while varying the dimension d. The running time of Mokrov et al. (2021) becomes prohibitively +long (5 hours for d = 30), so we only include its result for d ≤ 30. In Figure 3, we plot the same +metrics as functions of t for d = 30 and d = 60. We see that SCVM-NODE achieves far lower +metrics in all dimensions considered. We notice the gradient flow computed by JKO methods might +not result in monotonically decreasing KL divergence (first column in Figure 3), likely because the +6 + +Under peer review. +Figure 1: Qualitative comparison between the target mixture of 10 Gaussians in dimension 60 and +the probability flow solution of SCVM-NODE at t = 5. Samples are projected onto the first two +PCA components and kernel density estimation is used to generate the contours. +optimization at each JKO step has yet to reach the minimum even though we use 2000 gradient +updates for each step. +20 +40 +60 +Dimension d +100 +101 +Symmetric KL +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +20 +40 +60 +Dimension d +101 +Energy distance +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +20 +40 +60 +Dimension d +102 +Wassserstein-2 distance +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +Figure 2: Quantitative comparison for the mixture of Gaussians experiment across dimension d at +t = 5. +0 +2 +4 +Time t (d = 30) +100 +101 +102 +Symmetric KL +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +0 +2 +4 +Time t (d = 30) +101 +Energy distance +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +0 +2 +4 +Time t (d = 30) +102 +2 × 102 +3 × 102 +4 × 102 +Wassserstein-2 distance +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +0 +2 +4 +Time t (d = 60) +101 +102 +Symmetric KL +SCVM-NODE +JKO-ICNN-PD +0 +2 +4 +Time t (d = 60) +102 +2 × 101 +3 × 101 +4 × 101 +6 × 101 +Energy distance +SCVM-NODE +JKO-ICNN-PD +0 +2 +4 +Time t (d = 60) +103 +3 × 102 +4 × 102 +6 × 102 +Wassserstein-2 distance +SCVM-NODE +JKO-ICNN-PD +Figure 3: Quantitative comparison for the mixture of Gaussians experiment for varying t in dimen- +sion 30 (top row) and 60 (bottom row). +To illustrate the computational bottleneck of JKO-based methods, in Figure 4, we plot the run time +(in seconds) of each JKO step for the JKO-ICNN and JKO-ICNN-PD for dimension 20. For both +methods, the running time for each JKO step increases linearly because samples (and for JKO- +ICNN also log det terms) need to be pushed through a growing chain of ICNNs; as a result, the total +running time scales quadratically with the number of JKO steps. The memory consumption scales +linearly with the number of JKO steps as well which can become prohibitive. For d = 20, training +SCVM-NODE took only 6.78 minutes, while JKO-ICNN and JKO-ICNN-PD with 40 JKO steps +took 29.28 and 137.66 minutes respectively. JKO methods also take about 10x as long evaluation +time as SCVM-NODE in dimension 20 (and more in higher dimensions) due to density access which +requires solving an optimization problem for each JKO step. On top of the computational advantage +and the better results, our method also does not have temporal discretization: after being trained, the +flow can be accessed at any time t (Figure 14). +4.2 +ORNSTEIN-UHLENBECK PROCESS +To compare the accuracy of the obtained solution at all time t, we consider the Ornstein-Uhlenbeck +(OU) process following the same experimental setup as in Mokrov et al. (2021); Fan et al. (2021). +The OU process is the Wasserstein gradient flow of the KL divergence with respect to a Gaussian +µ∗ = N(β, Γ−1) where β ∈ Rd and Γ is a d × d positive-definite matrix. When the initial distribu- +tion is µ∗ +0 = N(0, Id), the gradient flow at time t is known to be a Gaussian distribution G(t) with +mean (Id − e−tΓ)β and covariance Γ−1(Id − e−2tΓ) + e−2tΓ. We set the total time T = 2. We +consider both SCVM-TIPF and SCVM-NODE. +In Figure 5, for each method, we compute the symmetric KL and the symmetric f-divergence be- +tween the recovered measure at time t and G(t) as functions of t in dimension d = 5 and d = 10. We +7 + +Target measure +Probability flow at t = 5.00 +20 +20 +10 +10 +0 +0 +-10 +-10 +-20 +20 +-10 +0 +10 +20 +-20 +-10 +0 +10 +20Under peer review. +0 +5 +10 +15 +20 +25 +30 +35 +40 +JKO step +0 +100 +200 +300 +400 +Running time (seconds) +JKO-ICNN-PD +JKO-ICNN +Figure 4: Running time for each JKO step in dimension 20 of a particular run for the mixture of +Gaussians experiment. +found that JKO methods result in much higher errors for small t compared to both SCVM-TIPF and +SCVM-NODE: this is expected because the dependency of G(t) on t is exponential, so convergence +to µ∗ is faster in the beginning, yet a constant step size is used for JKO methods. In Figure 6, we +compute the same metrics at final time t = T as functions of dimension d for d = 2, 3, . . . , 10 (top +row) and d = 10, 20, . . . , 60 (bottom row). We see that SCVM-TIPF gives the best results in low +dimensions; however, scaling it to d ≥ 10 is difficult as many coupling layers are needed. In high +dimensions, both JKO-ICNN methods achieve good results. We suspect this is because the network +architecture for ICNN has convex quadratic skip connections, the gradients of which are linear maps +so ICNN methods excel at learning linear maps which are sufficient for recovering the OU process. +Indeed, if we replace the convex quadratic skip connections with linear connections—this is closer +to the original ICNN (Amos et al., 2017)—, then the performance of JKO-ICNN and JKO-ICNN-PD +drops drastically and results in numbers worse than those of SCVM-NODE (Figure 15). +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Time t (d = 5) +10−4 +10−3 +10−2 +Symmetric KL +SCVM-TIPF +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Time t (d = 5) +10−5 +10−4 +10−3 +10−2 +Symmetric f-divergence +SCVM-TIPF +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Time t (d = 10) +10−5 +10−4 +10−3 +10−2 +Symmetric KL +SCVM-TIPF +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Time t (d = 10) +10−3 +10−2 +10−1 +Symmetric f-divergence +SCVM-TIPF +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +Figure 5: Symmetric KL and f-divergence for the OU process experiment as functions of t in +dimension 5 (top row) and 10 (bottom row). +2 +3 +4 +5 +6 +7 +8 +9 +10 +Dimension d +10−5 +10−4 +10−3 +10−2 +Symmetric KL +SCVM-TIPF +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +2 +3 +4 +5 +6 +7 +8 +9 +10 +Dimension d +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +Symmetric f-divergence +SCVM-TIPF +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +10 +20 +30 +40 +50 +60 +Dimension d +10−3 +10−2 +10−1 +Symmetric KL +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +10 +20 +30 +40 +50 +60 +Dimension d +10−2 +10−1 +Symmetric f-divergence +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +Figure 6: Symmetric KL and f-divergence at t = T for the OU process experiment as functions of +dimension. Top: d = 2, 3, . . . , 10. Bottom: d = 10, 20, . . . , 60. +4.3 +POROUS MEDIUM EQUATION +Following Fan et al. (2021), we consider the porous medium equation with only diffusion: ∂tpt = +∆pm +t +with m > 1. It is the Wasserstein gradient flow of F(µ) = +� +1 +m−1p(x)m dx where p is +the density of µ. The corresponding Wasserstein gradient is ∇W2F(µ)(x) = mpm−2(x). This +flow has as closed-form solution given by the Barenblatt profile V´azquez (2007) when initialized +accordingly: +p∗ +t (x)=(t + t0)−α � +C − β∥x∥2(t + t0) +−2α +d +� +1 +m−1 ++ +, +(11) +where t0 > 0 is the starting time, α = +m +d(m−1)+2, β = +(m−1)α +2dm , and C > 0 is a free constant. +We do not consider SCVM-NODE here because the integration-by-part trick (Proposition 3.1) does +not apply. Similar to Fan et al. (2021), we choose m = 2 and total time T = 0.025. The initial +8 + +Under peer review. +measure follows a Barenblatt distribution supported in [−0.25, 0.25]d (C is chosen accordingly) +with t0 = 10−3. We use Metropolis-Hastings to sample from µ0. +We show the efficiency of SCVM-TIPF compared to JKO-ICNN in dimension d = 1, 2, . . . , 6. +We exclude JKO-ICNN-PD since it produces significantly worse results on this application. We +visualize the density pt of the recovered flow from SCVM-TIPF and JKO-ICNN in Figure 7 in +dimension 1 compared to p∗ +t . Both methods approximate p∗ +t well with SCVM-TIPF more precise +at the beginning of the flow; this is consistent with the observation in Figure 5 where JKO methods +result in bigger errors for small t. +−1 +0 +1 +t = 0.000 +0 +1 +2 +3 +4 +5 +pt(x) +JKO-ICNN +SCVM-TIPF +−1 +0 +1 +t = 0.004 +0 +1 +2 +3 +4 +5 +pt(x) +−1 +0 +1 +t = 0.006 +0 +1 +2 +3 +4 +5 +pt(x) +−1 +0 +1 +t = 0.015 +0 +1 +2 +3 +4 +5 +pt(x) +−1 +0 +1 +t = 0.025 +0 +1 +2 +3 +4 +5 +pt(x) +JKO-ICNN +SCVM-TIPF +Figure 7: Visualization of the densities of p∗ +t and pt for the porous medium equation in dimension 1 +at varying time steps t for SCVM-TIPF and JKO-ICNN. +In Figure 8, we plot the f-divergence, the Wasserstein-2 distance, and the total variation (TV) dis- +tance (details on the TV distance are given in Appendix B.4) between the recovered solution pt and +p∗ +t for both methods at t = 0.004 and t = 0.025. We also plot in Figure 16, for dimensions 3 and 6, +the evolution of the same metrics across time. Note that the values of all metrics are very low im- +plying that the solution from either method is very accurate, with SCVM-TIPF more precise in TV +distance and symmetric f-divergence, especially for d > 3. Like with the experiments in previous +sections, JKO-ICNN is much slower to train: in dimension 6, training JKO-ICNN took 102 minutes +compared to 21 minutes for SCVM-TIPF. +2 +3 +4 +5 +6 +Dimension d +10−6 +10−5 +10−4 +10−3 +10−2 +TV distance +JKO-ICNN t = 0.004 +JKO-ICNN t = 0.025 +SCVM-TIPF t = 0.004 +SCVM-TIPF t = 0.025 +2 +3 +4 +5 +6 +Dimension d +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +Symmetric f-divergence +JKO-ICNN t = 0.004 +JKO-ICNN t = 0.025 +SCVM-TIPF t = 0.004 +SCVM-TIPF t = 0.025 +2 +3 +4 +5 +6 +Dimension d +10−3 +10−2 +Wassserstein-2 distance +JKO-ICNN t = 0.004 +JKO-ICNN t = 0.025 +SCVM-TIPF t = 0.004 +SCVM-TIPF t = 0.025 +Figure 8: Total variation distance, symmetric f-divergence, and Wasserstein-2 distances across di- +mensions at t = 0.004 and t = 0.025 between pt and p∗ +t for solving the porous medium equation. +4.4 +TIME-DEPENDENT ORNSTEIN-UHLENBECK +In this section, we qualitatively evaluate our method for solving a PDE that is not a Wasserstein +gradient flow. In this case, JKO-based methods cannot be applied. Consider the OU process from +Section 4.2 when the mean β and the covariance matrix Γ become time-dependent as βt and Γt. The +resulting PDE is a time-dependent Fokker-Planck equation of the form (4) with a velocity field +ft(X, µt) = Γt(βt − X) − D∇ log pt(X). +(12) +In this configuration, when the initial measure p0 is Gaussian, the solution µt can again be +shown to be Gaussian with mean and covariance following an ODE. More details are given in +Appendix C.1. Following Boffi & Vanden-Eijnden (2022), we consider, in dimension 2 and 3, +time-dependent attraction towards a harmonic mean βt = a(sin(πωt), cos(πωt)), augmented to +βt = a(sin(πωt), cos(πωt), t) in dimension 3. +We apply both SCVM-TIPF and SCVM-NODE to this problem and compare our results with those +of SDE-EM, the particle-based Euler-Maruyama discretization of the stochastic differential equation +associated with the Fokker-Planck equation. We did not compare against Boffi & Vanden-Eijnden +(2022) because their method uses only 50 particles. To compute metrics for SDE-EM, we use +kernel density estimation on the evolving particles. Similar to what has been observed for the static +OU process, SCVM-TIPF outperforms SCVM-NODE in these low dimensions. SCVM-TIPF also +obtains better results than SDE-EM. Visual simulations of the evolution of a few sampled particles +are given Figure 17 and Figure 18. +9 + +Under peer review. +2 +4 +6 +8 +10 +Time t (d = 2) +10−5 +10−4 +10−3 +10−2 +10−1 +Symmetric KL divergence +SCVM-TIPF +SCVM-NODE +SDE-EM +2 +4 +6 +8 +10 +Time t (d = 2) +10−2 +Wassserstein-2 distance +SCVM-TIPF +SCVM-NODE +SDE-EM +2 +4 +6 +8 +10 +Time t (d = 3) +10−4 +10−3 +10−2 +10−1 +100 +Symmetric KL divergence +SCVM-TIPF +SCVM-NODE +SDE-EM +2 +4 +6 +8 +10 +Time t (d = 3) +10−1 +2 × 10−2 +3 × 10−2 +4 × 10−2 +6 × 10−2 +Wassserstein-2 distance +SCVM-TIPF +SCVM-NODE +SDE-EM +Figure 9: Symmetric KL divergence and Wasserstein-2 distances across time for d = 2, 3 between +the recovered flows and the ground truth for the time-dependent OU process. +4.5 +ADDITIONAL QUALITATIVE EXPERIMENTS +To demonstrate the flexibility of our method, we apply our algorithm to model more general mass- +conserving dynamics than the ones considered in the previous sections. Animated GIFs of these +dynamics can be found at this link. +Flock of birds. +We first propose to model the dynamics of a flock of birds by augmenting the +time-dependent Fokker-Planck equation (12) with an interaction term: +ft(X, µt)=Γt(βt − X) +αt(X − E[µt])−D∇ log pt(X). +Since ft needs to access E[µt], the resulting PDE is not a Fokker-Planck equation (4) but can +be solved with our method by estimating E[µt] using Monte Carlo on samples from µt. +We +use a similar setup as in Section 4.4, except we now use an “infinity sign” attraction βt = +a(cos(2πωt), 0.5 sin(2πωt)) along with a sinusoidal αt = 2 sin(πwt). Depending on the sign of +αt, particles are periodically attracted towards or repulsed from their mean. Both SCVM-TIPF and +SCVM-NODE produce similar visual results as shown in Figure 10 and Figure 20. +t = 0.0 +t = 1.0 +t = 2.0 +t = 3.5 +t = 5.0 +t = 6.0 +Figure 10: Flow at a few sampled particles of SCVM-TIPF which simulates a flock of birds. See +Figure 19 for visualization with more time steps. +Flow splashing against obstacles. +We now model the phenomenon of a 2-dimensional flow +splashing against obstacles using a Fokker-Planck equation (4) where bt encodes the configura- +tion of three obstacles that repel the flow (See Appendix C.3 for details). We solve this PDE using +SCVM-NODE for T = 5 and visualize the recovered flow in (11). When solving the same PDE +using SDE-EM, the flow incorrectly crosses the bottom right obstacle due to a finite time step size +(Figure 22) whereas our method has no such issue and results in continuous sample paths (Fig- +ure 21). +Smooth interpolation of measures. +We formulate the problem of smoothly interpolating a list +of measures as a time-dependent Fokker-Planck equation and use it to interpolate MNIST digits 1, +2, and 3, starting from a Gaussian (Figure 12). We use a mixture of small-variance Gaussians to +represent each digit given as an image. See Appendix C.4 for more details. +10 + +Under peer review. +Figure 11: A flow splashing against three obstacles (in purple). Particles are colored based on the +initial y coordinates. +Figure 12: Flow at selected time steps that interpolate MNIST digits 1, 2, 3 starting from a Gaussian, +computed using SCVM-NODE. +5 +CONCLUSION +By extending the concept of self-consistency from Shen et al. (2022), we present an iterative op- +timization method for solving a wide class of mass-conserving PDEs without temporal or spatial +discretization. Our method achieves strong quantitative results in computing Wasserstein gradient +flows compared to recent JKO-based methods while requiring far less computation time. +Below we highlight three directions for future work. First, as discussed, the two ways to param- +eterize a probability flow, TIPF and NODE, both have their specific limitations. 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Optimal transport for applied mathematicians. Birk¨auser, NY, 55(58-63):94, +2015. +Zebang Shen, Zhenfu Wang, Satyen Kale, Alejandro Ribeiro, Aim Karbasi, and Hamed Hassani. +Self-consistency of the fokker-planck equation. arXiv preprint arXiv:2206.00860, 2022. +G´abor J Sz´ekely and Maria L Rizzo. Energy statistics: A class of statistics based on distances. +Journal of statistical planning and inference, 143(8):1249–1272, 2013. +Juan Luis V´azquez. The porous medium equation: mathematical theory. Oxford University Press +on Demand, 2007. +Michael Westdickenberg and Jon Wilkening. Variational particle schemes for the porous medium +equation and for the system of isentropic euler equations. ESAIM: Mathematical Modelling and +Numerical Analysis, 44(1):133–166, 2010. +13 + +Under peer review. +A +INTEGRATION-BY-PARTS TRICK +This is a common trick used in score-matching literature (Hyv¨arinen & Dayan, 2005). +Proof of Proposition 3.1. Fix t > 0. The form of ft in (4) is +ft(x; µt) = bt(x) − Dt(x)∇ log pt(x). +Hence +EX∼µθ′ +t +� +vθ +t (X)⊤ft(X; µθ′ +t ) +� += EX∼µθ′ +t +� +vθ +t (X)⊤bt(X) +� +− EX∼µθ′ +t +� +vθ +t (X)⊤Dt(x)∇ log pθ′ +t (x) +� +. +The second term can be written as +EX∼µθ′ +t +� +vθ +t (X)⊤Dt(x)∇ log pt(x) +� += +� +vθ +t (X)⊤Dt(x)∇ log pθ′ +t (x) dpθ′ +t (x) += +� +vθ +t (X)⊤Dt(x)∇pθ′ +t (x)/pθ′ +t (x) · pθ′ +t (x) dx += +� +vθ +t (X)⊤Dt(x)∇pθ′ +t (x) dx += − +� +∇ · +� +Dt(x)⊤vθ +t (X) +� +pθ′ +t (x) dx += −EX∼µθ′ +t +� +∇ · +� +Dt(x)⊤vθ +t (X) +�� +, +where we use integration-by-parts to get the second last equation and the assumption that vθ +t , Dt are +bounded and pθ +t (x) → 0 as ∥x∥ → ∞. +B +IMPLEMENTATION DETAILS +B.1 +NETWORK ARCHITECTURES FOR SCVM. +For TIPF, our implementation follows Dinh et al. (2016). Each coupled layer uses 3-layer fully- +connected networks with layer size 64, 128, 128 for both scale and translation prediction. We use +twice as many coupling layers as the dimension of the problem while each coupling layer updates +one coordinate; we found using fewer layers with random masking gives much worse results. +For NODE, we use a 4-layer fully-connected network for modeling the velocity field with layer size +128. +We use CELU activation (Barron, 2017) which is continuously differentiable for all layers for both +TIPF and NODE. +We also add a sinusoidal embedding for the time input t plus two fully-connected layers of size 64 +before concatenating it with the spatial input. The numerical integration for NODE is done using +the built-in odeint from JAX with a relative and absolute tolerance of 10−3; we did not find +considerable improvement when using a lower tolerance. +We always use the integration-by-parts trick for SCVM-NODE whenever possible. Since TIPF has +tractable log density, we do not use such a trick and optimize (9) directly for SCVM-TIPF which we +found to produce better results. +B.2 +HYPERPARAMETERS. +Unless mentioned otherwise, we choose the following hyperparameters for Algorithm 1. We set +Ntrain = 105, B = 1000, L = 20. We use Ninner = 1 in all experiments: while using a bigger Ninner +results in faster convergence for the outer iteration, each inner iteration takes a longer time—see a +comparison in Figure 13. We use Adam (Kingma & Ba, 2014) with a cosine decay learning rate +scheduler, with initial learning rate 10−3, the number of decay steps same as Ntrain, and α = 0.01 +(so the final learning rate is 10−5). +14 + +Under peer review. +B.3 +IMPLEMENTATION OF JKO METHODS. +We base our JAX implementation of ICNN on the codebase by the original ICNN author: +https://github.com/facebookresearch/w2ot. Compared to the original ICNN im- +plementation by Amos et al. (2017), we add an additional convex quadratic skip connections used +by Mokrov et al. (2021), which as discussed at the end of Section 4.2, might contribute to the excel- +lent performance of JKO methods for the OU process experiment. For ICNNs, we use hidden layer +sizes 64, 128, 128, 64. The quadratic rank for the convex quadratic skip connections is set to 20. +The activation layer is taken to be CELU. +To implement the method by Fan et al. (2021), we model the dual potential as a 4-layer fully- +connected network with layer size 128, with CELU activation. For the gradient flow of KL diver- +gence and generalized entropy (used in Section 4.3), we follow closely the variational formulation +and the necessary change of variables detailed in Fan et al. (2021, Corollary 3.3, Corollary 3.4). +In order to compute the log density at any JKO step, following Mokrov et al. (2021), we need to +solve a convex optimization to find the inverse of the gradient of an ICNN. We use the LBFGS +algorithm from JAXopt (Blondel et al., 2021) to solve the optimization with tolerance 10−2 (except +for Section 4.3 we use a tolerance of 10−3 to obtain finer inverses, but it takes 6x longer compared +to 10−2). +For each JKO step, we perform 1000 stochastic gradient descent using Adam optimizer with a +learning rate of 10−3, except for the mixture of Gaussians experiment, we use 2000 steps—using +fewer steps will result in worse results. We have tested with the learning rate schedules used in Fan +et al. (2021); Mokrov et al. (2021) and did not notice any improvement. +B.4 +EVALUATION METRICS +For all our experiments, calculations of the symmetric KL divergence, the symmetric f-divergence, +the energy distance, and the Wasserstein-2 distance are repeated 10 times on 1000 samples from +each distribution. Our plots show both the average and the standard deviation calculated over these +10 repetitions. +For the porous medium equation (Section 4.3), the total variation distance is used in Figure 8 and +Figure 16 to compare the estimated and ground-truth solutions. It is approximated by the L1 dis- +tance between the densities calculated over 50000 samples uniformly distributed on the compact +[−1.25xmax, 1.25xmax] with xmax = C/ +� +β(t + t0) +−2α +d +� +being the bound of the support of p∗ +t . +C +ADDITIONAL EXPERIMENTAL DETAILS +2000 +4000 +6000 +8000 +10000 +Training step +1.0 +1.5 +2.0 +Symmetric KL +Ntrain = 1 +Ntrain = 2 +Ntrain = 5 +2000 +4000 +6000 +8000 +10000 +Training step +1.4 +1.6 +1.8 +Energy distance +Ntrain = 1 +Ntrain = 2 +Ntrain = 5 +2000 +4000 +6000 +8000 +10000 +Training step +2.4 +2.5 +2.6 +Wassserstein-2 distance +Ntrain = 1 +Ntrain = 2 +Ntrain = 5 +Figure 13: Metrics (after taking log10) as functions of the training step for SCVM-NODE applied +to the mixture of Gaussians experiment in dimension 60. While higher values of Ninner converge in +fewer training steps, they also take Ninner times longer on each step. We found different values of +Ninner do not make a considerable difference for the same amount of training time. +C.1 +TIME-DEPENDANT FOKKER-PLANCK +We give more details on the time-dependent OU process experiments presented Section 4.4. We +consider a time-dependent Fokker-Planck equation of the form (4) with the velocity field +ft(X, µt) = Γt(X − βt) − Dt∇ log pt(X). +(13) +15 + +Under peer review. +Figure 14: Probability flow at irregular time steps of our method with the same setup as in Figure 1 +with SCVM-NODE. Note the gradient flow becomes almost stationary after t = 2.5. +10 +20 +30 +40 +50 +60 +Dimension d +10−3 +10−2 +10−1 +100 +101 +102 +Symmetric KL +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +10 +20 +30 +40 +50 +60 +Dimension d +10−1 +100 +101 +102 +103 +Symmetric f-divergence +SCVM-NODE +JKO-ICNN-PD +JKO-ICNN +Figure 15: Symmetric KL and f-divergence at t = T for the OU process experiment as functions of +dimension d = 10, 20, . . . , 60 where the ICNN architecture used in the JKO methods is modified so +that the convex quadratic skip connections are replaced with linear layers. +When the initial measure p0 is Gaussian, the solution µt can again be shown to be Gaussian with +mean mt and covariance Σt solutions of the differential equations: +� m′ +t += −Γt(mt − βt) +Σ′ +t += −ΓtΣt − ΣtΓ⊤ +t + 2Dt. +(14) +In practice, we experiment with constant Γt = diag(1, 3) and Dt = σ2Id. We also experience in +dimension 3 by considering and Γt = diag(1, 3, 1). We set a = 3, ω = 1, σ = +√ +0.25 and pick as +initial distribution p0 a Gaussian with mean b0 and covariance σ2Id. We set the total time to T = 10. +We plot in Figure 17, for dimension 2, snapshots at different time steps of particles following the +flow given by our method with TIPF parametrization. We only show SCVM-TIPF because SCVM- +NODE gives visually indistinguishable trajectories. We also plot in Figure 18 the evolution of +particles simulated by Euler-Maruyama (EM-SDE) discretization of the Fokker-Planck equation. +Corresponding animated GIFs be found at this link. +0.005 +0.010 +0.015 +0.020 +0.025 +Time t (d = 3) +10−3 +10−2 +TV distance +JKO-ICNN +SCVM-TIPF +0.005 +0.010 +0.015 +0.020 +0.025 +Time t (d = 3) +10−4 +10−3 +10−2 +10−1 +100 +Symmetric f-divergence +JKO-ICNN +SCVM-TIPF +0.005 +0.010 +0.015 +0.020 +0.025 +Time t (d = 3) +10−2 +3 × 10−3 +4 × 10−3 +6 × 10−3 +Wassserstein-2 distance +JKO-ICNN +SCVM-TIPF +0.005 +0.010 +0.015 +0.020 +0.025 +Time t (d = 6) +10−6 +10−5 +10−4 +TV distance +JKO-ICNN +SCVM-TIPF +0.005 +0.010 +0.015 +0.020 +0.025 +Time t (d = 6) +10−5 +10−3 +10−1 +101 +Symmetric f-divergence +JKO-ICNN +SCVM-TIPF +0.005 +0.010 +0.015 +0.020 +0.025 +Time t (d = 6) +3 × 10−2 +4 × 10−2 +Wassserstein-2 distance +JKO-ICNN +SCVM-TIPF +Figure 16: Metrics (TV, Symmetric f-divergence and Wasserstein-2 distance) across time for dimen- +sions 3 and 6 between the estimated µt and the ground-truth µ∗ +t when solving the Porous Medium +Equation. +16 + +t = 0.04 +t = 0.32 +t = 1.08 +t = 2.56 +t = 5.00Under peer review. +t = 0.0 +t = 0.5 +t = 1.0 +t = 1.5 +t = 2.0 +t = 2.5 +t = 3.0 +t = 3.5 +t = 4.0 +t = 4.5 +Figure 17: Evolution of particles (in blue) following the flow learned with SCVM-TIPF for the time +dependant OU process (Section 4.4). In red is the moving attraction trap. +t = 0.0 +t = 0.5 +t = 1.0 +t = 1.5 +t = 2.0 +t = 2.5 +t = 3.0 +t = 3.5 +t = 4.0 +t = 4.5 +Figure 18: Evolution of particles (in blue) obtained by SDE-EM discretization for the time depen- +dant OU process (Section 4.4). In red is the moving attraction trap. +C.2 +FLOCK OF BIRDS +For this application, we use a constant Γt = Id and a constant diffusion matrix D = σ2Id. We set +a = 3, ω = 0.5, and σ = +√ +0.25. We pick as initial distribution p0 a Gaussian with mean (0, 0) and +covariance σ2Id. We set the total time to T = 10. +We respectively show in Figure 19 and Figure 20 simulations of particles following the flow learned +with SCVM-TIPF and SCVM-NODE. Corresponding animated GIFs be found at this link. +C.3 +FLOW SPLASHING AGAINST OBSTACLES +We use the following formulation for modeling the flow. Each obstacle is modeled as a line segment. +The endpoints of the three obstacles are: +((0, 3), (3, 0.5)), ((1, 0), (1.5, 0)), ((−2, −4), (6, 0)). +We model the dynamics as a Fokker-Planck equation where ft of the form (4) is defined as +bt(x) = (qsink − x) + 20 +3 +� +i=1 +x − πOi(x) +∥x − πOi(x)∥pN (0,0.04)(∥x − πOi(x)∥), +Dt(x) = I2, +where qsink = (4, 0), and πOi(x) is the projection of x onto obstacle i represented as a line segment, +and pN (0,0.04) is the density of an 1-dimensional Gaussian with variance 0.04. +17 + +4 +3 +2 +1 +0 +-1 +-2 +3 +4 + +-2 +0 +t3 +2 +1 +0 +-1 +-2 +3 +2 +04 +3 +2 +1 +-1 +-2 +3 +-4 +-4 +U4 +3 +2 +1 +0 +-1 +-2 +3 +-4 +4 +2 +0 +23 +2 +1 +0 +-1 +-2 +3 +-4 +2 +0 +2 +43 +2 +1 +0 +-1 +-2 +3 +-4 +2 +0+ +3 +2 +1 - +0 +-1 +-2 +3 +-4 +2 +22 +1 - +0 +-1 +-2 +-3 +22 +1 - +-1 +-2 +3 +22 +1 +0 +-1 +-2 +-3 +24 +3 +2 +1 +1 +2 +3 +4 +-43 +2 +1 +0 +-1 +2 +-3 +-24 +3 +2 +1 +1 +-2 +3 +4 +-2 +23 +2 +1 - +1 +2 +3 +-4 +-2 +0 +23 +2 +1 +0 +-1 +2 +-3 +-4 +-2 +0 +23 +2 +1 +0 +-1 +2 +-3 +-2 +24 +3 +2 +1 +0 +-1 +2 +3 +-4 +-2 +42 +1 +0 +-1 +-2 +-31 +0 +-1 +2 +-3 +-4 +21 +0 +-1 +-2 +-3Under peer review. +t = 0.0 +t = 0.5 +t = 1.0 +t = 1.5 +t = 2.0 +t = 2.5 +t = 3.0 +t = 3.5 +t = 4.0 +t = 4.5 +Figure 19: Evolution of particles following the flow trained with TIPF parametrization on the flock +of birds PDE (Section 4.5). In red the moving attraction mean. +t = 0.0 +t = 0.5 +t = 1.0 +t = 1.5 +t = 2.0 +t = 2.5 +t = 3.0 +t = 3.5 +t = 4.0 +t = 4.5 +Figure 20: Evolution of particles following the flow trained with NODE parametrization on the flock +of birds PDE (Section 4.5). In red the moving attraction mean. +The initial distribution is chosen to be N(0, 0.25I2). We train SCVM-NODE for 104 with an initial +learning rate of 10−4. Training takes 5.4 minutes. The time step size for SDE-EM used to produce +Figure 22 is 0.005. +Figure 21: Trajectory of 200 random particles across time for the same setup as in Figure 11. +C.4 +SMOOTH INTERPOLATION OF MEASURES +Suppose we are to interpolate M measures ν1, . . . , νM with densities q1, . . . , qM, and we want the +flow to approximate νi at time ri. We model the dynamics as a Fokker-Planck equation where ft of +18 + +4 +3 +2 +1 +-1 +-2 +3 +4 +24 +3 +0 +1 +2 +-3 +-4 +-2 +2m +2 +0 +-1 +-2 +-3 +-4 +-2 +0 +24 +3 +2 +1 - +1 +2 +-3 +4 +-2 +0 +23 +2 +1 +0 +-1 +2 +-3 +-4 +-2 +23 +2 +1 +0 +-1 +2 +-3 +-4 +-2 +0 +24 +3 +2 +1 +1 +2 +3 +-4 +4 +-2 +0 +23 +2 +0 +-1 +2 +-3 +-4 +-2 +01 +-1 +-2 +-3 +-2 +21 +2 +-3 +4 +-2 +24 +3 +2 +0 +-1 +2 +3 +4 +4 +2 +0 +21 +0 +-1 +2 +-3 +-2 +0 +24 +3 +2 +-1 +2 +3 +4 - +-4 +-2 +0 +24 +3 +2 +-1 +-2 +-3 +-4 +t +-2 +0 +23 +2 +1 +0 +-1 +2 +3 +-4 +-2 +23 +1 +0 +-1 +2 +-3 +-4 +-2 +24 +3 +2 +1 +1 +2 +3 +-4 +-2 +0 +24 +3 +2 +0 +1 +2 +3 +-4 +-2 +2 +4-1 +-2 +-3 +-2 +2-1 +2 +-3 +-2 +2Under peer review. +Figure 22: Same setup as in Figure 11 but with SDE-EM. We see the paths of the particles are not +continuous. Moreover, the particles spill over the obstacle on the bottom right due to a finite time +step size. In comparison, SCVM-NODE does not have such a problem. +the form (4) is taken to be +bt(x) = +M +� +i=1 +φ(t − ri)(∇ log qi(x) − ∇ log pt(x)) +Dt(x) = I2, +where φ(t) is defined as the continuous bump function +φ(t) = +� 1.0 +|t| < 0.5h +(0.6h − |t|)/(0.1h) +|t| < 0.6h +0.0 +otherwise, +for bandwidth h = 1.0. +We use the first three images of 1, 2, 3 from the MNIST dataset. To construct νi from a digit image, +we use a mixture of Gaussians where we put one equally-weighted Gaussian with covariance 0.022I2 +on the pixels with values greater than 0.5 (images are first normalized to have values in [0, 1]). The +initial distribution is N((0.5, 0.5), 0.04I2). To train SCVM-NODE, we use an initial learning rate +of 10−4 with cosine decay for a total of 5 × 105 iterations. This takes 7 hours to train, although +comparable results can be obtained after the 1-hour mark. +19 + +t = 0.05 +t = 0.2 +t = 0.5 +t = 1.0 +t = 2.0 +t = 4.0 \ No newline at end of file diff --git a/mtFST4oBgHgl3EQfKjji/content/tmp_files/load_file.txt b/mtFST4oBgHgl3EQfKjji/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..88c7e979f3f254b196e929981cde94ea6f7f4c76 --- /dev/null +++ b/mtFST4oBgHgl3EQfKjji/content/tmp_files/load_file.txt @@ -0,0 +1,1420 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf,len=1419 +page_content='Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' SELF-CONSISTENT VELOCITY MATCHING OF PROBA- BILITY FLOWS Lingxiao Li MIT CSAIL lingxiao@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='edu Samuel Hurault Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Bordeaux, Bordeaux INP, CNRS, IMB samuel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='hurault@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='u-bordeaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='fr Justin Solomon MIT CSAIL jsolomon@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='edu ABSTRACT We present a discretization-free scalable framework for solving a large class of mass-conserving partial differential equations (PDEs), including the time- dependent Fokker-Planck equation and the Wasserstein gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The main observation is that the time-varying velocity field of the PDE solution needs to be self-consistent: it must satisfy a fixed-point equation involving the flow character- ized by the same velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' By parameterizing the flow as a time-dependent neural network, we propose an end-to-end iterative optimization framework called self-consistent velocity matching to solve this class of PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Compared to existing approaches, our method does not suffer from temporal or spatial discretization, covers a wide range of PDEs, and scales to high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Experimentally, our method recovers analytical solutions accurately when they are available and achieves comparable or better performance in high dimensions with less train- ing time compared to recent large-scale JKO-based methods that are designed for solving a more restrictive family of PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 1 INTRODUCTION Mass conservation is a ubiquitous phenomenon in dynamical systems arising from fluid dynamics, electromagnetism, thermodynamics, and stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Mathematically, mass conservation is formulated as the continuity equation: ∂tpt(x) = −∇ · (vtpt), ∀x, t ∈ [0, T] (1) where pt : Rd → R is a scalar quantity such that the total mass � pt(x) is conserved with respect to t, vt : Rd → Rd is a velocity field, and T > 0 is total time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We will assume, for all t ∈ [0, T], pt ≥ 0 and � pt(x) dx = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', pt is a probability density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We use µt to denote the probability measure with density pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Once a pair (pt, vt) satisfies (1), the density pt is coupled with vt in the sense that the evolution of pt in time is characterized by vt (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We consider the subclass of mass-conserving PDEs that can be written in a single equation of the form ∂tpt(x) = −∇ · (ft(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µt)pt), ∀x, t ∈ [0, T] (2) where ft(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µt) : Rd → Rd is a given function depending on µt, with initial condition µ0 = µ∗ 0 for a given initial probability measure µ∗ 0 with density p∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Different choices of ft lead to a large class of mass-conserving PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For instance, given a func- tional F : P2(Rd) → R on the space of probability distributions with finite second moments, if we take ft(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µt) := −∇W2F(µt)(x), (3) where ∇W2F(µ) : Rd → Rd is the Wasserstein gradient of F, then the solution to (2) is the Wasserstein gradient flow of F (Santambrogio, 2015, Chapter 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Thus, solving (2) efficiently 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='13737v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='LG] 31 Jan 2023 Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' allows us to optimize in the probability measure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' If we take ft(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µt) := bt(x) − Dt(x)∇ log pt(x), (4) where bt is a velocity field and Dt(x) is a positive-semidefinite matrix, then we obtain the time- dependent Fokker-Planck equation Risken & Risken (1996), which describes the time evolution of the probability flow undergoing drift bt and diffusion with coefficient Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The predominant strategy to solve (2) is to use an Eulerian representation of the density field pt on a discretized mesh or as a neural network (Raissi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' However, these approaches do not fully exploit the mass-conservation principle and are difficult to scale to high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2022) recently introduced the notion of self-consistency for the Fokker-Planck equation, a Lagrangian formulation of (2) involving the velocity field of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In this work, we extend their notion of self-consistency to a more general class of mass-conserving PDEs of the form (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' To this end, we develop an iterative optimization scheme called self-consistent velocity matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' With the probability flow parameterized as a neural network, at each iteration, we refine the velocity field vt of the current flow to match an estimate of ft evaluated using the network weights from the previous iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' This iterative formulation allows us to rewrite the velocity-matching objectives for certain PDEs to get rid of the computationally expensive quantities such as ∇ log pt in the Fokker-Planck equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Moreover, our method is agnostic to the probability flow parameterization: we have empir- ically found that the two popular ways of parameterizing the flow—as a time-varying pushforward map (Biloˇs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2021) and as a time-varying velocity field (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2018)—both have merits in different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Our method tackles mass-conserving PDEs of the form (2) in a unified manner without tempo- ral or spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Experimentally, it can recover true solutions faithfully for PDEs with analytically-known solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Only recent neural JKO-based methods (Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Alvarez-Melis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2021) are capable of solving PDEs of the form (2) in high dimen- sions, and these methods are specialized to Wasserstein gradient flows (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Our algorithm achieves comparable or better performance in our test cases compared to these JKO methods while using a lower computational budget and without discretizing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We further demonstrate the flexibility of our method on a series of qualitative experiments for modeling flocks of birds, flows splashing against obstacles, and computing smooth interpolation of measures, all without discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 2 RELATED WORKS Classical PDE solvers for mass-conserving PDEs such as the Fokker-Planck equation and the Wasserstein gradient flow either use an Eulerian representation of the density and discretize space as a grid or mesh Burger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Carrillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Peyr´e (2015) or use a Lagrangian represen- tation, which discretizes the flow as a collection of interacting particles simulated forward in time Crisan & Lyons (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Westdickenberg & Wilkening (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Due to spatial discretization, these methods struggle with high-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Hence, the rest of the section focuses solely on recent neural network-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Physics-informed neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Physics-informed neural networks (PINNs) are prominent methods that solve PDEs using deep learning (Raissi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Karniadakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The main idea is to minimize the residual of the PDE along with loss terms to enforce the boundary conditions and to match observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Our notion of self-consistency is a Lagrangian analog of the residual in PINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Our velocity matching only occurs along the flow of the current solution where interesting dynamics happen, while in PINNs the residual is evaluated on collocation points that occupy the entire domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Hence our method is particularly suitable for high-dimensional problems where the dynamics have a low-dimensional structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Neural JKO methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Recent works (Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Alvarez-Melis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2021) apply deep learning to the time-discretized JKO scheme (Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 1998) to solve Wasser- stein gradient flow (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' By pushing a reference measure through a chain of neural networks, usually parameterized as input-convex neural networks (ICNNs) (Amos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2017), these methods avoid discretizing the space and are thus capable of solving high-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021) optimize one ICNN to minimize Kullback-Leibler (KL) divergence plus a Wasserstein-2 dis- tance term at each JKO step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' This method is extended to other functionals by Alvarez-Melis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 2 Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021) use the variational formulation of f-divergence to obtain a faster primal- dual approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' An often overlooked problem of neural JKO methods is that the total training time scales quadrati- cally with the number of JKO steps: to draw samples for the current step, initial samples from the reference measure must be passed through a long chain of neural networks, along with expensive quantities like densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' However, using too few JKO steps results in large temporal discretization errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Moreover, the optimization at each step might not have fully converged before the next step begins, resulting in an unpredictable accumulation of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In contrast, our method does not suf- fer from temporal discretization and can be trained end-to-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' It outperforms these neural JKO methods with less training time in most experiments we considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Velocity matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' A few recent papers employ the idea of velocity matching to construct a flow that follows a learned velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' di Langosco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021) simulate the Wasserstein gradient flow of the KL divergence by learning a velocity field that drives a set of particles forward in time for Bayesian posterior inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The velocity field is refined on the fly based on the current positions of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Boffi & Vanden-Eijnden (2022) propose a similar method that applies to a more general class of time-dependent Fokker-Planck equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' These two methods can only approximate probability measures using finite particles and can have large temporal discretization errors similar to JKO methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Two recent methods (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Lipman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2022) use flow matching for generative modeling by learning a velocity field that generates a probability path connecting a reference distribution to the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Yet these two methods are not designed for solving PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Most relevant to our work, Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2022) propose the concept of self-consistency for the Fokker- Planck equation, that the velocity field recovering the velocity field of the flow solution to the Fokker-Planck equation must satisfy a fixed-point equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' They theoretically show that, under certain regularity conditions, the Wasserstein-2 distance between the current solution and the true solution is bounded by a term measuring the violation of the fixed-point equation (including up to second-order spatial derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Their algorithm minimizes such violation using neural ODE parameterization (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2018) and the adjoint method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Our work extends the concept of self- consistency to a wider class of PDEs in the form of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Unlike Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2022), our method does not optimize a fixed objective but instead carries out infinite-dimensional fixed-point iterations on the self-consistency condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' While the experiments of Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2022) are limited to a simple 2D example, presumably due to the computational cost of the higher-order spatial derivatives in their objective, our method excels at solving a variety of large-scale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 3 SELF-CONSISTENT VELOCITY MATCHING 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1 PROBABILITY FLOW OF THE CONTINUITY EQUATION A key property of the continuity equation (1) is that any solution (pt, vt)t∈[0,T ] (provided pt is con- tinuous with respect to t and vt is bounded) corresponds to a unique flow map {Φt(·) : Rd → Rd}t∈[0,T ] that solves the ordinary differential equations (ODEs) (Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2005, Proposi- tion 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='8) Φ0(x) = x, d dtΦt(x) = vt(Φt(x)), ∀x, t ∈ [0, T], (5) and the flow map satisfies µt = (Φt)#µ0 for all t ∈ [0, T], where (Φt)#µ0 to denote the push- forward measure of µ0 by Φt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Moreover, the converse is true: any solution (Φt, vt) of (5) with Lipschitz continuous and bounded vt is a solution of (1) with µt = (Φt)#µ0 (Ambrosio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2005, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Thus the Eulerian viewpoint of (1) is equivalent to the Lagrangian viewpoint of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We next exploit this equivalence by modeling the probability flow using the Lagrangian viewpoint so that it automatically satisfies the continuity equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 PARAMETRIZING PROBABILITY FLOWS Our algorithm will be agnostic to the exact parameterization used to represent the probability flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' As such, we need a way to parameterize the flow to access the following quantities for all t ∈ [0, T]: 3 Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Φt : Rd → Rd, the flow map at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Φt(x0) is the location of a particle at time t if it is at x0 at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We assume Φt is invertible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' vt : Rd → Rd, the velocity field of the flow at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µt ∈ P(Rd), the probability measure at time t from which we can access samples and its density pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We will assume all these quantities are sufficiently continuous and bounded to ensure the Eulerian and Lagrangian viewpoints in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1 are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' This can be achieved by using continuously differentiable activation functions in the network architectures and assuming the network weights are finite similar to the uniqueness arguments given in (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We will use the following two ways to parameterize the flow, modeling either the flow map Φt or the velocity field vt as a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Time-dependent Invertible Push Forward (TIPF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We first parameterize a probability flow by modeling Φt : Rd → Rd as an invertible network for every t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The network architecture is chosen so that Φt has an analytical inverse with a tractable Jacobian determinant, similar to (Biloˇs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We augment RealNVP (Dinh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2016) so that the network for predicting scale and translation takes t as an additional input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' To enforce the initial condition, we need Φ0 to be the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' This condition can be baked into the network architecture (Biloˇs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2021) or enforced by adding an additional loss term EX∼µ∗ 0∥Φ0(X) − X∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For brevity, we will from now on omit in the text this additional loss term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The velocity field can be recovered via vt(x) = ∂tΦt(Φ−1 t (x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' To recover the density pt of µt = (Φt)#µ0, we use the change-of-variable formula log pt(x) = log p∗ 0(Φ−1 t (x)) + log det ��JΦ−1 t (x) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Neural ODE (NODE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We also parameterize a flow by modeling vt : Rd → Rd as a neural net- work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' this is used in Neural ODE (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The network only needs to satisfy the minimum requirement of being continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The flow map and the density can be recovered via numerical in- tegration: Φt(x) = x + � t 0 vs(Φs(x)) ds and log pt(Φt(x)) = log p∗ 0(x) − � t 0 ∇ · vs(Φs(x)) ds, a direct consequence of (1) also known as the instantaneous change-of-variable formula (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' To obtain the inverse of the flow map, we integrate along −vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' With NODE, the initial condition µ0 = µ∗ 0 is obtained for free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' While the use of invertible coupling layers in TIPF allows efficient access to samples and densities, TIPF becomes less effective in higher dimensions as many couple layers are needed to retain good expressive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In contrast, NODE puts little constraints on the network architecture, but nu- merical integration can be slow and have errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Handling the initial condition is trivial for NODE while an additional loss term or special architecture is needed for TIPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' As we will show in the experiments, both strategies have merits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 FORMULATION We now describe our algorithm for solving mass-conserving PDEs (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' A PDE of this form is determined by ft(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µt) : Rd → Rd plus the initial condition µ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' If a probability flow µt with flow map Φt and velocity field vt satisfies the following self-consistency condition, vt(x) = ft(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µt), ∀x in the support of µt, (6) then the continuity equation of this flow implies the corresponding PDE (2) is solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Conversely, the velocity field of any solution of (2) will satisfy (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2022) develop this concept for the Fokker-Planck equation, and here we generalize it to a wider class of PDEs of the form (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Hence, instead of solving (2) which is a condition on the density pt that might be hard to access, we can solve (6) which is a more tractable condition on the velocity field vt that is readily accessible using TIPF or NODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Let θ be the network weights that parameterize the probability flow using TIPF or NODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The flow’s measure, velocity field, and flow map at time t are denoted as µθ t , vθ t , Φθ t respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' One option to solve (6) would be to minimize min θ � T 0 EX∼µθ t ���vθ t (X) − ft(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µθ t ) ��2� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (7) This formulation is reminiscent of PINNs (Raissi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2019) where a residual of the original PDE is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Direct optimization of (7) is challenging: while the integration over [0, T] and µθ t can be 4 Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' approximated using Monte Carlo, to apply stochastic gradient descent, we must differentiate through the µθ t and ft: this can be either expensive or intractable depending on the network parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The algorithm by Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2022) uses the adjoint method specialized to Fokker-Planck equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' extending their approach to more general PDEs requires a closed-form formula for the time evolution of the quantities within ft, which can only be obtained on a case-by-case basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Instead, we propose the following iterative optimization algorithm to solve (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Let θk denote the network weights at iteration k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We define iterates θk+1 := arg min θ F(θ, θk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (8) where F(θ, θk):= � T 0 EX∼µ θk t ����vθ t (X) − ft(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µθk t ) ��� 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (9) Effectively, in (9), we only match the velocity field vθ t to what it should be according to ft based on the network weights θk from the previous iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' This scheme is an infinite-dimensional analog to fixed-point iterations as vt is a continuous vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Since θk is fixed, minimizing (9) over θ is a lot easier than directly minimizing (7), as vθ t only needs to match a constant velocity field ft(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µθk t );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' we found a few steps of stochastic gradient descent sufficient for the optimization in (8) (see a comparison in Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We call this iterative algorithm self-consistent velocity matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' If ft depends on the density of µt only through the score ∇ log pt (corresponding to a diffusion term in the PDE), then we can apply an integration-by-parts trick (Hyv¨arinen & Dayan, 2005) to get rid of this density dependency by adding a divergence term of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Suppose ft is from the Fokker-Planck equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Then the cross term in (9) after expanding the squared norm has the following alternative expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For every t ∈ [0, T], for ft defined in (4), assume vθ t , Dt are bounded and continu- ously differentiable, and µθ′ t is a measure with a continuously differentiable density pθ′ t that vanishes in infinity and not at finite points, then we have EX∼µθ′ t � vθ t (X)⊤ft(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µθ′ t ) � = EX∼µθ′ t � vθ t (X)⊤bt(X) + ∇ · � D⊤ t (x)vθ t (X) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (10) We provide the derivation in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Minimizing (9) is then equivalent to minimizing the ex- pectation of the squared norm of vθ t plus the cross term (10), and access to pt is no longer needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' This is useful for NODE parameterization since obtaining the score would otherwise require addi- tional numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 PRACTICAL ALGORITHM We apply stochastic gradient descent to solve (9) using the Adam optimizer (Kingma & Ba, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Our algorithm is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For sampling time steps t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , tL in [0, T], we use stratified sampling where tl is uniformly sampled from [(l−1)T/L, lT/L];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' such a sampling strategy results in more stable training in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We retain the optimizer state of Adam from iteration k to iteration k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We implemented our method using JAX (Bradbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2018) and FLAX (Heek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' See Appendix B for the implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 4 EXPERIMENTS We show the efficiency and accuracy of our method on several PDEs of the form (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We start with three Wasserstein gradient flow experiments (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3) and compare against JKO methods by Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021) and Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We will not compare against Alvarez-Melis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021) since it is the same as JKO-ICNN except with a log det approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' we will not use such approximation to ensure accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Next, we consider the time-dependent 5 Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Algorithm 1 Self-consistent velocity matching Input: ft(·, ·), µ∗ 0, T, Ntrain, Ninner, B, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Initialize network weights θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , Ntrain do θ′ ← θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , Ninner do Sample x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , xB ∼ µ∗ 0, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , tL ∼ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' yb,l ← Φθ′ tl (xb), ∀b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , B, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Minimize 1 BL � b,l ���vθ t (yb,l) − ft(yb,l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (Φθ′ tl )#µ∗ 0) ��� 2 over θ for one gradient step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' end for end for Output: optimized θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Fokker-Planck equation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 and compare it against the Euler-Maruyama method for sim- ulating stochastic differential equations (Higham, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Finally, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 we show that our framework is capable of generating complicated dynamics in dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We will use SCVM- TIPF and SCVM-NODE to denote our method with TIPF and NODE parameterization respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We use JKO-ICNN to denote the method by Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021) and JKO-ICNN-PD to denote the method by Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021) (PD for “primal-dual”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We use SDE-EM to denote the Euler-Maruyama method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We implemented all competing methods in JAX—see more details in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For JKO methods, we always use 40 JKO steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For quantitative evaluation, we use the following metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' To compare mea- sures with density access, following Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021), we use the symmetric Kullback-Leibler (symmetric KL) divergence, defined as SymKL(ρ1, ρ2) := KL(ρ1 ∥ ρ2) + KL(ρ2 ∥ ρ1), where KL(ρ1 ∥ ρ2) := EX∼ρ1[log dρ1(X)/dρ2(X)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' When estimating symmetric KL divergence using sam- ples, due to the finite sample size and the numerical error in estimating the log density, the estimated divergence can be negative when it is close to zero—when this occurs we take absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We also consider an alternative f-divergence Df(ρ1 ∥ ρ2) := EX∼ρ2[(log ρ1(X)−log ρ2(X))2/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Com- pared to KL divergence, sample estimates of Df are always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We similarly define the sym- metric f-divergence SymDf(ρ1, ρ2) := Df(ρ1 ∥ ρ2) + Df(ρ2 ∥ ρ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' To compare measures with only sample access, we consider the energy distance (Sz´ekely & Rizzo, 2013) and the Wasserstein-2 distance (Bonneel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' More details on the metric calculations are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1 SAMPLING FROM MIXTURES OF GAUSSIANS We consider computing the Wasserstein gradient flow of the KL divergence F(µ) = KL(µ ∥ µ∗) where we have density access to the target measure µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' To fit into our framework, we set ft(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µt) = ∇ log p∗(x)−∇ log pt(x) which matches (4) with bt(x) = ∇ log p∗(x) and Dt(x) = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Following the experimental setup in Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021) and Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021), we take µ∗ to be a mixture of 10 Gaussians with identity covariance and means sampled uniformed in [−5, 5]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The initial measure is µ∗ 0 = N(0, 16Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We solve the corresponding Fokker-Planck PDE for a total time of T = 5 and for d = 10, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' As TIPF parameterization does not scale to high dimensions, we only consider SCVM-NODE in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Figure 1 shows the samples produced by SCVM-NODE align well with those from the target mea- sure in dimension 60 at t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In Figure 14, we visualize µt produced by our method at irregular time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We quantitatively compare our solutions with those from Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021) and Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In Figure 2, we plot various metrics for all methods at t = 5 (compared against the target distribu- tion) while varying the dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The running time of Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021) becomes prohibitively long (5 hours for d = 30), so we only include its result for d ≤ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In Figure 3, we plot the same metrics as functions of t for d = 30 and d = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We see that SCVM-NODE achieves far lower metrics in all dimensions considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We notice the gradient flow computed by JKO methods might not result in monotonically decreasing KL divergence (first column in Figure 3), likely because the 6 Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Figure 1: Qualitative comparison between the target mixture of 10 Gaussians in dimension 60 and the probability flow solution of SCVM-NODE at t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Samples are projected onto the first two PCA components and kernel density estimation is used to generate the contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' optimization at each JKO step has yet to reach the minimum even though we use 2000 gradient updates for each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 20 40 60 Dimension d 100 101 Symmetric KL SCVM-NODE JKO-ICNN-PD JKO-ICNN 20 40 60 Dimension d 101 Energy distance SCVM-NODE JKO-ICNN-PD JKO-ICNN 20 40 60 Dimension d 102 Wassserstein-2 distance SCVM-NODE JKO-ICNN-PD JKO-ICNN Figure 2: Quantitative comparison for the mixture of Gaussians experiment across dimension d at t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Time t (d = 30) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Symmetric KL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN-PD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Time t (d = 30) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Energy distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN-PD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Time t (d = 30) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 × 102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 × 102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 × 102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Wassserstein-2 distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN-PD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Time t (d = 60) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Symmetric KL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN-PD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Time t (d = 60) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 × 101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 × 101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 × 101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6 × 101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Energy distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN-PD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Time t (d = 60) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 × 102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 × 102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6 × 102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Wassserstein-2 distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN-PD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Figure 3: Quantitative comparison for the mixture of Gaussians experiment for varying t in dimen- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='sion 30 (top row) and 60 (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' To illustrate the computational bottleneck of JKO-based methods, in Figure 4, we plot the run time (in seconds) of each JKO step for the JKO-ICNN and JKO-ICNN-PD for dimension 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For both methods, the running time for each JKO step increases linearly because samples (and for JKO- ICNN also log det terms) need to be pushed through a growing chain of ICNNs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' as a result, the total running time scales quadratically with the number of JKO steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The memory consumption scales linearly with the number of JKO steps as well which can become prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For d = 20, training SCVM-NODE took only 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='78 minutes, while JKO-ICNN and JKO-ICNN-PD with 40 JKO steps took 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='28 and 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='66 minutes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' JKO methods also take about 10x as long evaluation time as SCVM-NODE in dimension 20 (and more in higher dimensions) due to density access which requires solving an optimization problem for each JKO step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' On top of the computational advantage and the better results, our method also does not have temporal discretization: after being trained, the flow can be accessed at any time t (Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ORNSTEIN-UHLENBECK PROCESS To compare the accuracy of the obtained solution at all time t, we consider the Ornstein-Uhlenbeck (OU) process following the same experimental setup as in Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The OU process is the Wasserstein gradient flow of the KL divergence with respect to a Gaussian µ∗ = N(β, Γ−1) where β ∈ Rd and Γ is a d × d positive-definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' When the initial distribu- tion is µ∗ 0 = N(0, Id), the gradient flow at time t is known to be a Gaussian distribution G(t) with mean (Id − e−tΓ)β and covariance Γ−1(Id − e−2tΓ) + e−2tΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We set the total time T = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We consider both SCVM-TIPF and SCVM-NODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In Figure 5, for each method, we compute the symmetric KL and the symmetric f-divergence be- tween the recovered measure at time t and G(t) as functions of t in dimension d = 5 and d = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We 7 Target measure Probability flow at t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='00 20 20 10 10 0 0 10 10 20 20 10 0 10 20 20 10 0 10 20Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 0 5 10 15 20 25 30 35 40 JKO step 0 100 200 300 400 Running time (seconds) JKO-ICNN-PD JKO-ICNN Figure 4: Running time for each JKO step in dimension 20 of a particular run for the mixture of Gaussians experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' found that JKO methods result in much higher errors for small t compared to both SCVM-TIPF and SCVM-NODE: this is expected because the dependency of G(t) on t is exponential, so convergence to µ∗ is faster in the beginning, yet a constant step size is used for JKO methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In Figure 6, we compute the same metrics at final time t = T as functions of dimension d for d = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , 10 (top row) and d = 10, 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , 60 (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We see that SCVM-TIPF gives the best results in low dimensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' however, scaling it to d ≥ 10 is difficult as many coupling layers are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In high dimensions, both JKO-ICNN methods achieve good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We suspect this is because the network architecture for ICNN has convex quadratic skip connections, the gradients of which are linear maps so ICNN methods excel at learning linear maps which are sufficient for recovering the OU process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Indeed, if we replace the convex quadratic skip connections with linear connections—this is closer to the original ICNN (Amos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2017)—, then the performance of JKO-ICNN and JKO-ICNN-PD drops drastically and results in numbers worse than those of SCVM-NODE (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='00 Time t (d = 5) 10−4 10−3 10−2 Symmetric KL SCVM-TIPF SCVM-NODE JKO-ICNN-PD JKO-ICNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='00 Time t (d = 5) 10−5 10−4 10−3 10−2 Symmetric f-divergence SCVM-TIPF SCVM-NODE JKO-ICNN-PD JKO-ICNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='00 Time t (d = 10) 10−5 10−4 10−3 10−2 Symmetric KL SCVM-TIPF SCVM-NODE JKO-ICNN-PD JKO-ICNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='00 Time t (d = 10) 10−3 10−2 10−1 Symmetric f-divergence SCVM-TIPF SCVM-NODE JKO-ICNN-PD JKO-ICNN Figure 5: Symmetric KL and f-divergence for the OU process experiment as functions of t in dimension 5 (top row) and 10 (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Dimension d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Symmetric KL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-TIPF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN-PD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Dimension d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Symmetric f-divergence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-TIPF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN-PD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Dimension d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Symmetric KL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN-PD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Dimension d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Symmetric f-divergence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN-PD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='JKO-ICNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Figure 6: Symmetric KL and f-divergence at t = T for the OU process experiment as functions of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Top: d = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Bottom: d = 10, 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 POROUS MEDIUM EQUATION Following Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021), we consider the porous medium equation with only diffusion: ∂tpt = ∆pm t with m > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' It is the Wasserstein gradient flow of F(µ) = � 1 m−1p(x)m dx where p is the density of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The corresponding Wasserstein gradient is ∇W2F(µ)(x) = mpm−2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' This flow has as closed-form solution given by the Barenblatt profile V´azquez (2007) when initialized accordingly: p∗ t (x)=(t + t0)−α � C − β∥x∥2(t + t0) −2α d � 1 m−1 + , (11) where t0 > 0 is the starting time, α = m d(m−1)+2, β = (m−1)α 2dm , and C > 0 is a free constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We do not consider SCVM-NODE here because the integration-by-part trick (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1) does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Similar to Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021), we choose m = 2 and total time T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The initial 8 Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' measure follows a Barenblatt distribution supported in [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25]d (C is chosen accordingly) with t0 = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We use Metropolis-Hastings to sample from µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We show the efficiency of SCVM-TIPF compared to JKO-ICNN in dimension d = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We exclude JKO-ICNN-PD since it produces significantly worse results on this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We visualize the density pt of the recovered flow from SCVM-TIPF and JKO-ICNN in Figure 7 in dimension 1 compared to p∗ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Both methods approximate p∗ t well with SCVM-TIPF more precise at the beginning of the flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' this is consistent with the observation in Figure 5 where JKO methods result in bigger errors for small t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' −1 0 1 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='000 0 1 2 3 4 5 pt(x) JKO-ICNN SCVM-TIPF −1 0 1 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='004 0 1 2 3 4 5 pt(x) −1 0 1 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='006 0 1 2 3 4 5 pt(x) −1 0 1 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='015 0 1 2 3 4 5 pt(x) −1 0 1 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 0 1 2 3 4 5 pt(x) JKO-ICNN SCVM-TIPF Figure 7: Visualization of the densities of p∗ t and pt for the porous medium equation in dimension 1 at varying time steps t for SCVM-TIPF and JKO-ICNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In Figure 8, we plot the f-divergence, the Wasserstein-2 distance, and the total variation (TV) dis- tance (details on the TV distance are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4) between the recovered solution pt and p∗ t for both methods at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='004 and t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We also plot in Figure 16, for dimensions 3 and 6, the evolution of the same metrics across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Note that the values of all metrics are very low im- plying that the solution from either method is very accurate, with SCVM-TIPF more precise in TV distance and symmetric f-divergence, especially for d > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Like with the experiments in previous sections, JKO-ICNN is much slower to train: in dimension 6, training JKO-ICNN took 102 minutes compared to 21 minutes for SCVM-TIPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 2 3 4 5 6 Dimension d 10−6 10−5 10−4 10−3 10−2 TV distance JKO-ICNN t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='004 JKO-ICNN t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 SCVM-TIPF t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='004 SCVM-TIPF t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 2 3 4 5 6 Dimension d 10−6 10−5 10−4 10−3 10−2 10−1 100 Symmetric f-divergence JKO-ICNN t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='004 JKO-ICNN t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 SCVM-TIPF t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='004 SCVM-TIPF t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 2 3 4 5 6 Dimension d 10−3 10−2 Wassserstein-2 distance JKO-ICNN t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='004 JKO-ICNN t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 SCVM-TIPF t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='004 SCVM-TIPF t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 Figure 8: Total variation distance, symmetric f-divergence, and Wasserstein-2 distances across di- mensions at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='004 and t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 between pt and p∗ t for solving the porous medium equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 TIME-DEPENDENT ORNSTEIN-UHLENBECK In this section, we qualitatively evaluate our method for solving a PDE that is not a Wasserstein gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In this case, JKO-based methods cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Consider the OU process from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 when the mean β and the covariance matrix Γ become time-dependent as βt and Γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The resulting PDE is a time-dependent Fokker-Planck equation of the form (4) with a velocity field ft(X, µt) = Γt(βt − X) − D∇ log pt(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (12) In this configuration, when the initial measure p0 is Gaussian, the solution µt can again be shown to be Gaussian with mean and covariance following an ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' More details are given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Following Boffi & Vanden-Eijnden (2022), we consider, in dimension 2 and 3, time-dependent attraction towards a harmonic mean βt = a(sin(πωt), cos(πωt)), augmented to βt = a(sin(πωt), cos(πωt), t) in dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We apply both SCVM-TIPF and SCVM-NODE to this problem and compare our results with those of SDE-EM, the particle-based Euler-Maruyama discretization of the stochastic differential equation associated with the Fokker-Planck equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We did not compare against Boffi & Vanden-Eijnden (2022) because their method uses only 50 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' To compute metrics for SDE-EM, we use kernel density estimation on the evolving particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Similar to what has been observed for the static OU process, SCVM-TIPF outperforms SCVM-NODE in these low dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' SCVM-TIPF also obtains better results than SDE-EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Visual simulations of the evolution of a few sampled particles are given Figure 17 and Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 9 Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Time t (d = 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Symmetric KL divergence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-TIPF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SDE-EM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Time t (d = 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Wassserstein-2 distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-TIPF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SDE-EM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Time t (d = 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Symmetric KL divergence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-TIPF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SDE-EM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Time t (d = 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 × 10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 × 10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 × 10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6 × 10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Wassserstein-2 distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-TIPF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SCVM-NODE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='SDE-EM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='Figure 9: Symmetric KL divergence and Wasserstein-2 distances across time for d = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 3 between the recovered flows and the ground truth for the time-dependent OU process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 ADDITIONAL QUALITATIVE EXPERIMENTS To demonstrate the flexibility of our method, we apply our algorithm to model more general mass- conserving dynamics than the ones considered in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Animated GIFs of these dynamics can be found at this link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Flock of birds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We first propose to model the dynamics of a flock of birds by augmenting the time-dependent Fokker-Planck equation (12) with an interaction term: ft(X, µt)=Γt(βt − X) +αt(X − E[µt])−D∇ log pt(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Since ft needs to access E[µt], the resulting PDE is not a Fokker-Planck equation (4) but can be solved with our method by estimating E[µt] using Monte Carlo on samples from µt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We use a similar setup as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4, except we now use an “infinity sign” attraction βt = a(cos(2πωt), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 sin(2πωt)) along with a sinusoidal αt = 2 sin(πwt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Depending on the sign of αt, particles are periodically attracted towards or repulsed from their mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Both SCVM-TIPF and SCVM-NODE produce similar visual results as shown in Figure 10 and Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 Figure 10: Flow at a few sampled particles of SCVM-TIPF which simulates a flock of birds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' See Figure 19 for visualization with more time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Flow splashing against obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We now model the phenomenon of a 2-dimensional flow splashing against obstacles using a Fokker-Planck equation (4) where bt encodes the configura- tion of three obstacles that repel the flow (See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We solve this PDE using SCVM-NODE for T = 5 and visualize the recovered flow in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' When solving the same PDE using SDE-EM, the flow incorrectly crosses the bottom right obstacle due to a finite time step size (Figure 22) whereas our method has no such issue and results in continuous sample paths (Fig- ure 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Smooth interpolation of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We formulate the problem of smoothly interpolating a list of measures as a time-dependent Fokker-Planck equation and use it to interpolate MNIST digits 1, 2, and 3, starting from a Gaussian (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We use a mixture of small-variance Gaussians to represent each digit given as an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' See Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 10 Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Figure 11: A flow splashing against three obstacles (in purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Particles are colored based on the initial y coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Figure 12: Flow at selected time steps that interpolate MNIST digits 1, 2, 3 starting from a Gaussian, computed using SCVM-NODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 5 CONCLUSION By extending the concept of self-consistency from Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2022), we present an iterative op- timization method for solving a wide class of mass-conserving PDEs without temporal or spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Our method achieves strong quantitative results in computing Wasserstein gradient flows compared to recent JKO-based methods while requiring far less computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Below we highlight three directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' First, as discussed, the two ways to param- eterize a probability flow, TIPF and NODE, both have their specific limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Finding a new parameterization that combines the advantages of both TIPF and NODE is an important next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We also hope to extend our approach to incorporate more complicated boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Finally, from a theoretical perspective, it would be interesting to explore the convergence properties of the proposed iterative procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' REFERENCES David Alvarez-Melis, Yair Schiff, and Youssef Mroueh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Optimizing functionals on the space of probabilities with input convex neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='00774, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Gradient flows: in metric spaces and in the space of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Springer Science & Business Media, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Brandon Amos, Lei Xu, and J Zico Kolter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Input convex neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In International Confer- ence on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 146–155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Jonathan T Barron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Continuously differentiable exponential linear units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' arXiv preprint arXiv:1704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='07483, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Marin Biloˇs, Johanna Sommer, Syama Sundar Rangapuram, Tim Januschowski, and Stephan G¨unnemann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Neural flows: Efficient alternative to neural odes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Advances in Neural Informa- tion Processing Systems, 34:21325–21337, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 11 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='05 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='37 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='38 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='40 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='41 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='44 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='72 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='10 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='40 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='71 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='91 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='21 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='45 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='63 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='81 t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='41Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Mathieu Blondel, Quentin Berthet, Marco Cuturi, Roy Frostig, Stephan Hoyer, Felipe Llinares- L´opez, Fabian Pedregosa, and Jean-Philippe Vert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Efficient and modular implicit differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} 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+page_content=' G´abor J Sz´ekely and Maria L Rizzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Energy statistics: A class of statistics based on distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Journal of statistical planning and inference, 143(8):1249–1272, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Juan Luis V´azquez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The porous medium equation: mathematical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Oxford University Press on Demand, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Michael Westdickenberg and Jon Wilkening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Variational particle schemes for the porous medium equation and for the system of isentropic euler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' ESAIM: Mathematical Modelling and Numerical Analysis, 44(1):133–166, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 13 Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' A INTEGRATION-BY-PARTS TRICK This is a common trick used in score-matching literature (Hyv¨arinen & Dayan, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Fix t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The form of ft in (4) is ft(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µt) = bt(x) − Dt(x)∇ log pt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Hence EX∼µθ′ t � vθ t (X)⊤ft(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' µθ′ t ) � = EX∼µθ′ t � vθ t (X)⊤bt(X) � − EX∼µθ′ t � vθ t (X)⊤Dt(x)∇ log pθ′ t (x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The second term can be written as EX∼µθ′ t � vθ t (X)⊤Dt(x)∇ log pt(x) � = � vθ t (X)⊤Dt(x)∇ log pθ′ t (x) dpθ′ t (x) = � vθ t (X)⊤Dt(x)∇pθ′ t (x)/pθ′ t (x) · pθ′ t (x) dx = � vθ t (X)⊤Dt(x)∇pθ′ t (x) dx = − � ∇ · � Dt(x)⊤vθ t (X) � pθ′ t (x) dx = −EX∼µθ′ t � ∇ · � Dt(x)⊤vθ t (X) �� , where we use integration-by-parts to get the second last equation and the assumption that vθ t , Dt are bounded and pθ t (x) → 0 as ∥x∥ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' B IMPLEMENTATION DETAILS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1 NETWORK ARCHITECTURES FOR SCVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For TIPF, our implementation follows Dinh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Each coupled layer uses 3-layer fully- connected networks with layer size 64, 128, 128 for both scale and translation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We use twice as many coupling layers as the dimension of the problem while each coupling layer updates one coordinate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' we found using fewer layers with random masking gives much worse results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For NODE, we use a 4-layer fully-connected network for modeling the velocity field with layer size 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We use CELU activation (Barron, 2017) which is continuously differentiable for all layers for both TIPF and NODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We also add a sinusoidal embedding for the time input t plus two fully-connected layers of size 64 before concatenating it with the spatial input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The numerical integration for NODE is done using the built-in odeint from JAX with a relative and absolute tolerance of 10−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' we did not find considerable improvement when using a lower tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We always use the integration-by-parts trick for SCVM-NODE whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Since TIPF has tractable log density, we do not use such a trick and optimize (9) directly for SCVM-TIPF which we found to produce better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 HYPERPARAMETERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Unless mentioned otherwise, we choose the following hyperparameters for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We set Ntrain = 105, B = 1000, L = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We use Ninner = 1 in all experiments: while using a bigger Ninner results in faster convergence for the outer iteration, each inner iteration takes a longer time—see a comparison in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We use Adam (Kingma & Ba, 2014) with a cosine decay learning rate scheduler, with initial learning rate 10−3, the number of decay steps same as Ntrain, and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='01 (so the final learning rate is 10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 14 Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 IMPLEMENTATION OF JKO METHODS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We base our JAX implementation of ICNN on the codebase by the original ICNN author: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='com/facebookresearch/w2ot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Compared to the original ICNN im- plementation by Amos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2017), we add an additional convex quadratic skip connections used by Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021), which as discussed at the end of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2, might contribute to the excel- lent performance of JKO methods for the OU process experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For ICNNs, we use hidden layer sizes 64, 128, 128, 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The quadratic rank for the convex quadratic skip connections is set to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The activation layer is taken to be CELU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' To implement the method by Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021), we model the dual potential as a 4-layer fully- connected network with layer size 128, with CELU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For the gradient flow of KL diver- gence and generalized entropy (used in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3), we follow closely the variational formulation and the necessary change of variables detailed in Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In order to compute the log density at any JKO step, following Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021), we need to solve a convex optimization to find the inverse of the gradient of an ICNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We use the LBFGS algorithm from JAXopt (Blondel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=', 2021) to solve the optimization with tolerance 10−2 (except for Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 we use a tolerance of 10−3 to obtain finer inverses, but it takes 6x longer compared to 10−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For each JKO step, we perform 1000 stochastic gradient descent using Adam optimizer with a learning rate of 10−3, except for the mixture of Gaussians experiment, we use 2000 steps—using fewer steps will result in worse results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We have tested with the learning rate schedules used in Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Mokrov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (2021) and did not notice any improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 EVALUATION METRICS For all our experiments, calculations of the symmetric KL divergence, the symmetric f-divergence, the energy distance, and the Wasserstein-2 distance are repeated 10 times on 1000 samples from each distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Our plots show both the average and the standard deviation calculated over these 10 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' For the porous medium equation (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3), the total variation distance is used in Figure 8 and Figure 16 to compare the estimated and ground-truth solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' It is approximated by the L1 dis- tance between the densities calculated over 50000 samples uniformly distributed on the compact [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25xmax, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25xmax] with xmax = C/ � β(t + t0) −2α d � being the bound of the support of p∗ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' C ADDITIONAL EXPERIMENTAL DETAILS 2000 4000 6000 8000 10000 Training step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 Symmetric KL Ntrain = 1 Ntrain = 2 Ntrain = 5 2000 4000 6000 8000 10000 Training step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='8 Energy distance Ntrain = 1 Ntrain = 2 Ntrain = 5 2000 4000 6000 8000 10000 Training step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6 Wassserstein-2 distance Ntrain = 1 Ntrain = 2 Ntrain = 5 Figure 13: Metrics (after taking log10) as functions of the training step for SCVM-NODE applied to the mixture of Gaussians experiment in dimension 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' While higher values of Ninner converge in fewer training steps, they also take Ninner times longer on each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We found different values of Ninner do not make a considerable difference for the same amount of training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1 TIME-DEPENDANT FOKKER-PLANCK We give more details on the time-dependent OU process experiments presented Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We consider a time-dependent Fokker-Planck equation of the form (4) with the velocity field ft(X, µt) = Γt(X − βt) − Dt∇ log pt(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (13) 15 Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Figure 14: Probability flow at irregular time steps of our method with the same setup as in Figure 1 with SCVM-NODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Note the gradient flow becomes almost stationary after t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 10 20 30 40 50 60 Dimension d 10−3 10−2 10−1 100 101 102 Symmetric KL SCVM-NODE JKO-ICNN-PD JKO-ICNN 10 20 30 40 50 60 Dimension d 10−1 100 101 102 103 Symmetric f-divergence SCVM-NODE JKO-ICNN-PD JKO-ICNN Figure 15: Symmetric KL and f-divergence at t = T for the OU process experiment as functions of dimension d = 10, 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , 60 where the ICNN architecture used in the JKO methods is modified so that the convex quadratic skip connections are replaced with linear layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' When the initial measure p0 is Gaussian, the solution µt can again be shown to be Gaussian with mean mt and covariance Σt solutions of the differential equations: � m′ t = −Γt(mt − βt) Σ′ t = −ΓtΣt − ΣtΓ⊤ t + 2Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' (14) In practice, we experiment with constant Γt = diag(1, 3) and Dt = σ2Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We also experience in dimension 3 by considering and Γt = diag(1, 3, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We set a = 3, ω = 1, σ = √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25 and pick as initial distribution p0 a Gaussian with mean b0 and covariance σ2Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We set the total time to T = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We plot in Figure 17, for dimension 2, snapshots at different time steps of particles following the flow given by our method with TIPF parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We only show SCVM-TIPF because SCVM- NODE gives visually indistinguishable trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We also plot in Figure 18 the evolution of particles simulated by Euler-Maruyama (EM-SDE) discretization of the Fokker-Planck equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Corresponding animated GIFs be found at this link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 Time t (d = 3) 10−3 10−2 TV distance JKO-ICNN SCVM-TIPF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 Time t (d = 3) 10−4 10−3 10−2 10−1 100 Symmetric f-divergence JKO-ICNN SCVM-TIPF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 Time t (d = 3) 10−2 3 × 10−3 4 × 10−3 6 × 10−3 Wassserstein-2 distance JKO-ICNN SCVM-TIPF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 Time t (d = 6) 10−6 10−5 10−4 TV distance JKO-ICNN SCVM-TIPF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 Time t (d = 6) 10−5 10−3 10−1 101 Symmetric f-divergence JKO-ICNN SCVM-TIPF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='025 Time t (d = 6) 3 × 10−2 4 × 10−2 Wassserstein-2 distance JKO-ICNN SCVM-TIPF Figure 16: Metrics (TV, Symmetric f-divergence and Wasserstein-2 distance) across time for dimen- sions 3 and 6 between the estimated µt and the ground-truth µ∗ t when solving the Porous Medium Equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 16 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='04 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='32 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='08 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='56 t = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='00Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 Figure 17: Evolution of particles (in blue) following the flow learned with SCVM-TIPF for the time dependant OU process (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In red is the moving attraction trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 Figure 18: Evolution of particles (in blue) obtained by SDE-EM discretization for the time depen- dant OU process (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In red is the moving attraction trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 FLOCK OF BIRDS For this application, we use a constant Γt = Id and a constant diffusion matrix D = σ2Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We set a = 3, ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5, and σ = √ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We pick as initial distribution p0 a Gaussian with mean (0, 0) and covariance σ2Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We set the total time to T = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We respectively show in Figure 19 and Figure 20 simulations of particles following the flow learned with SCVM-TIPF and SCVM-NODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Corresponding animated GIFs be found at this link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 FLOW SPLASHING AGAINST OBSTACLES We use the following formulation for modeling the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Each obstacle is modeled as a line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The endpoints of the three obstacles are: ((0, 3), (3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5)), ((1, 0), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5, 0)), ((−2, −4), (6, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We model the dynamics as a Fokker-Planck equation where ft of the form (4) is defined as bt(x) = (qsink − x) + 20 3 � i=1 x − πOi(x) ∥x − πOi(x)∥pN (0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='04)(∥x − πOi(x)∥), Dt(x) = I2, where qsink = (4, 0), and πOi(x) is the projection of x onto obstacle i represented as a line segment, and pN (0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='04) is 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 Figure 19: Evolution of particles following the flow trained with TIPF parametrization on the flock of birds PDE (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In red the moving attraction mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 Figure 20: Evolution of particles following the flow trained with NODE parametrization on the flock of birds PDE (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In red the moving attraction mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The initial distribution is chosen to be N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='25I2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We train SCVM-NODE for 104 with an initial learning rate of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Training takes 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The time step size for SDE-EM used to produce Figure 22 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Figure 21: Trajectory of 200 random particles across time for the same setup as in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 SMOOTH INTERPOLATION OF MEASURES Suppose we are to interpolate M measures ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , νM with densities q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' , qM, and we want the flow to approximate νi at time ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We model the dynamics as a Fokker-Planck equation where ft of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='4 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2Under peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Figure 22: Same setup as in Figure 11 but with SDE-EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We see the paths of the particles are not continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' Moreover, the particles spill over the obstacle on the bottom right due to a finite time step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' In comparison, SCVM-NODE does not have such a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' the form (4) is taken to be bt(x) = M � i=1 φ(t − ri)(∇ log qi(x) − ∇ log pt(x)) Dt(x) = I2, where φ(t) is defined as the continuous bump function φ(t) = � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 |t| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5h (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6h − |t|)/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='1h) |t| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='6h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 otherwise, for bandwidth h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' We use the first three images of 1, 2, 3 from the MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' To construct νi from a digit image, we use a mixture of Gaussians where we put one equally-weighted Gaussian with covariance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='022I2 on the pixels with values greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 (images are first normalized to have values in [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' The initial distribution is N((0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='04I2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' To train SCVM-NODE, we use an initial learning rate of 10−4 with cosine decay for a total of 5 × 105 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' This takes 7 hours to train, although comparable results can be obtained after the 1-hour mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content=' 19 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='05 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='2 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='5 t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0 t = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} +page_content='0' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtFST4oBgHgl3EQfKjji/content/2301.13737v1.pdf'} diff --git 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100644 index 0000000000000000000000000000000000000000..60a25a14b45b9e130b0f2953a5cf139635e55fbd --- /dev/null +++ b/ntE4T4oBgHgl3EQfug2d/content/tmp_files/2301.05233v1.pdf.txt @@ -0,0 +1,603 @@ +Identifying chromophore fingerprints of brain tumor tissue on +hyperspectral imaging using principal component analysis +Ivan Ezhov1,*, Luca Giannoni2,3, Suprosanna Shit1, Frederic Lange6, Florian Kofler1,4, +Bjoern Menze5, Ilias Tachtsidis6, Daniel Rueckert1,7 +1 Klinikum rechts der Isar, Technical University of Munich, Munich +2 University of Florence, Florence +3 European Laboratory for Non-Linear Spectroscopy, Florence +4 Helmholtz Zentrum M¨unchen, Munich +5 University of Zurich, Zurich +6 University College London, London +7 Imperial College London, London +* ivan.ezhov@tum.de +Abstract +Hyperspectral imaging (HSI) is an optical technique that processes the electromagnetic +spectrum at a multitude of monochromatic, adjacent frequency bands. The +wide-bandwidth spectral signature of a target object’s reflectance allows fingerprinting +its physical, biochemical, and physiological properties. HSI has been applied for various +applications, such as remote sensing and biological tissue analysis. Recently, HSI was +also used to differentiate between healthy and pathological tissue under operative +conditions in a surgery room on patients diagnosed with brain tumors. In this article, +we perform a statistical analysis of the brain tumor patients’ HSI scans from the +HELICoiD dataset with the aim of identifying the correlation between reflectance +spectra and absorption spectra of tissue chromophores. By using the principal +component analysis (PCA), we determine the most relevant spectral features for intra- +and inter-tissue class differentiation. Furthermore, we demonstrate that such spectral +features are correlated with the spectra of cytochrome, i.e., the chromophore highly +involved in (hyper) metabolic processes. Identifying such fingerprints of chromophores +in reflectance spectra is a key step for automated molecular profiling and, eventually, +expert-free biomarker discovery. +Introduction +Hyperspectral imaging is a noninvasive optical sensing technique that uses a broad +range of narrow wavelength bands to analyze a target object [11]. A collection of +reflections from all the bands can serve as a fingerprint of various physical, chemical, +and physiological properties of matter. HSI has been successfully explored for remote +sensing [9], drug screening [13], and medical applications [12]. Recently the imaging +technique has also been used to identify functional and pathological biomarkers of brain +tissue [4,7]. Applied to biological tissues, HSI facilitates identifying biomarkers such as +tissue metabolic activity or oxygenation. In turn, the biomarkers can shed light on the +functional and pathological state of the examined tissue. +1/7 +arXiv:2301.05233v1 [q-bio.QM] 12 Jan 2023 + +Differentiation between tissue biomarkers from the reflection spectra would in large +benefit by relating the spectra with absorption and scattering of the incoming light. +Absorption and scattering are the two main processes for light energy dissipation. +Mathematically, one can describe their effect on the incoming electromagnetic wave via +the Beer-Lambert law: +IR(λ) = I0(λ)e−(µa(λ)+µs(λ)) +(1) +where I0(λ) is the intensity of the incoming light, IR(λ) is the intensity of the +reflected light captured by the detector camera, µa and µs are the absorption and +scattering coefficients, and λ is the wavelength. The scattering originates from different +structural inhomogeneities in living tissue. Analytical descriptions of the process +typically model the scattering coefficient with a low-degree polynomial dependency on +the wavelength: µs ∼ λ(−n), with n being the degree [10]. Different from the scattering, +the absorption is not a bulk effect but rather occurs at the level of light interaction with +single tissue molecules (or, more precisely, with molecules’ chromophores). The energy +dissipation due to absorption is transformed into the excitation of molecules. Since the +excitation happens when light energy matches the distance between quantum energy +states (which is a unique molecular property), chromophores’ absorption spectra possess +characteristic peaks. +Brain tissues vary in their content of chromophores. For example, blood vessels have +a relatively larger concentration of hemoglobin, whereas glioma tissue presumably has a +higher percentage of cytochrome (a protein actively involved in metabolic +processes) [1,16]. Thus the total absorption spectra (∼ exp(−(� +i ciµi +a)) should +manifest varying spectral signatures (here ci defines the concentration of a particular +chromophore). Correspondingly, the captured reflection spectrum varies across tissue +types as it is inversely proportional to the total absorption. The open question is +whether one can solve the inverse problem, i.e., retrieve from the reflectance spectrum +the composition of chromophores. +Several works exist attempting to perform unmixing of a reflection spectrum into a +composition of chromophores spectra [2,3]. However, the main bottleneck of recovering +a physiologically complete chromophore set is the ill-posedness of the inverse problem. +Despite having characteristic peaks, the absorption spectra do not form an orthogonal +basis within the HSI operation range of wavelengths. Thus, mathematically speaking, +the mapping between reflection spectra and chromophores set is not bijective, i.e., +different combinations of chromophores absorption can equally fit the reflectance. This +is one of the reasons why existing works test unmixing algorithms with a limited +number of chromophores in a composition. To bypass the ill-posedness of the problem, +one would at least need to understand the relative distribution of chromophores in each +tissue. +Inspired by previous works [5,15], in this article, we aim to identify chromophores +spectra from glioma HSI images in a model-agnostic fashion by using statistical analysis +means. Namely, we perform a PCA study to identify correlations between the principal +components and the absorption spectra of various chromophores constituting brain +tissues. +1 +Method +For our study, we used HSI images from the HELICoiD dataset [4]. The HELICoiD +dataset consists of glioma patients which underwent HSI monitoring during surgical +operations. The image dimensions are of varying spatial size across the dataset but with +a fixed spectral size of 826 bands. The images were sparsely labeled (less than 25% of +2/7 + +Synthetic RGB image +Tumor tissue +Normal tissue +Blood vessels +Figure 1. An example of a synthetic RGB image from the HELICoiD dataset. The +image was obtained by merging three bands corresponding to red, green, and blue +wavelengths from an HSI cube. +The segmentations overlayed on top of the image +represent three classes: tumor tissue, normal tissue, and blood vessels. +450 +500 +550 +600 +650 +700 +750 +800 +850 +900 +Wavelength, nm +50 +100 +150 +200 +Reflectance +HSI spectrogram +tumor +normal +blood +450 +500 +550 +600 +650 +700 +750 +800 +850 +900 +Wavelength, nm +50 +100 +150 +200 +Reflectance +0 +2 +4 +6 +8 +10 +Absorption coefficient, OD/cm/mM +HSI spectrogram +redCCO +HHb +HbO2 +Figure 2. HSI spectra for three HELICoiD classes: tumor tissue, normal tissue, and +blood vessels (left). HSI spectra for the three HELICoiD classes and absorption spectra +of typical chromophores: reduced cytochrome aa3, oxy- and deoxy-hemoglobin (right). +the image area) into four classes: tumor tissue, normal healthy tissue, blood vessels, and +background, Figure 1. We preselected twelve patients which were diagnosed with grade +IV glioblastoma as the primary tumor. From each of the preselected patients, we +extracted spectra that belong to three classes (all HELICoiD classes, except the +background). In total, we collected 30k spectra equally distributed over the three +semantic classes. Figure 2 demonstrates typical spectral profiles for each class. The +absorption spectra were taken from the BORL GitHub repository [14] +Next, we performed the PCA for all 30k spectra in a high-dimensional space (R826) +to identify axes of the highest variance. Our reasoning here is that, on one side, the +projections of the first principal component (or a few first ones) into the original basis +would inform us on how each HSI spectral band is important for capturing the data +variance. On the other side, the spectral variance between the tissue classes originates +from the different distribution of chromophores concentration. Therefore, we expect to +observe a correlation between the principal components and the absorption spectra of +chromophores. +2 +Results and discussion +We performed PCA in two different settings: +1. First, we wanted to identify the principal components for a mixed dataset +composed of spectra from different tissue classes. Such a test would allow determining +3/7 + +spectral bands that best differentiate between the classes. Figures 3 and 4 show the +results of such PCA tests. Here, ntb denotes the 1st principal component for a dataset +composed of all three classes, nt - for normal and tumor tissue samples, nb - for normal +tissue and blood vessels. We visualize only the 1st component weights since it explains +more than 98% percent of the variance. +500 +600 +700 +800 +900 +Wavelength, nm +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +Weight +ntb +nt +nb +bt +10 +20 +30 +40 +50 +oxCCO +500 +600 +700 +800 +900 +Wavelength, nm +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +ntb +nt +nb +bt +0 +25 +50 +75 +100 +125 +150 +175 +ox_cyt-b +500 +600 +700 +800 +900 +Wavelength, nm +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +ntb +nt +nb +bt +0 +5 +10 +15 +20 +25 +Absorption coefficient, OD/cm/mM +ox_cyt-c +500 +600 +700 +800 +900 +Wavelength, nm +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +Weight +ntb +nt +nb +bt +0 +20 +40 +60 +80 +100 +redCCO +500 +600 +700 +800 +900 +Wavelength, nm +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +ntb +nt +nb +bt +0 +50 +100 +150 +200 +250 +red_cyt-b +500 +600 +700 +800 +900 +Wavelength, nm +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +ntb +nt +nb +bt +0 +10 +20 +30 +40 +50 +60 +70 +Absorption coefficient, OD/cm/mM +red_cyt-c +Figure 3. +Absorption spectra of cytochromes: oxidized (upper row) and reduced +(bottom row) of three prosthetic group types (CCO, cyt-b, and cyt-c). In solid line, we +show the 1st principal component for four different datasets: ”ntb” denotes a dataset +composed of all three classes (normal tissue, tumor, blood vessels), ”nt” - is for normal +and tumor tissue samples, ”nb” - for normal tissue and blood vessels, and ”bt” - for +blood vessels and tumor. Reduced cytochrome-c-oxidase (redCCO) reveals the highest +correlation with the principal component. +As evident from the figures, the range between 500 and 600 nm brings the highest +correlation. Particularly absorption profile of the reduced cytochrome-c-oxidase +(redCCO) reveals a very close match with the principal component. This confirms our +original hypothesis, as this is the interval of wavelengths where cytochromes have +characteristic absorption peaks. Cytochromes are present in a high concentration in the +tumor microenvironment, less so in normal tissue, and only marginally in the +endothelial cells of the inner walls of blood vessels. Our statistical analysis accurately +captures this biological fact - the 1st principal component has the highest weights for +separation between tumor tissue and blood vessels (bt) and tumor against normal tissue +(nt) while having smaller weights for the blood versus normal tissue (nb) separation. +500 +600 +700 +800 +900 +Wavelength, nm +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +Weight +ntb +nt +nb +bt +0 +20 +40 +60 +80 +100 +120 +HHb +500 +600 +700 +800 +900 +Wavelength, nm +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +ntb +nt +nb +bt +0 +20 +40 +60 +80 +100 +120 +140 +HbO2 +500 +600 +700 +800 +900 +Wavelength, nm +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +ntb +nt +nb +bt +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +Absorption coefficient, OD/cm/mM +Water +Figure 4. Absorption spectra of hemoglobin, deoxy- (left) and oxy- (middle), and +water(right). +In solid line, we show the 1st principal component for four different +datasets: ”ntb” denotes a dataset composed of all three classes (normal tissue, tumor, +blood vessels), ”nt” - is for normal and tumor tissue samples, ”nb” - for normal tissue +and blood vessels, and ”bt” - for blood vessels and tumor. +4/7 + +500 +600 +700 +800 +900 +Wavelength, nm +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +Weight +normal +blood +tumor +0 +20 +40 +60 +80 +100 +Absorption coefficient, OD/cm/mM +redCCO +Figure 5. Absorption spectra of hemoglobin: deoxy- (left) and oxy- (right). In solid +line, we show the 1st principal component for four different datasets: ”ntb” denotes a +dataset composed of all three classes (normal tissue, tumor, blood vessels), ”nt” - is for +normal and tumor tissue samples, ”nb” - for normal tissue and blood vessels, and ”bt” - +for blood vessels and tumor. +2. We wanted to test whether PCA can reveal spectral signatures correlated with +molecular absorption within a single class. This test is motivated by the fact that +glioma tissue possesses high variability of cytochrome concentration since the tumor is +vastly heterogeneous. In enhancing actively proliferating tumor, the hypermetabolism +should be accompanied by an abnormal cytochrome amount [1]. In contrast, no +proliferation is expected in the necrotic core area, and thus, the concentration of +cytochrome should be minimal. Therefore the weights of the 1st principal component +are expected to be aligned with the cytochrome absorption and be more pronounced for +the tumor class than for healthy tissue and vessels. This was also confirmed by the +PCA, as seen in Figure 5 - the 1st principal component of the dataset composed of +tumor samples has the highest weight in the 500-600 nm range. We want to point out +that total absorption from any tissue class is a combination of absorption from a set of +chromophores. For example, Figure 4 illustrates that oxy- and deoxyhemoglobin also +have characteristic peaks in this interval. Hemoglobin concentration in tissues, though, +has a contrary distribution to cytochrome, being higher in blood vessels and less in +tumor and normal tissues. However, as discussed just above, the intra-class PCA rather +captures the relation between tissues in its dependency on cytochrome. This poses the +question of whether the observation is due to a much higher concentration of +cytochrome than hemoglobin in tissues or a much larger variance in cytochrome within +a tissue class. We did not find evidence for the former in literature, rather opposite is +typically observed [8]. And if the latter is true, the difference between the magnitude of +the principal components for the three tissue classes can quantify the intra-class +chromophore variance. Such knowledge helps in understanding plausible ranges of +concentration which is in turn valuable for deciphering the molecular profiling of tissue. +Finally, we would like to discuss another interesting observation. If before 650 nm we +observe the correlation between principal components and absorption spectra, then +there are no apparent signs of it after that range. Also, the original reflectance spectra +for all classes, Figure 2, behave identically in the lower-frequency range, monotonically +decaying. One explanation can be that reflectance spectra are dominated in this range +by scattering. However, the scattering has tissue-dependent behavior [10], while we +observe identical decay (especially for blood vessels and normal tissue reflectance). +Instead, our explanation of such behavior is the following. The abscissa axis on the +spectrogram represents wavelengths. But in a way, one can view this axis as also +representing time. Within the HELICoiD project, a push broom scanner was used to +acquire the HSI images. It operates by scanning line-by-line the two-dimensional field of +5/7 + +view. Such a method provides notably slower image acquisition time compared to +snapshot imaging. Assuming that the acquisition of the HSI cube starts at 450 nm, +there is a high probability that by the time the scanner reaches 650 nm, the field of +view, i.e., living brain tissue under surgical operation, is filled with a significant amount +of blood and it becomes infeasible to identify any tissue-specific chromophore. In this +case, the reflectance would be dominated by the absorption of hemoglobin +chromophores, the main constituent of blood. The absorption spectrum of +oxyhemoglobin, Figure 2, reinforces this explanation further since, compared to other +chromophores, it manifests monotonically growing behavior in this range. It is worth +noting that it is important to look at both reflectance PCA weights to explain this +observation. In the range between 450 and 600nm reflection of tumor and blood vessels +is also identical, but PCA reveals that in this range, the two tissue types differ in terms +of cytochrome absorption. In the low-frequency range above 650 nm, both reflectance +spectra and PCA show identical behavior that suggests the dominance of hemoglobin +absorption for all three classes. To end the discussion, we envision that the presence of +high sampling rate data might be able to help here as one can focus on the heart +frequency to identify more precisely the contribution from the blood components. +3 +Conclusion +In this work, we analyze HSI glioma images made available within the HELICoiD +project. We perform a PCA-based statistical analysis of this dataset to identify +chromophore absorption signatures in the HSI reflection spectra. PCA revealed the +correlation of chromophores, especially cytochrome, with the principal components. We +discuss the possibility of using such analysis to decypher relative chromophore +concentration in brain tissues. Furthermore, we argue that in the low-frequency range of +electromagnetic range used by the HELICoiD optical system, there is evidence of +hemoglobin dominance for all tissue types. This suggests a high presence of blood in the +field of view, which limits spectral fingerprinting in this wavelength range. We believe +such analysis is vital for understanding the correctness of data acquisition experiments +and current HSI limitations [6]. +4 +Acknowledgments +The authors have received funding from the European Union’s Horizon Europe research +and innovation program under grant agreement number 101071040. F.L. and I.T. are +supported by UCL, which, as a UK participant in the EU HyperProbe project, is +supported by UKRI grant number 10048387. B.M. is supported by Helmut Horten +Foundation. +References +1. H. Abramczyk, J. M. Surmacki, B. Brozek-Pluska, and M. Kopec. Revision of +commonly accepted warburg mechanism of cancer development: Redox-sensitive +mitochondrial cytochromes in breast and brain cancers by raman imaging. +Cancers, 13(11):2599, 2021. +2. N. Dobigeon, Y. Altmann, N. Brun, and S. Moussaoui. Linear and nonlinear +unmixing in hyperspectral imaging. In Data Handling in Science and Technology, +volume 30, pages 185–224. Elsevier, 2016. +6/7 + +3. N. Dobigeon, J.-Y. Tourneret, C. Richard, J. C. M. Bermudez, S. McLaughlin, +and A. O. Hero. Nonlinear unmixing of hyperspectral images: Models and +algorithms. IEEE Signal processing magazine, 31(1):82–94, 2013. +4. H. Fabelo, S. Ortega, A. Szolna, D. Bulters, J. F. Pi˜neiro, S. Kabwama, +A. JO’Shanahan, H. Bulstrode, S. Bisshopp, B. R. Kiran, et al. In-vivo +hyperspectral human brain image database for brain cancer detection. IEEE +Access, 7:39098–39116, 2019. +5. A. O. Gerstner, W. Laffers, F. Bootz, D. L. Farkas, R. Martin, J. Bendix, and +B. Thies. Hyperspectral imaging of mucosal surfaces in patients. Journal of +biophotonics, 5(3):255–262, 2012. +6. L. Giannoni and et al. Hyperprobe consortium: innovate tumour neurosurgery +with innovative photonic solutions. ECBO, EB102-52, 2023. +7. L. Giannoni, F. Lange, and I. Tachtsidis. Hyperspectral imaging solutions for +brain tissue metabolic and hemodynamic monitoring: past, current and future +developments. Journal of Optics, 20(4):044009, 2018. +8. L. Giannoni, F. Lange, and I. Tachtsidis. Investigation of the quantification of +hemoglobin and cytochrome-c-oxidase in the exposed cortex with near-infrared +hyperspectral imaging: a simulation study. Journal of biomedical optics, +25(4):046001, 2020. +9. M. Govender, K. Chetty, and H. Bulcock. A review of hyperspectral remote +sensing and its application in vegetation and water resource studies. Water Sa, +33(2):145–151, 2007. +10. S. L. Jacques. Optical properties of biological tissues: a review. Physics in +Medicine & Biology, 58(11):R37, 2013. +11. M. J. Khan, H. S. Khan, A. Yousaf, K. Khurshid, and A. Abbas. Modern trends +in hyperspectral image analysis: A review. Ieee Access, 6:14118–14129, 2018. +12. G. Lu and B. Fei. Medical hyperspectral imaging: a review. Journal of +biomedical optics, 19(1):010901, 2014. +13. T. Puchert, D. Lochmann, J. Menezes, and G. Reich. Near-infrared chemical +imaging (nir-ci) for counterfeit drug identification—a four-stage concept with a +novel approach of data processing (linear image signature). Journal of +pharmaceutical and biomedical analysis, 51(1):138–145, 2010. +14. I. Tachtsidis and P. Pinti. UCL NIR spectra. +https://github.com/multimodalspectroscopy/UCL-NIR-Spectra#readme. +15. K. Yokoyama, M. Watanabe, and E. Okada. Estimation of the optical path +length factor for functional imaging of an exposed cortex by principal component +analysis. In European Conference on Biomedical Optics, page 5138 168. Optica +Publishing Group, 2003. +16. W.-X. Zong, J. D. Rabinowitz, and E. White. Mitochondria and cancer. +Molecular cell, 61(5):667–676, 2016. +7/7 + diff --git a/ntE4T4oBgHgl3EQfug2d/content/tmp_files/load_file.txt b/ntE4T4oBgHgl3EQfug2d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0d31c938d72d09ca7deb419804c24d716272a3ad --- /dev/null +++ b/ntE4T4oBgHgl3EQfug2d/content/tmp_files/load_file.txt @@ -0,0 +1,386 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf,len=385 +page_content='Identifying chromophore fingerprints of brain tumor tissue on hyperspectral imaging using principal component analysis Ivan Ezhov1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Luca Giannoni2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Suprosanna Shit1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Frederic Lange6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Florian Kofler1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Bjoern Menze5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Ilias Tachtsidis6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Daniel Rueckert1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='7 1 Klinikum rechts der Isar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Technical University of Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Munich 2 University of Florence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Florence 3 European Laboratory for Non-Linear Spectroscopy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Florence 4 Helmholtz Zentrum M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Munich 5 University of Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Zurich 6 University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' London 7 Imperial College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' London ivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='ezhov@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='de Abstract Hyperspectral imaging (HSI) is an optical technique that processes the electromagnetic spectrum at a multitude of monochromatic, adjacent frequency bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' The wide-bandwidth spectral signature of a target object’s reflectance allows fingerprinting its physical, biochemical, and physiological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' HSI has been applied for various applications, such as remote sensing and biological tissue analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Recently, HSI was also used to differentiate between healthy and pathological tissue under operative conditions in a surgery room on patients diagnosed with brain tumors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' In this article, we perform a statistical analysis of the brain tumor patients’ HSI scans from the HELICoiD dataset with the aim of identifying the correlation between reflectance spectra and absorption spectra of tissue chromophores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' By using the principal component analysis (PCA), we determine the most relevant spectral features for intra- and inter-tissue class differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Furthermore, we demonstrate that such spectral features are correlated with the spectra of cytochrome, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=', the chromophore highly involved in (hyper) metabolic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Identifying such fingerprints of chromophores in reflectance spectra is a key step for automated molecular profiling and, eventually, expert-free biomarker discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Introduction Hyperspectral imaging is a noninvasive optical sensing technique that uses a broad range of narrow wavelength bands to analyze a target object [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' A collection of reflections from all the bands can serve as a fingerprint of various physical, chemical, and physiological properties of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' HSI has been successfully explored for remote sensing [9], drug screening [13], and medical applications [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Recently the imaging technique has also been used to identify functional and pathological biomarkers of brain tissue [4,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Applied to biological tissues, HSI facilitates identifying biomarkers such as tissue metabolic activity or oxygenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' In turn, the biomarkers can shed light on the functional and pathological state of the examined tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' 1/7 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='05233v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='QM] 12 Jan 2023 Differentiation between tissue biomarkers from the reflection spectra would in large benefit by relating the spectra with absorption and scattering of the incoming light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Absorption and scattering are the two main processes for light energy dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Mathematically, one can describe their effect on the incoming electromagnetic wave via the Beer-Lambert law: IR(λ) = I0(λ)e−(µa(λ)+µs(λ)) (1) where I0(λ) is the intensity of the incoming light, IR(λ) is the intensity of the reflected light captured by the detector camera, µa and µs are the absorption and scattering coefficients, and λ is the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' The scattering originates from different structural inhomogeneities in living tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Analytical descriptions of the process typically model the scattering coefficient with a low-degree polynomial dependency on the wavelength: µs ∼ λ(−n), with n being the degree [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Different from the scattering, the absorption is not a bulk effect but rather occurs at the level of light interaction with single tissue molecules (or, more precisely, with molecules’ chromophores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' The energy dissipation due to absorption is transformed into the excitation of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Since the excitation happens when light energy matches the distance between quantum energy states (which is a unique molecular property), chromophores’ absorption spectra possess characteristic peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Brain tissues vary in their content of chromophores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' For example, blood vessels have a relatively larger concentration of hemoglobin, whereas glioma tissue presumably has a higher percentage of cytochrome (a protein actively involved in metabolic processes) [1,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Thus the total absorption spectra (∼ exp(−(� i ciµi a)) should manifest varying spectral signatures (here ci defines the concentration of a particular chromophore).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Correspondingly, the captured reflection spectrum varies across tissue types as it is inversely proportional to the total absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' The open question is whether one can solve the inverse problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=', retrieve from the reflectance spectrum the composition of chromophores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Several works exist attempting to perform unmixing of a reflection spectrum into a composition of chromophores spectra [2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' However, the main bottleneck of recovering a physiologically complete chromophore set is the ill-posedness of the inverse problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Despite having characteristic peaks, the absorption spectra do not form an orthogonal basis within the HSI operation range of wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Thus, mathematically speaking, the mapping between reflection spectra and chromophores set is not bijective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=', different combinations of chromophores absorption can equally fit the reflectance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' This is one of the reasons why existing works test unmixing algorithms with a limited number of chromophores in a composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' To bypass the ill-posedness of the problem, one would at least need to understand the relative distribution of chromophores in each tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Inspired by previous works [5,15], in this article, we aim to identify chromophores spectra from glioma HSI images in a model-agnostic fashion by using statistical analysis means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Namely, we perform a PCA study to identify correlations between the principal components and the absorption spectra of various chromophores constituting brain tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' 1 Method For our study, we used HSI images from the HELICoiD dataset [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' The HELICoiD dataset consists of glioma patients which underwent HSI monitoring during surgical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' The image dimensions are of varying spatial size across the dataset but with a fixed spectral size of 826 bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' The images were sparsely labeled (less than 25% of 2/7 Synthetic RGB image Tumor tissue Normal tissue Blood vessels Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' An example of a synthetic RGB image from the HELICoiD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' The image was obtained by merging three bands corresponding to red, green, and blue wavelengths from an HSI cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' The segmentations overlayed on top of the image represent three classes: tumor tissue, normal tissue, and blood vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' 450 500 550 600 650 700 750 800 850 900 Wavelength, nm 50 100 150 200 Reflectance HSI spectrogram tumor normal blood 450 500 550 600 650 700 750 800 850 900 Wavelength, nm 50 100 150 200 Reflectance 0 2 4 6 8 10 Absorption coefficient, OD/cm/mM HSI spectrogram redCCO HHb HbO2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' HSI spectra for three HELICoiD classes: tumor tissue, normal tissue, and blood vessels (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' HSI spectra for the three HELICoiD classes and absorption spectra of typical chromophores: reduced cytochrome aa3, oxy- and deoxy-hemoglobin (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' the image area) into four classes: tumor tissue, normal healthy tissue, blood vessels, and background, Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' We preselected twelve patients which were diagnosed with grade IV glioblastoma as the primary tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' From each of the preselected patients, we extracted spectra that belong to three classes (all HELICoiD classes, except the background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' In total, we collected 30k spectra equally distributed over the three semantic classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Figure 2 demonstrates typical spectral profiles for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' The absorption spectra were taken from the BORL GitHub repository [14] Next, we performed the PCA for all 30k spectra in a high-dimensional space (R826) to identify axes of the highest variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Our reasoning here is that, on one side, the projections of the first principal component (or a few first ones) into the original basis would inform us on how each HSI spectral band is important for capturing the data variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' On the other side, the spectral variance between the tissue classes originates from the different distribution of chromophores concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Therefore, we expect to observe a correlation between the principal components and the absorption spectra of chromophores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' 2 Results and discussion We performed PCA in two different settings: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' First, we wanted to identify the principal components for a mixed dataset composed of spectra from different tissue classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Such a test would allow determining 3/7 spectral bands that best differentiate between the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Figures 3 and 4 show the results of such PCA tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Here, ntb denotes the 1st principal component for a dataset composed of all three classes, nt - for normal and tumor tissue samples, nb - for normal tissue and blood vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' We visualize only the 1st component weights since it explains more than 98% percent of the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' 500 600 700 800 900 Wavelength, nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='07 Weight ntb nt nb bt 10 20 30 40 50 oxCCO 500 600 700 800 900 Wavelength, nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='07 ntb nt nb bt 0 25 50 75 100 125 150 175 ox_cyt-b 500 600 700 800 900 Wavelength, nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='07 ntb nt nb bt 0 5 10 15 20 25 Absorption coefficient, OD/cm/mM ox_cyt-c 500 600 700 800 900 Wavelength, nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='07 Weight ntb nt nb bt 0 20 40 60 80 100 redCCO 500 600 700 800 900 Wavelength, nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='07 ntb nt nb bt 0 50 100 150 200 250 red_cyt-b 500 600 700 800 900 Wavelength, nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='07 ntb nt nb bt 0 10 20 30 40 50 60 70 Absorption coefficient, OD/cm/mM red_cyt-c Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Absorption spectra of cytochromes: oxidized (upper row) and reduced (bottom row) of three prosthetic group types (CCO, cyt-b, and cyt-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' In solid line, we show the 1st principal component for four different datasets: ”ntb” denotes a dataset composed of all three classes (normal tissue, tumor, blood vessels), ”nt” - is for normal and tumor tissue samples, ”nb” - for normal tissue and blood vessels, and ”bt” - for blood vessels and tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Reduced cytochrome-c-oxidase (redCCO) reveals the highest correlation with the principal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' As evident from the figures, the range between 500 and 600 nm brings the highest correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Particularly absorption profile of the reduced cytochrome-c-oxidase (redCCO) reveals a very close match with the principal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' This confirms our original hypothesis, as this is the interval of wavelengths where cytochromes have characteristic absorption peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Cytochromes are present in a high concentration in the tumor microenvironment, less so in normal tissue, and only marginally in the endothelial cells of the inner walls of blood vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Our statistical analysis accurately captures this biological fact - the 1st principal component has the highest weights for separation between tumor tissue and blood vessels (bt) and tumor against normal tissue (nt) while having smaller weights for the blood versus normal tissue (nb) separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' 500 600 700 800 900 Wavelength, nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='07 Weight ntb nt nb bt 0 20 40 60 80 100 120 HHb 500 600 700 800 900 Wavelength, nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='07 ntb nt nb bt 0 20 40 60 80 100 120 140 HbO2 500 600 700 800 900 Wavelength, nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='07 ntb nt nb bt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='06 Absorption coefficient, OD/cm/mM Water Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Absorption spectra of hemoglobin, deoxy- (left) and oxy- (middle), and water(right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' In solid line, we show the 1st principal component for four different datasets: ”ntb” denotes a dataset composed of all three classes (normal tissue, tumor, blood vessels), ”nt” - is for normal and tumor tissue samples, ”nb” - for normal tissue and blood vessels, and ”bt” - for blood vessels and tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' 4/7 500 600 700 800 900 Wavelength, nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='07 Weight normal blood tumor 0 20 40 60 80 100 Absorption coefficient, OD/cm/mM redCCO Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Absorption spectra of hemoglobin: deoxy- (left) and oxy- (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' In solid line, we show the 1st principal component for four different datasets: ”ntb” denotes a dataset composed of all three classes (normal tissue, tumor, blood vessels), ”nt” - is for normal and tumor tissue samples, ”nb” - for normal tissue and blood vessels, and ”bt” - for blood vessels and tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' We wanted to test whether PCA can reveal spectral signatures correlated with molecular absorption within a single class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' This test is motivated by the fact that glioma tissue possesses high variability of cytochrome concentration since the tumor is vastly heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' In enhancing actively proliferating tumor, the hypermetabolism should be accompanied by an abnormal cytochrome amount [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' In contrast, no proliferation is expected in the necrotic core area, and thus, the concentration of cytochrome should be minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Therefore the weights of the 1st principal component are expected to be aligned with the cytochrome absorption and be more pronounced for the tumor class than for healthy tissue and vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' This was also confirmed by the PCA, as seen in Figure 5 - the 1st principal component of the dataset composed of tumor samples has the highest weight in the 500-600 nm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' We want to point out that total absorption from any tissue class is a combination of absorption from a set of chromophores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' For example, Figure 4 illustrates that oxy- and deoxyhemoglobin also have characteristic peaks in this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Hemoglobin concentration in tissues, though, has a contrary distribution to cytochrome, being higher in blood vessels and less in tumor and normal tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' However, as discussed just above, the intra-class PCA rather captures the relation between tissues in its dependency on cytochrome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' This poses the question of whether the observation is due to a much higher concentration of cytochrome than hemoglobin in tissues or a much larger variance in cytochrome within a tissue class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' We did not find evidence for the former in literature, rather opposite is typically observed [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' And if the latter is true, the difference between the magnitude of the principal components for the three tissue classes can quantify the intra-class chromophore variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Such knowledge helps in understanding plausible ranges of concentration which is in turn valuable for deciphering the molecular profiling of tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Finally, we would like to discuss another interesting observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' If before 650 nm we observe the correlation between principal components and absorption spectra, then there are no apparent signs of it after that range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Also, the original reflectance spectra for all classes, Figure 2, behave identically in the lower-frequency range, monotonically decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' One explanation can be that reflectance spectra are dominated in this range by scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' However, the scattering has tissue-dependent behavior [10], while we observe identical decay (especially for blood vessels and normal tissue reflectance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Instead, our explanation of such behavior is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' The abscissa axis on the spectrogram represents wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' But in a way, one can view this axis as also representing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Within the HELICoiD project, a push broom scanner was used to acquire the HSI images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' It operates by scanning line-by-line the two-dimensional field of 5/7 view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Such a method provides notably slower image acquisition time compared to snapshot imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Assuming that the acquisition of the HSI cube starts at 450 nm, there is a high probability that by the time the scanner reaches 650 nm, the field of view, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=', living brain tissue under surgical operation, is filled with a significant amount of blood and it becomes infeasible to identify any tissue-specific chromophore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' In this case, the reflectance would be dominated by the absorption of hemoglobin chromophores, the main constituent of blood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' The absorption spectrum of oxyhemoglobin, Figure 2, reinforces this explanation further since, compared to other chromophores, it manifests monotonically growing behavior in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' It is worth noting that it is important to look at both reflectance PCA weights to explain this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' In the range between 450 and 600nm reflection of tumor and blood vessels is also identical, but PCA reveals that in this range, the two tissue types differ in terms of cytochrome absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' In the low-frequency range above 650 nm, both reflectance spectra and PCA show identical behavior that suggests the dominance of hemoglobin absorption for all three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' To end the discussion, we envision that the presence of high sampling rate data might be able to help here as one can focus on the heart frequency to identify more precisely the contribution from the blood components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' 3 Conclusion In this work, we analyze HSI glioma images made available within the HELICoiD project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' We perform a PCA-based statistical analysis of this dataset to identify chromophore absorption signatures in the HSI reflection spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' PCA revealed the correlation of chromophores, especially cytochrome, with the principal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' We discuss the possibility of using such analysis to decypher relative chromophore concentration in brain tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' Furthermore, we argue that in the low-frequency range of electromagnetic range used by the HELICoiD optical system, there is evidence of hemoglobin dominance for all tissue types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' This suggests a high presence of blood in the field of view, which limits spectral fingerprinting in this wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' We believe such analysis is vital for understanding the correctness of data acquisition experiments and current HSI limitations [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' 4 Acknowledgments The authors have received funding from the European Union’s Horizon Europe research and innovation program under grant agreement number 101071040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' are supported by UCL, which, as a UK participant in the EU HyperProbe project, is supported by UKRI grant number 10048387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE4T4oBgHgl3EQfug2d/content/2301.05233v1.pdf'} +page_content=' B.' metadata={'source': 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v3.0 +Galaxy quenching timescales from a forensic reconstruction of their +colour evolution +Matías Bravo1,2★, Aaron S. G. Robotham1,3, Claudia del P. Lagos1,3, +Luke J. M. Davies1, Sabine Bellstedt1 and Jessica E. Thorne1 +1International Centre for Radio Astronomy Research (ICRAR), M468, University of Western Australia, 35 Stirling Hwy, Crawley, +WA 6009, Australia. +2Department of Physics & Astronomy, McMaster University, 1280 Main Street W, Hamilton, ON, L8S 4M1, Canada +3ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D). +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +The timescales on which galaxies move out of the blue cloud to the red sequence (𝜏Q) provide +insight into the mechanisms driving quenching. Here, we build upon previous work, where +we showcased a method to reconstruct the colour evolution of observed low-redshift galaxies +from the Galaxy And Mass Assembly (GAMA) survey based on spectral energy distribution +(SED) fitting with ProSpect, together with a statistically-driven definition for the blue and red +populations. We also use the predicted colour evolution from the shark semi-analytic model, +combined with SED fits of our simulated galaxy sample, to study the accuracy of the measured +𝜏Q and gain physical insight into the colour evolution of galaxies. In this work, we measure 𝜏Q +in a consistent approach for both observations and simulations. After accounting for selection +bias, we find evidence for an increase in 𝜏Q in GAMA as a function of cosmic time (from 𝜏Q∼ 1 +Gyr to 𝜏Q∼ 2 Gyr in the lapse of ∼ 4 Gyr), but not in shark (𝜏Q≲ 1 Gyr). Our observations +and simulations disagree on the effect of stellar mass, with GAMA showing massive galaxies +transitioning faster, but is the opposite in shark. We find that environment only impacts +galaxies below ∼1010 M⊙ in GAMA, with satellites having shorter 𝜏Q than centrals by ∼ 0.4 +Gyr, with shark only in qualitative agreement. Finally, we compare to previous literature, +finding consistency with timescales in the order of couple Gyr, but with several differences +that we discuss. +Key words: galaxies: evolution – software: simulations – techniques: photometric +1 +INTRODUCTION +One of the most striking features of galaxies in the local Universe +is the optical colour bimodality (e.g., Strateva et al. 2001; Blanton +et al. 2003; Baldry et al. 2004; Driver et al. 2006), with most galaxies +being either blue or red. Compared to these populations, there are +comparatively few galaxies in the intermediate region, often referred +to as the "green valley", (e.g., Martin et al. 2007; Wyder et al. +2007; Schawinski et al. 2014). Stars are the dominant source of +the light emitted by most galaxies (at low redshift), suggesting that +this bimodality is a consequence of the presence of two dominant +stellar populations for galaxies. As the (intrinsic) colour of stars is +mainly driven by their age, the colour bimodality is a reflection of +a bimodality in the recent star formation in galaxies. +★ E-mail:bravosam@mcmaster.ca +These populations are also characterised by intrinsically differ- +ent galaxy properties. Red galaxies are preferentially of early-type +morphology (e.g., Bershady et al. 2000; Mignoli et al. 2009; Schaw- +inski et al. 2014), more massive (e.g., Baldry et al. 2004; Peng et al. +2010; Taylor et al. 2015), and found in denser environments (e.g., +Kauffmann et al. 2004; Baldry et al. 2006; Peng et al. 2010). Stud- +ies have also shown that this bimodality is seen across cosmic time, +with the fraction of galaxies in the red population increasing towards +recent times (e.g., Wolf et al. 2003; Bell et al. 2004; Williams et al. +2009). It has also been found that the first galaxies that joined the +red population are more massive than those that have joined at more +recent times, a process called downsizing (e.g., Cowie et al. 1996; +Brinchmann & Ellis 2000; Heavens et al. 2004). Combined, these +observations present a broad picture where galaxies grow as part +of the star-forming blue population, with some of them eventually +ceasing to form stars and joining the red population (commonly +© 2023 The Authors +arXiv:2301.03702v1 [astro-ph.GA] 9 Jan 2023 + +2 +Bravo et al. +referred to as quenching, e.g., Bell et al. 2004; Blanton 2006; Faber +et al. 2007). +The relative lack of galaxies located in the green valley implies +short timescales to transition in colour (quench) for the galaxies +that join the red population (e.g., Schawinski et al. 2014; Bremer +et al. 2018). Different mechanisms to quench star formation are +expected to do so on different timescales (e.g., Kaviraj et al. 2011; +Wetzel et al. 2013; Schawinski et al. 2014; Wheeler et al. 2014), +hence, studying these timescales can offer a view into the physical +processes that govern galaxy evolution. Theoretical models are a +critical tool to explore these mechanism, as we gain insight by testing +their predictions against results from observations. A well-known +example in the literature is that a quenching mechanism capable of +stopping gas accretion onto galaxies is required to produce massive +red galaxies, usually assumed to be driven by active galactic nuclei +or shock heating of the halo gas (e.g., Bower et al. 2006; Cattaneo +et al. 2006; Croton et al. 2006; Lagos et al. 2008). +A historical challenge for simulations has been their inability +to produce colour distributions well-matched to observations (e.g., +Weinmann et al. 2006; Font et al. 2008; Coil et al. 2008), though re- +cent advances have largely ameliorated this tensions (e.g., Trayford +et al. 2015; Nelson et al. 2018; Lagos et al. 2019; Bravo et al. 2020). +These advances now enable the exploration of the colour evolution +of galaxies with theoretical models, leading to the prediction of the +timescales on which galaxies transition from being blue to red (e.g., +Trayford et al. 2016; Nelson et al. 2018; Wright et al. 2019). This +colour evolution is not directly measurable from observations and +can only be inferred (e.g., Schawinski et al. 2014; Smethurst et al. +2015; Rowlands et al. 2018; Phillipps et al. 2019). This means that +results cannot be directly compared to the predictions of theoretical +models. Further complicating comparisons is the lack of a unified +definition for how to measure the colour transition timescales, or +even what galaxies should be classified as blue or red (e.g., see +classifications by Martin et al. 2007; Schawinski et al. 2014; Taylor +et al. 2015; Bremer et al. 2018; Wright et al. 2019). +In Bravo et al. (2022, hereafter Paper I), we described a novel +method to reconstruct the colour evolution of low-redshift observed +galaxies from the Galaxy And Mass Assembly (GAMA; Driver et al. +2011; Liske et al. 2015) survey, using the ProSpect spectral energy +distribution (SED) fitting tool (Robotham et al. 2020). We tested +this recovery by performing the same procedure with a comparable +sample of galaxies generated with the shark semi-analytic model +(SAM; Lagos et al. 2018, 2019), finding that we can accurately +recover the colour evolution of the last ∼ 6 Gyr for galaxies with +current masses above ∼109 M⊙. Finally, we provided a statistically- +motivated definition for the blue and red populations, their evolution +through cosmic time, and demonstrated the resulting probabilities +of galaxies belonging to either blue or red population. +In this work we now utilise these results to explore how quickly +galaxies transition from being blue to red, in a novel approach that +is consistent and directly comparable for both observations and sim- +ulations. In Section 3 we explore the distribution of probabilities of +galaxies being red, to construct statistically-motivated definitions +for when a galaxy is certainly a member of of either population, +and when is transitioning between both. We then use that classifi- +cation to explore the timescale on which galaxies transitioned from +blue to red in Section 4, exploring possible time, mass, and envi- +ronmental effects. For conciseness, we will refer to this blue-to-red +transition timescale as 𝜏Q throughout this work. In Section 5 we dis- +cuss our results, both for the physical implications of the timescales +we measure and to compare with the existing literature. Finally, we +present our conclusions in Section 6. In this work, we adopt the +Planck Collaboration et al. (2016) ΛCDM cosmology, with values +of matter, baryon, and dark energy densities of Ω𝑏 = 0.0488, and +ΩΛ = 0.6879, respectively, and a Hubble parameter of H0 = 67.51 +km s−1 Mpc−1. +2 +GALAXY CATALOGUES +In this work, we use the data set presented in Paper I, which we +briefly outline. This data set is composed of the intrinsic colour +(i.e., not attenuated by dust) and stellar mass histories for three +low-redshift galaxy samples, both derived from their star formation +and metallicity histories (SFH and 𝑍H, respectively). The first one +is comprised of ∼ 7, 000 galaxies from the GAMA survey used in +Bellstedt et al. (2020, 2021). The other two are each comprised of +∼ 30, 000 GAMA-like galaxies (i.e., 𝑟apparent < 19.8 mag) from +the shark SAM (Lagos et al. 2018, 2019). For the GAMA sample, +we reconstructed their colour evolution from the star formation and +metallicity histories inferred from the SED fitting by Bellstedt et al. +(2020)1 by combining these histories with the stellar population +synthesis model used for the fitting (Bruzual & Charlot 2003). The +two shark samples contain the same galaxies, the difference is how +we constructed the colour and stellar mass evolution: one sample +is the predicted evolution from the simulation itself (which we will +refer as shark); the other presents the inferred evolution from SED +fitting the shark galaxies with the same method as with the GAMA +sample (we will refer to this sample as sharkfit). Section 2 of Paper +I contains the detailed description of these three samples, with +Appendix A offering a deeper exploration of our SED modelling +choices for the interested reader. +Inspired by Taylor et al. (2015), in Paper I we modelled the +colour-mass distribution for each sample with a time-and-mass- +dependent Gaussian Mixture Model (GMM). Consistent with the +modelling by Baldry et al. (2004) and Taylor et al. (2015), we de- +scribed the colour-mass distribution of galaxies in Paper I with two +evolving populations: blue and red2 . These GMMs are described +by five parameters: the relative fraction of blue (or red) galax- +ies, and the means and standard deviations of each population. In +Paper I we presented a two-step parameterisation of these param- +eters, first as a function of stellar mass, and second as a function +of lookback time. For the stellar mass parameterisation, based on +the distributions of the GMM parameters as a function of stellar +mass, we chose to parameterise the relative blue/red fractions with +a logistic curve, and with first-order polynomials for the means and +standard deviations. We then parameterised the time evolution of +the stellar mass-dependent Gaussian parameters, using second- and +third-order polynomials. Section 3 of Paper I provides the complete +description of this modelling, with section 3.1, figure 2 and table 1 +offering an simple overview. +With a complete parameterisation of the evolution of the colour +1 The SFH model adopted by Bellstedt et al. (2020) is a skewed Gaussian, +a parametric model but significantly more flexible than other common para- +metric models in the literature (e.g., da Cunha et al. 2008; Noll et al. 2009; +Carnall et al. 2018; Boquien et al. 2019), but still unable to model rejuve- +nation episodes. While not unique among SED fitting models (e.g., Carnall +et al. 2018; Johnson et al. 2021), the use of ProSpect in the literature has +been unique in the assumption that gas metallicities evolves, modelling it as +a linear scaling of the mass growth of galaxies (Bellstedt et al. 2020; Thorne +et al. 2021, 2022). +2 We did test using three components, but we found no statistical evidence +for a third (green) population. See section 3.3 of Paper I for further details. +MNRAS 000, 1–16 (2023) + +Forensic quenching timescales +3 +10.0 +10.2 +10.4 +M⋆ [M⊙] +1.0 +1.5 +2.0 +u − r [mag] +tLB = 1.4 Gyr +10.0 +10.2 +10.4 +M⋆ [M⊙] +tLB = 2.7 Gyr +10.0 +10.2 +10.4 +M⋆ [M⊙] +tLB = 4.0 Gyr +Figure 1. The colour evolution of a single galaxy and its transition from the blue to the red population from the results Paper I, with the galaxy CATAID=92739 +from the GAMA survey used for this example. Each panels shows a small section of the colour-mass plane for the GAMA survey at three different lookback +times, with the coloured contours showing the probability of being red for a a galaxy in any given position in this space. The complete evolution track of the +galaxy is shown by the black and white line running from the bottom left to the top right of each panel, with the position of the galaxy in the corresponding +lookback time of each panel shown with a star marker. +populations for all three samples, we can calculate the probability +for any galaxy belonging to the blue or red population at any given +time. As in figure 12 of Paper I, in this work we choose to show +the probability of being red, 𝑃R +3 , which is calculated from the +Gaussian Mixture Model with which we model the galaxy colour +distribution (see sections 3.3 and 3.4 of Paper I for further details). +We also showed that our colour-based classification leads to +a sensible separation in specific star formation rate as a function +of stellar mass. The transition zone is not cleanly defined in this +space, as expected from the scatter between colour and specific star +formation rate. In Paper I, through the comparison of the colour +evolution of the three samples (GAMA, shark, and sharkfit), we +found that the reconstruction of the colour evolution becomes biased +by the modelling choices in the SED fitting for lookback times above +≳ 6 Gyr. For this reason, while we will measure colour evolution +from a lookback time of 10 Gyr onward and show some of our +results at higher lookback times, we mainly focus on the 𝜏Q we +measure below a lookback time of 6 Gyr. +In this work, we use 𝑃R to calculate 𝜏Q, defining the lookback +times when a galaxy leaves the blue population (𝑡LB,B) and when it +joins the red population (𝑡LB,R), which are related to 𝜏Q as: +𝜏Q = 𝑡LB,B − 𝑡LB,R, +(1) +where we define these lookback times such that 𝑡LB,B > 𝑡LB,R (i.e., +𝜏Q> 0). In Paper I we chose a time step of 100 Myr to reconstruct the +colour evolution of galaxies, meaning that the shortest measurable +𝜏Q is 100 Myr. Figure 1 shows an example of the data set we +constructed in Paper I and use in this work to measure 𝜏Q, with +both the evolutionary tracks in the colour-mass plane of individual +galaxies and our model to calculate 𝑃R at any point in the 𝑡LB–𝑀★– +(𝑢 − 𝑟) space. Our example galaxy moves from being likely blue +3 Formally the probability of being red is a function of lookback time, +stellar mass, and colour, but for brevity we will refer to it as 𝑃R instead of +𝑃R(𝑡LB, 𝑀★, 𝑢 − 𝑟). +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +PR +10−2 +10−1 +100 +101 +102 +PDF +GAMA +Shark +Sharkfit +Figure 2. Distribution of probability of being red for all galaxies and all +time steps below 6 Gyr. As in Paper I, the orange line shows the distribution +for GAMA, cyan for shark, and purple for sharkfit. Each bin spans 1% +in probability. Highlighted in blue/red are the probability ranges where we +define a galaxy as being blue/red (𝑃R= 0.02 and 𝑃R= 0.98, respectively). +until a lookback time of ∼ 4, to have a similar probability of being +either blue or red at ∼ 2.7 Gyr (𝑃R∼ 0.5), to likely becoming red at +∼ 1.5 Gyr. What is not immediately obvious from this Figure alone +is what values of 𝑃R best define 𝑡LB,B and 𝑡LB,R, which is the first +aspect we will address in Section 3. +3 +DISTRIBUTION AND EVOLUTION OF THE +PROBABILITY OF GALAXIES BEING RED +With the probabilistic blue/red classification from Paper I, the last +step needed to measure 𝜏Q is the choice of probabilities at which a +MNRAS 000, 1–16 (2023) + +4 +Bravo et al. +10−2 +10−1 +100 +101 +102 +PDF +GAMA +109.0−9.5 M⊙ +109.5−10.0 M⊙ +1010.0−10.5 M⊙ +1010.5−11.0 M⊙ +1011.0−11.5 M⊙ +Shark +Sharkfit +0.1 +0.3 +0.5 +0.7 +0.9 +PR +10−2 +10−1 +100 +101 +102 +PDF +0.1 +0.3 +0.5 +0.7 +0.9 +PR +tLB ∈ [1, 2) Gyr +tLB ∈ [2, 3) Gyr +tLB ∈ [3, 4) Gyr +tLB ∈ [4, 5) Gyr +tLB ∈ [5, 6) Gyr +tLB ∈ [6, 7) Gyr +tLB ∈ [7, 8) Gyr +tLB ∈ [8, 9) Gyr +tLB ∈ [9, 10) Gyr +0.1 +0.3 +0.5 +0.7 +0.9 +PR +Figure 3. Distribution of the probability of galaxies being red. Each column shows the probability distribution for single sample, from left to right: GAMA, +shark, and sharkfit. The top row shows the distribution at all time steps below 6 Gyr binned by stellar mass at observation time, with bins of increasing mass +shown with lighter colours. The bottom row shows the distribution at all stellar masses binned by lookback time, with bins of increasing lookback time in +lighter colours. Highlighted in blue (red) are the probability ranges where we define a galaxy as being blue (red), as in Figure 2. +galaxy is considered to be a part of the blue or red populations. While +ultimately this is an arbitrary choice, we will use the distribution +of our calculated probabilities to inform this choice, just like our +choice of a GMM to describe the colour populations in Paper I was +informed by the reconstructed colour distributions. As the green +valley is sparsely populated, most of the mass of the PDF will be +near the edges (i.e., near 𝑃R= 0 and 𝑃R= 1). We use the second +derivative of the decrease of the PDF from the edges towards the +centre as a guide for our choice. Figure 2 shows the distribution +of probabilities of being red (𝑃R), stacked from all time steps and +stellar masses given our selection criteria. The transition from the +extremes of the probability range is dramatic, with a ∼ 2 dex (∼ 1 +dex) decrease in the PDF from the 0-1% bin (99-100%) to the 1- +2% bin (98-99%). This would suggest 𝑃R> 0.99 (𝑃R< 0.01) is a +reasonable criterion for a galaxy to be confidently classified as blue +(red). +While such a selection will work on average, the distribution +of probabilities may depend on stellar mass and/or lookback time. +To examine this, Figure 3 shows the probability distributions for all +samples binned by both stellar mass and lookback time. GAMA and +shark show opposite trends, with the former exhibiting a probabil- +ity distribution that is mass-independent but time-dependent, and +the latter being mass-dependent but time-independent. This differ- +ence suggest that we expect a stellar mass trend for 𝜏Q in shark, +and a lookback time trend in GAMA. sharkfit exhibits a mix of the +trends in GAMA and shark, suggesting that our modelling choices +in ProSpect may be impacting our 𝜏Q measurements, but also that +they are not completely dictated by them. +While for most of the PDFs shown, a choice of 𝑃R< 0.01 +(𝑃R> 0.99) would still lead to a strong blue (red) classification, this +is not true for the two highest mass bins in shark. For this reason +we will use a more conservative classification of 𝑃R< 0.02 for blue +galaxies, 𝑃R> 0.98 for red galaxies, for the rest of this work (see +also the bottom row of Table 2). We note that small variations to +these limits do not affect the qualitative nature of our results, nor do +they lead to strong quantitative changes. +3.1 +Time evolution of the fractions of blue, transitional, and +red galaxies +We explore the fractions of galaxies in blue/transitional/red regions +across cosmic time in Figure 4. We also show in Figure 4 the frac- +tions for both central and satellite galaxies, according to their classi- +fication at observation time. For this, we follow the same convention +adopted in Bravo et al. (2020) of treating all isolated and central +group galaxies4 from GAMA as centrals, and the remaining galaxies +as satellites. Since we demonstrated in that work that central/satellite +4 Those with RankIterCen = 1 in the GAMA Galaxy Group Catalogue, +from the iterative ranking procedure defined in section 4.2.1 of Robotham +et al. (2011). +MNRAS 000, 1–16 (2023) + +Forensic quenching timescales +5 +1 +2 +3 +4 +5 +6 +7 +8 +9 +tLB [Gyr] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +f +Blue galaxies +1 +2 +3 +4 +5 +6 +7 +8 +9 +tLB [Gyr] +True central/satellite classification +Transition galaxies +2 +4 +6 +8 +10 +tLB [Gyr] +Red galaxies +1 +2 +3 +4 +5 +6 +7 +8 +9 +tLB [Gyr] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +f +Blue galaxies +1 +2 +3 +4 +5 +6 +7 +8 +9 +tLB [Gyr] +GAMA-like central/satellite classification +Transition galaxies +GAMA +all galaxies +centrals +satellites +Shark +all galaxies +centrals +satellites +Sharkfit +all galaxies +centrals +satellites +2 +4 +6 +8 +10 +tLB [Gyr] +Red galaxies +Figure 4. Time evolution of the fraction of galaxies classified as blue/transitional/red, as a function of the central/satellite classification for shark and sharkfit. +Galaxies included correspond to those above the evolving mass completeness limits at 𝑧 ∼ 0.06 defined in Paper I (see their section 3.2), corresponding to +𝑀★(𝑧 ∼ 0.06) ≥109.1 M⊙ for GAMA and 𝑀★(𝑧 ∼ 0.06) ≥109.0 M⊙ for shark/sharkfit. Each column shows a different population: blue in the left column, +transitional in the middle, and red in the right. In each panel the corresponding population is shown for our three samples, with the combined central+satellite +fraction is shown in solid lines, centrals only with dashed lines, and satellites with dotted lines. shark and sharkfit are shown in the top row using the +central/satellite classification from the simulation, and using a GAMA-like classification in the bottom row (23% confusion, following the results from Bravo +et al. 2020; Chauhan et al. 2021). Line colours are as in Figure 2. Columns are as in Figure 3. The results for GAMA are identical in each column, they are +repeated for easier comparison with both classifications from the simulations. +Selected galaxies +GAMA +Shark +Sharkfit +Are red by 𝑡LB = 1 Gyr +22.9% +26.7% +30.2% +Became red at 𝑡LB < 10 Gyr +15.0% +< 26.0% +21.2% +Were blue at 𝑡LB = 10 Gyr and are red at 𝑡LB = 1 Gyr +14.3% +22.2% +21.0% +Table 1. Percentage of the total population of galaxies that are currently red, that became red after 𝑡LB = 10 Gyr, and that transitioned from blue to red after +𝑡LB = 10 Gyr, from all three samples. The bottom row is the sample selected to measure 𝜏Q. +confusion plays is an important factor, we show the results for both +the true central/satellite classification in shark/sharkfit and a con- +fused classification. We note that we use a higher level of confusion +than in Bravo et al. (2020), 23% instead of 15%, because the sample +we use from GAMA in this work is limited to a significantly lower +redshift (𝑧 < 0.06 instead of 𝑧 < 0.6). This elevated confusion can +be seen in figure 3 of Bravo et al. (2020), and we first presented and +tested this higher value in Chauhan et al. (2021). +The time dependence (independence) of the density for inter- +mediate values of 𝑃R seen for GAMA and sharkfit (shark) in +Figure 3 is clearly reflected in the time evolution of the transitional +fraction of galaxies shown in Figure 4. The red fraction is in excel- +MNRAS 000, 1–16 (2023) + +6 +Bravo et al. +PDF +GAMA +Shark +Sharkfit +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +2 +4 +6 +8 +tLB,R [Gyr] +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +Figure 5. The lookback time when red galaxies joined the red population (𝑡LB,R) as a function of current (𝑧 ∼ 0.06) stellar mass, together with the stellar mass +distribution, for each of our samples selected following Table 1. For each individual sample, the solid lines indicates the 𝑡LB,R running median, the dashed +lines the running 16-84th percentiles, both using the same bins as the stellar mass histograms of the top panels. The black markers indicate the stellar mass +and 𝑡LB,R median point for each bin. The background contours indicate the smooth distribution obtained using the gaussian_kde Gaussian Kernel Density +Estimator (KDE) function from scipy, to avoid visualisation artefacts due to the discreteness of our data in 𝑡LB,R–𝜏Q space. The lightest contour shows the +highest-density region containing 99% of the mass of the Gaussian KDE, with the rest of the contours evenly spaced in percentage of the mass contained. +lent agreement between shark and sharkfit, which indicates that +we are accurately recovering this fraction with ProSpect. In con- +trast, the transitional fractions in sharkfit are in better agreement +with GAMA than shark, suggesting that this is to some degree +affected by the modelling choices in ProSpect. +Central galaxies show a higher blue fraction than satellites +in all samples, but there are differences across samples. shark +and sharkfit predict a higher fraction of blue centrals relative to +GAMA, at lookback times of ≲ 5 Gyr for the former and at all +lookback times for the latter, though including central/satellite clas- +sification confusion lessens this tension. The opposite trend is true +for the red population, being under-estimated by shark/sharkfit +compared to GAMA. Interestingly, while sharkfit shows a strong +under-prediction of the transitional fraction of centrals at ≳ 3 Gyr +relative to shark, the difference is mostly absorbed by the blue +fraction, which is overestimated (underestimated) in sharkfit above +(below) a lookback time of ∼ 2 Gyr. This points to the 𝑡LB,B recov- +ered for centrals being biased towards later times. +shark and sharkfit exhibit a significantly higher fraction of +red satellites compared to GAMA, reaching ∼ 80% at 𝑡LB = 1 +Gyr, a factor of ∼ 3 larger than observations, but this tension is +strongly reduced when accounting for central/satellite classification +confusion. The transitional satellites in sharkfit show a similar +difference to those in shark as previously mentioned for centrals, +but unlike centrals, the under-abundance of transitional satellites +in sharkfit is balanced out by an over-abundance of both blue +and red satellites. This difference suggests that the SFH model +parameterisation adopted in ProSpect may cause the transition +measured to be too fast. GAMA and sharkfit exhibit a qualitatively +similar evolution for the transition fraction, and they come into +quantitative agreement at lookback times of ≳ 6 Gyr, in line with +the results from Paper I. +4 +DISTRIBUTION AND TIME EVOLUTION OF 𝜏Q +Defining both 𝑡LB,B and 𝑡LB,R is straightforward for GAMA and +sharkfit, as 𝑃R is monotonically-increasing due to our choice of +SFH in ProSpect5 , with 𝑡LB,B (𝑡LB,R) simply being the last (first) +time the galaxy was a member of the blue (red) population. This +is not true for shark, and while rejuvenation in shark is not a +common occurrence in general (see Appendix A2 of Paper I), the +measurement of 𝜏Q for galaxies that do rejuvenate presents a chal- +lenge (mostly massive centrals). To measure 𝜏Q in shark, we first +select galaxies that are blue at 𝑡LB = 10 Gyr and that are red 𝑡LB = 1 +Gyr (to ensure measurable timescales), trace the continuous time +span during which the galaxy was red, and the latest time before +this period that the galaxy was blue. Table 1 shows the fraction of +red galaxies in our three samples, indicating that selecting galaxies +being blue at least at a lookback time of 10 Gyr discards ∼20–40% +of the current red galaxies, with GAMA seeing the largest reduction +in sample size and shark the smallest. +Before we explore the measured 𝜏Q from our samples, we first +discuss the 𝑡LB,R distributions in Figure 5. GAMA exhibits a clear +trend in 𝑡LB,R with stellar mass, with more massive galaxies be- +coming red at earlier times. In contrast, the 𝑡LB,R distribution in +both shark and sharkfit are broadly consistent for stellar masses +above ≳ 9.3 Gyr. The overall good agreement between both shark +5 The caveat to this statement is that, depending on the details of the evolu- +tion of the galaxy colour populations as a whole, a galaxy may see a decrease +in 𝑃R without changing its colour. We do observe this behaviour in both +GAMA and sharkfit, being particularly clear for galaxies above ∼1010.5 +M⊙ in the former. For this reason, we force a monotonic time evolution of +𝑃R for galaxies that cross our 𝑃R= 0.98 threshold in these two samples. We +find the maximum 𝑃R for all galaxies after they become red, and then set all +subsequent values of 𝑃R to this maximum value. +MNRAS 000, 1–16 (2023) + +Forensic quenching timescales +7 +1 +2 +3 +4 +τQ [Gyr] +GAMA +1 ≤ tLB,R/Gyr < 2.5 +4 ≤ tLB,R/Gyr < 5.5 +7 ≤ tLB,R/Gyr < 8.5 +1 +2 +3 +4 +τQ [Gyr] +Shark +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +1 +2 +3 +4 +τQ [Gyr] +Sharkfit +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +Figure 6. Blue-to-red transition timescales (𝜏Q) as a function of current stellar mass. The distributions are shown for three 𝑡LB,R bins: 1 ≤𝑡LB,R< 2.5 (left +column), 4 ≤𝑡LB,R< 5.5 (middle column), and 7 ≤𝑡LB,R< 8.5 (right column). Each row corresponds to a different sample, from top to bottom: GAMA, shark, +and sharkfit. Solid lines, dashed lines, markers and contours as in Figure 5. The diagonally-hatched region indicates where the 𝜏Q measurements become +incomplete in the corresponding 𝑡LB,R bin, and the cross-hatched where no 𝜏Q measurement is possible (only visible on the right-most column due to our +choice of limits for the 𝑦-axis). Note that the 𝑡LB,R bin shown in the left column lies in the range of lookback times that we found affected by SED-fitting-related +biases in Paper I. +and sharkfit indicates that there are no strong biases in our GAMA +measurements, hence the strong difference between GAMA and +shark/sharkfit is not a consequence of our colour evolution recon- +struction. In other words, the fact that we do not find a downsizing +trend in shark/sharkfit as strong as in GAMA is a short-coming +of the physical models in shark. Figure 5 also shows the overall +stellar mass distribution of the three samples, which are in good +agreement, thought shark/sharkfit show a bimodality that is not +clear in GAMA. +4.1 +𝜏Q distribution of the overall galaxy population +In Figure 6 we show the distribution of 𝜏Q as a function of stel- +lar mass, divided into three lookback time bins. The distributions +are roughly consistent across cosmic time in shark, the largest +difference being the increased dispersion above ∼1010 M⊙, which +suggests that there is no strong time evolution of the 𝜏Q distribution. +In contrast, the 𝜏Q distribution of GAMA changes as a function of +𝑡LB,R, with the median 𝜏Q increasing by a factor of ∼ 3 in the span +of 4.5 Gyr. +GAMA and sharkfit display similar timescale-mass relations +in the highest 𝑡LB,R bin. This is, in line with the results in Paper +I, where we found a strong similarity in the galaxy distributions of +both GAMA and sharkfit in colour-mass space at high lookback +times (𝑡LB > 6 Gyr), likely driven by dust parameter degeneracies +(see Appendix A of Paper I for further details). The timescale-mass +relations measured in shark and sharkfit are in good agreement +in the other two lookback time bins, indicating that we can recover +MNRAS 000, 1–16 (2023) + +8 +Bravo et al. +PDF +GAMA (centrals) +Shark (centrals) +Sharkfit (centrals) +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +1 +2 +3 +4 +τQ [Gyr] +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +PDF +GAMA (satellites) +Shark (satellites) +Sharkfit (satellites) +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +1 +2 +3 +4 +τQ [Gyr] +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +Confused central/satellite classification +True central/satellite classification +Figure 7. 𝜏Q as a function of current stellar mass, together with the stellar mass distribution, divided between centrals and satellites. Only galaxies with +1 ≤𝑡LB,R< 6 Gyr are shown. For each individual sample, the solid/dashed (dash-dotted/dotted) lines indicate the 𝜏Q running median/16-84th percentiles for +the GAMA-like (true) central-satellite classification, both using the same bins as the stellar mass histograms. The black cross (open) markers indicate the stellar +mass and 𝑡LB,R median point for the GAMA-like (true) central-satellite classification in each bin. Contours as in Figure 5. The coloured (black) histograms +show the measured stellar mass distribution for the GAMA-like (true) central-satellite classification. Note that the gap seen in the running median for the true +centrals in shark corresponds to a mass bin where no galaxies are present (see histogram above). +this with ProSpect, which validates the difference between both +and GAMA as real. The only significant difference between shark +and sharkfit at these lower lookback times is that we do not re- +cover the longest 𝜏Q from shark, possibly due to episodes of weak +rejuvenation extending the time period galaxies remain in the tran- +sitional region. The 𝜏Q–𝑀★ relation that we observe in GAMA is +in clear tension with that we predict in shark, which also exhibit +opposite trends with stellar mass, i.e., low-mass galaxies in GAMA +take much longer to quench at recent times than in shark. We ex- +plore the effect of selection biases in the measured 𝜏Q evolution in +Appendix B, where we find that GAMA does exhibit a strong time +evolution of the 𝜏Q distribution, a mild evolution in sharkfit, and +that the 𝜏Q distribution in shark is independent of cosmic time. +MNRAS 000, 1–16 (2023) + +Forensic quenching timescales +9 +1011 +1012 +1013 +1014 +Mhalo [M⊙/h] +0 +250 +500 +750 +1000 +1250 +n [dex−1] +Shark +1011 +1012 +1013 +1014 +Mhalo [M⊙/h] +0 +100 +200 +300 +400 +n [dex−1] +GAMA +All haloes with satellites +Haloes with red satellites (true) +Haloes with red satellites (confused) +Haloes with Ng ≥ 5 +Figure 8. Comparison of the halo mass distribution between GAMA and shark. shark is shown on the left panel, GAMA is shown on the right. Three +selections are shown here: all haloes with at least one satellite (light grey), all haloes with at least one red satellite (light red, shaded area for GAMA-like +central/satellite classification, dashed line for the true shark classification), and haloes with at least five galaxies (dark magenta). Note that both panels have +different scales for the y-axis, to focus on the effect of the different selections rather the different sizes of our samples. +4.2 +Environmental effects on 𝜏Q +We now explore how 𝜏Q compares between central and satellite +galaxies. Figure 7 shows both the stellar mass distribution and 𝜏Q +as a function of their current stellar mass, divided into centrals and +satellites. For shark and sharkfit we show the results for both true +and GAMA-like central/satellite classifications. +True centrals and satellites in shark show a significant differ- +ence in 𝜏Q for galaxies below ∼1010.5 M⊙, with centrals exhibiting +longer timescales than satellites. Satellites in sharkfit are in good +agreement with those from shark. Centrals show less of an agree- +ment for 𝜏Q, in particular the timescales in sharkfit for centrals of +∼109.5 M⊙ (∼1010.5 M⊙) are shorter than in shark by a factor +of ∼ 4 (∼ 2). In comparison, GAMA centrals and satellites only +differ at 𝑀★ ≲1010 M⊙, with 𝜏Q of satellites being ∼ 0.4 Gyr than +for centrals. The difference between centrals and satellites is re- +duced when using a GAMA-like classification for both shark and +sharkfit, suggesting the possibility of a larger difference between +GAMA centrals and satellites than what we measure. +To further explore the differences for satellites between GAMA +and shark, Figure 8 shows the mass distribution of haloes that host +satellites in both samples. Chauhan et al. (2021) found that the +recovery of shark halo masses with the Robotham et al. (2011) +group finder, which was used to infer halo masses for GAMA, is +reasonable for the high-multiplicity groups (𝑁𝑔 ≥ 5) but the quality +of the recovery noticeably decreases for low-multiplicity groups. +Those results can account for the difference between GAMA and +shark in the distribution of halo masses for haloes hosting at least +one satellite. The majority of haloes in shark that host at least +one satellite also host at least one red satellite, in strong contrast to +GAMA, but the difference becomes smaller when accounting for +central/satellite classification confusion. +Figure 8 shows that the halo mass distribution of 𝑁𝑔 ≥ 5 +groups in GAMA and shark are comparable, hence this should be a +strong probe for the treatment and evolution of the satellites residing +in such haloes. For centrals, we ignore 𝑁𝑔 ≥ 5 group red centrals, +as there are ≲40 of the latter in GAMA and shark6 . Overall, +we find little difference between true central (all true satellite) and +isolated (𝑁𝑔 ≥ 5 satellite) galaxies in all three samples, with the +most important finding being that shark/sharkfit lack the observed +numbers of ≲1010 M⊙ red isolated galaxies we find in GAMA (see +Figure included in the supplemental material). +4.3 +Connection between 𝜏Q and satellite infall in shark +One of the powerful features of using simulations is that they enable +us to further explore the evolution of red galaxies. In particular, here +we explore how 𝜏Q is linked to the central-to-satellite transition. +Naturally, we can only do this study in shark and not GAMA7 , +given that we do not know the infall time for the satellite galaxies in +GAMA. Nonetheless, this will give us an indication of whether 𝜏Q +is clearly linked to the physical models included shark. We classify +satellite galaxies in shark in three groups, based on the lookback +time of infall (𝑡LB,infall) relative to its transition from blue to red, +those that: +6 There are significantly more in sharkfit, factor of ∼ 4 more than in shark, +but this is a consequence of the delayed formation of the red population from +our ProSpect fits to shark. This could be a result of the relative fractions +of galaxies that undergo rejuvenation, as defined in Paper I, as ∼ 20% of +the red centrals in shark had at least one rejuvenation episode, compared +to only ∼ 4% of satellites. We should also note that we are using a subset +of the full simulation box for shark/sharkfit, so it is possible to increase +this number by a factor of ∼30. The low number from GAMA is the limit to +study 𝑁𝑔 ≥ 5 group red centrals. +7 We can also explore this aspect with sharkfit, but as based on the results +in Appendix A, there are reasons not to. Both 𝑡LB,B and 𝑡LB,R are not as +well-recovered than 𝜏Q, which leads to a significant confusion on whether +a satellite became red before, during, or after infall, e.g., the percentage of +the former increases from ∼ 4% to ∼ 15%. Classifying whether a sharkfit +satellite became red based on 𝑡LB,R measured in shark would reduce the +analysis to how well-recovered is the evolution of centrals and satellites, +which is the topic of Appendix A. +MNRAS 000, 1–16 (2023) + +10 +Bravo et al. +PDF +Shark (before infall, 4.5%) +Shark (during infall, 13.8%) +Shark (after infall, 81.7%) +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +1 +2 +3 +4 +τQ [Gyr] +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +Figure 9. 𝜏Q of satellites in shark as a function of current stellar mass, and the stellar mass distribution, divided by when a galaxy became red relative to the +time when the galaxy became a satellite. The percentage that each sample represents of the total of red satellites in shark is shown on the top labels of each +column. Solid lines, dashed lines, markers and contours as in Figure 5. +(i) became red before they became a satellite, i.e., transitioned +in colour before infall, 𝑡LB,B>𝑡LB,R> 𝑡LB,infall; +(ii) were a central the last time they were blue, but were a satellite +by the time they became red, i.e., infall happened during the colour +transition, 𝑡LB,B> 𝑡LB,infall >𝑡LB,R; +(iii) were still blue when they became a satellite, i.e., transitioned +in colour after infall, 𝑡LB,infall >𝑡LB,B>𝑡LB,R. +The distribution of stellar mass and 𝜏Q for these categories are +shown in Figure 9. It is clear that becoming a satellite is the main +driver for galaxies to become red, as 81.7% of the red satellites in +shark became red before after infall (iii). The stellar mass distri- +bution of galaxies that became red before (i) and after infall show +a marked difference, with those that became a satellite while in +transition (ii) showing a distribution intermediate between the other +two. shark galaxies galaxies that became red after infall have the +shortest 𝜏Q, while those galaxies that became red before infall have +the longest 𝜏Q. Galaxies that became a satellite while in transition +show intermediate 𝜏Q relative to the other two groups. +5 +DISCUSSION +5.1 +Comparing our 𝜏Q definition with previous literature +Fundamental to any measurement of 𝜏Q is how it is defined. Mea- +surements of 𝜏Q in the literature include derivation from star for- +mation rates (e.g., Wetzel et al. 2013; Belli et al. 2019; Tacchella +et al. 2022), galaxy colours (e.g., Schawinski et al. 2014; Trayford +et al. 2016; Bremer et al. 2018; McNab et al. 2021), and spectral +properties (e.g., Wheeler et al. 2014; Rowlands et al. 2018). Defini- +tions include timescales to cross specific thresholds (e.g., Trayford +et al. 2016; Bremer et al. 2018; Tacchella et al. 2022), 𝑒-folding +timescales (e.g., Wetzel et al. 2013; Schawinski et al. 2014; Wheeler +et al. 2014; Bremer et al. 2018), and inference from population den- +sities (e.g., Rowlands et al. 2018; McNab et al. 2021). Establishing +how these different measurements compare is outside the scope of +this work, and in this work we only compare our 𝜏Q definition and +measurements to those in the literature that are derived from galaxy +colours. +While definitions from observations abound (e.g., Schawinski +et al. 2014; Smethurst et al. 2015; Bremer et al. 2018; Phillipps +et al. 2019), differences in the recovery of the intrinsic stellar light +result in significant differences in the loci of the colour populations. +Table 2 provides a description of these selections, and the top row of +Figure 10 shows how those from Schawinski et al. (2014) and Bre- +mer et al. (2018); Phillipps et al. (2019) compare to ours. The issue +of definitions that adopt a fixed parameterisation instead of being +a function of population properties are clear here as the match to +ours is strongly sample-driven. The Schawinski et al. (2014) green +valley selection covers almost exclusively the red population for +shark/sharkfit, and the Phillipps et al. (2019) covering mostly the +blue population for GAMA8. Furthermore, (to the author’s knowl- +edge) there are no observational selections that account for colour +evolution with cosmic time, i.e., the selections are at fixed lookback +time/redshift. +We now compare the classifications used for simulations by +Trayford et al. (2016); Nelson et al. (2018); Wright et al. (2019) +to the one we adopt for this work, shown in the bottom row of +Figure 10. Trayford et al. (2016) classifies galaxies from the EAGLE +simulation (Schaye et al. 2015) between red, green and blue using +straight lines in the colour-mass plane, where only the normalisation +evolves with time. While it does overlap most of the region we +classify as transitional in GAMA, it is clearly slanted as a function +of stellar mass compared to our statistically-based selection. The +fixed nature of the selection limits also makes it a poor choice +for shark/sharkfit, as it strongly overlaps the red population. The +Trayford et al. (2016) selection limits are also consistently wider +8 This is not to say that these are poor selections for the samples for which +they were designed (see figures 4 and 1 of Schawinski et al. 2014; Phillipps +et al. 2019, respectively), which we could only assess by implementing our +method to their data sets. +MNRAS 000, 1–16 (2023) + +Forensic quenching timescales +11 +Reference +Data type +Parameter space +Transition region lower limit +Transition region upper limit +Schawinski et al. (2014) +Observation +(𝑢 − 𝑟)–𝑀★ +0.25 log10(𝑀★/M⊙) − 0.75 +0.25 log10(𝑀★/M⊙) − 0.24 +Smethurst et al. (2015) +Observation +(𝑢 − 𝑟)–𝑟 +−0.244tanh +� +𝑟+20.07 +1.09 +� ++ 20.6 − 𝜎 +−0.244tanh +� +𝑟+20.07 +1.09 +� ++ 20.6 + 𝜎 +Trayford et al. (2016) +Simulation +(𝑢 − 𝑟)–𝑀★ +0.2 log10(𝑀★/M⊙) − 0.25𝑧0.6 − 0.3 +0.2 log10(𝑀★/M⊙) − 0.25𝑧0.6 + 0.24 +Bremer et al. (2018) +Observation +(𝑢 − 𝑟)–𝑀★ +0.1 log10(𝑀★/M⊙) + 0.3 +0.2 log10(𝑀★/M⊙) − 0.5 +Nelson et al. (2018) +Simulation +(𝑔 − 𝑟)–𝑀★ +𝜇B + 𝜎B +𝜇R − 𝜎R +Phillipps et al. (2019) +Observation +(𝑢 − 𝑟)–𝑀★ +0.1 log10(𝑀★/M⊙) + 0.3 +0.2 log10(𝑀★/M⊙) − 0.5 +Wright et al. (2019) +Simulation +(𝑢 − 𝑟)–𝑀★ +𝜇B + 1.5𝜎B +𝜇R − 1.5𝜎R +This work +Both +(𝑢 − 𝑟)–𝑀★ +𝑃R> 0.02 +𝑃R< 0.98 +Table 2. Sample of literature criteria to define the green valley/transition region. Our classification is at the end of the table for comparison purposes. Several +remarks need to be made for a fair comparison. All literature definitions using observations include no time evolution and are valid only at low redshift +(𝑧 ≤ 0.25). The definition by Smethurst et al. (2015) references a dispersion (𝜎), but it is not clear what dispersion they are using, besides that it seems to +be independent of stellar mass (see their figure 3). Bremer et al. (2018) and Phillipps et al. (2019) use the same definition, but the former limits it to a narrow +stellar mass range (1010.25 < 𝑀 ★ /M⊙ < 1010.75), while the latter expands it to their full sample. Nelson et al. (2018) and Wright et al. (2019) use the same +type of criteria, but they chose difference factors for the standard deviation, whether this is a consequence of their different choices for colour or not is not clear. +Note that 𝜇{B,R}, 𝜎{B,R}, and 𝑃R all are functions of lookback time, stellar mass and colour, see Section 3 for details. +0 +1 +2 +(u − r)ab [mag] +GAMA +Shark +Sharkfit +109 +1010 +1011 +M⋆ [M⊙] +0 +1 +2 +(u − r)ab [mag] +109 +1010 +1011 +M⋆ [M⊙] +109 +1010 +1011 +1012 +M⋆ [M⊙] +tLB = 1.0 Gyr +25–50–75% +Schawinski+14 +Phillipps+19 +Trayford+2016 +Nelson+2018 +Wright+2019 +Figure 10. Comparison of the colour classification we adopt to two observational (Schawinski et al. 2014; Phillipps et al. 2019, top row; in orange and teal, +respectively) and three theoretical (Trayford et al. 2016; Nelson et al. 2018; Wright et al. 2019, bottom row; in brown, green, and cyan, respectively) literature +examples. The comparison is made at the lowest lookback time of our data (1 Gyr), as a rough middle point between the redshift ranges of the literature +classifications. Each column shows the classification for one of our samples, following the same thresholds as in Figure 2, with the underlying histogram +bins being coloured in blue/grey/red if they median 𝑃R of the galaxies classifies them as being blue/transitional/red. The dashed/dash-dotted/dotted contours +indicate the highest density regions in each panel that contain 25/50/75% of the galaxies of each sample. The transitional region from the literature examples +are shown by coloured bands in each panel. The black region in the left column indicates the stellar mass below the stellar mass completeness limit for GAMA +(lies below 109 M⊙ for shark and sharkfit, further details in section 3.2 of Paper I). The classifications from Nelson et al. (2018) and Wright et al. (2019) are +defined as functions of the means and standard deviations of each population, which is the reason why they are different across our samples. +MNRAS 000, 1–16 (2023) + +12 +Bravo et al. +than our transitional region, save for shark above ∼1010.5 M⊙, +which would lead to an over-estimation of 𝜏Q. +The classifications from Nelson et al. (2018) for IllustrisTNG +and Wright et al. (2019) for EAGLE are more directly comparable +to our classification, as they are all based on GMM fits to the colour +population. Both define their green selection as: +𝐺upper = 𝜇R − 𝑓 𝜎R, +(2) +𝐺lower = 𝜇B + 𝑓 𝜎B, +(3) +where 𝜇{B,R} and 𝜎{B,R} are the mean and standard deviation of the +blue/red population, respectively, and 𝑓 is a constant value. Nelson +et al. (2018) adopts a value of 𝑓 = 1, while Wright et al. (2019) +adopts 𝑓 = 1.5. For a fair comparison, we have used the 𝜇{B,R} +and 𝜎{B,R} we measured in Paper I to implement their selections, +as the colour evolution in both EAGLE and IllustrisTNG may well +not match those in GAMA and shark. +Both are in reasonable agreement with our classification for +sharkfit, with Nelson et al. (2018) being a remarkable match, but +the results for GAMA and shark show that this is just a lucky +coincidence. In particular, in shark both criteria fail to reproduce +our probabilistically-based classification, and at low masses they +lead to an over-estimation of the transitional region. At high masses +both criteria under-estimate 𝜏Q. +5.2 +Comparing our 𝜏Q measurements between observations +and simulations +A fundamental question to answer when exploring the 𝜏Q distri- +bution is how it evolves with cosmic time. This can provide an +insight into the physical processes behind the colour transformation +of galaxies. A 𝜏Q distribution invariant with time would suggest +that the different mechanisms and their relative prevalence are also +time invariant. If 𝜏Q evolves with cosmic time instead, that suggests +some combination of the following: different mechanisms operate +at different times, the mechanisms’ efficiency change with time, or +that the relative mix of these mechanisms evolves with time. +As described in Section 4, the challenge in establishing whether +𝜏Q evolves with time is selection bias, irrespective of the chosen lim- +its. Furthermore, galaxies of different current stellar masses could +have transitioned in colour at different lookback times. When ac- +counting for both lookback time and stellar mass dependencies, we +find that GAMA shows a clear evolution towards longer timescales +at more recent cosmic times, while the same is not evident in shark. +We do find a time evolution for 𝜏Q in sharkfit, unlike shark, which +suggests that the difference between GAMA and shark may be +attributable in part to ProSpect. The difference in evolution of 𝜏Q +as a function of stellar mass between GAMA and sharkfit suggests +that some of this evolution might be real, instead of induced by our +models in ProSpect, though more work would be required to make +a more conclusive statement. +Since the 𝜏Q distribution is broadly consistent for all three +samples for 𝑡LB,R< 4 Gyr, we can use that as a selection to study the +connection between 𝜏Q and other galaxy properties. This selection +means that we are roughly halving the number of galaxies that +we can explore in shark/sharkfit, as galaxies above ∼109.3 M⊙ +exhibit a consistent median 𝑡LB,R of ∼ 4 Gyr in these samples, +but for GAMA we will only be able to explore a small fraction +of ∼1010 M⊙ galaxies. Comparing the 𝜏Q–𝑀★ distribution for all +three samples, shown in Figure 11, shows that shark under-predicts +𝜏Q by up to ∼ 2 Gyr for 109–1010.5 M⊙, with the tension being +larger for lower stellar masses. Since the 𝜏Q distribution in sharkfit +is in good agreement with that from shark, and that for masses +above ∼1010.5 M⊙ all three samples show a good agreement, the +difference between GAMA and shark at lower masses suggests that +the mechanisms that quench these lower mass galaxies in shark are +too efficient. +The red population in shark is strongly dominated by satel- +lites (71.4% of the sample), in apparent disagreement with GAMA +(50.8%), but as shown in Paper I this is alleviated by accounting +for observational confusion between centrals and satellites. Centrals +and satellites below ∼1010.5 M⊙ in shark exhibit different 𝜏Q, but +using a GAMA-like classification reduces the difference between +both, suggesting that the similarity between centrals and satellites +in GAMA could be due classification confusion. The choice of cen- +tral/satellite classification cannot account for the strong difference +in 𝜏Q between GAMA and shark/sharkfit, as satellites exhibit +shorter transition timescales in shark/sharkfit than GAMA. +Since 83.2% of shark satellites became red after infall, this +points to the quenching mechanisms for satellites in shark, the in- +stantaneous stripping all halo gas of the galaxy, being too aggressive. +A more gradual stripping of gas would allow satellites to replenish +their interstellar medium (ISM) gas, increasing 𝜏Q (e.g., Font et al. +2008). Other SAMs have adopted gradual halo gas stripping models, +usually combined with the inclusion of ISM stripping to balance the +availability of gas for satellites (e.g., Font et al. 2008; Croton et al. +2006, 2016; Stevens & Brown 2017; Cora et al. 2018). More ex- +treme models can also be found in the literature. E.g., in Henriques +et al. (2015) satellites fully retain their halo gas and their ISM in +haloes of masses lower than 1014 M⊙, leading to slow exhaustion +of the gas to star formation. The latter was required by that model +to reproduce the observed red fraction of galaxies. Based on these +results, we will explore in future work if the combination of gradual +halo gas stripping and ISM stripping leads to longer timescales than +instantaneous halo gas stripping without ISM stripping. +shark satellites that became red before infall (i.e., that became +red while being centrals) exhibit a similar 𝜏Q distribution as those of +(current) central galaxies, but they show markedly different stellar +mass distribution, with the former having a significant contribution +from ≲109.5 M⊙ galaxies. These low-mass satellites that became +red as centrals exhibit starburst episodes prior to infall, indicating +that the otherwise temporary gas exhaustion was made permanent +by the instantaneous hot gas stripping in shark. Both low-mass +central and isolated red galaxies in GAMA exhibit long 𝜏Q relative +to shark, suggesting that this is not the mechanism to reduce the +tension between both. A possible improvement could be to extend +the effectiveness of the AGN radio-mode feedback to lower halo +masses in shark, as reducing the amount of gas available for cooling +into the galaxy would lead to a gradual decrease of star formation +(i.e., a long 𝜏Q). +5.3 +Comparing our 𝜏Q measurements with literature results +What follows now is a series of comparisons with a representative +collection of literature results, both from observations and simu- +lations, which are also shown in Figure 11 as comparison to our +results. Since the calculation of 𝜏Q could have a strong effect on +the measured values, we include an overview of how they were de- +rived. We remark that this means that good quantitative (or even +qualitative) agreement should not be expected to be a natural re- +sult, as disagreements may well stem from methodological (or even +conceptual) differences. Also, little discussion is presented on the +MNRAS 000, 1–16 (2023) + +Forensic quenching timescales +13 +109 +1010 +1011 +1012 +M⋆(z ∼ 0.06) [M⊙] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +τQ [Gyr] +Schawinski+2014 +Smethurst+2015 +Trayford+2016 +Bremer+2018 +Nelson+2018 +Rowlands+2018 +Wright+2019 +GAMA +Shark +Sharkfit +Figure 11. Comparison between the 𝜏Q we have measured in this work to a variety of literature results. Solid and hatched areas indicate results that only +provide a range of values for 𝜏Q, together with the stellar mass range used in the work. Segmented lines indicate singular 𝜏Q values, with the extend of the line +and markers on the edge the stellar mass range used to measure it. Solid lines indicate running medians of 𝜏Q as a function of stellar mass. Multiple lines may +appear if the respective work provided more than one result. The results from this work are shown in colour, those from the literature in different shades of +grey. Note that we show only one of the two measured 𝜏Q values from Rowlands et al. (2018), as the other lies outside our choice of range for 𝜏Q (∼ 6.6 Gyr +for 1011 < 𝑀★/M⊙ < 1011.5) +possible lookback time dependence of 𝜏Q in the literature, so we +will omit that aspect. +Starting with results from observations, Schawinski et al. +(2014) inferred colour transition timescales combining colours and +stellar masses of galaxies from the Sloan Digital Sky Survey (SDSS; +York et al. 2000) with the morphological classification from Galaxy- +Zoo (Lintott et al. 2008). For this, they first divided their galaxy +sample between "blue", "green" and "red" in the colour-stellar mass +plane, with the limits being straight lines chosen by visual inspec- +tion (see Figure 10 and Table 2). They generated simple colour his- +tories with a exponentially-decaying SFH with different 𝑒-folding +timescales, and then compared the colour evolution from these SFHs +to the colour-colour distribution of both early- and late-type galax- +ies. From these comparisons they inferred a fast transition for early- +type galaxies (𝜏Q≲ 250 Myr), with late-type galaxies transitioning +slowly (𝜏Q≳ 1 Gyr). Since only ∼ 20% of the GAMA sample +shown in Figure 11 correspond to elliptical galaxies (according to +the Driver et al. (2022) morphological classification), the expecta- +tion would be for our sample to better match the timescales that +Schawinski et al. (2014) found for late-type galaxies, which is in- +deed the case. In contrast, shark/sharkfit show 𝜏Q in agreement +with Schawinski et al. (2014) only for galaxies above ∼1010.5 M⊙, +with galaxies below this mass exhibiting a median 𝜏Q neither consis- +tent with early-type nor late-type galaxies. While Schawinski et al. +(2014) find a dependence of 𝜏Q on halo mass for late-type galaxies, +they do not find the same for the early-types that dominate the red +population, which agrees with the similarity we find between low- +and high-multiplicity groups in GAMA. +Smethurst et al. (2015) also used GALEX/SDSS photometry +plus GalaxyZoo morphological classification (though the more re- +cent GalaxyZoo2 release, Willett et al. 2013) to infer quenching +timescales. They also classify galaxies as blue, green or red, but use +instead the definition from Baldry et al. (2004), where everything +< 1𝜎 from the local minimum in colour-magnitude as green. Fur- +thermore, they also adopt a similar approach of generating sample +colour evolution tracks from exponentially-decaying SFHs, though +they use a Bayesian approach to find the timescale and quenching +onset that best matches every galaxy in their sample. For bulge- +and disc-dominated galaxies they find median timescales of ∼ 1 +and ∼ 2 Gyr respectively, which is in good agreement with our re- +sults from GAMA. Their figures 8 and 11 suggest that these values +may be strongly driven by galaxies quenching early in the Universe +(𝑡LB,R> 6 Gyr), which we do not explore in this work due to biases +in the recovery. It is interesting to note that they seem to find a +strong evolution toward longer timescales at more recent times (see +their figures 8 through 11), though it is not clear if this is just a se- +lection effect. Regardless, this qualitatively agrees with our findings +in GAMA, where galaxies that become red more recently do it on +longer timescales than those becoming red earlier. +Rowlands et al. (2018) used the strength of the 4000 Å break +and the excess Balmer absorption from GAMA and the VIMOS +Public Extragalactic Redshift Survey (VIPERS). They first divided +galaxies as either post-starburst or not based on being above or +below a Balmer absorption limit, and then further divided the +non post-starburst between blue, green and red based on two ad +hoc 4000 Å break values. They then measured the number den- +sity evolution of these classifications at a wide range of redshifts +(0.05 < 𝑧 < 1.0) to infer transition timescales at two partly overlap- +ping stellar mass ranges (>1010.6 M⊙ and >1011 M⊙). They found +transition timescale for green valley galaxies of ∼ 2.6/6.6 Gyr for +their mid/high-mass selection, independent of lookback time. This +positive trend for timescales with stellar mass is opposite to our +findings in GAMA, but it is likely driven by those being a different +type of timescales, with the values themselves being significantly +higher than seen in either GAMA or shark. This is likely because +Rowlands et al. (2018) measure the timescale for 𝑧 ∼ 0.7 green +valley galaxies would join the 𝑧 = 0 red population, whereas we +measure the timescale over which observed red galaxies became +red, which Schawinski et al. (2014) shows are dominated by differ- +ent morphological types. +MNRAS 000, 1–16 (2023) + +14 +Bravo et al. +Bremer et al. (2018) measured the fraction of galaxies in the +green valley as a function of environment from GAMA, defined in +colour-stellar mass space (see Figure 10 and Table 2), and com- +bined it with the stellar ages of Taylor et al. (2011) to infer 𝜏Q +9 +. They limited their analysis to galaxies with stellar masses in the +1010.25–1010.75 M⊙ range, 0.1 < 𝑧 < 0.2 and 𝑟-band axial ra- +tio 𝑏/𝑎 > 0.5. They found a 𝜏Q of ∼1–2 Gyr, in good agreement +with our measurement of 𝜏Q for GAMA. Also similar to our re- +sults from GAMA, they found no evidence for environmental effect +from the near-constant fraction of green valley galaxies as a func- +tion of group multiplicity10, but our results from shark show that +central/satellite confusion can strongly diminish any environmental +signature present in 𝜏Q. +We now focus on a comparison with literature results pre- +sented for galaxy formation simulations. Trayford et al. (2016) used +galaxies with stellar masses of 1010–1011 M⊙ from the EAGLE +simulation (Schaye et al. 2015), and classifying them as blue, green +or red by ad hoc colour selections, defined as a function of both +stellar mass and redshift (see Figure 10 and Table 2). From these, +they selected 𝑧 = 0 red galaxies and measured the timescale over +which they transition from blue to red. They found a median 𝜏Q of +∼ 2 Gyr, though with a distribution strongly skewed towards shorter +timescales, with a peak closer to ∼ 1.5 Gyr. While they do not find a +strong difference in the median 𝜏Q for centrals and satellites (order +of a few hundred Myr), the distributions shown in their figure 10 +indicate that centrals are less skewed to short timescales. shark +does display the same trend, though we find shorter 𝜏Q (factor of +∼ 3). Their results are in agreement with our results from GAMA, +but it is not clear if this holds for earlier times, as they do not explore +the time evolution of 𝜏Q. They do not find evidence of a strong de- +pendence with stellar mass, which seems in better agreement with +shark than GAMA. Finally, we find in shark the same results that +they do with regards to satellites: the majority become red when +becoming a satellite. +Nelson et al. (2018) employed a similar method to ours to mea- +sure 𝜏Q from the IllustrisTNG simulation (Pillepich et al. 2018), +first characterising the colour population with two Gaussian com- +ponents, with parameters as function of stellar mass and redshift. +They then defined the limits for each population, set at 1𝜎 from the +mean of each population, which is the most significant difference +with our probability-based approach (see Figure 10 and Table 2). +Like Trayford et al. (2016), they also found an asymmetrical 𝜏Q +distribution, skewed to shorter values, finding similar median and +peak values (∼ 2 and ∼ 1.6 Gyr, respectively). They found a depen- +dence with stellar mass, with 𝜏Q peaking for galaxies of ∼1010 M⊙. +While the range of median 𝜏Q values they measure coincides with +that from GAMA, we find a different trend with stellar mass, with +the best agreement being for galaxies ≳1010.5 M⊙. They also found +a weak trend for centrals below 1010 M⊙ to take longer to become +red than satellites. +Also using galaxies from the EAGLE simulation, Wright et al. +9 A similar method was also presented in Phillipps et al. (2019), but instead +they used the 𝑒-folding time from the Taylor et al. (2011) fits to infer how long +will current green galaxies take to become red. This is the reason why we +do not discuss their results, to avoid comparisons between our reconstructed +evolution to their predicted evolution. +10 They do find evidence that galaxies in high density environment have +shorter lifespans as part of the blue population than in less dense environ- +ment, suggesting that a richer environment will trigger an earlier transition +to red. +(2019) used a classification close to that used by Nelson et al. +(2018), finding 𝜏Q to be in the ∼2–4 Gyr range, depending on +both stellar mass and environment. They found different 𝜏Q for +centrals and satellites for galaxies below ∼1010.5 M⊙ (∼ 4 and ∼ 2 +Gyr respectively), with timescales showing a inverted U-shape and +both centrals and satellites peaking at ∼109.7 M⊙. For larger stellar +masses they found all galaxies to have similar 𝜏Q (∼ 2 Gyr). Their +𝜏Q measurements are in strong agreement with those of Nelson et al. +(2018), despite using different thresholds to measure 𝜏Q (see Table +2), which suggests that the stellar mass trend of 𝜏Q found in both +works may be a consequence of how they define the limits of the +blue and red populations. +6 +CONCLUSIONS +In this work, we have used the characterisation of the colour evolu- +tion of the blue and red galaxy populations we presented in Paper I, +to calculate upper limits for 𝜏Q on which red galaxies transitioned +from being blue to red. For this, we first calculated the probability +of all galaxies in our three samples (GAMA, shark and sharkfit) +to belong to the red population, then used the distribution of this +probability to define the values between which we will measure 𝜏Q. +Accounting for selection biases, we find evidence that 𝜏Q evolves +with time only in GAMA, with 𝜏Q increasing from ∼ 1 to ∼ 3 +Gyr in a time span of ∼ 4 Gyr (in shark/sharkfit 𝜏Q remains sta- +ble at ≲ 1 Gyr). Our observations and simulations do not agree +on whether there is a stellar mass dependence on the lookback time +when they became red, with the former strongly suggesting that cur- +rent high-mass galaxies became red before low-mass galaxies (i.e., +downsizing), while the latter show no such trend. We find a differ- +ence between centrals and satellites in GAMA only for 𝑀★ ≲1010 +M⊙, with satellites showing 𝜏Q ∼ 0.4 Gyr shorter than centrals. +The results from shark suggest the possibility of a larger differ- +ence being hidden by observational central/satellite classification +confusion. Finally, we find that assuming an instantaneous halo gas +stripping in shark is the likely driver for the shorter-than-observed +𝜏Q for satellites. +ACKNOWLEDGEMENTS +We thank Chris Power and Pascal Elahi for their role in completing +the SURFS 𝑁-body DM-only simulations suite, Rodrigo Tobar for +his contributions to shark, Andrea Cattaneo and Benjamin Johnson +for the comments and feedback provided to the doctoral thesis on +which this work is based, Ruby Wright for providing the data from +Wright et al. (2019) for Figure 11. +MB acknowledges the support of the University of Western +Australia through a Scholarship for International Research Fees and +Ad Hoc Postgraduate Scholarship. LJMD and ASGR acknowledge +support from the Australian Research Councils Future Fellowship +scheme (FT200100055 and FT200100375, respectively). CdPL is +funded by the ARC Centre of Excellence for All Sky Astrophysics in +3 Dimensions (ASTRO 3D), through project number CE170100013. +CdPL also thanks the MERAC Foundation for a Postdoctoral Re- +search Award. SB acknowledges support by the Australian Research +Council’s funding scheme DP180103740. JET is supported by the +Australian Government Research Training Program (RTP) Scholar- +ship. +This work was supported by resources provided by the Pawsey +MNRAS 000, 1–16 (2023) + +Forensic quenching timescales +15 +Supercomputing Centre with funding from the Australian Govern- +ment and the Government of Western Australia. We gratefully ac- +knowledge DUG Technology for their support and HPC services. +GAMA is a joint European-Australasian project based around +a spectroscopic campaign using the Anglo-Australian Telescope. +The GAMA input catalogue is based on data taken from the +Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Sur- +vey. Complementary imaging of the GAMA regions is being ob- +tained by a number of independent survey programmes includ- +ing GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel- +ATLAS, GMRT and ASKAP providing UV to radio coverage. +GAMA is funded by the STFC (UK), the ARC (Australia), the +AAO, and the participating institutions. The GAMA website is +http://www.gama-survey.org/. Based on observations made +with ESO Telescopes at the La Silla Paranal Observatory under +programme ID 179.A-2004. Based on observations made with ESO +Telescopes at the La Silla Paranal Observatory under programme +ID 177.A-3016. +The analysis on this work was performed using the program- +ming languages Python v3.8 (https://www.python.org), with +the open source packages matplotlib (Hunter 2007), NumPy (Har- +ris et al. 2020), pandas (pandas development team 2022), SciCM +(https://github.com/MBravoS/scicm), SciPy (Virtanen et al. +2020), and splotch (https://github.com/MBravoS/splotch), +in addition of the software previously described. +DATA AVAILABILITY +The 𝑃R tracks and 𝜏Q catalogues generated for this work will be +shared on reasonable request to the corresponding author. For all +other data, see the Data Availability statement in Paper I. +REFERENCES +Baldry I. K., Glazebrook K., Brinkmann J., Ivezić Ž., Lupton R. H., Nichol +R. C., Szalay A. S., 2004, ApJ, 600, 681 +Baldry I. K., Balogh M. L., Bower R. G., Glazebrook K., Nichol R. C., +Bamford S. 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K., et al., 2007, ApJS, 173, 293 +York D. G., et al., 2000, AJ, 120, 1579 +MNRAS 000, 1–16 (2023) + +16 +Bravo et al. +0.00 +0.25 +0.50 +0.75 +1.00 +PDF [Gyr−1] +All galaxies +∆τQ +∆tLB,B +∆tLB,R +0.00 +0.25 +0.50 +0.75 +1.00 +PDF [Gyr−1] +Centrals +−4 +−2 +0 +2 +4 +Sharkfit-Shark +0.00 +0.25 +0.50 +0.75 +1.00 +PDF [Gyr−1] +Satellites +Figure A1. Recovery of 𝑡LB,B (in hatched blue), 𝑡LB,R (hatched red), and 𝜏Q +(solid black) from shark red galaxies with ProSpect. To avoid visualisation +artefacts due to the discreteness of all three values shown (Δ𝑡LB,B, Δ𝑡LB,R, +and Δ𝜏Q), the PDFs shown have been constructed using the gaussian_kde +Gaussian Kernel Density Estimator (KDE) function from scipy. +da Cunha E., Charlot S., Elbaz D., 2008, MNRAS, 388, 1595 +pandas +development +team +T., +2022, +pandas-dev/pandas: +Pandas, +doi:10.5281/zenodo.7093122, +https://doi.org/10.5281/ +zenodo.7093122 +APPENDIX A: RECOVERY OF 𝜏Q FROM shark WITH +ProSpect +Figure A1 shows the recovery of 𝑡LB,B, 𝑡LB,R, and 𝜏Q of shark +galaxies using ProSpect. In general, we find small median biases +in the recovery of 𝜏Q (≲ 0.03 Gyr) but we find a large scatter in the +recovery (16th–84th percentile range of ∼ 1.1 Gyr), indicating that +the population as a whole is reasonable recovered but not individual +galaxies. The most striking feature shown is that 𝜏Q is better re- +covered than either 𝑡LB,B or 𝑡LB,R, e.g., we better recover how fast +galaxies become red rather than when they leave (enter) the blue +(red) population. While it is possible that 𝜏Q is intrinsically more +constraining than 𝑡LB,{B,R}, we believe that this is more likely a +consequence of 𝜏Q being more easily recovered with the chosen +SFH/𝑍H models in ProSpect (see section 2 of Paper I for more +details). +We find that Δ𝜏Q trends with other properties, like stellar mass +or infall category (see Section 4.3), are almost completely accounted +for by the central/satellite classification, with centrals showing a +worse recovery than satellites. E.g., we find that 𝜏Q recovery wors- +ens with increasing mass, and that category (iii) satellites are better +recovered than those in category (i), but those are a consequence of +the dominating type of galaxies as a function of stellar mass and that +category (iii) ((i)) galaxies quenched as satellites (centrals), respec- +tively. The one exception to this are ∼1010.5 M⊙ centrals, which are +the main driver for the skew towards under-estimated 𝜏Q values. The +difference between centrals and satellites is likely a consequence of +the SFH model we use in ProSpect, a skewed-Gaussian SFH, be- +ing better suited to model the quenching of the latter. This should +not be necessarily understood as rejuvenation being a key factor, as +few red galaxies undergo a rejuvenation episode (see appendix A +of Paper I), but rather that limited gas replenishment can extend the +time quenching in a manner that is not well-captured by a skewed +Gaussian. +APPENDIX B: CORROBORATION OF THE TIME +EVOLUTION OF 𝜏Q, OR LACK OF THEREOF +To explore if any of our samples display a time-dependent 𝜏Q dis- +tribution, when comparing two lookback time bins set a limit to +the maximum 𝜏Q included in the comparison. This is to remove +the possible bias due to the larger span of 𝜏Q values that we can +measure at the lower lookback time bin. I.e., when comparing the +1 ≤𝑡LB,R/Gyr < 2.5 and 2.5 ≤𝑡LB,R/Gyr < 4 bins, we set the +upper 𝜏Q limit for both bins at 6 Gyr, as that is the largest 𝜏Q we can +measure at a looback time of 4 Gyr given our chosen starting point +of 10 Gyr (Section 2). This process ensures an equal 𝜏Q complete- +ness for the two lookback time bins being compared, enabling us to +study whether the 𝜏Q distribution evolves with cosmic time or not. +Figure B1 shows the measured 𝜏Q as a function of stel- +lar mass at two different 𝑡LB,R bins (2.5 ≤𝑡LB,R/Gyr < 4 and +4 ≤𝑡LB,R/Gyr < 5.5) and compares them to the lowest 𝑡LB,R bin +(1 ≤𝑡LB,R/Gyr < 2.5). shark shows a strong consistency when +comparing similarly-selected samples at different lookback times, +indicating that 𝜏Q does not depend on lookback time for this sam- +ple. In contrast, GAMA exhibits a strong evolution. Galaxies with +𝑀★ <∼1010.5 M⊙ that became red at a lookback time of 4–5.5 +Gyr have 𝜏Q values that are a factor of ∼ 2 shorter than those of +similarly-selected galaxies that transitioned in the 1–2.5 Gyr range, +with galaxies above that stellar mass showing a smaller evolution +in 𝜏Q. A similar decrease by a factor of ∼ 2 is evident in sharkfit, +though without the stellar mass dependence seen in GAMA. While +this suggests that this evolution is at least partially due to our mod- +elling choices in ProSpect, it is not obvious that this can fully +explain it, as GAMA and sharkfit display different mass dependen- +cies and measured timescales. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–16 (2023) + +Forensic quenching timescales +17 +M⋆(z ∼ 0.06) [M⊙] +0 +1 +2 +3 +4 +τQ [Gyr] +GAMA +M⋆(z ∼ 0.06) [M⊙] +Shark +M⋆(z ∼ 0.06) [M⊙] +Sharkfit +2.5 ≤ tLB,R/Gyr < 4 +1 ≤ tLB,R/Gyr < 2.5 +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +0 +1 +2 +3 +4 +τQ [Gyr] +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +109 +1010 +1011 +M⋆(z ∼ 0.06) [M⊙] +4 ≤ tLB,R/Gyr < 5.5 +1 ≤ tLB,R/Gyr < 2.5 +Figure B1. Comparison between the 𝜏Q measured at two different lookback time bins with a comparable 𝑡LB,R–𝜏Q selection. The top row compares the 𝜏Q +distribution between the 1 ≤𝑡LB,R/Gyr < 2.5 and 2.5 ≤𝑡LB,R/Gyr < 4 bins, the bottom row between 1 ≤𝑡LB,R/Gyr < 2.5 and 4 ≤𝑡LB,R/Gyr < 5.5. The +solid lines indicate the 𝜏Q running median, and the dashed lines and shaded areas the 16-84th percentiles, with those in colour being measured at the respective +𝑡LB,R bins and those in black from the lowest 𝑡LB,R bin. Each column shows the results for a different sample, left to right: GAMA, shark, and sharkfit. +MNRAS 000, 1–16 (2023) + diff --git a/ptE2T4oBgHgl3EQfKgYC/content/tmp_files/load_file.txt b/ptE2T4oBgHgl3EQfKgYC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0e0b6aa0bb2fc3e0a143222ef858f9dfc36594f2 --- /dev/null +++ b/ptE2T4oBgHgl3EQfKgYC/content/tmp_files/load_file.txt @@ -0,0 +1,1407 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf,len=1406 +page_content='MNRAS 000, 1–16 (2023) Preprint 11 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 Galaxy quenching timescales from a forensic reconstruction of their colour evolution Matías Bravo1,2★, Aaron S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Robotham1,3, Claudia del P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Lagos1,3, Luke J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Davies1, Sabine Bellstedt1 and Jessica E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Thorne1 1International Centre for Radio Astronomy Research (ICRAR), M468, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2Department of Physics & Astronomy, McMaster University, 1280 Main Street W, Hamilton, ON, L8S 4M1, Canada 3ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' in original form ZZZ ABSTRACT The timescales on which galaxies move out of the blue cloud to the red sequence (𝜏Q) provide insight into the mechanisms driving quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Here, we build upon previous work, where we showcased a method to reconstruct the colour evolution of observed low-redshift galaxies from the Galaxy And Mass Assembly (GAMA) survey based on spectral energy distribution (SED) fitting with ProSpect, together with a statistically-driven definition for the blue and red populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We also use the predicted colour evolution from the shark semi-analytic model, combined with SED fits of our simulated galaxy sample, to study the accuracy of the measured 𝜏Q and gain physical insight into the colour evolution of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In this work, we measure 𝜏Q in a consistent approach for both observations and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' After accounting for selection bias, we find evidence for an increase in 𝜏Q in GAMA as a function of cosmic time (from 𝜏Q∼ 1 Gyr to 𝜏Q∼ 2 Gyr in the lapse of ∼ 4 Gyr), but not in shark (𝜏Q≲ 1 Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Our observations and simulations disagree on the effect of stellar mass, with GAMA showing massive galaxies transitioning faster, but is the opposite in shark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We find that environment only impacts galaxies below ∼1010 M⊙ in GAMA, with satellites having shorter 𝜏Q than centrals by ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='4 Gyr, with shark only in qualitative agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Finally, we compare to previous literature, finding consistency with timescales in the order of couple Gyr, but with several differences that we discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Key words: galaxies: evolution – software: simulations – techniques: photometric 1 INTRODUCTION One of the most striking features of galaxies in the local Universe is the optical colour bimodality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Strateva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Blanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Baldry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Driver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2006), with most galaxies being either blue or red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Compared to these populations, there are comparatively few galaxies in the intermediate region, often referred to as the "green valley", (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Wyder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Stars are the dominant source of the light emitted by most galaxies (at low redshift), suggesting that this bimodality is a consequence of the presence of two dominant stellar populations for galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' As the (intrinsic) colour of stars is mainly driven by their age, the colour bimodality is a reflection of a bimodality in the recent star formation in galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' ★ E-mail:bravosam@mcmaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='ca These populations are also characterised by intrinsically differ- ent galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Red galaxies are preferentially of early-type morphology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Bershady et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Mignoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Schaw- inski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014), more massive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Baldry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2015), and found in denser environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Baldry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Stud- ies have also shown that this bimodality is seen across cosmic time, with the fraction of galaxies in the red population increasing towards recent times (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' It has also been found that the first galaxies that joined the red population are more massive than those that have joined at more recent times, a process called downsizing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Cowie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Brinchmann & Ellis 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Heavens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Combined, these observations present a broad picture where galaxies grow as part of the star-forming blue population, with some of them eventually ceasing to form stars and joining the red population (commonly © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='03702v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='GA] 9 Jan 2023 2 Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' referred to as quenching, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Blanton 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Faber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The relative lack of galaxies located in the green valley implies short timescales to transition in colour (quench) for the galaxies that join the red population (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Bremer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Different mechanisms to quench star formation are expected to do so on different timescales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Kaviraj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Wetzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Wheeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014), hence, studying these timescales can offer a view into the physical processes that govern galaxy evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Theoretical models are a critical tool to explore these mechanism, as we gain insight by testing their predictions against results from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' A well-known example in the literature is that a quenching mechanism capable of stopping gas accretion onto galaxies is required to produce massive red galaxies, usually assumed to be driven by active galactic nuclei or shock heating of the halo gas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Cattaneo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Croton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Lagos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' A historical challenge for simulations has been their inability to produce colour distributions well-matched to observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Weinmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Font et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Coil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2008), though re- cent advances have largely ameliorated this tensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Trayford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Lagos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' These advances now enable the exploration of the colour evolution of galaxies with theoretical models, leading to the prediction of the timescales on which galaxies transition from being blue to red (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Trayford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This colour evolution is not directly measurable from observations and can only be inferred (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Smethurst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Rowlands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Phillipps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This means that results cannot be directly compared to the predictions of theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Further complicating comparisons is the lack of a unified definition for how to measure the colour transition timescales, or even what galaxies should be classified as blue or red (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', see classifications by Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Bremer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2022, hereafter Paper I), we described a novel method to reconstruct the colour evolution of low-redshift observed galaxies from the Galaxy And Mass Assembly (GAMA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Driver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Liske et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2015) survey, using the ProSpect spectral energy distribution (SED) fitting tool (Robotham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We tested this recovery by performing the same procedure with a comparable sample of galaxies generated with the shark semi-analytic model (SAM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Lagos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018, 2019), finding that we can accurately recover the colour evolution of the last ∼ 6 Gyr for galaxies with current masses above ∼109 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Finally, we provided a statistically- motivated definition for the blue and red populations, their evolution through cosmic time, and demonstrated the resulting probabilities of galaxies belonging to either blue or red population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In this work we now utilise these results to explore how quickly galaxies transition from being blue to red, in a novel approach that is consistent and directly comparable for both observations and sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In Section 3 we explore the distribution of probabilities of galaxies being red, to construct statistically-motivated definitions for when a galaxy is certainly a member of of either population, and when is transitioning between both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We then use that classifi- cation to explore the timescale on which galaxies transitioned from blue to red in Section 4, exploring possible time, mass, and envi- ronmental effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For conciseness, we will refer to this blue-to-red transition timescale as 𝜏Q throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In Section 5 we dis- cuss our results, both for the physical implications of the timescales we measure and to compare with the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Finally, we present our conclusions in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In this work, we adopt the Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2016) ΛCDM cosmology, with values of matter, baryon, and dark energy densities of Ω𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0488, and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6879, respectively, and a Hubble parameter of H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='51 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2 GALAXY CATALOGUES In this work, we use the data set presented in Paper I, which we briefly outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This data set is composed of the intrinsic colour (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', not attenuated by dust) and stellar mass histories for three low-redshift galaxy samples, both derived from their star formation and metallicity histories (SFH and 𝑍H, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The first one is comprised of ∼ 7, 000 galaxies from the GAMA survey used in Bellstedt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The other two are each comprised of ∼ 30, 000 GAMA-like galaxies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', 𝑟apparent < 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='8 mag) from the shark SAM (Lagos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For the GAMA sample, we reconstructed their colour evolution from the star formation and metallicity histories inferred from the SED fitting by Bellstedt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2020)1 by combining these histories with the stellar population synthesis model used for the fitting (Bruzual & Charlot 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The two shark samples contain the same galaxies, the difference is how we constructed the colour and stellar mass evolution: one sample is the predicted evolution from the simulation itself (which we will refer as shark);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' the other presents the inferred evolution from SED fitting the shark galaxies with the same method as with the GAMA sample (we will refer to this sample as sharkfit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Section 2 of Paper I contains the detailed description of these three samples, with Appendix A offering a deeper exploration of our SED modelling choices for the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Inspired by Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2015), in Paper I we modelled the colour-mass distribution for each sample with a time-and-mass- dependent Gaussian Mixture Model (GMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Consistent with the modelling by Baldry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2004) and Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2015), we de- scribed the colour-mass distribution of galaxies in Paper I with two evolving populations: blue and red2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' These GMMs are described by five parameters: the relative fraction of blue (or red) galax- ies, and the means and standard deviations of each population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In Paper I we presented a two-step parameterisation of these param- eters, first as a function of stellar mass, and second as a function of lookback time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For the stellar mass parameterisation, based on the distributions of the GMM parameters as a function of stellar mass, we chose to parameterise the relative blue/red fractions with a logistic curve, and with first-order polynomials for the means and standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We then parameterised the time evolution of the stellar mass-dependent Gaussian parameters, using second- and third-order polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Section 3 of Paper I provides the complete description of this modelling, with section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1, figure 2 and table 1 offering an simple overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' With a complete parameterisation of the evolution of the colour 1 The SFH model adopted by Bellstedt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2020) is a skewed Gaussian, a parametric model but significantly more flexible than other common para- metric models in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', da Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Noll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Carnall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Boquien et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2019), but still unable to model rejuve- nation episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' While not unique among SED fitting models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Carnall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2021), the use of ProSpect in the literature has been unique in the assumption that gas metallicities evolves, modelling it as a linear scaling of the mass growth of galaxies (Bellstedt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2 We did test using three components, but we found no statistical evidence for a third (green) population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' See section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3 of Paper I for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' MNRAS 000, 1–16 (2023) Forensic quenching timescales 3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='4 M⋆ [M⊙] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 u − r [mag] tLB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='4 Gyr 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='4 M⋆ [M⊙] tLB = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='7 Gyr 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='4 M⋆ [M⊙] tLB = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 Gyr Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The colour evolution of a single galaxy and its transition from the blue to the red population from the results Paper I, with the galaxy CATAID=92739 from the GAMA survey used for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Each panels shows a small section of the colour-mass plane for the GAMA survey at three different lookback times, with the coloured contours showing the probability of being red for a a galaxy in any given position in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The complete evolution track of the galaxy is shown by the black and white line running from the bottom left to the top right of each panel, with the position of the galaxy in the corresponding lookback time of each panel shown with a star marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' populations for all three samples, we can calculate the probability for any galaxy belonging to the blue or red population at any given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' As in figure 12 of Paper I, in this work we choose to show the probability of being red, 𝑃R 3 , which is calculated from the Gaussian Mixture Model with which we model the galaxy colour distribution (see sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='4 of Paper I for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We also showed that our colour-based classification leads to a sensible separation in specific star formation rate as a function of stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The transition zone is not cleanly defined in this space, as expected from the scatter between colour and specific star formation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In Paper I, through the comparison of the colour evolution of the three samples (GAMA, shark, and sharkfit), we found that the reconstruction of the colour evolution becomes biased by the modelling choices in the SED fitting for lookback times above ≳ 6 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For this reason, while we will measure colour evolution from a lookback time of 10 Gyr onward and show some of our results at higher lookback times, we mainly focus on the 𝜏Q we measure below a lookback time of 6 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In this work, we use 𝑃R to calculate 𝜏Q, defining the lookback times when a galaxy leaves the blue population (𝑡LB,B) and when it joins the red population (𝑡LB,R), which are related to 𝜏Q as: 𝜏Q = 𝑡LB,B − 𝑡LB,R, (1) where we define these lookback times such that 𝑡LB,B > 𝑡LB,R (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', 𝜏Q> 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In Paper I we chose a time step of 100 Myr to reconstruct the colour evolution of galaxies, meaning that the shortest measurable 𝜏Q is 100 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Figure 1 shows an example of the data set we constructed in Paper I and use in this work to measure 𝜏Q, with both the evolutionary tracks in the colour-mass plane of individual galaxies and our model to calculate 𝑃R at any point in the 𝑡LB–𝑀★– (𝑢 − 𝑟) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Our example galaxy moves from being likely blue 3 Formally the probability of being red is a function of lookback time, stellar mass, and colour, but for brevity we will refer to it as 𝑃R instead of 𝑃R(𝑡LB, 𝑀★, 𝑢 − 𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 PR 10−2 10−1 100 101 102 PDF GAMA Shark Sharkfit Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Distribution of probability of being red for all galaxies and all time steps below 6 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' As in Paper I, the orange line shows the distribution for GAMA, cyan for shark, and purple for sharkfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Each bin spans 1% in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Highlighted in blue/red are the probability ranges where we define a galaxy as being blue/red (𝑃R= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='02 and 𝑃R= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='98, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' until a lookback time of ∼ 4, to have a similar probability of being either blue or red at ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='7 Gyr (𝑃R∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5), to likely becoming red at ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' What is not immediately obvious from this Figure alone is what values of 𝑃R best define 𝑡LB,B and 𝑡LB,R, which is the first aspect we will address in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 3 DISTRIBUTION AND EVOLUTION OF THE PROBABILITY OF GALAXIES BEING RED With the probabilistic blue/red classification from Paper I, the last step needed to measure 𝜏Q is the choice of probabilities at which a MNRAS 000, 1–16 (2023) 4 Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 10−2 10−1 100 101 102 PDF GAMA 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙ 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 M⊙ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 M⊙ 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙ Shark Sharkfit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='9 PR 10−2 10−1 100 101 102 PDF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='9 PR tLB ∈ [1, 2) Gyr tLB ∈ [2, 3) Gyr tLB ∈ [3, 4) Gyr tLB ∈ [4, 5) Gyr tLB ∈ [5, 6) Gyr tLB ∈ [6, 7) Gyr tLB ∈ [7, 8) Gyr tLB ∈ [8, 9) Gyr tLB ∈ [9, 10) Gyr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='9 PR Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Distribution of the probability of galaxies being red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Each column shows the probability distribution for single sample, from left to right: GAMA, shark, and sharkfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The top row shows the distribution at all time steps below 6 Gyr binned by stellar mass at observation time, with bins of increasing mass shown with lighter colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The bottom row shows the distribution at all stellar masses binned by lookback time, with bins of increasing lookback time in lighter colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Highlighted in blue (red) are the probability ranges where we define a galaxy as being blue (red), as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' galaxy is considered to be a part of the blue or red populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' While ultimately this is an arbitrary choice, we will use the distribution of our calculated probabilities to inform this choice, just like our choice of a GMM to describe the colour populations in Paper I was informed by the reconstructed colour distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' As the green valley is sparsely populated, most of the mass of the PDF will be near the edges (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', near 𝑃R= 0 and 𝑃R= 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We use the second derivative of the decrease of the PDF from the edges towards the centre as a guide for our choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Figure 2 shows the distribution of probabilities of being red (𝑃R), stacked from all time steps and stellar masses given our selection criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The transition from the extremes of the probability range is dramatic, with a ∼ 2 dex (∼ 1 dex) decrease in the PDF from the 0-1% bin (99-100%) to the 1- 2% bin (98-99%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This would suggest 𝑃R> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='99 (𝑃R< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='01) is a reasonable criterion for a galaxy to be confidently classified as blue (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' While such a selection will work on average, the distribution of probabilities may depend on stellar mass and/or lookback time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' To examine this, Figure 3 shows the probability distributions for all samples binned by both stellar mass and lookback time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' GAMA and shark show opposite trends, with the former exhibiting a probabil- ity distribution that is mass-independent but time-dependent, and the latter being mass-dependent but time-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This differ- ence suggest that we expect a stellar mass trend for 𝜏Q in shark, and a lookback time trend in GAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' sharkfit exhibits a mix of the trends in GAMA and shark, suggesting that our modelling choices in ProSpect may be impacting our 𝜏Q measurements, but also that they are not completely dictated by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' While for most of the PDFs shown, a choice of 𝑃R< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='01 (𝑃R> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='99) would still lead to a strong blue (red) classification, this is not true for the two highest mass bins in shark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For this reason we will use a more conservative classification of 𝑃R< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='02 for blue galaxies, 𝑃R> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='98 for red galaxies, for the rest of this work (see also the bottom row of Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We note that small variations to these limits do not affect the qualitative nature of our results, nor do they lead to strong quantitative changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1 Time evolution of the fractions of blue, transitional, and red galaxies We explore the fractions of galaxies in blue/transitional/red regions across cosmic time in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We also show in Figure 4 the frac- tions for both central and satellite galaxies, according to their classi- fication at observation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For this, we follow the same convention adopted in Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2020) of treating all isolated and central group galaxies4 from GAMA as centrals, and the remaining galaxies as satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Since we demonstrated in that work that central/satellite 4 Those with RankIterCen = 1 in the GAMA Galaxy Group Catalogue, from the iterative ranking procedure defined in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1 of Robotham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' MNRAS 000, 1–16 (2023) Forensic quenching timescales 5 1 2 3 4 5 6 7 8 9 tLB [Gyr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 f Blue galaxies 1 2 3 4 5 6 7 8 9 tLB [Gyr] True central/satellite classification Transition galaxies 2 4 6 8 10 tLB [Gyr] Red galaxies 1 2 3 4 5 6 7 8 9 tLB [Gyr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 f Blue galaxies 1 2 3 4 5 6 7 8 9 tLB [Gyr] GAMA-like central/satellite classification Transition galaxies GAMA all galaxies centrals satellites Shark all galaxies centrals satellites Sharkfit all galaxies centrals satellites 2 4 6 8 10 tLB [Gyr] Red galaxies Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Time evolution of the fraction of galaxies classified as blue/transitional/red, as a function of the central/satellite classification for shark and sharkfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Galaxies included correspond to those above the evolving mass completeness limits at 𝑧 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06 defined in Paper I (see their section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2), corresponding to 𝑀★(𝑧 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) ≥109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1 M⊙ for GAMA and 𝑀★(𝑧 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) ≥109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 M⊙ for shark/sharkfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Each column shows a different population: blue in the left column, transitional in the middle, and red in the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In each panel the corresponding population is shown for our three samples, with the combined central+satellite fraction is shown in solid lines, centrals only with dashed lines, and satellites with dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' shark and sharkfit are shown in the top row using the central/satellite classification from the simulation, and using a GAMA-like classification in the bottom row (23% confusion, following the results from Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Chauhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Line colours are as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Columns are as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The results for GAMA are identical in each column, they are repeated for easier comparison with both classifications from the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Selected galaxies GAMA Shark Sharkfit Are red by 𝑡LB = 1 Gyr 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='9% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='7% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2% Became red at 𝑡LB < 10 Gyr 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0% < 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2% Were blue at 𝑡LB = 10 Gyr and are red at 𝑡LB = 1 Gyr 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3% 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0% Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Percentage of the total population of galaxies that are currently red, that became red after 𝑡LB = 10 Gyr, and that transitioned from blue to red after 𝑡LB = 10 Gyr, from all three samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The bottom row is the sample selected to measure 𝜏Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' confusion plays is an important factor, we show the results for both the true central/satellite classification in shark/sharkfit and a con- fused classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We note that we use a higher level of confusion than in Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2020), 23% instead of 15%, because the sample we use from GAMA in this work is limited to a significantly lower redshift (𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06 instead of 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This elevated confusion can be seen in figure 3 of Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2020), and we first presented and tested this higher value in Chauhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The time dependence (independence) of the density for inter- mediate values of 𝑃R seen for GAMA and sharkfit (shark) in Figure 3 is clearly reflected in the time evolution of the transitional fraction of galaxies shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The red fraction is in excel- MNRAS 000, 1–16 (2023) 6 Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' PDF GAMA Shark Sharkfit 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 2 4 6 8 tLB,R [Gyr] 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The lookback time when red galaxies joined the red population (𝑡LB,R) as a function of current (𝑧 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) stellar mass, together with the stellar mass distribution, for each of our samples selected following Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For each individual sample, the solid lines indicates the 𝑡LB,R running median, the dashed lines the running 16-84th percentiles, both using the same bins as the stellar mass histograms of the top panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The black markers indicate the stellar mass and 𝑡LB,R median point for each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The background contours indicate the smooth distribution obtained using the gaussian_kde Gaussian Kernel Density Estimator (KDE) function from scipy, to avoid visualisation artefacts due to the discreteness of our data in 𝑡LB,R–𝜏Q space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The lightest contour shows the highest-density region containing 99% of the mass of the Gaussian KDE, with the rest of the contours evenly spaced in percentage of the mass contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' lent agreement between shark and sharkfit, which indicates that we are accurately recovering this fraction with ProSpect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In con- trast, the transitional fractions in sharkfit are in better agreement with GAMA than shark, suggesting that this is to some degree affected by the modelling choices in ProSpect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Central galaxies show a higher blue fraction than satellites in all samples, but there are differences across samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' shark and sharkfit predict a higher fraction of blue centrals relative to GAMA, at lookback times of ≲ 5 Gyr for the former and at all lookback times for the latter, though including central/satellite clas- sification confusion lessens this tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The opposite trend is true for the red population, being under-estimated by shark/sharkfit compared to GAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Interestingly, while sharkfit shows a strong under-prediction of the transitional fraction of centrals at ≳ 3 Gyr relative to shark, the difference is mostly absorbed by the blue fraction, which is overestimated (underestimated) in sharkfit above (below) a lookback time of ∼ 2 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This points to the 𝑡LB,B recov- ered for centrals being biased towards later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' shark and sharkfit exhibit a significantly higher fraction of red satellites compared to GAMA, reaching ∼ 80% at 𝑡LB = 1 Gyr, a factor of ∼ 3 larger than observations, but this tension is strongly reduced when accounting for central/satellite classification confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The transitional satellites in sharkfit show a similar difference to those in shark as previously mentioned for centrals, but unlike centrals, the under-abundance of transitional satellites in sharkfit is balanced out by an over-abundance of both blue and red satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This difference suggests that the SFH model parameterisation adopted in ProSpect may cause the transition measured to be too fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' GAMA and sharkfit exhibit a qualitatively similar evolution for the transition fraction, and they come into quantitative agreement at lookback times of ≳ 6 Gyr, in line with the results from Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 4 DISTRIBUTION AND TIME EVOLUTION OF 𝜏Q Defining both 𝑡LB,B and 𝑡LB,R is straightforward for GAMA and sharkfit, as 𝑃R is monotonically-increasing due to our choice of SFH in ProSpect5 , with 𝑡LB,B (𝑡LB,R) simply being the last (first) time the galaxy was a member of the blue (red) population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This is not true for shark, and while rejuvenation in shark is not a common occurrence in general (see Appendix A2 of Paper I), the measurement of 𝜏Q for galaxies that do rejuvenate presents a chal- lenge (mostly massive centrals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' To measure 𝜏Q in shark, we first select galaxies that are blue at 𝑡LB = 10 Gyr and that are red 𝑡LB = 1 Gyr (to ensure measurable timescales), trace the continuous time span during which the galaxy was red, and the latest time before this period that the galaxy was blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Table 1 shows the fraction of red galaxies in our three samples, indicating that selecting galaxies being blue at least at a lookback time of 10 Gyr discards ∼20–40% of the current red galaxies, with GAMA seeing the largest reduction in sample size and shark the smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Before we explore the measured 𝜏Q from our samples, we first discuss the 𝑡LB,R distributions in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' GAMA exhibits a clear trend in 𝑡LB,R with stellar mass, with more massive galaxies be- coming red at earlier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In contrast, the 𝑡LB,R distribution in both shark and sharkfit are broadly consistent for stellar masses above ≳ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The overall good agreement between both shark 5 The caveat to this statement is that, depending on the details of the evolu- tion of the galaxy colour populations as a whole, a galaxy may see a decrease in 𝑃R without changing its colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We do observe this behaviour in both GAMA and sharkfit, being particularly clear for galaxies above ∼1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙ in the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For this reason, we force a monotonic time evolution of 𝑃R for galaxies that cross our 𝑃R= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='98 threshold in these two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We find the maximum 𝑃R for all galaxies after they become red, and then set all subsequent values of 𝑃R to this maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' MNRAS 000, 1–16 (2023) Forensic quenching timescales 7 1 2 3 4 τQ [Gyr] GAMA 1 ≤ tLB,R/Gyr < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 4 ≤ tLB,R/Gyr < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 7 ≤ tLB,R/Gyr < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 1 2 3 4 τQ [Gyr] Shark 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 1 2 3 4 τQ [Gyr] Sharkfit 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Blue-to-red transition timescales (𝜏Q) as a function of current stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The distributions are shown for three 𝑡LB,R bins: 1 ≤𝑡LB,R< 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 (left column), 4 ≤𝑡LB,R< 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 (middle column), and 7 ≤𝑡LB,R< 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 (right column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Each row corresponds to a different sample, from top to bottom: GAMA, shark, and sharkfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Solid lines, dashed lines, markers and contours as in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The diagonally-hatched region indicates where the 𝜏Q measurements become incomplete in the corresponding 𝑡LB,R bin, and the cross-hatched where no 𝜏Q measurement is possible (only visible on the right-most column due to our choice of limits for the 𝑦-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Note that the 𝑡LB,R bin shown in the left column lies in the range of lookback times that we found affected by SED-fitting-related biases in Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' and sharkfit indicates that there are no strong biases in our GAMA measurements, hence the strong difference between GAMA and shark/sharkfit is not a consequence of our colour evolution recon- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In other words, the fact that we do not find a downsizing trend in shark/sharkfit as strong as in GAMA is a short-coming of the physical models in shark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Figure 5 also shows the overall stellar mass distribution of the three samples, which are in good agreement, thought shark/sharkfit show a bimodality that is not clear in GAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1 𝜏Q distribution of the overall galaxy population In Figure 6 we show the distribution of 𝜏Q as a function of stel- lar mass, divided into three lookback time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The distributions are roughly consistent across cosmic time in shark, the largest difference being the increased dispersion above ∼1010 M⊙, which suggests that there is no strong time evolution of the 𝜏Q distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In contrast, the 𝜏Q distribution of GAMA changes as a function of 𝑡LB,R, with the median 𝜏Q increasing by a factor of ∼ 3 in the span of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' GAMA and sharkfit display similar timescale-mass relations in the highest 𝑡LB,R bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This is, in line with the results in Paper I, where we found a strong similarity in the galaxy distributions of both GAMA and sharkfit in colour-mass space at high lookback times (𝑡LB > 6 Gyr), likely driven by dust parameter degeneracies (see Appendix A of Paper I for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The timescale-mass relations measured in shark and sharkfit are in good agreement in the other two lookback time bins, indicating that we can recover MNRAS 000, 1–16 (2023) 8 Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' PDF GAMA (centrals) Shark (centrals) Sharkfit (centrals) 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 1 2 3 4 τQ [Gyr] 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] PDF GAMA (satellites) Shark (satellites) Sharkfit (satellites) 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 1 2 3 4 τQ [Gyr] 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] Confused central/satellite classification True central/satellite classification Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 𝜏Q as a function of current stellar mass, together with the stellar mass distribution, divided between centrals and satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Only galaxies with 1 ≤𝑡LB,R< 6 Gyr are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For each individual sample, the solid/dashed (dash-dotted/dotted) lines indicate the 𝜏Q running median/16-84th percentiles for the GAMA-like (true) central-satellite classification, both using the same bins as the stellar mass histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The black cross (open) markers indicate the stellar mass and 𝑡LB,R median point for the GAMA-like (true) central-satellite classification in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Contours as in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The coloured (black) histograms show the measured stellar mass distribution for the GAMA-like (true) central-satellite classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Note that the gap seen in the running median for the true centrals in shark corresponds to a mass bin where no galaxies are present (see histogram above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' this with ProSpect, which validates the difference between both and GAMA as real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The only significant difference between shark and sharkfit at these lower lookback times is that we do not re- cover the longest 𝜏Q from shark, possibly due to episodes of weak rejuvenation extending the time period galaxies remain in the tran- sitional region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The 𝜏Q–𝑀★ relation that we observe in GAMA is in clear tension with that we predict in shark, which also exhibit opposite trends with stellar mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', low-mass galaxies in GAMA take much longer to quench at recent times than in shark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We ex- plore the effect of selection biases in the measured 𝜏Q evolution in Appendix B, where we find that GAMA does exhibit a strong time evolution of the 𝜏Q distribution, a mild evolution in sharkfit, and that the 𝜏Q distribution in shark is independent of cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' MNRAS 000, 1–16 (2023) Forensic quenching timescales 9 1011 1012 1013 1014 Mhalo [M⊙/h] 0 250 500 750 1000 1250 n [dex−1] Shark 1011 1012 1013 1014 Mhalo [M⊙/h] 0 100 200 300 400 n [dex−1] GAMA All haloes with satellites Haloes with red satellites (true) Haloes with red satellites (confused) Haloes with Ng ≥ 5 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Comparison of the halo mass distribution between GAMA and shark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' shark is shown on the left panel, GAMA is shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Three selections are shown here: all haloes with at least one satellite (light grey), all haloes with at least one red satellite (light red, shaded area for GAMA-like central/satellite classification, dashed line for the true shark classification), and haloes with at least five galaxies (dark magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Note that both panels have different scales for the y-axis, to focus on the effect of the different selections rather the different sizes of our samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 Environmental effects on 𝜏Q We now explore how 𝜏Q compares between central and satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Figure 7 shows both the stellar mass distribution and 𝜏Q as a function of their current stellar mass, divided into centrals and satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For shark and sharkfit we show the results for both true and GAMA-like central/satellite classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' True centrals and satellites in shark show a significant differ- ence in 𝜏Q for galaxies below ∼1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙, with centrals exhibiting longer timescales than satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Satellites in sharkfit are in good agreement with those from shark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Centrals show less of an agree- ment for 𝜏Q, in particular the timescales in sharkfit for centrals of ∼109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙ (∼1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙) are shorter than in shark by a factor of ∼ 4 (∼ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In comparison, GAMA centrals and satellites only differ at 𝑀★ ≲1010 M⊙, with 𝜏Q of satellites being ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='4 Gyr than for centrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The difference between centrals and satellites is re- duced when using a GAMA-like classification for both shark and sharkfit, suggesting the possibility of a larger difference between GAMA centrals and satellites than what we measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' To further explore the differences for satellites between GAMA and shark, Figure 8 shows the mass distribution of haloes that host satellites in both samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Chauhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2021) found that the recovery of shark halo masses with the Robotham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2011) group finder, which was used to infer halo masses for GAMA, is reasonable for the high-multiplicity groups (𝑁𝑔 ≥ 5) but the quality of the recovery noticeably decreases for low-multiplicity groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Those results can account for the difference between GAMA and shark in the distribution of halo masses for haloes hosting at least one satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The majority of haloes in shark that host at least one satellite also host at least one red satellite, in strong contrast to GAMA, but the difference becomes smaller when accounting for central/satellite classification confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Figure 8 shows that the halo mass distribution of 𝑁𝑔 ≥ 5 groups in GAMA and shark are comparable, hence this should be a strong probe for the treatment and evolution of the satellites residing in such haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For centrals, we ignore 𝑁𝑔 ≥ 5 group red centrals, as there are ≲40 of the latter in GAMA and shark6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Overall, we find little difference between true central (all true satellite) and isolated (𝑁𝑔 ≥ 5 satellite) galaxies in all three samples, with the most important finding being that shark/sharkfit lack the observed numbers of ≲1010 M⊙ red isolated galaxies we find in GAMA (see Figure included in the supplemental material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3 Connection between 𝜏Q and satellite infall in shark One of the powerful features of using simulations is that they enable us to further explore the evolution of red galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In particular, here we explore how 𝜏Q is linked to the central-to-satellite transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Naturally, we can only do this study in shark and not GAMA7 , given that we do not know the infall time for the satellite galaxies in GAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Nonetheless, this will give us an indication of whether 𝜏Q is clearly linked to the physical models included shark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We classify satellite galaxies in shark in three groups, based on the lookback time of infall (𝑡LB,infall) relative to its transition from blue to red, those that: 6 There are significantly more in sharkfit, factor of ∼ 4 more than in shark, but this is a consequence of the delayed formation of the red population from our ProSpect fits to shark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This could be a result of the relative fractions of galaxies that undergo rejuvenation, as defined in Paper I, as ∼ 20% of the red centrals in shark had at least one rejuvenation episode, compared to only ∼ 4% of satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We should also note that we are using a subset of the full simulation box for shark/sharkfit, so it is possible to increase this number by a factor of ∼30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The low number from GAMA is the limit to study 𝑁𝑔 ≥ 5 group red centrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 7 We can also explore this aspect with sharkfit, but as based on the results in Appendix A, there are reasons not to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Both 𝑡LB,B and 𝑡LB,R are not as well-recovered than 𝜏Q, which leads to a significant confusion on whether a satellite became red before, during, or after infall, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', the percentage of the former increases from ∼ 4% to ∼ 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Classifying whether a sharkfit satellite became red based on 𝑡LB,R measured in shark would reduce the analysis to how well-recovered is the evolution of centrals and satellites, which is the topic of Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' MNRAS 000, 1–16 (2023) 10 Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' PDF Shark (before infall, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5%) Shark (during infall, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='8%) Shark (after infall, 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='7%) 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 1 2 3 4 τQ [Gyr] 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 𝜏Q of satellites in shark as a function of current stellar mass, and the stellar mass distribution, divided by when a galaxy became red relative to the time when the galaxy became a satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The percentage that each sample represents of the total of red satellites in shark is shown on the top labels of each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Solid lines, dashed lines, markers and contours as in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (i) became red before they became a satellite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', transitioned in colour before infall, 𝑡LB,B>𝑡LB,R> 𝑡LB,infall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (ii) were a central the last time they were blue, but were a satellite by the time they became red, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', infall happened during the colour transition, 𝑡LB,B> 𝑡LB,infall >𝑡LB,R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (iii) were still blue when they became a satellite, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', transitioned in colour after infall, 𝑡LB,infall >𝑡LB,B>𝑡LB,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The distribution of stellar mass and 𝜏Q for these categories are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' It is clear that becoming a satellite is the main driver for galaxies to become red, as 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='7% of the red satellites in shark became red before after infall (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The stellar mass distri- bution of galaxies that became red before (i) and after infall show a marked difference, with those that became a satellite while in transition (ii) showing a distribution intermediate between the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' shark galaxies galaxies that became red after infall have the shortest 𝜏Q, while those galaxies that became red before infall have the longest 𝜏Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Galaxies that became a satellite while in transition show intermediate 𝜏Q relative to the other two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 5 DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1 Comparing our 𝜏Q definition with previous literature Fundamental to any measurement of 𝜏Q is how it is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Mea- surements of 𝜏Q in the literature include derivation from star for- mation rates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Wetzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Belli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Tacchella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2022), galaxy colours (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Trayford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Bremer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' McNab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2021), and spectral properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Wheeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Rowlands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Defini- tions include timescales to cross specific thresholds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Trayford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Bremer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Tacchella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2022), 𝑒-folding timescales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Wetzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Wheeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Bremer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018), and inference from population den- sities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Rowlands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' McNab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Establishing how these different measurements compare is outside the scope of this work, and in this work we only compare our 𝜏Q definition and measurements to those in the literature that are derived from galaxy colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' While definitions from observations abound (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Smethurst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Bremer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Phillipps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2019), differences in the recovery of the intrinsic stellar light result in significant differences in the loci of the colour populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Table 2 provides a description of these selections, and the top row of Figure 10 shows how those from Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2014) and Bre- mer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Phillipps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019) compare to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The issue of definitions that adopt a fixed parameterisation instead of being a function of population properties are clear here as the match to ours is strongly sample-driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2014) green valley selection covers almost exclusively the red population for shark/sharkfit, and the Phillipps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019) covering mostly the blue population for GAMA8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Furthermore, (to the author’s knowl- edge) there are no observational selections that account for colour evolution with cosmic time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', the selections are at fixed lookback time/redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We now compare the classifications used for simulations by Trayford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019) to the one we adopt for this work, shown in the bottom row of Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Trayford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2016) classifies galaxies from the EAGLE simulation (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2015) between red, green and blue using straight lines in the colour-mass plane, where only the normalisation evolves with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' While it does overlap most of the region we classify as transitional in GAMA, it is clearly slanted as a function of stellar mass compared to our statistically-based selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The fixed nature of the selection limits also makes it a poor choice for shark/sharkfit, as it strongly overlaps the red population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The Trayford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2016) selection limits are also consistently wider 8 This is not to say that these are poor selections for the samples for which they were designed (see figures 4 and 1 of Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Phillipps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2019, respectively), which we could only assess by implementing our method to their data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' MNRAS 000, 1–16 (2023) Forensic quenching timescales 11 Reference Data type Parameter space Transition region lower limit Transition region upper limit Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2014) Observation (𝑢 − 𝑟)–𝑀★ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='25 log10(𝑀★/M⊙) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='25 log10(𝑀★/M⊙) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='24 Smethurst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2015) Observation (𝑢 − 𝑟)–𝑟 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='244tanh � 𝑟+20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='09 � + 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6 − 𝜎 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='244tanh � 𝑟+20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='09 � + 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6 + 𝜎 Trayford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2016) Simulation (𝑢 − 𝑟)–𝑀★ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 log10(𝑀★/M⊙) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='25𝑧0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 log10(𝑀★/M⊙) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='25𝑧0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='24 Bremer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018) Observation (𝑢 − 𝑟)–𝑀★ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1 log10(𝑀★/M⊙) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 log10(𝑀★/M⊙) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018) Simulation (𝑔 − 𝑟)–𝑀★ 𝜇B + 𝜎B 𝜇R − 𝜎R Phillipps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019) Observation (𝑢 − 𝑟)–𝑀★ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1 log10(𝑀★/M⊙) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 log10(𝑀★/M⊙) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019) Simulation (𝑢 − 𝑟)–𝑀★ 𝜇B + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5𝜎B 𝜇R − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5𝜎R This work Both (𝑢 − 𝑟)–𝑀★ 𝑃R> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='02 𝑃R< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='98 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Sample of literature criteria to define the green valley/transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Our classification is at the end of the table for comparison purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Several remarks need to be made for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' All literature definitions using observations include no time evolution and are valid only at low redshift (𝑧 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The definition by Smethurst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2015) references a dispersion (𝜎), but it is not clear what dispersion they are using, besides that it seems to be independent of stellar mass (see their figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Bremer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018) and Phillipps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019) use the same definition, but the former limits it to a narrow stellar mass range (1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='25 < 𝑀 ★ /M⊙ < 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='75), while the latter expands it to their full sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018) and Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019) use the same type of criteria, but they chose difference factors for the standard deviation, whether this is a consequence of their different choices for colour or not is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Note that 𝜇{B,R}, 𝜎{B,R}, and 𝑃R all are functions of lookback time, stellar mass and colour, see Section 3 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 0 1 2 (u − r)ab [mag] GAMA Shark Sharkfit 109 1010 1011 M⋆ [M⊙] 0 1 2 (u − r)ab [mag] 109 1010 1011 M⋆ [M⊙] 109 1010 1011 1012 M⋆ [M⊙] tLB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 Gyr 25–50–75% Schawinski+14 Phillipps+19 Trayford+2016 Nelson+2018 Wright+2019 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Comparison of the colour classification we adopt to two observational (Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Phillipps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2019, top row;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' in orange and teal, respectively) and three theoretical (Trayford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2019, bottom row;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' in brown, green, and cyan, respectively) literature examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The comparison is made at the lowest lookback time of our data (1 Gyr), as a rough middle point between the redshift ranges of the literature classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Each column shows the classification for one of our samples, following the same thresholds as in Figure 2, with the underlying histogram bins being coloured in blue/grey/red if they median 𝑃R of the galaxies classifies them as being blue/transitional/red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The dashed/dash-dotted/dotted contours indicate the highest density regions in each panel that contain 25/50/75% of the galaxies of each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The transitional region from the literature examples are shown by coloured bands in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The black region in the left column indicates the stellar mass below the stellar mass completeness limit for GAMA (lies below 109 M⊙ for shark and sharkfit, further details in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 of Paper I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The classifications from Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018) and Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019) are defined as functions of the means and standard deviations of each population, which is the reason why they are different across our samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' MNRAS 000, 1–16 (2023) 12 Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' than our transitional region, save for shark above ∼1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙, which would lead to an over-estimation of 𝜏Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The classifications from Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018) for IllustrisTNG and Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019) for EAGLE are more directly comparable to our classification, as they are all based on GMM fits to the colour population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Both define their green selection as: 𝐺upper = 𝜇R − 𝑓 𝜎R, (2) 𝐺lower = 𝜇B + 𝑓 𝜎B, (3) where 𝜇{B,R} and 𝜎{B,R} are the mean and standard deviation of the blue/red population, respectively, and 𝑓 is a constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018) adopts a value of 𝑓 = 1, while Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019) adopts 𝑓 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For a fair comparison, we have used the 𝜇{B,R} and 𝜎{B,R} we measured in Paper I to implement their selections, as the colour evolution in both EAGLE and IllustrisTNG may well not match those in GAMA and shark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Both are in reasonable agreement with our classification for sharkfit, with Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018) being a remarkable match, but the results for GAMA and shark show that this is just a lucky coincidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In particular, in shark both criteria fail to reproduce our probabilistically-based classification, and at low masses they lead to an over-estimation of the transitional region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' At high masses both criteria under-estimate 𝜏Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 Comparing our 𝜏Q measurements between observations and simulations A fundamental question to answer when exploring the 𝜏Q distri- bution is how it evolves with cosmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This can provide an insight into the physical processes behind the colour transformation of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' A 𝜏Q distribution invariant with time would suggest that the different mechanisms and their relative prevalence are also time invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' If 𝜏Q evolves with cosmic time instead, that suggests some combination of the following: different mechanisms operate at different times, the mechanisms’ efficiency change with time, or that the relative mix of these mechanisms evolves with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' As described in Section 4, the challenge in establishing whether 𝜏Q evolves with time is selection bias, irrespective of the chosen lim- its.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Furthermore, galaxies of different current stellar masses could have transitioned in colour at different lookback times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' When ac- counting for both lookback time and stellar mass dependencies, we find that GAMA shows a clear evolution towards longer timescales at more recent cosmic times, while the same is not evident in shark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We do find a time evolution for 𝜏Q in sharkfit, unlike shark, which suggests that the difference between GAMA and shark may be attributable in part to ProSpect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The difference in evolution of 𝜏Q as a function of stellar mass between GAMA and sharkfit suggests that some of this evolution might be real, instead of induced by our models in ProSpect, though more work would be required to make a more conclusive statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Since the 𝜏Q distribution is broadly consistent for all three samples for 𝑡LB,R< 4 Gyr, we can use that as a selection to study the connection between 𝜏Q and other galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This selection means that we are roughly halving the number of galaxies that we can explore in shark/sharkfit, as galaxies above ∼109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3 M⊙ exhibit a consistent median 𝑡LB,R of ∼ 4 Gyr in these samples, but for GAMA we will only be able to explore a small fraction of ∼1010 M⊙ galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Comparing the 𝜏Q–𝑀★ distribution for all three samples, shown in Figure 11, shows that shark under-predicts 𝜏Q by up to ∼ 2 Gyr for 109–1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙, with the tension being larger for lower stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Since the 𝜏Q distribution in sharkfit is in good agreement with that from shark, and that for masses above ∼1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙ all three samples show a good agreement, the difference between GAMA and shark at lower masses suggests that the mechanisms that quench these lower mass galaxies in shark are too efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The red population in shark is strongly dominated by satel- lites (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='4% of the sample), in apparent disagreement with GAMA (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='8%), but as shown in Paper I this is alleviated by accounting for observational confusion between centrals and satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Centrals and satellites below ∼1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙ in shark exhibit different 𝜏Q, but using a GAMA-like classification reduces the difference between both, suggesting that the similarity between centrals and satellites in GAMA could be due classification confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The choice of cen- tral/satellite classification cannot account for the strong difference in 𝜏Q between GAMA and shark/sharkfit, as satellites exhibit shorter transition timescales in shark/sharkfit than GAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Since 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2% of shark satellites became red after infall, this points to the quenching mechanisms for satellites in shark, the in- stantaneous stripping all halo gas of the galaxy, being too aggressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' A more gradual stripping of gas would allow satellites to replenish their interstellar medium (ISM) gas, increasing 𝜏Q (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Font et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Other SAMs have adopted gradual halo gas stripping models, usually combined with the inclusion of ISM stripping to balance the availability of gas for satellites (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Font et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Croton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2006, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Stevens & Brown 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Cora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' More ex- treme models can also be found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', in Henriques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2015) satellites fully retain their halo gas and their ISM in haloes of masses lower than 1014 M⊙, leading to slow exhaustion of the gas to star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The latter was required by that model to reproduce the observed red fraction of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Based on these results, we will explore in future work if the combination of gradual halo gas stripping and ISM stripping leads to longer timescales than instantaneous halo gas stripping without ISM stripping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' shark satellites that became red before infall (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', that became red while being centrals) exhibit a similar 𝜏Q distribution as those of (current) central galaxies, but they show markedly different stellar mass distribution, with the former having a significant contribution from ≲109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙ galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' These low-mass satellites that became red as centrals exhibit starburst episodes prior to infall, indicating that the otherwise temporary gas exhaustion was made permanent by the instantaneous hot gas stripping in shark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Both low-mass central and isolated red galaxies in GAMA exhibit long 𝜏Q relative to shark, suggesting that this is not the mechanism to reduce the tension between both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' A possible improvement could be to extend the effectiveness of the AGN radio-mode feedback to lower halo masses in shark, as reducing the amount of gas available for cooling into the galaxy would lead to a gradual decrease of star formation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', a long 𝜏Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3 Comparing our 𝜏Q measurements with literature results What follows now is a series of comparisons with a representative collection of literature results, both from observations and simu- lations, which are also shown in Figure 11 as comparison to our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Since the calculation of 𝜏Q could have a strong effect on the measured values, we include an overview of how they were de- rived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We remark that this means that good quantitative (or even qualitative) agreement should not be expected to be a natural re- sult, as disagreements may well stem from methodological (or even conceptual) differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Also, little discussion is presented on the MNRAS 000, 1–16 (2023) Forensic quenching timescales 13 109 1010 1011 1012 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0 τQ [Gyr] Schawinski+2014 Smethurst+2015 Trayford+2016 Bremer+2018 Nelson+2018 Rowlands+2018 Wright+2019 GAMA Shark Sharkfit Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Comparison between the 𝜏Q we have measured in this work to a variety of literature results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Solid and hatched areas indicate results that only provide a range of values for 𝜏Q, together with the stellar mass range used in the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Segmented lines indicate singular 𝜏Q values, with the extend of the line and markers on the edge the stellar mass range used to measure it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Solid lines indicate running medians of 𝜏Q as a function of stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Multiple lines may appear if the respective work provided more than one result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The results from this work are shown in colour, those from the literature in different shades of grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Note that we show only one of the two measured 𝜏Q values from Rowlands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018), as the other lies outside our choice of range for 𝜏Q (∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6 Gyr for 1011 < 𝑀★/M⊙ < 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5) possible lookback time dependence of 𝜏Q in the literature, so we will omit that aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Starting with results from observations, Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2014) inferred colour transition timescales combining colours and stellar masses of galaxies from the Sloan Digital Sky Survey (SDSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2000) with the morphological classification from Galaxy- Zoo (Lintott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For this, they first divided their galaxy sample between "blue", "green" and "red" in the colour-stellar mass plane, with the limits being straight lines chosen by visual inspec- tion (see Figure 10 and Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They generated simple colour his- tories with a exponentially-decaying SFH with different 𝑒-folding timescales, and then compared the colour evolution from these SFHs to the colour-colour distribution of both early- and late-type galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' From these comparisons they inferred a fast transition for early- type galaxies (𝜏Q≲ 250 Myr), with late-type galaxies transitioning slowly (𝜏Q≳ 1 Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Since only ∼ 20% of the GAMA sample shown in Figure 11 correspond to elliptical galaxies (according to the Driver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2022) morphological classification), the expecta- tion would be for our sample to better match the timescales that Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2014) found for late-type galaxies, which is in- deed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In contrast, shark/sharkfit show 𝜏Q in agreement with Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2014) only for galaxies above ∼1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙, with galaxies below this mass exhibiting a median 𝜏Q neither consis- tent with early-type nor late-type galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' While Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2014) find a dependence of 𝜏Q on halo mass for late-type galaxies, they do not find the same for the early-types that dominate the red population, which agrees with the similarity we find between low- and high-multiplicity groups in GAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Smethurst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2015) also used GALEX/SDSS photometry plus GalaxyZoo morphological classification (though the more re- cent GalaxyZoo2 release, Willett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2013) to infer quenching timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They also classify galaxies as blue, green or red, but use instead the definition from Baldry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2004), where everything < 1𝜎 from the local minimum in colour-magnitude as green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Fur- thermore, they also adopt a similar approach of generating sample colour evolution tracks from exponentially-decaying SFHs, though they use a Bayesian approach to find the timescale and quenching onset that best matches every galaxy in their sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For bulge- and disc-dominated galaxies they find median timescales of ∼ 1 and ∼ 2 Gyr respectively, which is in good agreement with our re- sults from GAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Their figures 8 and 11 suggest that these values may be strongly driven by galaxies quenching early in the Universe (𝑡LB,R> 6 Gyr), which we do not explore in this work due to biases in the recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' It is interesting to note that they seem to find a strong evolution toward longer timescales at more recent times (see their figures 8 through 11), though it is not clear if this is just a se- lection effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Regardless, this qualitatively agrees with our findings in GAMA, where galaxies that become red more recently do it on longer timescales than those becoming red earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Rowlands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018) used the strength of the 4000 Å break and the excess Balmer absorption from GAMA and the VIMOS Public Extragalactic Redshift Survey (VIPERS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They first divided galaxies as either post-starburst or not based on being above or below a Balmer absorption limit, and then further divided the non post-starburst between blue, green and red based on two ad hoc 4000 Å break values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They then measured the number den- sity evolution of these classifications at a wide range of redshifts (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='05 < 𝑧 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='0) to infer transition timescales at two partly overlap- ping stellar mass ranges (>1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6 M⊙ and >1011 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They found transition timescale for green valley galaxies of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6 Gyr for their mid/high-mass selection, independent of lookback time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This positive trend for timescales with stellar mass is opposite to our findings in GAMA, but it is likely driven by those being a different type of timescales, with the values themselves being significantly higher than seen in either GAMA or shark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This is likely because Rowlands et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018) measure the timescale for 𝑧 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='7 green valley galaxies would join the 𝑧 = 0 red population, whereas we measure the timescale over which observed red galaxies became red, which Schawinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2014) shows are dominated by differ- ent morphological types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' MNRAS 000, 1–16 (2023) 14 Bravo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Bremer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018) measured the fraction of galaxies in the green valley as a function of environment from GAMA, defined in colour-stellar mass space (see Figure 10 and Table 2), and com- bined it with the stellar ages of Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2011) to infer 𝜏Q 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They limited their analysis to galaxies with stellar masses in the 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='25–1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='75 M⊙ range, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1 < 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='2 and 𝑟-band axial ra- tio 𝑏/𝑎 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They found a 𝜏Q of ∼1–2 Gyr, in good agreement with our measurement of 𝜏Q for GAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Also similar to our re- sults from GAMA, they found no evidence for environmental effect from the near-constant fraction of green valley galaxies as a func- tion of group multiplicity10, but our results from shark show that central/satellite confusion can strongly diminish any environmental signature present in 𝜏Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We now focus on a comparison with literature results pre- sented for galaxy formation simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Trayford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2016) used galaxies with stellar masses of 1010–1011 M⊙ from the EAGLE simulation (Schaye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2015), and classifying them as blue, green or red by ad hoc colour selections, defined as a function of both stellar mass and redshift (see Figure 10 and Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' From these, they selected 𝑧 = 0 red galaxies and measured the timescale over which they transition from blue to red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They found a median 𝜏Q of ∼ 2 Gyr, though with a distribution strongly skewed towards shorter timescales, with a peak closer to ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' While they do not find a strong difference in the median 𝜏Q for centrals and satellites (order of a few hundred Myr), the distributions shown in their figure 10 indicate that centrals are less skewed to short timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' shark does display the same trend, though we find shorter 𝜏Q (factor of ∼ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Their results are in agreement with our results from GAMA, but it is not clear if this holds for earlier times, as they do not explore the time evolution of 𝜏Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They do not find evidence of a strong de- pendence with stellar mass, which seems in better agreement with shark than GAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Finally, we find in shark the same results that they do with regards to satellites: the majority become red when becoming a satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018) employed a similar method to ours to mea- sure 𝜏Q from the IllustrisTNG simulation (Pillepich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2018), first characterising the colour population with two Gaussian com- ponents, with parameters as function of stellar mass and redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They then defined the limits for each population, set at 1𝜎 from the mean of each population, which is the most significant difference with our probability-based approach (see Figure 10 and Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Like Trayford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2016), they also found an asymmetrical 𝜏Q distribution, skewed to shorter values, finding similar median and peak values (∼ 2 and ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='6 Gyr, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They found a depen- dence with stellar mass, with 𝜏Q peaking for galaxies of ∼1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' While the range of median 𝜏Q values they measure coincides with that from GAMA, we find a different trend with stellar mass, with the best agreement being for galaxies ≳1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They also found a weak trend for centrals below 1010 M⊙ to take longer to become red than satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Also using galaxies from the EAGLE simulation, Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 9 A similar method was also presented in Phillipps et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019), but instead they used the 𝑒-folding time from the Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2011) fits to infer how long will current green galaxies take to become red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This is the reason why we do not discuss their results, to avoid comparisons between our reconstructed evolution to their predicted evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 10 They do find evidence that galaxies in high density environment have shorter lifespans as part of the blue population than in less dense environ- ment, suggesting that a richer environment will trigger an earlier transition to red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019) used a classification close to that used by Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018), finding 𝜏Q to be in the ∼2–4 Gyr range, depending on both stellar mass and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' They found different 𝜏Q for centrals and satellites for galaxies below ∼1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙ (∼ 4 and ∼ 2 Gyr respectively), with timescales showing a inverted U-shape and both centrals and satellites peaking at ∼109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='7 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For larger stellar masses they found all galaxies to have similar 𝜏Q (∼ 2 Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Their 𝜏Q measurements are in strong agreement with those of Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2018), despite using different thresholds to measure 𝜏Q (see Table 2), which suggests that the stellar mass trend of 𝜏Q found in both works may be a consequence of how they define the limits of the blue and red populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 6 CONCLUSIONS In this work, we have used the characterisation of the colour evolu- tion of the blue and red galaxy populations we presented in Paper I, to calculate upper limits for 𝜏Q on which red galaxies transitioned from being blue to red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' For this, we first calculated the probability of all galaxies in our three samples (GAMA, shark and sharkfit) to belong to the red population, then used the distribution of this probability to define the values between which we will measure 𝜏Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Accounting for selection biases, we find evidence that 𝜏Q evolves with time only in GAMA, with 𝜏Q increasing from ∼ 1 to ∼ 3 Gyr in a time span of ∼ 4 Gyr (in shark/sharkfit 𝜏Q remains sta- ble at ≲ 1 Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Our observations and simulations do not agree on whether there is a stellar mass dependence on the lookback time when they became red, with the former strongly suggesting that cur- rent high-mass galaxies became red before low-mass galaxies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', downsizing), while the latter show no such trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We find a differ- ence between centrals and satellites in GAMA only for 𝑀★ ≲1010 M⊙, with satellites showing 𝜏Q ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='4 Gyr shorter than centrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The results from shark suggest the possibility of a larger differ- ence being hidden by observational central/satellite classification confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Finally, we find that assuming an instantaneous halo gas stripping in shark is the likely driver for the shorter-than-observed 𝜏Q for satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank Chris Power and Pascal Elahi for their role in completing the SURFS 𝑁-body DM-only simulations suite, Rodrigo Tobar for his contributions to shark, Andrea Cattaneo and Benjamin Johnson for the comments and feedback provided to the doctoral thesis on which this work is based, Ruby Wright for providing the data from Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' (2019) for Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' MB acknowledges the support of the University of Western Australia through a Scholarship for International Research Fees and Ad Hoc Postgraduate Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' LJMD and ASGR acknowledge support from the Australian Research Councils Future Fellowship scheme (FT200100055 and FT200100375, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' CdPL is funded by the ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' CdPL also thanks the MERAC Foundation for a Postdoctoral Re- search Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' SB acknowledges support by the Australian Research Council’s funding scheme DP180103740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' JET is supported by the Australian Government Research Training Program (RTP) Scholar- ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This work was supported by resources provided by the Pawsey MNRAS 000, 1–16 (2023) Forensic quenching timescales 15 Supercomputing Centre with funding from the Australian Govern- ment and the Government of Western Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We gratefully ac- knowledge DUG Technology for their support and HPC services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The GAMA input catalogue is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Sur- vey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Complementary imaging of the GAMA regions is being ob- tained by a number of independent survey programmes includ- ing GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel- ATLAS, GMRT and ASKAP providing UV to radio coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The GAMA website is http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='gama-survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='A-2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='A-3016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The analysis on this work was performed using the program- ming languages Python v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='8 (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='org), with the open source packages matplotlib (Hunter 2007), NumPy (Har- ris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2020), pandas (pandas development team 2022), SciCM (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='com/MBravoS/scicm), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' 2020), and splotch (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='com/MBravoS/splotch), in addition of the software previously described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' DATA AVAILABILITY The 𝑃R tracks and 𝜏Q catalogues generated for this work will be shared on reasonable request to the corresponding 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+page_content='00 PDF [Gyr−1] All galaxies ∆τQ ∆tLB,B ∆tLB,R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='00 PDF [Gyr−1] Centrals −4 −2 0 2 4 Sharkfit-Shark 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='00 PDF [Gyr−1] Satellites Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Recovery of 𝑡LB,B (in hatched blue), 𝑡LB,R (hatched red), and 𝜏Q (solid black) from shark red galaxies with ProSpect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' To avoid visualisation artefacts due to the discreteness of all three values shown (Δ𝑡LB,B, Δ𝑡LB,R, and Δ𝜏Q), the PDFs shown have been constructed using the gaussian_kde Gaussian Kernel Density Estimator (KDE) function from scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' da Cunha E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Charlot S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', Elbaz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', 2008, MNRAS, 388, 1595 pandas development team T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', 2022, pandas-dev/pandas: Pandas, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='7093122, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5281/ zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='7093122 APPENDIX A: RECOVERY OF 𝜏Q FROM shark WITH ProSpect Figure A1 shows the recovery of 𝑡LB,B, 𝑡LB,R, and 𝜏Q of shark galaxies using ProSpect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In general, we find small median biases in the recovery of 𝜏Q (≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='03 Gyr) but we find a large scatter in the recovery (16th–84th percentile range of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='1 Gyr), indicating that the population as a whole is reasonable recovered but not individual galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The most striking feature shown is that 𝜏Q is better re- covered than either 𝑡LB,B or 𝑡LB,R, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', we better recover how fast galaxies become red rather than when they leave (enter) the blue (red) population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' While it is possible that 𝜏Q is intrinsically more constraining than 𝑡LB,{B,R}, we believe that this is more likely a consequence of 𝜏Q being more easily recovered with the chosen SFH/𝑍H models in ProSpect (see section 2 of Paper I for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' We find that Δ𝜏Q trends with other properties, like stellar mass or infall category (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='3), are almost completely accounted for by the central/satellite classification, with centrals showing a worse recovery than satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', we find that 𝜏Q recovery wors- ens with increasing mass, and that category (iii) satellites are better recovered than those in category (i), but those are a consequence of the dominating type of galaxies as a function of stellar mass and that category (iii) ((i)) galaxies quenched as satellites (centrals), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The one exception to this are ∼1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙ centrals, which are the main driver for the skew towards under-estimated 𝜏Q values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The difference between centrals and satellites is likely a consequence of the SFH model we use in ProSpect, a skewed-Gaussian SFH, be- ing better suited to model the quenching of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This should not be necessarily understood as rejuvenation being a key factor, as few red galaxies undergo a rejuvenation episode (see appendix A of Paper I), but rather that limited gas replenishment can extend the time quenching in a manner that is not well-captured by a skewed Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' APPENDIX B: CORROBORATION OF THE TIME EVOLUTION OF 𝜏Q, OR LACK OF THEREOF To explore if any of our samples display a time-dependent 𝜏Q dis- tribution, when comparing two lookback time bins set a limit to the maximum 𝜏Q included in the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This is to remove the possible bias due to the larger span of 𝜏Q values that we can measure at the lower lookback time bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=', when comparing the 1 ≤𝑡LB,R/Gyr < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 ≤𝑡LB,R/Gyr < 4 bins, we set the upper 𝜏Q limit for both bins at 6 Gyr, as that is the largest 𝜏Q we can measure at a looback time of 4 Gyr given our chosen starting point of 10 Gyr (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This process ensures an equal 𝜏Q complete- ness for the two lookback time bins being compared, enabling us to study whether the 𝜏Q distribution evolves with cosmic time or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Figure B1 shows the measured 𝜏Q as a function of stel- lar mass at two different 𝑡LB,R bins (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 ≤𝑡LB,R/Gyr < 4 and 4 ≤𝑡LB,R/Gyr < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5) and compares them to the lowest 𝑡LB,R bin (1 ≤𝑡LB,R/Gyr < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' shark shows a strong consistency when comparing similarly-selected samples at different lookback times, indicating that 𝜏Q does not depend on lookback time for this sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' In contrast, GAMA exhibits a strong evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Galaxies with 𝑀★ <∼1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 M⊙ that became red at a lookback time of 4–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 Gyr have 𝜏Q values that are a factor of ∼ 2 shorter than those of similarly-selected galaxies that transitioned in the 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 Gyr range, with galaxies above that stellar mass showing a smaller evolution in 𝜏Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' A similar decrease by a factor of ∼ 2 is evident in sharkfit, though without the stellar mass dependence seen in GAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' While this suggests that this evolution is at least partially due to our mod- elling choices in ProSpect, it is not obvious that this can fully explain it, as GAMA and sharkfit display different mass dependen- cies and measured timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' MNRAS 000, 1–16 (2023) Forensic quenching timescales 17 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 0 1 2 3 4 τQ [Gyr] GAMA M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] Shark M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] Sharkfit 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 ≤ tLB,R/Gyr < 4 1 ≤ tLB,R/Gyr < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 0 1 2 3 4 τQ [Gyr] 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 109 1010 1011 M⋆(z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='06) [M⊙] 4 ≤ tLB,R/Gyr < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 1 ≤ tLB,R/Gyr < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Comparison between the 𝜏Q measured at two different lookback time bins with a comparable 𝑡LB,R–𝜏Q selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The top row compares the 𝜏Q distribution between the 1 ≤𝑡LB,R/Gyr < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 ≤𝑡LB,R/Gyr < 4 bins, the bottom row between 1 ≤𝑡LB,R/Gyr < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5 and 4 ≤𝑡LB,R/Gyr < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' The solid lines indicate the 𝜏Q running median, and the dashed lines and shaded areas the 16-84th percentiles, with those in colour being measured at the respective 𝑡LB,R bins and those in black from the lowest 𝑡LB,R bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' Each column shows the results for a different sample, left to right: GAMA, shark, and sharkfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} +page_content=' MNRAS 000, 1–16 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptE2T4oBgHgl3EQfKgYC/content/2301.03702v1.pdf'} diff --git a/qtE2T4oBgHgl3EQfKwZ0/content/2301.03706v1.pdf b/qtE2T4oBgHgl3EQfKwZ0/content/2301.03706v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0d47b8ba7a5d7c18cc1d4ffed99a2df7f8e76d5a --- /dev/null +++ b/qtE2T4oBgHgl3EQfKwZ0/content/2301.03706v1.pdf @@ -0,0 +1,3 @@ +version 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a/rNE1T4oBgHgl3EQfPwMX/content/tmp_files/2301.03030v1.pdf.txt b/rNE1T4oBgHgl3EQfPwMX/content/tmp_files/2301.03030v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..960155c310f8e2fd1e9f67a0cfcac0603e9b6c51 --- /dev/null +++ b/rNE1T4oBgHgl3EQfPwMX/content/tmp_files/2301.03030v1.pdf.txt @@ -0,0 +1,3478 @@ +1 +Exploring High Thermal Conductivity Polymers via +Interpretable Machine Learning with Physical Descriptors +Xiang Huang1,†, Shengluo Ma1,†, C. Y. Zhao1, Hong Wang2, and Shenghong Ju1, 2, * +1 China-UK Low Carbon College, Shanghai Jiao Tong University, Shanghai, China +2 Materials Genome Initiative Center, School of Material Science and Engineering, Shanghai Jiao Tong +University, Shanghai, China +†Equal contribution + +ABSTRACT +The efficient and economical exploitation of polymers with high thermal conductivity is essential to +solve the issue of heat dissipation in organic devices. Currently, the experimental preparation of functional +thermal conductivity polymers remains a trial and error process due to the multi-degrees of freedom during +the synthesis and characterization process. Polymer informatics, which efficiently combines data science, +machine learning, and polymer experiment/simulation, leading to the efficient design of polymer materials +with desired properties. However, available polymer thermal conductivity databases are rare, and +establishing appropriate polymer representation is still challenging. In this work, we have proposed a high- +throughput screening framework for polymer chains with high thermal conductivity via interpretable +machine learning and physical-feature engineering. The polymer thermal conductivity datasets for training +were first collected by molecular dynamics simulation. Inspired by the drug-like small molecule +representation and molecular force field, 320 polymer monomer descriptors were calculated and the 20 +optimized descriptors with physical meaning were extracted by hierarchical down-selection. All the machine +learning models achieve a prediction accuracy R2 greater than 0.80, which is superior to that of represented +by traditional graph descriptors. Further, the cross-sectional area and dihedral stiffness descriptors were +identified for positive/negative contribution to thermal conductivity, and 107 promising polymer structures +with thermal conductivity greater than 20.00 W/mK were obtained. Mathematical formulas for predicting +the polymer thermal conductivity were also constructed by using symbolic regression. The high thermal +conductivity polymer structures are mostly π-conjugated, whose overlapping p-orbitals enable easily to +maintain strong chain stiffness and large group velocities. The proposed data-driven framework should +facilitate the theoretical and experimental design of polymers with desirable properties. + +* Corresponding author: shenghong.ju@sjtu.edu.cn. + +2 +1. INTRODUCTION +Polymers are extensively used in industry and daily life, owing to various advantages of chemical +inertness, mechanical flexibility and light weight 1. As the organic electronics are becoming smaller +while the power density keeps increasing, the thermal management and heat dissipation capability have +attracted significant attention 2. However, conventional polymers are thermal insulators with reported +thermal conductivity in the range from 0.1 to 0.5 W/mK, preventing the development of organic +electronics 3. Polymers with high thermal conductivity are urgently demanded in organic energy storage +and electronic devices to accommodate revolutionary innovations in organic electronics and +optoelectronics 4. The polymer morphology and topology were found to be closely related to thermal +conductivity 5. Increasing the crystallite orientation and crystallinity can significantly reduce the phonon +scattering and enhance the thermal conductivity along the chain directions, which has been +demonstrated by both experiments 6-8 and theoretical simulations 9-12. A recent study has fabricated +polyethylene (PE) films by disentanglement and alignment of amorphous chains with a metal-like +thermal conductivity of 62 W/mK, over two orders of magnitude greater than that of classical +amorphous polymers 6. Moreover, molecular dynamics simulations have suggested that individual +crystalline PE chains have a very high or even divergent thermal conductivity 11. These findings provide +opportunities for solving the heat dissipation problem of polymer devices. +Despite the fact that chain alignment, crystallinity, polymer fibers or even single-chain polymer +structures exhibit great influence on the thermal characteristic 13-15, the polymer library is quite large, +with as many as 108 monomeric organic molecules known to exist in chemical space 16. Current research +on the thermal conductivity of polymers is still an Edisonian process, guided by intuition or experience +in a trial-and-error approach that is time-consuming and expensive 17. Moreover, most of the studies are +conducted on simple structures such as PE 4,6,11, which makes it difficult to grasp the general rule of the +factors affecting the thermal conductivity of polymers and to discover polymer molecular structures +with high thermal conductivity in huge chemical space. +The field of polymer informatics 18, associated with the development of artificial intelligence and +machine learning (ML) methods, attempts to utilize the data-driven centric method for physical property +regulation or device development of organic materials to resolve the conflict between structural freedom +and efficiency/cost in the traditional trial-and-error approach. The research on polymer informatics has +attracted extensive attention and succeeded in recent years 19-21, involving the prediction of organic +optical 22-24, electrical 25-27 and thermal properties 28-32. Particularly, several efforts emerged in the search + +3 +or design of structures with high TC as related to crystalline polymers 30, amorphous polymers 31,32 and +copolymers 33. Most of these studies have employed graph descriptors 31 or polymer chemistry fragment +statistics 30,32,33 to describe monomer structures in informatics algorithms, also called fingerprints or +representations. The graph descriptors generated rely on molecular/monomer graph information, +formulated by knowledge domain feature engineering 34 or by attempting to form general descriptors 35. +Moreover, descriptors such as molecular access systems (MACCS) 36 are obtained through statistics of +different chemical fragments, and are closely related to molecular graphs. Subsequently, they are +collectively referred to as graph descriptors. The fingerprint is required for the unique, complete, +minimal representation of each candidate, and the successful fingerprint is a challenging task 37. Besides, +polymers are composed of many repeating units, which are more complex than organic small molecules +and require accurate capture of information on monomer connection sites 26. The graph descriptions +have long been applied and validated in the development of drug-like small molecules 38, and the +availability of open-source toolkits such as RDKit 39 and Mol2vec 34 has facilitated their accessibility, +which is also one reason that graph descriptions are popular in polymer informatics. However, the graph +descriptor is in the form of a string of numeric vectors. The completeness of the molecular structure +determines the coupling association between the digits. Hence, the relationship between molecular +monomers and material properties is difficult to grasp. +Exploring the ensemble of physically independent descriptors for the representation of molecular +structures is important in qualitative structure-property relationship (QSPR) modeling and enables more +intuitive guidelines for molecular structure evaluation 40. Feature engineering for the collection and +reduction of physical descriptors are critical steps in determining effective capabilities in polymer +informatics. The development of automatic, universal and efficient tools for the calculation of +descriptors of organic molecules is of interest to researchers, which translates the chemical information +encoded in the symbolic representation of molecules into useful numbers or some standardized +experimental results 41. Several open-source and commercial software 41-43 are available to calculate +various types of molecular descriptors such as carbon atomic number, molecular weight and Extended +Topochemical Atom (ETA) 44, which have been successfully applied in organic chemistry synthesis 45, +molecular antibacterial activity prediction 46 and so on. In addition, the parameter conditions in +experiments or simulations affect the molecular properties. For instance, the force-field-inspired +descriptors such as types of bond, angle and dihedral have been validated for the prediction of the +specific heat of polymers, even if the datasets are from experiments 29. The number reduction of polymer + +4 +features is another concern, as some descriptors may have little relevance to the target property, and a +low-dimensional descriptor space is much easier to build up for the ML model 47. Feature extraction +and selection are the dominant approaches to reduce the dimensionality of features. Feature extraction +creates subsets from the original data space, such as principal component analysis (PCA), where the +specific meaning of the new features obtained is difficult to understand 48. Feature selection retains the +physical meaning of individual descriptors, while filters based on correlation evaluation have +dependencies on mathematical models, like the Pearson and Spearman coefficients that consider the +linear and monotonic relationships of the data, respectively 49. Further, the filter methods do not involve +ML models, which may lead to the inapplicability of the gained features. The wrapper-based feature +selection techniques combine ML models to eliminate redundant features, including recursive feature +elimination (RFE), sequential feature selection (SFS) and exhaustive feature selection (EFS) 50. Testing +different subsets of descriptors for informatics algorithms is the crucial feature of the wrapper +approaches, and the key is the strategy of combining different descriptors. Typical RFE seeks to improve +model performance by continuously reducing the low impact features from the remaining features in +iteratively constructed ML models, which refer to the ranking of feature weights assigned by models +such as random forests 48. Thus, the RFE relies on the feature weight evaluation mechanism of the ML +models. +Herein, focusing on the challenges of polymer monomer representation and feature selection, we +proposed an ML interpretable framework integrated with high-throughput MD simulations for the +discovery of polymer structures with high TC, as illustrated in Fig. 1. It consists of four components: 1) +polymer library construction, 2) MD simulation for the TC of polymers; 3) monomer feature +representation and hierarchical down-selection; 4) ML models construction for TC prediction. The +training data was collected from the literatures 51,52, and candidates from the databases of PolyInfo 53 +and PI1M 54 were applied for the virtual screening of high TC structures. All polymer monomers were +identified by the SMILES (simplified molecular input line entry system) strings and formed one- +dimensional polymer chains by replication. The TC of training datasets was calculated by MD +simulations with the second generation of the general AMBER force field – GAFF2 55. Inspired by drug- +like molecular representation and molecular force fields, we obtained 320 physical descriptors by +mordred software 41 calculation and force field parameter file extraction, and retained 20 optimized +descriptors by hierarchical down-selection. We then trained random forest (RF), extreme gradient +boosting (XGBoost) tree-based models, and multilayer perceptron (MLP) neural network model + +5 +separately to establish the relationship between the optimized descriptors and the TC of these +benchmark polymer datasets. Further, we analyzed the feature importance of each optimized descriptor +and extracted the chemical heuristic for high TC polymers design through SHAP analysis 56. Using the +trained ML models, 107 promising polymers with TC greater than 20.00 W/mK were identified, which are +served for symbolic regression to derive mathematical formulas for expressing the TC of promising polymers. +Last, we discussed the TC mechanisms of eight typical polymers. Overall, the proposed approach is +beneficial for theoretical or experimental investigations of high TC polymers. + +Fig. 1. Schematics of high-throughput screening of polymers with high TC via interpretable machine +learning. + +2. RESULTS AND DISCUSSION +2.1 Distribution of polymer datasets in chemical space +Polymer data from literatures 51,52 were utilized as the benchmark database for training machine +learning models, as well as PolyInfo 53 and PI1M 54 databases were used for the virtual screening of +polymer structures with high TC. The polymers are classified into 19 classes such as polyolefins, +polyethers and polyamides according to different elements and chemical functional groups 57. To +validate the distribution of the selected 1735 benchmark data over the other two datasets, their chemical +structures were visualized in 2D space by the uniform manifold approximation and projection (UAMP) +58, where the chemical structure of each monomer was transformed into the Morgan fingerprint 35 of a +1024 vector with a radius of 2 atoms. It is observed that the polymer structures in the selected benchmark +dataset (Fig. 2a) are well covered by the chemical space distribution of those in the PolyInfo (Fig. 2b) + +MomentSMILES +Polyolefins +Replication +Polysulfides +Polymerchains +Polyamides +ForcefieldAssignment +Date files with +FFParameters +di/dx +Totally19 classesof the +selected +Polylnfo +PI1M +structure relaxation +polymerstructures +(1735】 +(12043) +(>670000) +Polymel +informatics +Ensembletreesmodels +Neural network model +Feature Engineering +C:3- +Dataset +Q:7 +2 +Mordred &MD inspired (320) +Statistical and ML based +:0 +downselection +Mol2vec +MACCS +Optimized (20) +[Morgan +cMorgan6 +and PI1M (Fig. 2c) databases. Note that the PI1M dataset was generated by a generative model of a +recurrent neural network trained with data from PolyInfo, which fills the sparse region of the chemical +space of the PolyInfo dataset, but the distribution is consistent 54. Thus, the ML models trained with the +selected data are well able to learn the chemical features of all candidates and can be effectively adopted +for the virtual screening of polymer structures with high TC. + +Fig. 2. Visualization of polymer data distribution in a 2D space by UMAP. (a), (b) and (c) are +corresponding to the selected, PolyInfo and PI1M datasets, respectively. +2.2 Polymer descriptors hierarchical down-selection and ML Models Training +Polymer descriptors are hierarchically down-selected in three stages: removing features with low +variance, primary filtering referred to different correlation coefficients, and final selection assisted with +the ML model (shown in Supplementary Section S1). The collected initial monomer physical +descriptors are composed of 286 Mordred-based and 34 MD-inspired descriptors. The descriptors of +MD-inspired and Mordred-based descriptors are listed in Supplementary Section S2. The removal of +low variance descriptors is intended to eliminate descriptors with variance less than a specific threshold, +whose contribution to the target property of all polymer data (TC in this work) is considered to be nearly +consistent. After the variance threshold was set to 0.01, the 264 descriptors were reserved for the next +stage. We established the weight assignment mechanism (WAM) based on the different correlation +coefficients for further primary filtering of the descriptors, due to the various attentions of their +mathematical models. The Pearson, Spearman and Distance coefficients are used to evaluate linear, +monotonic and non-linear relationships between data respectively, while the maximum information +coefficient (MIC) reflects the association of two variables through information entropy, whether linear + +(a) +(b) +(c)7 +or nonlinear. The reliability of MIC depends on the data sample size and the value is reliable only with +large datasets. The four metric coefficients of Pearson, Spearman, Distance and MIC were incorporated +and each was assigned a weighting factor of 0.25, and the thresholds were set to 0.05, 0.05, 0.153 and +0.132, respectively. The 53 descriptors with a cumulative weight value of 1 were retained through VAM. +Random sequential feature selection (RFSF) combined with the RF model was then developed for +optimized descriptors determination. Considering all possible combinations of descriptors for ML +model training is time-consuming and expensive, so traditional SFS usually leads to sub-optimal +solutions, where the recommended ensemble of optimized descriptors is not unique, and is influenced +by the input order of the descriptors 59. Here, we disrupted the order of the input descriptors, combined +them with 100 RF model training cycles, and acquired the final optimized descriptors based on a +statistical approach. The threshold was set to 0.34, that is to maintain descriptors that occur more than +34 times in 100 RF model training runs. The results of the optimized descriptors based on VAM and +RSFS are shown in Fig. 3a and their detailed descriptions are listed in Supplementary Section S3. +Moreover, Fig. 3e exhibits the Pearson correlation matrices of the correlations among optimized +descriptors (Other metrics see in Fig. S1). It is found that most descriptors are positive correlated with +each other and negative correlated with TC. Only three descriptors are positive for TC, two of which +are MD-inspired descriptors. For example, the descriptor MW_ratio reflects the ratio of the molecular +weight of the mainchain to the molecular weight of the monomer, with values between 0 and 1. The +MW_ratio of 1 indicates that the polymer is without side chains, which reduces the loss of heat flux +along the chain and makes it possible to get large TC. +Figure 3b shows the results of the RF model trained with the optimized descriptors, with training +and test R2 of 0.87 and 0.84 respectively. To verify the extensibility of the optimized descriptors, +XGBoost and MLP models were deployed for training (see Fig. S2). The accuracy of the training and +test sets for XGBoost is 0.95 and 0.87, and that for MLP are 0.81 and 0.88 respectively, which is +comparable or even better than the RF model. Therefore, these three models are utilized in the +subsequent discussion. +The prediction accuracy of ML models at different down-selection stages illustrated in Figure 3c +(training and test data set prediction in Fig. S3). The extra PCA more than 95% variance was performed +to compare with RFSF technology. According to the relationship between the number of principal +components and the cumulative variance in Fig. S4, at least 19 components are required to exceed 95% +variance. This is close to the number of sets of optimized descriptors. As seen in Fig. 3c, the tree-based + +8 +models of the RF and XGBoost maintain relatively high accuracy even with large descriptor dimensions +because of their strong ability to prevent overfitting of the data. Moreover, the feature down-selection +process is usually accompanied by the loss of information, which results in the decrease of model +accuracy. However, the feature down-selection process also reduces the redundancy between data which +suppresses the overfitting and improves the accuracy of the MLP model. Overall, the accuracy of all +three models trained with the optimized descriptors from RFSF is higher than that of the models trained +with the PCA-derived descriptors, which demonstrates the effectiveness of our approach. +The ML models with different graph descriptors were applied for comparison in Fig. 3d (training +and test data set prediction in Fig. S5). The Mol2vec 34 is an unsupervised ML approach to learn vector +representations of molecular substructures, which requires a benchmark dataset for molecular structure +training. Here, the pre-trained polymer embedding model was from elsewhere 54, which was created +using the PolyInfo and PI1M datasets. The MACCS 36 descriptor is the structural key-based descriptor +with 166-bit keyset. The Morgan and Morgan count (cMorgan) 35 descriptors are the extended +connectivity fingerprints that capture molecular features relevant to molecular activity. The results in +Fig. 3d reflect the superiority of ML models trained with the optimized descriptor, no matter the models +of RF, XGBoost and MLP. The down-selection processes of physical descriptors examine +individual/combined descriptors in relation to TC, while the graph descriptors aim to represent +molecular/monomeric information as completely as possible. Whilst the elements or groups in the +molecular graph have been indicated to correlate with the TC of polymer chains 30, it is more intuitive +and effective to predict the TC of polymer chains using the associated physical descriptors. But not +absolute, which is also related to the parameters such as chain stiffness 60. + +9 + +Fig. 3. Polymer descriptors down-selection and ML models training. (a) Optimized descriptors acquired +by down-selection with four coefficients - Pearson, Spearman, Distance, and MIC coefficients - and RF +model. (b) Accuracy of RF model based on optimized descriptors, where training R2 is 0.875 and test +R2 is 0.844. (c) Accuracy of ML models at different down-selection processes, including initial (Init.), +mathematical correlation (Cor.) coefficients screening and RF model optimization (Opt.) stages. And, +an additional PCA approach was applied to compare. (d) Accuracy of ML models with different polymer +representation approaches. The violin plot represents the distribution of values, individual subsamples +are shown in gray, and the mean and standard of R2 in black. (e) Pearson correlation matrices showing +correlations among optimized descriptors and TC. The inset is the statistics of the Pearson coefficients +distribution. +2.3 Physical insights from interpretable ML model +Figure 4 summarizes the effect of the features using SHAP, for the RF model trained on optimized +descriptors. The SHAP approach attempts to address the unexplainable black-box challenge of ML +algorithms by calculating the marginal contribution of features to the model output 56. Hence, the +features of each polymer structure in training data sets are assigned the SHAP values separately. As +shown in Fig. 4a, the importance ranking of the optimized descriptors was referenced to the average +SHAP value. Among the top 8 optimized descriptors, the number of MD-inspired and Mordred-based +descriptors is equal, which reflects the fact that the construction of the RF model is a joint contribution +of these two types of descriptors. The distribution of SHAP values for each descriptor is displayed in +Fig.4b, and the depth of shade of datapoints in the beeswarm plot represents the magnitude of TC of + +Training data +Testdata +Init. +Cor. +PCA +Opt.10 +polymer structures in the training set. The distribution of SHAP values for the top-ranked features is +relatively wide, and is monotonic about the feature values overall (Fig. S6). +Here, we highlight the two MD-inspired descriptors of cross-sectional and Kd_average. The most +important descriptor of cross-sectional indicates the effective cross-sectional area of polymer chain, +which is intuitive in relation to the TC. From the Fig. 4c, the SHAP value for cross-sectional decreases +monotonically with the descriptor. According to the Fourier's law, the heat flow rate along 1-D polymer +chain can be expressed as 𝑄 𝑑𝑡 +⁄ += −𝑘𝐴 𝑑𝑇 𝑑𝑥 +⁄ +, where Q is the heat flow, 𝑑𝑡 is the time interval, 𝑘 +is the thermal conductivity, A is the cross-sectional area, and 𝑑𝑇 and 𝑑𝑥 are the temperature +difference and distance between the hot and cold ends, respectively 61. Therefore, the TC is negatively +related to the cross-sectional area, and polymers with high TC usually have a small cross-sectional area +(Fig.S7a). Moreover, the polymer chain structure is absent of disorder compared to the amorphous +structure, maintaining the symmetry of the crystal and reducing phonon scattering. However, the +polymer chains may rotate and become disordered due to temperature and other effects, resulting in a +rapid decrease in TC 62. The closely correlation between dihedral angle energy constant and polymer +chain stiffness has been demonstrated, and the dihedral angle force constant Kd has been artificially +increased in MD simulations to maintain PE chain stiffness and increase thermal conductivity 62,63. The +Kd_average is the average of all types of dihedral force constants from GAFF2 force field for polymer +chain, which is roughly proportional to the corresponding SHAP value in Fig. 4d. Especially for +polymer structures with great kd_average (>4 kcal/mol) usually have large SHAP values and TC (Fig +S7 b). Notably, the TC of polymer chains is influenced by multiple parameters and it is difficult to have +the individual descriptor to determine its value. One example is that crystalline polynorbornene has +been proved to be weakly sensitive with the chain stiffness, even if increasing the dihedral angular force +constant term in MD simulations 62. This confirms the significance of our proposed ML framework for +predicting the TC of polymers. + +11 + +Fig. 4. Analysis of feature importance using SHAP on RF model trained by optimized descriptors. (a) +Average SHAP values for 20 optimized descriptors. (b) Represent the SHAP values of each descriptor +related to training data set polymers in a beeswarm diagram. (c) and (d) SHAP values for the Cross- +section and Kd_average of the training data set polymers as a function of descriptor value. The Cross- +section is the effective cross-sectional area of polymer chain, and the Kd_average is the average value +of force constants of the dihedral angle from GAFF2 force field. +2.4 Discovery of high TC polymers +The optimized descriptors were validated reliability in combination with different ML models for +predicting the TC of polymer chains. Next, we applied these ML models to predict the TC of polymer +structures in the PolyInfo and PI1M databases, in order to virtually screen promising polymers with +high TC. The predicted polymer TC versus cross-sectional area from the ensemble of optimized +descriptors combined with RF, XGBoost and MLP are visualized in Fig. 5a-c, respectively. Where stars +indicate polyethylene with log2TC of 3.91, 4.66, and 5.30 predicted by RF, XGBoost, and MLP + +0.6 +0.4 +0.2 +ATSCOdV +0.0 +BCU +0.2 +0.4 +0.6 +0.8 +20 +40 +60 +80 +100 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +0.112 +respectively, and that calculated by MD simulation is 5.28. The dependence of TC on the cross-sectional +area is evident here, as almost all of the predicted high TC polymers have small cross-sectional areas. +Moreover, since PI1M has the same chemical distribution space as PolyInfo and fills the sparse area, +which covers most of the TC range of PolyInfo and enriches the polymer structures in the high TC +region. +Comparing the results from different ML models, the tree-based models of RF and XGBoost +predict the TC of polymers in a narrower space than that of the MLP. Though the excellent performance +of the tree-based models in preventing overfitting, the extrapolation of the models is usually inadequate +and the predictions are still limited to the range of TC of the polymer structures in the training set. In +contrast, neural network model of MLP usually has better extrapolation capability, and is superior in +finding small data such as high TC polymer structures, despite the relatively low training accuracy R2 +of the model. This finding is similar to previous study of predicting the permeability of gas separation +membranes using ML 17. As well, previous work has revealed the length dependence of the thermal +conductivity of polymer chains. Within a certain length range, the diverging thermal conductivity k and +chain length L can be fitted by k ∼ Lβ, where β indicates the relatively dominant phonon transport +mechanism 13. Here, we considered polymer chains with TC greater than 20.00 W/mK with an effective +length of 50 nm as the outstanding polymers with high TC. Then, balanced strategy to integrate the +three ML models performance were devised to recommend promising polymer structures for calculation +of TC by MD simulations. We identified the polymer structures in the PolyInfo dataset with RF, +XGBoost and MLP predictions of log2TC up to 3.51, 3.50 and 4.33, and only the polymer structures +with no less than 2 occurrences were picked for MD simulations. As a result, 24 polymer structures with +high TC were discovered and verified. Similarly, we implemented this method to identify 84 high TC +polymer structures in the PI1M database. After de-duplication, totally 107 high TC polymer structures +were found in this work, and the Synthetic accessibility (SA) scores were calculated as shown in Fig. +5d. The specific polymer structures can be seen in Supplementary Section S8. From Figure S8, we can +see that most of the high TC polymers are simple linear or contain aromatic rings in the mainchain, +which have small repeating unit lengths and no side chains. The SA score was initially utilized to +estimate the synthetic accessibility of drug-like molecules based on molecular complexity and fragment +contributions 64, and was subsequently adopted for polymers 31,32. The SA score values ranged from 1 +to 10, and synthesis is more difficult as the value increases. To take into account the effect of monomer +linkages, polymer molecules with a polymerization degree of 6 were calculated for the SA score. Among + +13 +them, +28 +polymer +structures +with +SA +no +more +than +3.00, +including +polyethylene, +polytetrafluoroethylene and poly(p-phenylene), and etc. Although it is currently difficult to fabricate +each of these structures, we believe that more polymers like PE chain will be prepared for exploring the +limits TC of polymers by combining advanced processes such as micromechanical stretching, +electrostatic spinning and nanotemplate preparation in the near future 4,6,11. + +Fig. 5. Prediction of high TC polymers in PolyInfo and PI1M databases using constructed ML models. +(a), (b) and (c) based on RF, XGBoost and MLP models, respectively. (d) Synthetic accessibility (SA) +score versus calculated log2K of screened high TC polymers (TC > 20.00 W/mK). The star indicates +PE, and the TC in this work is 38.98 W/mK. +2.5 Symbolic regression for prediction of promising polymers +Since the TC of polymer chains is influenced by complex multi-parameters, it is difficult to predict +trends in TC values for different polymers from any single descriptor. Symbolic regression (SR) +attempts to accelerate the discovery of materials with superior properties by relating available +descriptors through mathematical formulas to construct new combinatorial features 65. SR does not +require massive datasets, as long as a high consistency and accuracy 66,67. The 107 promising polymer +structures (TC > 20.00 W/mK) with optimized descriptors were utilized for SR, where the ratio of +training to test set was 3:1. The mathematical formula were acquired and selected using an efficient +stepwise strategy with SR based on genetic programming (GPSR) as implemented in the gplearn code + +Polylnfo +Polylnfo +PI1M +PI1M +Polylnfo +Polylnfo +PI1M +PI1M +Selected14 +68. The hyperparameters setup and the detailed formula determination process can be found in +Supplementary Section S9. Pearson coefficients are first applied to filter optimized descriptors and +create sub-descriptors, and a novel ensemble of 22 descriptors was obtained. The frequency of +occurrence of optimized descriptors in 158 mathematical formulas (PC values >=0.85 and complexity +<=10) is displayed in Fig. 6a, and first 8 descriptors were finally retained. It is worth emphasizing that +the MD-inspired descriptors of cross-sectional area (cross-sectional) and dihedral force constants +(Kd_average) appeared in each of the formulas. In Figure 6b, we calculated the Pearson coefficients of +the new set of descriptors with the TC, the results suggest these descriptors are closely associated with +the TC. Subsequently, we reset the grid search hyperparameters in gplearn and used R2 as the evaluation +metric. Only formulas with high R2 and low complexity (length of formula) are considered suitable for +the prediction the TC of polymer structures 69. Thus, 9073 mathematical formulas with complexity +within 30 and R2 over 0.6, which are characterized by complexity and accuracy R2 via density plot in +Fig. 6c. The four points of c, d, e and f at Pareto front were identified by Latin hypercube sampling +approach 70,71, and their corresponding formulas are expressed in Table S8. The complexities of the four +formulas are in the range of 20 to 30, and the fitting accuracies are all greater than 0.70. Moreover, the +training accuracy is mostly positive to complexity. For example, the formula represented by point c with +complexity of 20 has a relatively low accuracy R2 among the four points, but the fitting results are +consistent with the MD labeled log2K, as demonstrated in Fig. 6d. Meanwhile, all four identified +formulas include the descriptors of the Cross-sectional, Kd_average and Nd_average, which verified +that the TC of polymer chain is strongly correlated with the parameters such as cross-sectional area and +dihedral stiffness. These formulas are meaningful in the initial rapid screening of high TC polymer chain +structures. + +15 + +Fig. 6. GPSR for TC prediction of promising polymers. (a) Frequency of occurrence of optimized +descriptors in 158 mathematical formulas (PC values >=0.85 and complexity <=10). (b) Pearson +correlation matrices showing correlations among 22 descriptors and TC, where the descriptors d1~d8 +correspond to descriptors 1 to 8 in Fig. 6a. (c) Pareto front of accuracy R2 vs. complexity of 9073 +mathematical formulas shown via density plot. (d) MD labeled vs. fitting results of the formula (point +c) with complexity of 20 and training accuracy R2 of 0.71. +2.6 Thermal transport mechanism of promising crystal polymers +Taking into account factors such as TC and SA score, eight polymer structures (see in Fig. 7a) were +chosen for the analysis of phonon dispersion relations. Currently, polymer structures like [*]C=C[*] +and [*]N=N[*] are challenging to be synthesized experimentally, but are contributing to our +understanding of polymer thermal conductivity mechanisms. All of these polymer molecules are π- +conjugated structures except for the PE and the Polytetrafluoroethylene (PTFE), which are simple linear +structures. In π-conjugated polymer molecules, the overlap of p-orbitals has enhanced restraint in +inhibiting chain rotation and forming the rigid backbone 12. Figure 7b illustrates the phonon dispersion +relations, which were obtained by phonon spectral energy density (Phonon-SED) analysis 72, The +detailed description of the Phonon-SED approach can be found in the Method part. Since the acoustic +modes are dominated by the thermal transport of heat carriers in polymer crystals, phonon modes with +frequencies below 25 THz are demonstrated. Moreover, the phonon group velocity is approximated as + +d1 +11 +(2)2 +12 +(d3)3 +13 +(d4)4) +14 +(d5)5 +15 +(d6)6 +16 +17 +(d8)8 +18 +19 +6 +20 +13 +14 +8 +15 +16 +10 +17 +20 +5 +10 +15 +20 +11 +18 +21 +12 +19 +22 +Training data +Test data16 +the average of the slopes of all acoustic branches 12,60. In the one-dimensional polymer chain systems, +the TC is analyzed as 𝑘 = 𝑣𝑔𝐶𝑣𝑙, where 𝑣𝑔 is the phonon group velocity, 𝐶𝑣 is the volumetric heat +capacity and 𝑙 is the phonon mean free path. Due to the limitation of classical MD simulations, the +volumetric heat capacity can be expressed as 𝐶𝑣 = 3𝑘𝑏𝑁 𝑉 +⁄ , where 𝑘𝑏 is Boltzmann constant, N is +the number of atoms and V is the volume. Thus, phonon mean free path can be calculated by the ratio +of the TC to the multiplication of the phonon group velocity and the volumetric heat capacity. The +approximations of the above calculations allow the results to be rough, but it do help us to understand +the underlying thermal conductivity mechanisms of these promising polymer structures by comparing +the relative trends of the relevant parameters, as listed in Table 1. +The volumetric heat capacity of the eight polymer structures varies from 3.28 to 6.13, which is not +critical to the high TC of polymer chains. As for the phonon group velocity, the six π-conjugated +polymers have large values (more than 5900 m/s) due to overlapped p-orbital and delocalized electrons. +Additionally, the small atomic mass enables a large phonon group velocity. The PTFE has smaller +phonon group velocity than that of PE due to the relatively larger mass of fluorine atoms compared to +hydrogen atoms. The phonon mean free path provides valuable insights into phonon transport in the +polymer chains. Overall, simple linear polymer chains are easily to have long phonon mean free paths, +especially for linear π-conjugated polymers of [*]C=C[*] and [*]N=N[*]. These structures have large +chain stiffness and few atoms except for the backbone, thereby having weak phonon-disorder scattering. + +17 + +Fig. 7. Structure and phonon dispersion relations for the eight promising polymers. (a) Polymer chain +structures. (b) Phonon dispersion relations. The q is wavevector, the 𝜔 is phonon frequency and the +average phonon group velocity of one branch is estimated as the slope of the origin to the maximum +frequency point as shown in the red dashed line in the PHTC001 structure. + +C=55.94 W/mK +SA=1.05 +TC-38.98 W/mK +SA=1.12 + C=35.01 W/mk +SA=2.17 +1C=33.24 W/mK +SA=2.48 +IC=52.21 /mK +SA=2.51 +C=98.45 /mK +SA=2.62 +1C=147.68 Wmk +SA=3.12 +TC=1028.85 W/mK +SA=5.0518 +Table 1. Volumetric heat capacity, phonon group velocity, phonon mean free path and phonon thermal +conductivity for the eight promising polymers +Polymer +ID +SMILES +SA +Cv +(J/cm3K) +vg +(m/s) +l +(nm) +k +(W/mK) +PHTC001 +[*]c1ccc([*])cc1 +1.05 +5.27 +6822.21 +1.55 +55.94 +PHTC002 +[*]CC[*] +1.12 +6.13 +5240.91 +1.21 +38.98 +PHTC006 +[*]C=Cc1ccc([*])cc1 +2.17 +5.25 +6295.54 +1.06 +35.01 +PHTC014 +[*]c1ccc([*])nn1 +2.48 +5.03 +5927.05 +1.12 +33.24 +PHTC015 +[*]C(F)(F)C([*])(F)F +2.51 +3.84 +2952.11 +4.61 +52.21 +PHTC017 +[*]c1cnc([*])cn1 +2.62 +5.03 +7439.50 +2.63 +98.45 +PHTC034 +[*]C=C[*] +3.12 +4.37 +8380.09 +4.03 +147.68 +PHTC094 +[*]N=N[*] +5.05 +3.28 +6378.73 +49.22 +1028.85 +3. CONCLUSIONS +We have developed an interpretable ML framework for exploring high thermal conductivity +polymer chains via high-throughput MD simulations. Inspired by the drug-like small molecule +representation and the molecular force field, we reduced the initially calculated/collected 320 physical +descriptors to 20 optimized descriptors by hierarchical down-selection. The constructed ML models are +capable of effectively reflecting the relationship between optimized descriptors and property, and +exhibit high accuracy in TC prediction. All the models of RF, XGBoost and MLP achieved the R2 of +more than 0.80, which is superior to that of represented by conventional graph descriptors. Moreover, +the promotion or inhibition of TC by optimized descriptors like cross-sectional area and dihedral +stiffness was captured by RF model using SHAP analysis. +Using the trained ML models, we discovered 107 promising polymers with TC greater than 20.00 +W/mK, and 29 of which have SA scores no more than 3.00. These polymer structures have been +validated through high-fidelity MD simulations. Further, we used SR with optimized descriptors to fit +the TC of promising polymers, and the derived mathematical formulas enable a preliminary fast +screening of high TC polymers without relying on ML models, which is friendly for experimental +studies. In closing, we calculated phonon dispersion relations for eight typical polymer structures via +phonon spectral energy density analysis to reveal the underlying TC mechanisms. Notably, most of +these structures are π-conjugated polymers, whose overlapping p-orbitals enable easily to maintain +strong chain stiffness and large group velocities. Our approach may assist in the research of high- + +19 +performance polymers that are not limited to TC. +4. METHODS +4.1 Polymer modeling and cross-sectional area calculation +Polymer modeling is a monomer to chain process, implemented in the STK tool, with input +parameters of monomer SMILES and degree of polymerization 73. The length of the polymer chains +was set uniformly to 50 nm, and the degree of polymerization was obtained by dividing the chain length +by the monomer length and rounding up to an integer. Starting from the polymer SMILES, a molecular +chain with polymerization degree 2 was generated by RDKit and optimized using the MMFF force field +74. Then, the monomer length was determined by measuring the distance between equivalent atoms in +two repeating units in the heat transport direction. Following the modeling, a Python pipeline of +PYSIMM realized the assignment of GAFF2 force field parameters and the generation of MD +simulation input structure data files 75. +The cross-sectional area is one of the important parameters for thermal conductivity analysis. In +molecular dynamics simulations, the calculation of the cross-sectional area is difficult for systems that +do not occupy the entire simulation box. The cross-sectional area was estimated by the ratio of the van +der Waals volume to the length of the monomer 9. The Van der Waals volume of the monomer was +calculated by the sum of atomic and bond contributions, and has been successfully tested and applied +in previous drug compounds 76. +4.2 Calculation of TC by MD simulations +The TC of Polymer chains were obtained by non-equilibrium molecular dynamics simulations +(NEMD) performed in a Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) 75. +In terms of the NEMD method, the heat energy exchange was achieved by an enhanced version of the +heat exchange algorithm, which rescales and shifts the velocities of particles inside reservoirs to impose +a constant heat flux 77. The polymer chains were placed in a box of 540×60×60 (x×y×z) Å box, where +the dimension in the y and z directions was set to 60 Å to avoid interaction with the neighboring polymer +chains. Before TC calculation, the polymer chain structures were relaxed to reach a stable conformation. +Then, the polymer chain was divided into 50 slabs in the x direction, and the fixed regions at two ends +of the chain were set as a heat-insulating walls. In the NEMD simulation, the system was run under +NVT (constant number of atoms, volume, and temperature) and NVE (constant number of atoms, + +20 +volume, and energy) ensembles for 1 ns at 300 K sequentially to release chain stress 30,78. After that, the +heat was added/extracted to the heat source/sink regions (20 Å of each region) at the end of the polymer +chain in a regular rate to create a constant heat flux. The applied heat varies for different polymer chain +structures and ranged from 0.01 eV/ps to 0.08 eV/ps. At last, the temperature profile was averaged over +the last 2~3 ns and used for TC calculation based on Fourier’s law 𝑘 = −𝐽(𝑑𝑇 𝑑𝑥 +⁄ +), where 𝐽 is heat +flux, 𝑑𝑇 𝑑𝑥 +⁄ + is the temperature gradient. +4.3 Descriptors calculation and ML models construction +The ideal polymer descriptors are required to minimize and completely represent polymer +information, and is one of the key factors in determining the prediction accuracy of ML algorithms. The +physical descriptors for this work were sourced from both Mordred software calculations and GAFF2 +force field parameters extraction. The Mordred software was initially developed for small molecule +characteristics in cheminformatics, which can calculate more than 1800 descriptors 41. However, since +we consider two linkages of polymer monomers, only 286 valid descriptors were obtained. Therefore, +as a complement, we additionally extracted parameters from each polymer force field file as the +descriptor. For graph descriptors, MACCS, Morgan and cMorgan fingerprints were calculated in the +RDKit package 39. The Mol2vec fingerprints were embedded via Mol2vec 34. We referred the polymer +representation model trained using PoLyInfo and PI1M databases 54. +The ML models of RF, XGBoost and MLP were implemented by using Scikit-learn 79. +Hyperparametric optimization for RF, XGBoost and MLP was operated with the Bayesian Optimization +package 80 which is a global optimization tool to achieve good prediction accuracy R2. The Gaussian +regression process and acquisition function with 10 randomly pairs of parameters were selected for +initial training, and the ideal parameters for each ML model were determined after 100 optimization +iterations 46. +To explain the association of optimized descriptors with TC, we used the SHAP toolkit with RF +model to evaluate the feature importance 56. The SHAP analysis is based on a game-theoretic approach +that associates the optimal credit allocation with the local explanations of the model, which considers +the model performance by neglecting each feature and provides direction of each descriptor effect 46. +4.4 Mathematical formulas for TC fitted by symbolic regression (SR) +The mathematical formulae were acquired and selected using an efficient stepwise strategy with +SR based on genetic programming (GPSR) as implemented in gplearn 68. The 107 polymer structures + +21 +with TC greater than 20.00 W/mK were randomly divided into 3:1 as training and test sets respectively. +At first, Pearson coefficients were used as evaluation metrics of training fitness to filter optimized +descriptors and generate sub-descriptors, and a new dataset containing 22 descriptors was generated. +Further, the grid search strategy with the hyperparameters and metric R2 as listed in Table 2 was applied +to determine the mathematical formulas. We ultimately discussed four formulas at Pareto front were +identified by Latin hypercube sampling approach 70,71. More information about SR can be found in the +Supplementary Section S9. +Table 2. Setup of hyperparameters in gplearn for GPSR +Parameter +Value +Generations +300 +Population size in every generation +5000 +Probability of crossover (pc) +[0.30,0.90], step=0.05 +Subtree mutation (ps) +[(1-pc)/3,(1-pc)/2] (step= 0.01) +Hoist mutation (ph) +[(1-pc)/3,(1-pc)/2] (step = 0.01) +Point mutation (pp) +1-pc-ps-ph +Function set +{+, −,×,÷, √𝑥, ln 𝑥 , |𝑥|, −𝑥, 1/𝑥} +Parsimony coefficient +0.001, 0.003, 0.005 +Metric +R2 +Stopping criterial +0.900 +Random_state +0, 1, 2, 3, 4 +Init_depth +[2, 6], [4, 8], [6, 10], [2, 10] +4.5 Analysis of phonon dispersion relations by phonon spectral energy density (Phonon-SED) +To understand the TC mechanism of polymers, MD simulations coupled with Phonon-SED +approach 72 were employed to calculate the dispersion relations of polymers. The polymer chain with a +length of 100 Å was constructed as an input of SMILES and placed into a box with the cross section of +60×60 Å. After energy minimization, the system was run under the NVT (constant number of atoms, +volume, and temperature) ensemble for 0.25 ns at 2 K sequentially to release chain stress. Subsequently, +the system was run under the NVE (constant number of atoms, volume, and energy) ensemble for 2 +million steps with the timestep of 0.25 fs. During this period, the velocity and position of each atom in +the polymer backbone were recorded with intervals of 20 steps. The Phonon-SED converted the time +domain information of atomic velocities and positions into wave vectors versus angular frequencies via +two-dimensional Fourier transform, expressed as +𝛷(𝑞, 𝜔) = +1 +4𝜋𝜏0𝑁𝑇 +∑   +{𝑥,𝑦,𝑧} +𝛼 +∑   +𝐵 +𝑏 +𝑚𝑏 |∫   +𝜏0 +0 +∑   +𝑁𝑇 +𝑛 +𝑢̇ 𝛼(𝑛, 𝑏; 𝑡) × 𝑒𝑖𝑞⋅𝑟(𝑛,0;𝑡)−𝑖𝜔𝑡𝑑𝑡| +2 +(1) + +22 +Where 𝑞 is the wavevector, 𝜔 is the frequency, 𝜏0 is the simulation time, 𝑚𝑏 is the mass of atom +b, 𝛼 is the cartesian direction, 𝑁𝑇 is the number of the unit cell in the polymer chain, 𝑢̇ 𝛼(𝑛, 𝑏; 𝑡) is +the velocity of atom b in the unit cell n at time t in the 𝛼 direction, and 𝑟(𝑛, 0; 𝑡) is the equilibrium +position of unit cell n. + +Acknowledgements +This work was supported by Shanghai Pujiang Program (No. 20PJ1407500), the National Natural +Science Foundation of China (No. 52006134), Shanghai Key Fundamental Research Grant (No. +21JC1403300), SJTU-Warwick Joint Seed Fund. The computations in this paper were run on the π 2.0 +cluster supported by the Center for High Performance Computing at Shanghai Jiao Tong University. + +Author Contributions +Conceptualization: S. J. +Methodology: X. H., S. M. and S. J. +Investigation: X. H., S. M. and S. J. +Supervision: S. J., C. Y. Z. and H. W. +Writing—original draft: X. H., S. M. and S. J. +Writing—review & editing: X. H., S. M., S. J., C. Y. Z. and H. W. + +Competing interests +Authors declare that they have no competing interests. + +Supporting Information +The supporting information is available free of charge. + +REFERENCES +1 +Zhang, B. et al. Modulating Thermal Transport in Polymers and Interfaces: Theories, +Simulations, and Experiments. ES Energy & Environment 5, 37-55, doi:10.30919/esee8c306 +(2019). +2 +Ghaffari-Mosanenzadeh, S. et al. 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Zhao1, Hong Wang2, and Shenghong Ju1, 2, * +1 China-UK Low Carbon College, Shanghai Jiao Tong University, Shanghai, China +2 Materials Genome Initiative Center, School of Material Science and Engineering, Shanghai Jiao Tong +University, Shanghai, China +†Equal contribution + + + +* Corresponding author: shenghong.ju@sjtu.edu.cn. + +S1 +Section S1. Downselection of polymer descriptors +Polymer descriptors were downselected in three stages: removing features with low variance (Var.), +primary filtering referred to different correlation coefficients (Cor.), and final selection assisted with +ML model (Opt.), as shown in Fig. S1. + +Fig. S1. Downselection and analysis of descriptors. (a) Process of descriptors downselection. (b), (c) +and (d) Relationship between descriptors at different stages (Var., Cor. and Opt.) and thermal +conductivity based on various metrics or frequency of occurrence of descriptors though 100 Random +Forest (RF) model selections. (e), (f), (g) and (h) Pearson, Spearman, Distance and MIC correlation +matrices showing correlations among optimized descriptors and TC. The inset shows the distribution +statistics of the corresponding correlation coefficient values respectively. The description of the 20 +optimized descriptors is available in Section S3. + + +Initial (320) +Variance (264) +Correlations (53) +Optimized (20)S2 +Section S2. List of MD-inspired and Mordred-based fingerprints +The MD-inspired descriptors include force field related (FF-related) descriptors and thermal +conductivity related (TC- related) descriptors, listed in Table S1. +Table S1. Descriptions of all MD-inspired descriptors +Name +Description +Epsilon_max +Maximum value of the depth of the energy potential (Lennard–Jones parameter) +Epsilon_min +Minimum value of the depth of the energy potential (Lennard–Jones parameter) +Epsilon_average Average value of the depth of the energy potential (Lennard–Jones parameter) +Sigma_max +Maximum value of the equilibrium distance (Lennard–Jones parameter) +Sigma_min +Minimum value of the equilibrium distance (Lennard–Jones parameter) +Sigma_average +Average value of the equilibrium distance (Lennard–Jones parameter) +Kb_max +Maximum value of force constants of the bond +Kb_min +Minimum value of force constants of the bond +Kb_average +Average value of force constants of the bond +R0_max +Maximum value of equilibration structural parameters of the bond +R0_min +Minimum value of equilibration structural parameters of the bond +R0_average +Average value of equilibration structural parameters of the bond +Ka_max +Maximum value of force constants of the bond angle +Ka_min +Minimum value of force constants of the bond angle +Ka_average +Average value of force constants of the bond angle +Theta0_max +Maximum value of equilibration structural parameters of the bond angle +Theta0_min +Minimum value of equilibration structural parameters of the bond angle +Theta0_average +Average value of equilibration structural parameters of the bond angle +Kd_max +Maximum value of force constants of the dihedral angle +Kd_min +Minimum value of force constants of the dihedral angle +Kd_average +Average value of force constants of the dihedral angle +Nd_max +Maximum value of multiplicity for the torsional angle parameters +Nd_min +Minimum value of multiplicity for the torsional angle parameters +Nd_average +Average value of multiplicity for the torsional angle parameters +Delta_max +Maximum value of phase angle for the torsional angle parameters +Delta_min +Minimum value of phase angle for the torsional angle parameters +Delta_average +Average value of phase angle for the torsional angle parameters +Ki_max +Maximum value of force constants of the improper angle +Ki_min +Minimum value of force constants of the improper angle +Ki_average +Average value of force constants of the improper angle +MW +Molecular weight of monomer +MW_ratio +The ratio of the molecular weight of the main chain to that of the monomer +Vdw +Van der Waals volume of the monomer +Cross-sectional +Equivalent cross-sectional area of polymer chain + + + +S3 +The Mordred-based descriptors were calculated by Mordred software 1, the 286 Mordred-based +descriptors in this work are listed in Table S2, and the detailed meaning could be found at +https://mordred-descriptor.github.io/documentation/master/descriptors.html (accessed Aug 10, 2022 ). +Table S2. Descriptions of 286 Mordred-based descriptors +ABC +nAcid +nBase +SpMax_A +SpMAD_A +LogEE_A +VE1_A +VR1_A +VR3_A +nAromAtom +nAtom +nBridgehead +nHetero +nH +nN +nO +nS +nF +nCl +nBr +ATS0dv +ATS0Z +AATS0dv +AATS0d +AATS2d +AATS0Z +AATS1Z +AATS2Z +AATS3Z +ATSC0dv +ATSC1dv +ATSC2dv +ATSC3dv +ATSC4dv +ATSC5dv +ATSC6dv +ATSC7dv +ATSC8dv +ATSC1d +ATSC2d +ATSC3d +ATSC4d +ATSC5d +ATSC6d +ATSC7d +ATSC8d +ATSC0Z +ATSC1Z +ATSC2Z +ATSC3Z +ATSC4Z +ATSC5Z +ATSC6Z +ATSC7Z +ATSC8Z +AATSC0dv +AATSC1dv +AATSC2dv +AATSC3dv +AATSC0d +AATSC1d +AATSC2d +AATSC3d +AATSC0Z +AATSC1Z +AATSC2Z +AATSC3Z +MATS1dv +MATS2dv +MATS3dv +MATS1d +MATS2d +MATS3d +MATS1Z +MATS2Z +MATS3Z +GATS1dv +GATS2dv +GATS3dv +GATS1d +GATS2d +GATS3d +GATS1Z +GATS2Z +GATS3Z +BCUTdv-1h +BCUTdv-1l +BCUTd-1h +BCUTd-1l +BCUTZ-1h +BCUTZ-1l +BalabanJ +BertzCT +nBondsD +nBondsT +C1SP1 +C2SP1 +C1SP2 +C3SP2 +C1SP3 +C2SP3 +C3SP3 +C4SP3 +HybRatio +Xch-4d +Xch-5d +Xch-6d +Xch-4dv +Xc-3d +Xc-4d +Xc-5d +Xc-6d +Xpc-4d +AXp-0d +AXp-1d +AXp-2d +AXp-3d +NsCH3 +NdCH2 +NssCH2 +NdsCH +NsssCH +NddC +NdssC +NaaaC +NssssC +NsNH2 +NssNH +NaaNH +NdsN +NaaN +NsssN +NaasN +NsOH +NdO +NssO +NaaO +NdS +NssS +NaaS +NddssS +SsssCH +SdssC +SaasC +SaaaC +SsssN +SaasN +AETA_beta +AETA_beta_s +ETA_beta_ns_d +AETA_beta_ns_d AETA_eta_RL AETA_eta_BR ETA_epsilon_3 ETA_dBeta +fragCpx +fMF +nHBAcc +IC0 +IC1 +IC2 +IC3 +SIC0 +SIC1 +SIC2 +SIC3 +SIC4 +BIC5 +CIC0 +MIC0 +MIC1 +MIC2 +Kier2 +Kier3 +FilterItLogS +PEOE_VSA1 +PEOE_VSA2 +PEOE_VSA3 +PEOE_VSA4 +PEOE_VSA6 +PEOE_VSA7 +PEOE_VSA8 +PEOE_VSA9 +PEOE_VSA10 +PEOE_VSA11 +PEOE_VSA12 +PEOE_VSA13 SMR_VSA1 +SMR_VSA3 +SMR_VSA4 +SMR_VSA6 +SMR_VSA9 +SlogP_VSA1 +SlogP_VSA2 +SlogP_VSA3 + +S4 +SlogP_VSA4 +SlogP_VSA5 +SlogP_VSA7 +SlogP_VSA8 +SlogP_VSA10 +SlogP_VSA11 +EState_VSA1 +EState_VSA2 +EState_VSA3 +EState_VSA4 +EState_VSA5 +EState_VSA6 +EState_VSA7 +EState_VSA8 +EState_VSA9 +EState_VSA10 +VSA_EState1 +VSA_EState2 +VSA_EState3 +VSA_EState4 +VSA_EState5 +VSA_EState7 +VSA_EState8 +VSA_EState9 +AMID_h +AMID_N +AMID_O +AMID_X +nRing +n5Ring +n6Ring +n7Ring +nHRing +n4HRing +n5HRing +n6HRing +n5aRing +naHRing +n6aHRing +nARing +n5ARing +n6ARing +nAHRing +n5AHRing +n6AHRing +nFRing +n7FRing +n8FRing +n9FRing +n10FRing +n11FRing +n12FRing +nG12FRing +nFHRing +n7FHRing +n8FHRing +n10FHRing +nG12FHRing +nFaRing +n8FaRing +n9FaRing +n10FaRing +nG12FaRing +nFaHRing +nFARing +n10FARing +n12FARing +nG12FARing +nFAHRing +nG12FAHRing +RotRatio +SLogP +TopoPSA(NO) +TopoPSA +GGI4 +GGI5 +GGI6 +GGI7 +JGI1 +JGI2 +JGI3 +JGI4 +JGI5 +JGI6 +JGI7 +JGI8 +JGI9 +JGI10 +JGT10 +TopoShapeIndex +mZagreb1 + + + + + + + +S5 +Section S3. Descriptions of optimized descriptors for polymer thermal conductivity (TC) +prediction +Polymer optimized descriptors were obtained by downselection, listed in Table S3. +Table S3. Descriptions of all MD-inspired descriptors +No. Name +Description +Block +1 +ABC +Atom-bond connectivity index +Atom-bond connectivity index +descriptor +2 +LogEE_A +LogEE of adjacency matrix +Adjacency matrix descriptor +3 +Nh +Number of H atoms +Atom count descriptor +4 +ATSC0dv +Centered moreau-broto autocorrelation +of lag 0 weighted by valence electrons +Centered Autocorrelation of +Topological Structure descriptor +5 +BCUTdv-1h +First heighest eigenvalue of Burden +matrix weighted by valence electrons +BCUT descriptor +6 +Xc-4d +4-ordered Chi cluster weighted by +sigma electrons +Chi descriptor +7 +ETA_dBeta +ETA delta beta +ETA delta beta descriptor +8 +CIC0 +0-ordered complementary information +content +Complementary information +content descriptor +9 +Kier2 +kappa shape index 2 +Kappa shape index 2 descriptor +10 +SMR_VSA6 +MOE MR VSA Descriptor 6 ( 2.75 <= +x < 3.05) +MOE type descriptors using +Wildman-Crippen MR and +surface area contribution +11 +SlogP_VSA3 +MOE logP VSA Descriptor 3 (-0.20 <= +x < 0.00) +MOE type descriptors using +Wildman-Crippen LogP and +surface area contribution +12 +EState_VSA10 +EState VSA Descriptor 10 ( 9.17 <= x +< 15.00) +MOE type descriptors using +EState indices and surface area +contribution +13 +TopoPSA(NO) +Topological polar surface area (use +only nitrogen and oxygen) +Topological polar surface area +descriptor +14 +mZagreb1 +modified Zagreb index (version 1) +Zagreb index descriptor +15 +Kd_average +Average value of force constants of the +dihedral angle +FF-related descriptor +16 +Nd_average +Average value of multiplicity for the +torsional angle parameters +FF-related descriptor +17 +MW +Molecular weight of monomer +TC- related descriptor +18 +MW_ratio +The ratio of the molecular weight of +the main chain to that of the monomer +TC- related descriptor +19 +Vdw +Van der Waals volume of the monomer +TC- related descriptor +20 +Cross-sectional +Equivalent cross-sectional area of +polymer chain +TC- related descriptor + + + +S6 +Section S4. Construction of different ML models with optimized descriptors +Figure S2 shows the RF, XGBoost and MLP models trained with the optimized descriptors. The +test set is consisted of 100 randomly selected polymer structures from the benchmark dataset. + +Fig. S2. Construction of different ML models with optimized descriptors + + + + +Training R2: 0.87 +Training R2 : 0.95 +Training R2 : 0.81 +Test R2 +: 0.84 +Test R2 +: 0.87 +Test R2 +: 0.88 +%8 +Training data +Training data +Training data +Testdata +Testdata +TestdataS7 +Section S5. Accuracy of ML models at different descriptors downselection stages +Figure S3 shows the accuracy of ML models at different descriptors downselection stages. The +extra principal component analysis (PCA) 2 technique with 95% variance was used to compare. + +Fig. S3. Accuracy of ML models at different descriptors downselection stages +Figure S4 represents the relationship between the number of principal components and cumulative +variance. Here, the PCA with 19 principal components of 95% variance was performed. + +Fig. S4. Relationship between the number of principal components and cumulative variance + + + +Training R2: 0.91 +Training R2: 0.89 +Training R2: 0.91 +Test R? +:0.87 +Test R2 +: 0.88 +Test R2 +: 0.79 +80 +Training data +Training data +Trainingdata +Testdata +Test data +Testdata +Training R2: 0.97 +Training R2: 0.88 +Training R2: 0.89 +Test R2 +:0.89 +Test R2 +: 0.89 +Test R2 +: 0.78 +% +Training data +Training data +Training data +Testdata +Testdata +Testdata +Training R2: 0.84 +Training R2: 0.82 +Training R2: 0.87 +Test R2 +: 0.81 +Test R2 +:0.83 +Test R2 +: 0.81 +Training data +Training data +Training data +Testdata +Testdata +Testdata95% cut-off threshotdS8 +Section S6. Accuracy of ML models with various graph descriptors +Figure S5 shows the training and testing accuracy of ML models using Mol2vec 3, +MACCS 4, Morgan and Morgan count (cMorgan) 5 graph descriptors. + +Fig. S5. Accuracy of ML models with various graph descriptors + +Training R2: 0.90 +Training R2: 0.92 +Training R2: 0.75 +Test R2 +: 0.74 +Test R2 +: 0.79 +Test R2 +: 0.79 +Training data +Training data +Trainingdata +Testdata +Testdata +Testdata +Training R2: 0.77 +Training R2: 0.74 +Training R2: 0.82 +Test R2 +: 0.74 +Test R2 +: 0.74 +Test R? +: 0.71 +。 +Trainingdata +Trainingdata +Trainingdata +Testdata +Testdata +Test data +Training R2: 0.67 +Training R2: 0.78 +Training R2: 0.92 +Test R2 +: 0.70 +Test R2 +: 0.77 +Test R2 +: 0.66 +88 +800 +Training data +Training data +Training data +Test data +Testdata +Test data +Training R2: 0.70 +Training R2: 0.85 +Training R2: 0.92 +Test R? +: 0.68 +Test R2 +: 0.76 +Test R? +: 0.59 +0.0 +500 +Trainingdata +Training data +Training data +Testdata +Testdata +TestdataS9 +Section S7. Analysis of the feature importance +Figure S6 exhibits the SHAP values 6 for each optimized descriptor of the training data set +polymers as a function of corresponding descriptor value. The subplots are sorted by the average SHAP +value. The SHAP values of the top-ranked features corresponding to the training dataset polymers show +a monotonic relationship with the feature values overall, while the lower-ranked features have difficulty +in capturing this rule. Wherein, the relationships between two important MD-inspired descriptors and +TC are shown in Fig. S7. + +Fig. S6. Distribution of SHAP values for each optimized descriptor + + +Fig. S7. Relationships between two important MD-inspired descriptors and TC of 1735 benchmark +polymers. + +0.6 +0.5 +1 +0.4 +value +SHAP value +0.4 +SHAP value +0.3 +SHAP value +0.20 +0.4 +0.2 +0.3 +0.15 +0.0 +0.2 +0.2 +0.10 +SHAP +0.2 +0.1 +0.1 +0.05 +0.4 +0.0 +0.0 +0.00 +0.6 +0.1 +0.05 +. +0.8 +0.2 +0.1 +0.10 +20 +40 +60 +80 +100 +0 +10 +20 +30 +40 +50 +1.8 2.0 2.2 2.4 2.6 2.8 3.0 +0 +20304050 +60 +Cross-sectional +Kier2 +Nd_average +SlogP_VSA3 +0.15 +0.7 +0.6 +0.15 +0.10 +. +value +SHAP value +value +0.10 +0.5 +SHAP value +0.10 +0.05 +0.4 +0.05 +SHAP +0.00 +0.3 +SHAP +0.05 +0.2 +0.00 +0.05 +0.1 +0.00 +0.0 +0.05 +0.10 +0.1 +0.05 +-20 +-10 +0 +10 +20 +0 +1 +2 +3 +4 +5 +6 +0 +200 +400 +600 +800 +1000 +1 +2 +3 +4 +5 +6 +ETA dBeta +Kd_average +MW +CICO +0.100 +0.075 +.e.. +0.125 +0.25 +value +0.075 +SHAP value +0.050 +value +0.100 +SHAP value +0.20 +0.050 +0.025 +0.075 +0.025 +0.000 +0.15 +SHAP +0.025 +SHAP +0.050 +0.000 +0.10 +0.050 +0.025 +0.025 +0.000 +0.05 +0.050 +0.075 +0.025 +0.00 +0.075 +0.100 +2 +7 +6 +25 +0.05 +0 +20 +40 +60 +80 +3 +4 +5 +6 +10 +15 +20 +0 +200 +400 +600 +800 +1000 +SMRVSA6 +BCUTdv-1h +mZagreb1 +Vdw +0.10 +0.06 +0.02 +value +0.04 +value +0.08 +value +0.04 +value +0.00 +0.02 +0.06 +0.04 +0.02 +SHAP +SHAP +SHAP +0.02 +SHAP v +0.00 +0.02 +0.04 +0.00 +0.00 +-0.02 +0.06 +0.02 +0.02 +... +0.08 +. +0 +100 +200300400500 +0 +20 +40 +60 +80 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 0.2 +0.4 0.60.8 +1.0 +1.2 +ATSCOdv +nH +MW_ratio +Xc-4d +0.02 +0.03 +value +0.04 +value +value +0.04 +value +0.02 +0.01 +0.02 +0.00 +0.02 +0.01 +SHAP +0.00 +0.01 +SHAP +0.00 +0.00 +0.02 +0.02 +0.01 +0.03 +0.02 +0.02 +0.04 +0 +50 +100 +150200250 +01020 30 4050 607080 +2.53.03.54.04.55.0 +0 +10 +20 +30 +40 +50 +TopoPSA(NO) +EStateVSA10 +LogEE A +ABCS10 +Section S8. Discovery of polymers with high thermal conductivity (PHTC) by proposed ML +workflow +We performed virtual screening of polymer data in PolyInfo and PI1M based on the trained RF, +XGBoost and MLP machine learning models, respectively. In this work, totally of 107 polymer +structures with TC greater than 20 W/mK were identified and validated by MD simulations, which are +listed in Table S3. In this work, a total of 107 polymer structures with thermal conductivity greater than +20 W/mK were identified and validated by molecular dynamics simulations, as listed in Table 4, and +the relevant monomer structure can be viewed in Figure S8 by polymer ID (PID). The TC obtained from +RF, XGBoost and MLP model predictions and MD simulations are log2TC in W/mK. The effective +cross-sectional area of the polymer is given in Å2. +Table S4. MD-validated polymer structures with thermal conductivity greater than 20 W/mK +PID +SMILES +SA +score +Cross- +sectional +RF +XGBoost +MLP +MD +PHTC001 +[*]c1ccc([*])cc1 +1.05 +18.32 +5.00 +5.77 +6.28 +5.81 +PHTC002 +[*]CC[*] +1.12 +15.69 +3.91 +4.66 +5.30 +5.28 +PHTC003 +[*]c1cccc([*])c1 +1.50 +20.74 +4.02 +5.21 +5.86 +4.59 +PHTC004 +[*]CC(=O)N[*] +2.07 +15.13 +4.00 +4.34 +4.05 +4.41 +PHTC005 +[*]c1cccc([*])n1 +2.15 +19.72 +4.65 +4.98 +5.65 +5.06 +PHTC006 +[*]C=Cc1ccc([*])cc1 +2.17 +16.76 +4.29 +4.36 +5.48 +5.13 +PHTC007 +[*]c1ccc2cc([*])ccc2c1 +2.17 +18.28 +3.55 +4.37 +5.01 +5.12 +PHTC008 +[*]CC(=O)O[*] +2.21 +15.09 +4.01 +4.45 +4.36 +4.98 +PHTC009 +[*]c1cccc2c([*])cccc12 +2.22 +23.86 +3.17 +3.81 +4.36 +4.35 +PHTC010 +[*]c1ccc([*])c(Cl)c1 +2.23 +21.89 +3.73 +4.06 +5.34 +4.61 +PHTC011 +[*]c1ccc2ccc([*])cc2c1 +2.28 +19.21 +3.81 +4.49 +5.05 +4.62 +PHTC012 +[*]OCC([*])=O +2.28 +20.86 +3.31 +4.17 +3.70 +4.33 +PHTC013 +[*]c1ccc([*])nc1 +2.33 +16.98 +4.85 +5.08 +5.45 +4.50 +PHTC014 +[*]c1ccc([*])nn1 +2.48 +15.68 +4.87 +5.91 +4.97 +5.05 +PHTC015 +[*]C(F)(F)C([*])(F)F +2.51 +25.57 +4.05 +5.27 +5.07 +5.71 +PHTC016 +[*]NC(=O)N[*] +2.57 +14.34 +4.53 +4.91 +3.76 +5.47 +PHTC017 +[*]c1cnc([*])cn1 +2.62 +15.75 +5.11 +6.08 +5.85 +6.62 +PHTC018 +[*]c1ccc2nc([*])ccc2n1 +2.64 +16.79 +3.96 +4.00 +5.62 +5.93 +PHTC019 +[*]Oc1ccc([*])nc1 +2.65 +14.79 +4.16 +4.46 +4.17 +4.43 +PHTC020 +[*]Oc1ccc([*])cn1 +2.65 +14.74 +4.16 +4.52 +4.17 +4.76 +PHTC021 +[*]c1ccc2nc([*])ccc2c1 +2.78 +17.62 +3.96 +4.19 +5.59 +4.38 +PHTC022 +[*]c1cncc([*])n1 +2.80 +18.06 +5.08 +5.84 +5.53 +4.34 +PHTC023 +[*]c1ccnc([*])n1 +2.82 +18.06 +5.08 +5.82 +5.52 +5.00 +PHTC024 +[*]NNC([*])=O +2.86 +14.28 +4.19 +4.56 +3.32 +4.83 + +S11 +PID +SMILES +SA +score +Cross- +sectional +RF +XGBoost +MLP +MD +PHTC025 +[*]c1csc([*])n1 +2.87 +15.82 +5.11 +6.20 +5.76 +5.75 +PHTC026 +[*]c1cnc2cc([*])ccc2c1 +2.91 +17.29 +3.99 +4.24 +5.63 +4.82 +PHTC027 +[*]c1ccc2oc([*])nc2c1 +2.97 +15.17 +3.91 +3.55 +3.59 +4.34 +PHTC028 +[*]c1ccc(-c2ccc([*])s2)s1 +3.00 +15.98 +4.06 +4.26 +4.22 +5.64 +PHTC029 +[*]C=Cc1ccc([*])c(F)c1 +3.04 +17.60 +3.89 +4.34 +4.80 +4.41 +PHTC030 +[*]C=C/C=C\[*] +3.07 +15.68 +5.06 +5.35 +5.79 +5.33 +PHTC031 +[*]C=C/C=C/[*] +3.07 +14.06 +5.05 +5.10 +5.97 +7.55 +PHTC032 +[*]/C=C/C=C/[*] +3.07 +14.26 +5.05 +5.10 +5.95 +6.91 +PHTC033 +[*]c1nc([*])nc(Cl)n1 +3.09 +13.85 +4.53 +5.64 +5.63 +6.51 +PHTC034 +[*]C=C[*] +3.12 +15.35 +5.08 +5.44 +6.92 +7.21 +PHTC035 +[*]C=C/C=C/C=C/[*] +3.20 +13.73 +4.65 +4.34 +4.86 +7.42 +PHTC036 +[*]/C=C/C=C/C=C/[*] +3.20 +13.60 +4.65 +4.34 +4.87 +7.31 +PHTC037 +[*]c1nnc([*])s1 +3.27 +8.76 +5.09 +6.04 +6.14 +5.75 +PHTC038 +[*]CCOC(=O)NCCNC(=O)O[*] +3.33 +15.55 +3.20 +4.10 +3.65 +4.38 +PHTC039 +[*]C(=O)c1cnc([*])cn1 +3.34 +15.80 +4.11 +4.34 +4.35 +4.54 +PHTC040 +[*]c1cnc([*])s1 +3.35 +15.34 +4.86 +5.77 +5.40 +4.95 +PHTC041 +[*]C1=Cc2ccc([*])cc21 +3.37 +18.78 +4.37 +4.64 +5.50 +5.11 +PHTC042 +[*]OC(=O)O[*] +3.38 +11.93 +3.97 +4.05 +3.96 +5.61 +PHTC043 +[*]c1ccc2nc([*])ncc2n1 +3.39 +15.82 +3.98 +3.95 +5.54 +5.83 +PHTC044 +[*]c1cc(Cl)c([*])nn1 +3.45 +19.29 +4.19 +5.08 +4.97 +4.52 +PHTC045 +[*]c1cnc([*])nn1 +3.49 +14.33 +5.06 +5.97 +5.57 +6.46 +PHTC046 +[*]c1nnc([*])c(Cl)c1Cl +3.51 +15.39 +4.11 +5.12 +5.41 +4.60 +PHTC047 +[*]c1cnnc([*])n1 +3.53 +16.38 +5.04 +5.92 +5.33 +5.60 +PHTC048 +[*]c1cnc([*])n1C +3.55 +19.00 +4.25 +4.46 +5.63 +4.49 +PHTC049 +[*]c1ccc(-c2nnc([*])s2)s1 +3.63 +14.53 +4.15 +4.39 +4.83 +5.39 +PHTC050 +[*]c1nnc([*])c(F)c1F +3.64 +12.41 +4.40 +5.32 +4.51 +4.75 +PHTC051 +[*]c1ccc(-c2cnc([*])s2)s1 +3.76 +15.25 +4.17 +4.44 +4.85 +5.51 +PHTC052 +[*]C(Cl)=C([*])Cl +3.86 +20.41 +4.17 +4.38 +4.81 +5.13 +PHTC053 +[*]/C(Cl)=C(/[*])Cl +3.86 +20.56 +4.13 +4.42 +4.80 +5.08 +PHTC054 +[*]C#C[*] +3.87 +5.85 +4.23 +4.58 +5.15 +9.63 +PHTC055 +[*]C1=CC=C1[*] +3.90 +22.30 +4.36 +5.72 +5.65 +4.62 +PHTC056 +[*]c1nnc([*])c(O)n1 +3.90 +16.45 +4.50 +5.18 +4.20 +4.96 +PHTC057 +[*]c1noc([*])c1Br +3.90 +11.67 +4.61 +5.55 +4.77 +4.97 +PHTC058 +[*]C(=O)N1C(=O)N([*])C1=O +4.04 +16.85 +3.66 +4.36 +5.49 +5.37 +PHTC059 +[*]NC(=O)C(=S)C([*])=O +4.08 +21.85 +4.01 +5.07 +4.60 +5.91 +PHTC060 +[*]C(=O)c1nnc([*])nn1 +4.09 +11.89 +4.36 +5.35 +4.15 +5.95 +PHTC061 +[*]c1cc2nc([*])sc2nn1 +4.09 +14.51 +4.38 +4.81 +5.11 +4.35 +PHTC062 +[*]C(=O)C([*])=S +4.20 +16.81 +3.78 +4.92 +4.41 +5.03 +PHTC063 +[*]C1=C(F)C(Cl)=C1[*] +4.28 +17.13 +4.69 +4.59 +5.00 +5.84 + +S12 +PID +SMILES +SA +score +Cross- +sectional +RF +XGBoost +MLP +MD +PHTC064 +[*]C1=CC=C2C([*])=CC=C12 +4.32 +22.02 +3.94 +3.73 +4.63 +6.16 +PHTC065 +[*]NC(=O)C([*])=N +4.32 +18.67 +3.86 +5.17 +3.12 +4.37 +PHTC066 +[*]SC([*])=O +4.34 +8.22 +4.70 +5.29 +4.52 +6.73 +PHTC067 +[*]/C=C/C=C([*])Cl +4.36 +17.41 +4.65 +3.72 +4.63 +5.90 +PHTC068 +[*]/C(F)=C(/[*])C#N +4.36 +20.54 +3.77 +5.11 +3.47 +6.02 +PHTC069 +[*]C=Cc1csc(-c2cc([*])cs2)c1 +4.45 +15.88 +3.69 +3.61 +3.08 +4.33 +PHTC070 +[*]/C=C/C=C/C=C([*])Cl +4.45 +15.96 +4.28 +4.78 +3.99 +5.66 +PHTC071 +[*]/C=C/C=C/C=C(/[*])Cl +4.45 +15.96 +4.28 +4.78 +3.99 +5.66 +PHTC072 +[*]/C(F)=C(/[*])Cl +4.46 +18.39 +4.44 +5.10 +4.90 +6.18 +PHTC073 +[*]C(F)=C([*])Cl +4.46 +17.50 +4.69 +5.25 +4.94 +5.83 +PHTC074 +[*]OC([*])=S +4.53 +14.49 +4.63 +5.07 +3.76 +5.20 +PHTC075 +[*]C#CC=C[*] +4.54 +18.03 +4.51 +5.02 +5.18 +5.16 +PHTC076 +[*]OC([*])=C +4.54 +19.19 +4.07 +4.36 +3.57 +4.55 +PHTC077 +[*]/C=C(/[*])Cl +4.55 +21.62 +4.28 +4.80 +5.14 +6.03 +PHTC078 +[*]C=C([*])C#N +4.58 +24.93 +3.53 +4.00 +2.70 +4.88 +PHTC079 [*]C(=S)C(=O)C(F)(F)C([*])(F)F +4.59 +25.30 +3.19 +4.28 +3.92 +4.41 +PHTC080 +[*]SS[*] +4.66 +13.04 +3.77 +3.91 +0.95 +4.93 +PHTC081 +[*]/C=C/C=C/C=C([*])N +4.66 +15.44 +4.20 +4.41 +4.69 +5.53 +PHTC082 +[*]NNC([*])=N +4.67 +15.08 +4.45 +4.61 +4.41 +4.51 +PHTC083 +[*]C1=CC2=NC([*])=CC2=N1 +4.75 +17.77 +4.55 +4.07 +4.51 +6.10 +PHTC084 +[*]C#CC#CC=C[*] +4.78 +12.75 +4.11 +3.79 +4.61 +5.99 +PHTC085 +[*]C=C([*])F +4.82 +17.93 +4.71 +5.35 +4.81 +6.64 +PHTC086 +[*]/C=C/N[*] +4.83 +15.68 +5.02 +5.28 +6.52 +5.44 +PHTC087 +[*]C#C/C(C#N)=C(/[*])C#N +4.88 +15.14 +3.46 +5.00 +4.71 +5.37 +PHTC088 +[*]/N=C/C(=S)N[*] +4.88 +18.77 +4.37 +4.29 +5.00 +5.06 +PHTC089 +[*]/N=N/C([*])=O +4.90 +8.82 +4.55 +5.76 +6.04 +7.35 +PHTC090 +[*]/C=C/O[*] +4.92 +15.71 +3.94 +3.83 +3.43 +4.60 +PHTC091 +[*]=CC=NN=[*] +4.98 +5.95 +4.69 +4.92 +7.53 +4.71 +PHTC092 +[*]NN[*] +5.00 +11.88 +4.79 +5.37 +6.33 +4.60 +PHTC093 +[*]/N=N/[*] +5.05 +5.21 +4.83 +5.95 +6.81 +10.38 +PHTC094 +[*]N=N[*] +5.05 +5.93 +4.83 +5.95 +6.76 +10.01 +PHTC095 +[*]/N=C(/[*])Cl +5.08 +9.62 +5.06 +5.71 +6.50 +6.98 +PHTC096 +[*]C#C/C(C#N)=C(/[*])Cl +5.14 +14.06 +3.63 +5.06 +4.58 +5.79 +PHTC097 +[*]/C=N/C=C(/[*])C +5.14 +17.30 +4.02 +3.64 +4.36 +4.72 +PHTC098 +[*]/C=C/C=CS[*] +5.17 +14.14 +4.14 +3.28 +4.13 +4.36 +PHTC099 +[*]C1=CC2=CC([*])=NC2=N1 +5.28 +17.50 +4.62 +4.10 +4.55 +5.79 +PHTC100 +[*]/C=C/N=C([*])Cl +5.38 +19.65 +4.16 +4.37 +4.61 +5.60 +PHTC101 +[*]/C=C/S[*] +5.44 +14.40 +4.57 +4.20 +5.74 +4.35 +PHTC102 +[*]/C=N/[*] +5.53 +14.02 +5.10 +6.07 +6.07 +6.82 + +S13 +PID +SMILES +SA +score +Cross- +sectional +RF +XGBoost +MLP +MD +PHTC103 +[*]/N=N/N=C(/[*])C#N +5.66 +11.24 +3.69 +5.07 +4.33 +6.68 +PHTC104 +[*]/C=N/C=C/C=C(/[*])Cl +5.70 +15.50 +4.33 +4.46 +4.07 +5.70 +PHTC105 +[*]/C=C(/[*])[O-] +5.73 +18.54 +4.07 +4.66 +2.87 +6.10 +PHTC106 +[*]/C=C/C=C/N=N/[*] +5.76 +12.71 +4.74 +4.11 +7.01 +6.59 +PHTC107 +[*]C([2H])=C([*])[2H] +7.27 +11.30 +4.32 +4.33 +1.07 +7.48 + + + +Fig. S8. Continued + + +PHTC001 +PHTC002 +PHTC003 +PHTC004 +PHTC005 +PHTC006 +PHTC007 +PHTC008 +PHTC009 +PHTC010 +PHTC011 +PHTC012 +PHTC013 +PHTC014 +PHTC015 +PHTC016 +PHTC017 +PHTC018 +PHTC019 +PHTC020 +PHTC021 +PHTC022 +PHTC023 +PHTC024 +PHTC025 +PHTC026 +PHTC027 +PHTC028 +PHTC029 +PHTC030 +PHTC031 +PHTC032 +PHTC033 +PHTC034 +PHTC035 +PHTC036S14 + +Fig. S8. Continued + +人 +PHTC037 +PHTC038 +PHTC039 +PHTC040 +PHTC041 +PHTC042 +Y +PHTC043 +PHTC044 +PHTC045 +PHTC046 +PHTC047 +PHTC048 +a +X. +X +PHTC049 +PHTC050 +PHTC051 +PHTC052 +PHTC053 +PHTC054 +PHTC055 +PHTC056 +PHTC057 +PHTC058 +PHTC059 +PHTC060 +. +PHTC061 +PHTC062 +PHTC063 +PHTC064 +PHTC065 +PHTC066 +PHTC067 +PHTC068 +PHTC069 +PHTC070 +PHTC071 +PHTC072 +X +Y +PHTC073 +PHTC074 +PHTC075 +PHTC076 +PHTC077 +PHTC078 +PHTC079 +PHTC080 +PHTC081 +PHTC082 +PHTC083 +PHTC084S15 + +Fig. S8. Polymer monomer structures with TC greater than 20 W/mK verified by MD simulations. +Polymer properties can be found in Table S4 using the PID. + +PHTC085 +PHTC086 +PHTC087 +PHTC088 +PHTC089 +PHTC090 +PHTC091 +PHTC092 +PHTC093 +PHTC094 +PHTC095 +PHTC096 +PHTC097 +PHTC098 +PHTC099 +PHTC100 +PHTC101 +PHTC102 +X +PHTC103 +PHTC104 +PHTC105 +PHTC106 +PHTC107S16 +Section S9. Symbolic regression for characterizing the TC of promising polymers +The 107 promising polymer structures (TC > 20 W/mK) with optimized descriptors were utilized +for symbolic regression (SR), where 20% structures were randomly selected for testing set. The +mathematical formulae were acquired and selected using an efficient stepwise strategy with SR based +on a genetic programming (GPSR) as implemented in the gplearn code 7. At first, Pearson coefficients +were used as evaluation metrics of training fitness to filter optimized descriptors and generate sub- +descriptors, and the grid search strategy with the hyperparameters listed in Table S5 was applied in +GPSR. The hyperparametric combinations corresponding to the 22380 formulas are characterized by +complexity and training fitness of Pearson coefficients (PC) as shown in Fig. S9a. The complexity is +equivalent to the formula length in gplearn. Among these formulas, 4365 of them with PC values not +less than 0.85 are statistically by complexity in Fig. S9b. Generally, the formulas with large complexity +have a higher training fitness. But the formulas are also usually quite long and do not facilitate the +calculation and analysis. Only formulas with large fitness and low complexity are appropriate, so we +selected 158 formulas with the PC value not less than 0.85 and complexity within 10 for subsequent +analysis. By counting the frequency of occurrence of the 20 optimized descriptors in 158 formulas, the +first 8 descriptors were finally retained, as presented in Fig. S9c. It is worth emphasizing that the MD- +inspired descriptors of cross-sectional area (cross-sectional) and dihedral force constants (Kd_average) +appeared in each of the formulas. And, we additionally estimated the frequency of occurrence of sub- +descriptors created by taking the inverse (−𝑥), logarithm (ln 𝑥), and so on, as shown in Fig. S9d. As a +result, a new ensemble containing 22 descriptors which have a strong association with TC is listed in +Table S6 for retraining in gplearn. +Further, we reset the grid search hyperparameters listed in Table S7, and utilized the accuracy R2 +as the evaluation metric. The 19580 formulas with fitting accuracy greater than 0.6 versus complexity +are visualized in Fig. S10a. The effective accuracy is the average of the training and testing accuracy. +Despite the fact that the effective accuracy of some formulas exceeds 0.8, and their complexity is greater +than 50. The Pareto front of accuracy R2 vs. complexity (no more than 30) of 9073 mathematical +formulas shown via density plot in Fig. S10b, and the points of c, d, e and f were identified by Latin +hypercube sampling approach 8,9. The formulas for the four points are listed in Table S8, and their fitting +results compared with MD labeled log2TC are displayed in Fig. 10c-f. The definitions of the variables +𝑥0 to 𝑥21 can be found in Table S6. +Table S5. Setup of hyperparameters in gplearn for filter and generate new sub-descriptors +Parameter +Value +Combination +Generations +300 +1 +Population size in every generation +1000,2000 +2 +Probability of crossover (pc) +[0.30,0.90], step=0.05 +746 +Subtree mutation (ps) +[(1-pc)/3,(1-pc)/2] (step= 0.01) +Hoist mutation (ph) +[(1-pc)/3,(1-pc)/2] (step = 0.01) +Point mutation (pp) +1-pc-ps-ph +Function set +{+, −,×,÷, √𝑥, ln 𝑥 , |𝑥|, −𝑥, 1/𝑥} +1 +Parsimony coefficient +auto +1 +Metric +Pearson coefficient +1 +Stopping criterial +0.900 +1 +Random_state +0, 1, 2, 3, 4 +5 +Init_depth +[2, 6], [4, 8], [6, 10] +3 + +S17 + +Fig. S9. Filtering optimized descriptors and creating sub-descriptors in gplearn through Pearson +coefficient (PC) values. (a) Mathematical formula complexity versus PC values. (b) Statistics of +formulas with Pearson coefficient not less than 0.85 by complexity. (c) and (d) Frequency of occurrence +of optimized descriptors and sub-descriptors in 158 mathematical formulas (PC values >=0.85 and +complexity <=10). +Table S6. New descriptors ensemble for GPSR +Variable +Format +PC values +Variable +Format +PC values +𝑥0 +ABC +-0.37 +𝑥11 +-ETA_dBeta +0.11 +𝑥1 +10/ABC +0.44 +𝑥12 +Kd_average +0.20 +𝑥2 +√ABC +-0.39 +𝑥13 +√Kd_average +0.12 +𝑥3 +ATSC0dv +-0.32 +𝑥14 +LogEE_A +-0.40 +𝑥4 +Cross-sectional +-0.52 +𝑥15 +MW +-0.39 +𝑥5 +10/Cross-sectional +0.63 +𝑥16 +MW/100 +-0.39 +𝑥6 +1-10/Cross-sectional +0.59 +𝑥17 +100/MW +0.52 +𝑥7 +100/Cross-sectional +0.63 +𝑥18 +√MW +-0.43 +𝑥8 +√100/Cross − sectional +0.62 +𝑥19 +Nd_average +-0.31 +𝑥9 +-10/Cross-sectional +-0.63 +𝑥20 +log Nd_average +-0.35 +𝑥10 +ETA_dBeta +-0.11 +𝑥21 +− log Nd_average +0.44 + + + +inwsum +log_sum +ne_sum +Isort_sum +AC +OgEE +APO +de +Kier +Bet: +ectiona: +averag +ectional +averageS18 +Table S7. Reset of hyperparameters in gplearn for GPSR +Parameter +Value +Combination +Generations +300 +1 +Population size in every generation +5000 +1 +Probability of crossover (pc) +[0.30,0.90], step=0.05 +746 +Subtree mutation (ps) +[(1-pc)/3,(1-pc)/2] (step= 0.01) +Hoist mutation (ph) +[(1-pc)/3,(1-pc)/2] (step = 0.01) +Point mutation (pp) +1-pc-ps-ph +Function set +{+, −,×,÷, √𝑥, ln 𝑥 , |𝑥|, −𝑥, 1/𝑥} +1 +Parsimony coefficient +0.001, 0.003, 0.005 +3 +Metric +R2 +1 +Stopping criterial +0.900 +1 +Random_state +0, 1, 2, 3, 4 +5 +Init_depth +[2, 6], [4, 8], [6, 10], [2, 10] +4 + +Fig. S10. GPSR for TC prediction of promising polymers. (a) 19580 formulas with fitting accuracy +greater than 0.6 versus complexity. (b) Pareto front of accuracy R2 vs. complexity of 9073 mathematical +formulas shown via density plot. The four points of c, d, e and f represent the four formulas at the Pareto +front, whose fitting results vs. MD labeled TC are plotted in (c), (d), (e) and (f) respectively. + +888S19 +Table S8. The four mathematical formulas at the Pareto front in Fig. S10b +Point +Formulas +R2 +Complexity +c +𝑙𝑜𝑔2 𝐾 = √(𝑥12 − 𝑥8 − 𝑥16 + 0.49) × 𝑥9 × (𝑥6 + 𝑥20) + √𝑙𝑛⁡(𝑥12) ++ 𝑥21 +0.71 +20 +d +𝑙𝑜𝑔2 𝐾 = √𝑥12 − 𝑙𝑛 [(𝑥8 − 𝑥11) × +𝑥8 +𝑙𝑛(√𝑥19) +] × (𝑥6 + 𝑥20) × 𝑥7 ++ 𝑥21 + √𝑙𝑛 𝑥12 +0.74 +25 +e +𝑙𝑜𝑔2 𝐾 = 𝑙𝑛⁡[ +1 +𝑥10 × (𝑥1 − 𝑥2) − 𝑥12] × (𝑥5 − 0.824) ++ +𝑥17 +√ +𝑥3 +(𝑥12 − 𝑥11) × 𝑥5 ++ 𝑥21 + 𝑥13 +0.74 +28 +f +𝑙𝑜𝑔2 𝐾 = √𝑥8 + [ +𝑥10 +𝑙𝑛(𝑙𝑛 𝑥17 − 𝜒12) + 𝑥8 +− 𝑥12] × (𝑥6 + 𝑥20) × 𝑥8 +𝑥20 ++ 𝑥21 + √𝑥19 − 𝑥16 +0.77 +30 + + + +S20 +References +1 +Moriwaki, H., Tian, Y.-S., Kawashita, N. & Takagi, T. Mordred: a molecular descriptor +calculator. Journal of Cheminformatics 10, 4, doi:10.1186/s13321-018-0258-y (2018). +2 +Abdi, H. & Williams, L. J. Principal component analysis. Wiley interdisciplinary reviews: +computational statistics 2, 433-459 (2010). +3 +Jaeger, S., Fulle, S. & Turk, S. Mol2vec: Unsupervised Machine Learning Approach with +Chemical Intuition. Journal of Chemical Information and Modeling 58, 27-35, +doi:10.1021/acs.jcim.7b00616 (2018). +4 +Durant, J. L., Leland, B. A., Henry, D. R. & Nourse, J. G. Reoptimization of MDL Keys for +Use in Drug Discovery. Journal of Chemical Information and Computer Sciences 42, 1273- +1280, doi:10.1021/ci010132r (2002). +5 +Morgan, H. L. The generation of a unique machine description for chemical structures-a +technique developed at chemical abstracts service. Journal of chemical documentation 5, 107- +113 (1965). +6 +Lundberg, S. M. & Lee, S.-I. A unified approach to interpreting model predictions. Advances +in neural information processing systems 30 (2017). +7 +Stephens, T. Genetic Programming in Python, with a scikit-learn inspired API: gplearn, 2022). +8 +Paulson, N. H., Libera, J. A. & Stan, M. Flame spray pyrolysis optimization via statistics and +machine +learning. +Materials +& +Design +196, +108972, +doi:https://doi.org/10.1016/j.matdes.2020.108972 (2020). +9 +Agarwal, G., Doan, H. A., Robertson, L. A., Zhang, L. & Assary, R. S. Discovery of Energy +Storage Molecular Materials Using Quantum Chemistry-Guided Multiobjective Bayesian +Optimization. Chemistry of Materials 33, 8133-8144, doi:10.1021/acs.chemmater.1c02040 +(2021). + + diff --git a/rNE1T4oBgHgl3EQfPwMX/content/tmp_files/load_file.txt b/rNE1T4oBgHgl3EQfPwMX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d238394332f172b9480d8abdd52218f67aa0848 --- /dev/null +++ b/rNE1T4oBgHgl3EQfPwMX/content/tmp_files/load_file.txt @@ -0,0 +1,2746 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf,len=2745 +page_content='1 Exploring High Thermal Conductivity Polymers via Interpretable Machine Learning with Physical Descriptors Xiang Huang1,†, Shengluo Ma1,†, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Zhao1, Hong Wang2, and Shenghong Ju1, 2, * 1 China-UK Low Carbon College, Shanghai Jiao Tong University, Shanghai, China 2 Materials Genome Initiative Center, School of Material Science and Engineering, Shanghai Jiao Tong University, Shanghai, China †Equal contribution ABSTRACT The efficient and economical exploitation of polymers with high thermal conductivity is essential to solve the issue of heat dissipation in organic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Currently, the experimental preparation of functional thermal conductivity polymers remains a trial and error process due to the multi-degrees of freedom during the synthesis and characterization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Polymer informatics, which efficiently combines data science, machine learning, and polymer experiment/simulation, leading to the efficient design of polymer materials with desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' However, available polymer thermal conductivity databases are rare, and establishing appropriate polymer representation is still challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' In this work, we have proposed a high- throughput screening framework for polymer chains with high thermal conductivity via interpretable machine learning and physical-feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The polymer thermal conductivity datasets for training were first collected by molecular dynamics simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Inspired by the drug-like small molecule representation and molecular force field, 320 polymer monomer descriptors were calculated and the 20 optimized descriptors with physical meaning were extracted by hierarchical down-selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' All the machine learning models achieve a prediction accuracy R2 greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='80, which is superior to that of represented by traditional graph descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Further, the cross-sectional area and dihedral stiffness descriptors were identified for positive/negative contribution to thermal conductivity, and 107 promising polymer structures with thermal conductivity greater than 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00 W/mK were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Mathematical formulas for predicting the polymer thermal conductivity were also constructed by using symbolic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The high thermal conductivity polymer structures are mostly π-conjugated, whose overlapping p-orbitals enable easily to maintain strong chain stiffness and large group velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The proposed data-driven framework should facilitate the theoretical and experimental design of polymers with desirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Corresponding author: shenghong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='ju@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' INTRODUCTION Polymers are extensively used in industry and daily life, owing to various advantages of chemical inertness, mechanical flexibility and light weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' As the organic electronics are becoming smaller while the power density keeps increasing, the thermal management and heat dissipation capability have attracted significant attention 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' However, conventional polymers are thermal insulators with reported thermal conductivity in the range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='5 W/mK, preventing the development of organic electronics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Polymers with high thermal conductivity are urgently demanded in organic energy storage and electronic devices to accommodate revolutionary innovations in organic electronics and optoelectronics 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The polymer morphology and topology were found to be closely related to thermal conductivity 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Increasing the crystallite orientation and crystallinity can significantly reduce the phonon scattering and enhance the thermal conductivity along the chain directions, which has been demonstrated by both experiments 6-8 and theoretical simulations 9-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' A recent study has fabricated polyethylene (PE) films by disentanglement and alignment of amorphous chains with a metal-like thermal conductivity of 62 W/mK, over two orders of magnitude greater than that of classical amorphous polymers 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Moreover, molecular dynamics simulations have suggested that individual crystalline PE chains have a very high or even divergent thermal conductivity 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' These findings provide opportunities for solving the heat dissipation problem of polymer devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Despite the fact that chain alignment, crystallinity, polymer fibers or even single-chain polymer structures exhibit great influence on the thermal characteristic 13-15, the polymer library is quite large, with as many as 108 monomeric organic molecules known to exist in chemical space 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Current research on the thermal conductivity of polymers is still an Edisonian process, guided by intuition or experience in a trial-and-error approach that is time-consuming and expensive 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Moreover, most of the studies are conducted on simple structures such as PE 4,6,11, which makes it difficult to grasp the general rule of the factors affecting the thermal conductivity of polymers and to discover polymer molecular structures with high thermal conductivity in huge chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The field of polymer informatics 18, associated with the development of artificial intelligence and machine learning (ML) methods, attempts to utilize the data-driven centric method for physical property regulation or device development of organic materials to resolve the conflict between structural freedom and efficiency/cost in the traditional trial-and-error approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The research on polymer informatics has attracted extensive attention and succeeded in recent years 19-21, involving the prediction of organic optical 22-24, electrical 25-27 and thermal properties 28-32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Particularly, several efforts emerged in the search 3 or design of structures with high TC as related to crystalline polymers 30, amorphous polymers 31,32 and copolymers 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Most of these studies have employed graph descriptors 31 or polymer chemistry fragment statistics 30,32,33 to describe monomer structures in informatics algorithms, also called fingerprints or representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The graph descriptors generated rely on molecular/monomer graph information, formulated by knowledge domain feature engineering 34 or by attempting to form general descriptors 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Moreover, descriptors such as molecular access systems (MACCS) 36 are obtained through statistics of different chemical fragments, and are closely related to molecular graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Subsequently, they are collectively referred to as graph descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The fingerprint is required for the unique, complete, minimal representation of each candidate, and the successful fingerprint is a challenging task 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Besides, polymers are composed of many repeating units, which are more complex than organic small molecules and require accurate capture of information on monomer connection sites 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The graph descriptions have long been applied and validated in the development of drug-like small molecules 38, and the availability of open-source toolkits such as RDKit 39 and Mol2vec 34 has facilitated their accessibility, which is also one reason that graph descriptions are popular in polymer informatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' However, the graph descriptor is in the form of a string of numeric vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The completeness of the molecular structure determines the coupling association between the digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Hence, the relationship between molecular monomers and material properties is difficult to grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Exploring the ensemble of physically independent descriptors for the representation of molecular structures is important in qualitative structure-property relationship (QSPR) modeling and enables more intuitive guidelines for molecular structure evaluation 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Feature engineering for the collection and reduction of physical descriptors are critical steps in determining effective capabilities in polymer informatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The development of automatic, universal and efficient tools for the calculation of descriptors of organic molecules is of interest to researchers, which translates the chemical information encoded in the symbolic representation of molecules into useful numbers or some standardized experimental results 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Several open-source and commercial software 41-43 are available to calculate various types of molecular descriptors such as carbon atomic number, molecular weight and Extended Topochemical Atom (ETA) 44, which have been successfully applied in organic chemistry synthesis 45, molecular antibacterial activity prediction 46 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' In addition, the parameter conditions in experiments or simulations affect the molecular properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' For instance, the force-field-inspired descriptors such as types of bond, angle and dihedral have been validated for the prediction of the specific heat of polymers, even if the datasets are from experiments 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The number reduction of polymer 4 features is another concern, as some descriptors may have little relevance to the target property, and a low-dimensional descriptor space is much easier to build up for the ML model 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Feature extraction and selection are the dominant approaches to reduce the dimensionality of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Feature extraction creates subsets from the original data space, such as principal component analysis (PCA), where the specific meaning of the new features obtained is difficult to understand 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Feature selection retains the physical meaning of individual descriptors, while filters based on correlation evaluation have dependencies on mathematical models, like the Pearson and Spearman coefficients that consider the linear and monotonic relationships of the data, respectively 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Further, the filter methods do not involve ML models, which may lead to the inapplicability of the gained features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The wrapper-based feature selection techniques combine ML models to eliminate redundant features, including recursive feature elimination (RFE), sequential feature selection (SFS) and exhaustive feature selection (EFS) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Testing different subsets of descriptors for informatics algorithms is the crucial feature of the wrapper approaches, and the key is the strategy of combining different descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Typical RFE seeks to improve model performance by continuously reducing the low impact features from the remaining features in iteratively constructed ML models, which refer to the ranking of feature weights assigned by models such as random forests 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Thus, the RFE relies on the feature weight evaluation mechanism of the ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Herein, focusing on the challenges of polymer monomer representation and feature selection, we proposed an ML interpretable framework integrated with high-throughput MD simulations for the discovery of polymer structures with high TC, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' It consists of four components: 1) polymer library construction, 2) MD simulation for the TC of polymers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 3) monomer feature representation and hierarchical down-selection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 4) ML models construction for TC prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The training data was collected from the literatures 51,52, and candidates from the databases of PolyInfo 53 and PI1M 54 were applied for the virtual screening of high TC structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' All polymer monomers were identified by the SMILES (simplified molecular input line entry system) strings and formed one- dimensional polymer chains by replication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The TC of training datasets was calculated by MD simulations with the second generation of the general AMBER force field – GAFF2 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Inspired by drug- like molecular representation and molecular force fields, we obtained 320 physical descriptors by mordred software 41 calculation and force field parameter file extraction, and retained 20 optimized descriptors by hierarchical down-selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' We then trained random forest (RF), extreme gradient boosting (XGBoost) tree-based models, and multilayer perceptron (MLP) neural network model 5 separately to establish the relationship between the optimized descriptors and the TC of these benchmark polymer datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Further, we analyzed the feature importance of each optimized descriptor and extracted the chemical heuristic for high TC polymers design through SHAP analysis 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Using the trained ML models, 107 promising polymers with TC greater than 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00 W/mK were identified, which are served for symbolic regression to derive mathematical formulas for expressing the TC of promising polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Last, we discussed the TC mechanisms of eight typical polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Overall, the proposed approach is beneficial for theoretical or experimental investigations of high TC polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Schematics of high-throughput screening of polymers with high TC via interpretable machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' RESULTS AND DISCUSSION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='1 Distribution of polymer datasets in chemical space Polymer data from literatures 51,52 were utilized as the benchmark database for training machine learning models, as well as PolyInfo 53 and PI1M 54 databases were used for the virtual screening of polymer structures with high TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The polymers are classified into 19 classes such as polyolefins, polyethers and polyamides according to different elements and chemical functional groups 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' To validate the distribution of the selected 1735 benchmark data over the other two datasets, their chemical structures were visualized in 2D space by the uniform manifold approximation and projection (UAMP) 58, where the chemical structure of each monomer was transformed into the Morgan fingerprint 35 of a 1024 vector with a radius of 2 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' It is observed that the polymer structures in the selected benchmark dataset (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 2a) are well covered by the chemical space distribution of those in the PolyInfo (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 2b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='MomentSMILES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Polyolefins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Replication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Polysulfides ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Polymerchains ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Polyamides ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='ForcefieldAssignment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Date files with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='FFParameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='di/dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Totally19 classesof the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='selected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Polylnfo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='PI1M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='structure relaxation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='polymerstructures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='(1735】 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='(12043) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='(>670000) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Polymel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='informatics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Ensembletreesmodels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Neural network model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Feature Engineering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='C:3- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Q:7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Mordred &MD inspired (320) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Statistical and ML based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=':0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='downselection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Mol2vec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='MACCS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Optimized (20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='[Morgan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='cMorgan6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='and PI1M (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 2c) databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Note that the PI1M dataset was generated by a generative model of a recurrent neural network trained with data from PolyInfo, which fills the sparse region of the chemical space of the PolyInfo dataset, but the distribution is consistent 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Thus, the ML models trained with the selected data are well able to learn the chemical features of all candidates and can be effectively adopted for the virtual screening of polymer structures with high TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Visualization of polymer data distribution in a 2D space by UMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (a), (b) and (c) are corresponding to the selected, PolyInfo and PI1M datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='2 Polymer descriptors hierarchical down-selection and ML Models Training Polymer descriptors are hierarchically down-selected in three stages: removing features with low variance, primary filtering referred to different correlation coefficients, and final selection assisted with the ML model (shown in Supplementary Section S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The collected initial monomer physical descriptors are composed of 286 Mordred-based and 34 MD-inspired descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The descriptors of MD-inspired and Mordred-based descriptors are listed in Supplementary Section S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The removal of low variance descriptors is intended to eliminate descriptors with variance less than a specific threshold, whose contribution to the target property of all polymer data (TC in this work) is considered to be nearly consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' After the variance threshold was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01, the 264 descriptors were reserved for the next stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' We established the weight assignment mechanism (WAM) based on the different correlation coefficients for further primary filtering of the descriptors, due to the various attentions of their mathematical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The Pearson, Spearman and Distance coefficients are used to evaluate linear, monotonic and non-linear relationships between data respectively, while the maximum information coefficient (MIC) reflects the association of two variables through information entropy, whether linear (a) (b) (c)7 or nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The reliability of MIC depends on the data sample size and the value is reliable only with large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The four metric coefficients of Pearson, Spearman, Distance and MIC were incorporated and each was assigned a weighting factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='25, and the thresholds were set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='153 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='132, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The 53 descriptors with a cumulative weight value of 1 were retained through VAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Random sequential feature selection (RFSF) combined with the RF model was then developed for optimized descriptors determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Considering all possible combinations of descriptors for ML model training is time-consuming and expensive, so traditional SFS usually leads to sub-optimal solutions, where the recommended ensemble of optimized descriptors is not unique, and is influenced by the input order of the descriptors 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Here, we disrupted the order of the input descriptors, combined them with 100 RF model training cycles, and acquired the final optimized descriptors based on a statistical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The threshold was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='34, that is to maintain descriptors that occur more than 34 times in 100 RF model training runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The results of the optimized descriptors based on VAM and RSFS are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 3a and their detailed descriptions are listed in Supplementary Section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Moreover, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 3e exhibits the Pearson correlation matrices of the correlations among optimized descriptors (Other metrics see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' It is found that most descriptors are positive correlated with each other and negative correlated with TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Only three descriptors are positive for TC, two of which are MD-inspired descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' For example, the descriptor MW_ratio reflects the ratio of the molecular weight of the mainchain to the molecular weight of the monomer, with values between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The MW_ratio of 1 indicates that the polymer is without side chains, which reduces the loss of heat flux along the chain and makes it possible to get large TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Figure 3b shows the results of the RF model trained with the optimized descriptors, with training and test R2 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='87 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='84 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' To verify the extensibility of the optimized descriptors, XGBoost and MLP models were deployed for training (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The accuracy of the training and test sets for XGBoost is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='95 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='87, and that for MLP are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='81 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='88 respectively, which is comparable or even better than the RF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Therefore, these three models are utilized in the subsequent discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The prediction accuracy of ML models at different down-selection stages illustrated in Figure 3c (training and test data set prediction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The extra PCA more than 95% variance was performed to compare with RFSF technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' According to the relationship between the number of principal components and the cumulative variance in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S4, at least 19 components are required to exceed 95% variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' This is close to the number of sets of optimized descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 3c, the tree-based 8 models of the RF and XGBoost maintain relatively high accuracy even with large descriptor dimensions because of their strong ability to prevent overfitting of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Moreover, the feature down-selection process is usually accompanied by the loss of information, which results in the decrease of model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' However, the feature down-selection process also reduces the redundancy between data which suppresses the overfitting and improves the accuracy of the MLP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Overall, the accuracy of all three models trained with the optimized descriptors from RFSF is higher than that of the models trained with the PCA-derived descriptors, which demonstrates the effectiveness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The ML models with different graph descriptors were applied for comparison in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 3d (training and test data set prediction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The Mol2vec 34 is an unsupervised ML approach to learn vector representations of molecular substructures, which requires a benchmark dataset for molecular structure training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Here, the pre-trained polymer embedding model was from elsewhere 54, which was created using the PolyInfo and PI1M datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The MACCS 36 descriptor is the structural key-based descriptor with 166-bit keyset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The Morgan and Morgan count (cMorgan) 35 descriptors are the extended connectivity fingerprints that capture molecular features relevant to molecular activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 3d reflect the superiority of ML models trained with the optimized descriptor, no matter the models of RF, XGBoost and MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The down-selection processes of physical descriptors examine individual/combined descriptors in relation to TC, while the graph descriptors aim to represent molecular/monomeric information as completely as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Whilst the elements or groups in the molecular graph have been indicated to correlate with the TC of polymer chains 30, it is more intuitive and effective to predict the TC of polymer chains using the associated physical descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' But not absolute, which is also related to the parameters such as chain stiffness 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Polymer descriptors down-selection and ML models training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (a) Optimized descriptors acquired by down-selection with four coefficients - Pearson, Spearman, Distance, and MIC coefficients - and RF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (b) Accuracy of RF model based on optimized descriptors, where training R2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='875 and test R2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (c) Accuracy of ML models at different down-selection processes, including initial (Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' ), mathematical correlation (Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=') coefficients screening and RF model optimization (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=') stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' And, an additional PCA approach was applied to compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (d) Accuracy of ML models with different polymer representation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The violin plot represents the distribution of values, individual subsamples are shown in gray, and the mean and standard of R2 in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (e) Pearson correlation matrices showing correlations among optimized descriptors and TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The inset is the statistics of the Pearson coefficients distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='3 Physical insights from interpretable ML model Figure 4 summarizes the effect of the features using SHAP, for the RF model trained on optimized descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The SHAP approach attempts to address the unexplainable black-box challenge of ML algorithms by calculating the marginal contribution of features to the model output 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Hence, the features of each polymer structure in training data sets are assigned the SHAP values separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 4a, the importance ranking of the optimized descriptors was referenced to the average SHAP value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Among the top 8 optimized descriptors, the number of MD-inspired and Mordred-based descriptors is equal, which reflects the fact that the construction of the RF model is a joint contribution of these two types of descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The distribution of SHAP values for each descriptor is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='4b, and the depth of shade of datapoints in the beeswarm plot represents the magnitude of TC of Training data Testdata Init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' PCA Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='10 polymer structures in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The distribution of SHAP values for the top-ranked features is relatively wide, and is monotonic about the feature values overall (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Here, we highlight the two MD-inspired descriptors of cross-sectional and Kd_average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The most important descriptor of cross-sectional indicates the effective cross-sectional area of polymer chain, which is intuitive in relation to the TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' From the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 4c, the SHAP value for cross-sectional decreases monotonically with the descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=" According to the Fourier's law, the heat flow rate along 1-D polymer chain can be expressed as 𝑄 𝑑𝑡 ⁄ = −𝑘𝐴 𝑑𝑇 𝑑𝑥 ⁄ , where Q is the heat flow, 𝑑𝑡 is the time interval, 𝑘 is the thermal conductivity, A is the cross-sectional area, and 𝑑𝑇 and 𝑑𝑥 are the temperature difference and distance between the hot and cold ends, respectively 61." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Therefore, the TC is negatively related to the cross-sectional area, and polymers with high TC usually have a small cross-sectional area (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='S7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Moreover, the polymer chain structure is absent of disorder compared to the amorphous structure, maintaining the symmetry of the crystal and reducing phonon scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' However, the polymer chains may rotate and become disordered due to temperature and other effects, resulting in a rapid decrease in TC 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The closely correlation between dihedral angle energy constant and polymer chain stiffness has been demonstrated, and the dihedral angle force constant Kd has been artificially increased in MD simulations to maintain PE chain stiffness and increase thermal conductivity 62,63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The Kd_average is the average of all types of dihedral force constants from GAFF2 force field for polymer chain, which is roughly proportional to the corresponding SHAP value in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Especially for polymer structures with great kd_average (>4 kcal/mol) usually have large SHAP values and TC (Fig S7 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Notably, the TC of polymer chains is influenced by multiple parameters and it is difficult to have the individual descriptor to determine its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' One example is that crystalline polynorbornene has been proved to be weakly sensitive with the chain stiffness, even if increasing the dihedral angular force constant term in MD simulations 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' This confirms the significance of our proposed ML framework for predicting the TC of polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Analysis of feature importance using SHAP on RF model trained by optimized descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (a) Average SHAP values for 20 optimized descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (b) Represent the SHAP values of each descriptor related to training data set polymers in a beeswarm diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (c) and (d) SHAP values for the Cross- section and Kd_average of the training data set polymers as a function of descriptor value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The Cross- section is the effective cross-sectional area of polymer chain, and the Kd_average is the average value of force constants of the dihedral angle from GAFF2 force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='4 Discovery of high TC polymers The optimized descriptors were validated reliability in combination with different ML models for predicting the TC of polymer chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Next, we applied these ML models to predict the TC of polymer structures in the PolyInfo and PI1M databases, in order to virtually screen promising polymers with high TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The predicted polymer TC versus cross-sectional area from the ensemble of optimized descriptors combined with RF, XGBoost and MLP are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 5a-c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Where stars indicate polyethylene with log2TC of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='91, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='66, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='30 predicted by RF, XGBoost, and MLP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='2 ATSCOdV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='0 BCU 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='8 20 40 60 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='112 respectively, and that calculated by MD simulation is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The dependence of TC on the cross-sectional area is evident here, as almost all of the predicted high TC polymers have small cross-sectional areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Moreover, since PI1M has the same chemical distribution space as PolyInfo and fills the sparse area, which covers most of the TC range of PolyInfo and enriches the polymer structures in the high TC region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Comparing the results from different ML models, the tree-based models of RF and XGBoost predict the TC of polymers in a narrower space than that of the MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Though the excellent performance of the tree-based models in preventing overfitting, the extrapolation of the models is usually inadequate and the predictions are still limited to the range of TC of the polymer structures in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' In contrast, neural network model of MLP usually has better extrapolation capability, and is superior in finding small data such as high TC polymer structures, despite the relatively low training accuracy R2 of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' This finding is similar to previous study of predicting the permeability of gas separation membranes using ML 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' As well, previous work has revealed the length dependence of the thermal conductivity of polymer chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Within a certain length range, the diverging thermal conductivity k and chain length L can be fitted by k ∼ Lβ, where β indicates the relatively dominant phonon transport mechanism 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Here, we considered polymer chains with TC greater than 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00 W/mK with an effective length of 50 nm as the outstanding polymers with high TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Then, balanced strategy to integrate the three ML models performance were devised to recommend promising polymer structures for calculation of TC by MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' We identified the polymer structures in the PolyInfo dataset with RF, XGBoost and MLP predictions of log2TC up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='51, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='50 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='33, and only the polymer structures with no less than 2 occurrences were picked for MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' As a result, 24 polymer structures with high TC were discovered and verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Similarly, we implemented this method to identify 84 high TC polymer structures in the PI1M database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' After de-duplication, totally 107 high TC polymer structures were found in this work, and the Synthetic accessibility (SA) scores were calculated as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The specific polymer structures can be seen in Supplementary Section S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' From Figure S8, we can see that most of the high TC polymers are simple linear or contain aromatic rings in the mainchain, which have small repeating unit lengths and no side chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The SA score was initially utilized to estimate the synthetic accessibility of drug-like molecules based on molecular complexity and fragment contributions 64, and was subsequently adopted for polymers 31,32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The SA score values ranged from 1 to 10, and synthesis is more difficult as the value increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' To take into account the effect of monomer linkages, polymer molecules with a polymerization degree of 6 were calculated for the SA score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Among 13 them, 28 polymer structures with SA no more than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00, including polyethylene, polytetrafluoroethylene and poly(p-phenylene), and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Although it is currently difficult to fabricate each of these structures, we believe that more polymers like PE chain will be prepared for exploring the limits TC of polymers by combining advanced processes such as micromechanical stretching, electrostatic spinning and nanotemplate preparation in the near future 4,6,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Prediction of high TC polymers in PolyInfo and PI1M databases using constructed ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (a), (b) and (c) based on RF, XGBoost and MLP models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (d) Synthetic accessibility (SA) score versus calculated log2K of screened high TC polymers (TC > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00 W/mK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The star indicates PE, and the TC in this work is 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='98 W/mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='5 Symbolic regression for prediction of promising polymers Since the TC of polymer chains is influenced by complex multi-parameters, it is difficult to predict trends in TC values for different polymers from any single descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Symbolic regression (SR) attempts to accelerate the discovery of materials with superior properties by relating available descriptors through mathematical formulas to construct new combinatorial features 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' SR does not require massive datasets, as long as a high consistency and accuracy 66,67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The 107 promising polymer structures (TC > 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00 W/mK) with optimized descriptors were utilized for SR, where the ratio of training to test set was 3:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The mathematical formula were acquired and selected using an efficient stepwise strategy with SR based on genetic programming (GPSR) as implemented in the gplearn code Polylnfo Polylnfo PI1M PI1M Polylnfo Polylnfo PI1M PI1M Selected14 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The hyperparameters setup and the detailed formula determination process can be found in Supplementary Section S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Pearson coefficients are first applied to filter optimized descriptors and create sub-descriptors, and a novel ensemble of 22 descriptors was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The frequency of occurrence of optimized descriptors in 158 mathematical formulas (PC values >=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='85 and complexity <=10) is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 6a, and first 8 descriptors were finally retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' It is worth emphasizing that the MD-inspired descriptors of cross-sectional area (cross-sectional) and dihedral force constants (Kd_average) appeared in each of the formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' In Figure 6b, we calculated the Pearson coefficients of the new set of descriptors with the TC, the results suggest these descriptors are closely associated with the TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Subsequently, we reset the grid search hyperparameters in gplearn and used R2 as the evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Only formulas with high R2 and low complexity (length of formula) are considered suitable for the prediction the TC of polymer structures 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Thus, 9073 mathematical formulas with complexity within 30 and R2 over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='6, which are characterized by complexity and accuracy R2 via density plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The four points of c, d, e and f at Pareto front were identified by Latin hypercube sampling approach 70,71, and their corresponding formulas are expressed in Table S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The complexities of the four formulas are in the range of 20 to 30, and the fitting accuracies are all greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Moreover, the training accuracy is mostly positive to complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' For example, the formula represented by point c with complexity of 20 has a relatively low accuracy R2 among the four points, but the fitting results are consistent with the MD labeled log2K, as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 6d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Meanwhile, all four identified formulas include the descriptors of the Cross-sectional, Kd_average and Nd_average, which verified that the TC of polymer chain is strongly correlated with the parameters such as cross-sectional area and dihedral stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' These formulas are meaningful in the initial rapid screening of high TC polymer chain structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' GPSR for TC prediction of promising polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (a) Frequency of occurrence of optimized descriptors in 158 mathematical formulas (PC values >=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='85 and complexity <=10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (b) Pearson correlation matrices showing correlations among 22 descriptors and TC, where the descriptors d1~d8 correspond to descriptors 1 to 8 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (c) Pareto front of accuracy R2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' complexity of 9073 mathematical formulas shown via density plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (d) MD labeled vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' fitting results of the formula (point c) with complexity of 20 and training accuracy R2 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='6 Thermal transport mechanism of promising crystal polymers Taking into account factors such as TC and SA score, eight polymer structures (see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 7a) were chosen for the analysis of phonon dispersion relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Currently, polymer structures like [*]C=C[*] and [*]N=N[*] are challenging to be synthesized experimentally, but are contributing to our understanding of polymer thermal conductivity mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' All of these polymer molecules are π- conjugated structures except for the PE and the Polytetrafluoroethylene (PTFE), which are simple linear structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' In π-conjugated polymer molecules, the overlap of p-orbitals has enhanced restraint in inhibiting chain rotation and forming the rigid backbone 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Figure 7b illustrates the phonon dispersion relations, which were obtained by phonon spectral energy density (Phonon-SED) analysis 72, The detailed description of the Phonon-SED approach can be found in the Method part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Since the acoustic modes are dominated by the thermal transport of heat carriers in polymer crystals, phonon modes with frequencies below 25 THz are demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Moreover, the phonon group velocity is approximated as d1 11 (2)2 12 (d3)3 13 (d4)4) 14 (d5)5 15 (d6)6 16 17 (d8)8 18 19 6 20 13 14 8 15 16 10 17 20 5 10 15 20 11 18 21 12 19 22 Training data Test data16 the average of the slopes of all acoustic branches 12,60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' In the one-dimensional polymer chain systems, the TC is analyzed as 𝑘 = 𝑣𝑔𝐶𝑣𝑙, where 𝑣𝑔 is the phonon group velocity, 𝐶𝑣 is the volumetric heat capacity and 𝑙 is the phonon mean free path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Due to the limitation of classical MD simulations, the volumetric heat capacity can be expressed as 𝐶𝑣 = 3𝑘𝑏𝑁 𝑉 ⁄ , where 𝑘𝑏 is Boltzmann constant, N is the number of atoms and V is the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Thus, phonon mean free path can be calculated by the ratio of the TC to the multiplication of the phonon group velocity and the volumetric heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The approximations of the above calculations allow the results to be rough, but it do help us to understand the underlying thermal conductivity mechanisms of these promising polymer structures by comparing the relative trends of the relevant parameters, as listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The volumetric heat capacity of the eight polymer structures varies from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='28 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='13, which is not critical to the high TC of polymer chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' As for the phonon group velocity, the six π-conjugated polymers have large values (more than 5900 m/s) due to overlapped p-orbital and delocalized electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Additionally, the small atomic mass enables a large phonon group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The PTFE has smaller phonon group velocity than that of PE due to the relatively larger mass of fluorine atoms compared to hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The phonon mean free path provides valuable insights into phonon transport in the polymer chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Overall, simple linear polymer chains are easily to have long phonon mean free paths, especially for linear π-conjugated polymers of [*]C=C[*] and [*]N=N[*].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' These structures have large chain stiffness and few atoms except for the backbone, thereby having weak phonon-disorder scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Structure and phonon dispersion relations for the eight promising polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (a) Polymer chain structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (b) Phonon dispersion relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The q is wavevector, the 𝜔 is phonon frequency and the average phonon group velocity of one branch is estimated as the slope of the origin to the maximum frequency point as shown in the red dashed line in the PHTC001 structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' C=55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='94 W/mK SA=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05 TC-38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='98 W/mK SA=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='12 C=35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01 W/mk SA=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='17 1C=33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='24 W/mK SA=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='48 IC=52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='21 /mK SA=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='51 C=98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='45 /mK SA=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='62 1C=147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='68 Wmk SA=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='12 TC=1028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='85 W/mK SA=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='0518 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Volumetric heat capacity, phonon group velocity, phonon mean free path and phonon thermal conductivity for the eight promising polymers Polymer ID SMILES SA Cv (J/cm3K) vg (m/s) l (nm) k (W/mK) PHTC001 [*]c1ccc([*])cc1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='27 6822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='55 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='94 PHTC002 [*]CC[*] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='13 5240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='91 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='21 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='98 PHTC006 [*]C=Cc1ccc([*])cc1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='25 6295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='06 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01 PHTC014 [*]c1ccc([*])nn1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='03 5927.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='12 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='24 PHTC015 [*]C(F)(F)C([*])(F)F 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='84 2952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='61 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='21 PHTC017 [*]c1cnc([*])cn1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='62 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='03 7439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='63 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='45 PHTC034 [*]C=C[*] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='37 8380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='09 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='03 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='68 PHTC094 [*]N=N[*] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='28 6378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='73 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='22 1028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' CONCLUSIONS We have developed an interpretable ML framework for exploring high thermal conductivity polymer chains via high-throughput MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Inspired by the drug-like small molecule representation and the molecular force field, we reduced the initially calculated/collected 320 physical descriptors to 20 optimized descriptors by hierarchical down-selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The constructed ML models are capable of effectively reflecting the relationship between optimized descriptors and property, and exhibit high accuracy in TC prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' All the models of RF, XGBoost and MLP achieved the R2 of more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='80, which is superior to that of represented by conventional graph descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Moreover, the promotion or inhibition of TC by optimized descriptors like cross-sectional area and dihedral stiffness was captured by RF model using SHAP analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Using the trained ML models, we discovered 107 promising polymers with TC greater than 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00 W/mK, and 29 of which have SA scores no more than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' These polymer structures have been validated through high-fidelity MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Further, we used SR with optimized descriptors to fit the TC of promising polymers, and the derived mathematical formulas enable a preliminary fast screening of high TC polymers without relying on ML models, which is friendly for experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' In closing, we calculated phonon dispersion relations for eight typical polymer structures via phonon spectral energy density analysis to reveal the underlying TC mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Notably, most of these structures are π-conjugated polymers, whose overlapping p-orbitals enable easily to maintain strong chain stiffness and large group velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Our approach may assist in the research of high- 19 performance polymers that are not limited to TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' METHODS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='1 Polymer modeling and cross-sectional area calculation Polymer modeling is a monomer to chain process, implemented in the STK tool, with input parameters of monomer SMILES and degree of polymerization 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The length of the polymer chains was set uniformly to 50 nm, and the degree of polymerization was obtained by dividing the chain length by the monomer length and rounding up to an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Starting from the polymer SMILES, a molecular chain with polymerization degree 2 was generated by RDKit and optimized using the MMFF force field 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Then, the monomer length was determined by measuring the distance between equivalent atoms in two repeating units in the heat transport direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Following the modeling, a Python pipeline of PYSIMM realized the assignment of GAFF2 force field parameters and the generation of MD simulation input structure data files 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The cross-sectional area is one of the important parameters for thermal conductivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' In molecular dynamics simulations, the calculation of the cross-sectional area is difficult for systems that do not occupy the entire simulation box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The cross-sectional area was estimated by the ratio of the van der Waals volume to the length of the monomer 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The Van der Waals volume of the monomer was calculated by the sum of atomic and bond contributions, and has been successfully tested and applied in previous drug compounds 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='2 Calculation of TC by MD simulations The TC of Polymer chains were obtained by non-equilibrium molecular dynamics simulations (NEMD) performed in a Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' In terms of the NEMD method, the heat energy exchange was achieved by an enhanced version of the heat exchange algorithm, which rescales and shifts the velocities of particles inside reservoirs to impose a constant heat flux 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The polymer chains were placed in a box of 540×60×60 (x×y×z) Å box, where the dimension in the y and z directions was set to 60 Å to avoid interaction with the neighboring polymer chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Before TC calculation, the polymer chain structures were relaxed to reach a stable conformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Then, the polymer chain was divided into 50 slabs in the x direction, and the fixed regions at two ends of the chain were set as a heat-insulating walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' In the NEMD simulation, the system was run under NVT (constant number of atoms, volume, and temperature) and NVE (constant number of atoms, 20 volume, and energy) ensembles for 1 ns at 300 K sequentially to release chain stress 30,78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' After that, the heat was added/extracted to the heat source/sink regions (20 Å of each region) at the end of the polymer chain in a regular rate to create a constant heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The applied heat varies for different polymer chain structures and ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01 eV/ps to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='08 eV/ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' At last, the temperature profile was averaged over the last 2~3 ns and used for TC calculation based on Fourier’s law 𝑘 = −𝐽(𝑑𝑇 𝑑𝑥 ⁄ ), where 𝐽 is heat flux, 𝑑𝑇 𝑑𝑥 ⁄ is the temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='3 Descriptors calculation and ML models construction The ideal polymer descriptors are required to minimize and completely represent polymer information, and is one of the key factors in determining the prediction accuracy of ML algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The physical descriptors for this work were sourced from both Mordred software calculations and GAFF2 force field parameters extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The Mordred software was initially developed for small molecule characteristics in cheminformatics, which can calculate more than 1800 descriptors 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' However, since we consider two linkages of polymer monomers, only 286 valid descriptors were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Therefore, as a complement, we additionally extracted parameters from each polymer force field file as the descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' For graph descriptors, MACCS, Morgan and cMorgan fingerprints were calculated in the RDKit package 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The Mol2vec fingerprints were embedded via Mol2vec 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' We referred the polymer representation model trained using PoLyInfo and PI1M databases 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The ML models of RF, XGBoost and MLP were implemented by using Scikit-learn 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Hyperparametric optimization for RF, XGBoost and MLP was operated with the Bayesian Optimization package 80 which is a global optimization tool to achieve good prediction accuracy R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The Gaussian regression process and acquisition function with 10 randomly pairs of parameters were selected for initial training, and the ideal parameters for each ML model were determined after 100 optimization iterations 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' To explain the association of optimized descriptors with TC, we used the SHAP toolkit with RF model to evaluate the feature importance 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The SHAP analysis is based on a game-theoretic approach that associates the optimal credit allocation with the local explanations of the model, which considers the model performance by neglecting each feature and provides direction of each descriptor effect 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='4 Mathematical formulas for TC fitted by symbolic regression (SR) The mathematical formulae were acquired and selected using an efficient stepwise strategy with SR based on genetic programming (GPSR) as implemented in gplearn 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The 107 polymer structures 21 with TC greater than 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00 W/mK were randomly divided into 3:1 as training and test sets respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' At first, Pearson coefficients were used as evaluation metrics of training fitness to filter optimized descriptors and generate sub-descriptors, and a new dataset containing 22 descriptors was generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Further, the grid search strategy with the hyperparameters and metric R2 as listed in Table 2 was applied to determine the mathematical formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' We ultimately discussed four formulas at Pareto front were identified by Latin hypercube sampling approach 70,71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' More information about SR can be found in the Supplementary Section S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Setup of hyperparameters in gplearn for GPSR Parameter Value Generations 300 Population size in every generation 5000 Probability of crossover (pc) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='30,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='90], step=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05 Subtree mutation (ps) [(1-pc)/3,(1-pc)/2] (step= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01) Hoist mutation (ph) [(1-pc)/3,(1-pc)/2] (step = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01) Point mutation (pp) 1-pc-ps-ph Function set {+, −,×,÷, √𝑥, ln 𝑥 , |𝑥|, −𝑥, 1/𝑥} Parsimony coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='005 Metric R2 Stopping criterial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='900 Random_state 0, 1, 2, 3, 4 Init_depth [2, 6], [4, 8], [6, 10], [2, 10] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='5 Analysis of phonon dispersion relations by phonon spectral energy density (Phonon-SED) To understand the TC mechanism of polymers, MD simulations coupled with Phonon-SED approach 72 were employed to calculate the dispersion relations of polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The polymer chain with a length of 100 Å was constructed as an input of SMILES and placed into a box with the cross section of 60×60 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' After energy minimization, the system was run under the NVT (constant number of atoms, volume, and temperature) ensemble for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='25 ns at 2 K sequentially to release chain stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Subsequently, the system was run under the NVE (constant number of atoms, volume, and energy) ensemble for 2 million steps with the timestep of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='25 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' During this period, the velocity and position of each atom in the polymer backbone were recorded with intervals of 20 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The Phonon-SED converted the time domain information of atomic velocities and positions into wave vectors versus angular frequencies via two-dimensional Fourier transform, expressed as 𝛷(𝑞, 𝜔) = 1 4𝜋𝜏0𝑁𝑇 ∑ {𝑥,𝑦,𝑧} 𝛼 ∑ 𝐵 𝑏 𝑚𝑏 |∫ 𝜏0 0 ∑ 𝑁𝑇 𝑛 𝑢̇ 𝛼(𝑛, 𝑏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 𝑡) × 𝑒𝑖𝑞⋅𝑟(𝑛,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='𝑡)−𝑖𝜔𝑡𝑑𝑡| 2 (1) 22 Where 𝑞 is the wavevector, 𝜔 is the frequency, 𝜏0 is the simulation time, 𝑚𝑏 is the mass of atom b, 𝛼 is the cartesian direction, 𝑁𝑇 is the number of the unit cell in the polymer chain, 𝑢̇ 𝛼(𝑛, 𝑏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 𝑡) is the velocity of atom b in the unit cell n at time t in the 𝛼 direction, and 𝑟(𝑛, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 𝑡) is the equilibrium position of unit cell n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Acknowledgements This work was supported by Shanghai Pujiang Program (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 20PJ1407500), the National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 52006134), Shanghai Key Fundamental Research Grant (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 21JC1403300), SJTU-Warwick Joint Seed Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The computations in this paper were run on the π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='0 cluster supported by the Center for High Performance Computing at Shanghai Jiao Tong University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Author Contributions Conceptualization: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Methodology: X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Investigation: X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Supervision: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Writing—original draft: X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Writing—review & editing: X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Competing interests Authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' the Journal of machine Learning research 12, 2825-2830 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 80 Nogueira, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Bayesian Optimization: Open source constrained global optimization tool for Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' URL https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' com/fmfn/BayesianOptimization (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Supplemental Materials for Exploring High Thermal Conductivity Polymers via Interpretable Machine Learning with Physical Descriptors Xiang Huang1,†, Shengluo Ma1,†, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Zhao1, Hong Wang2, and Shenghong Ju1, 2, * 1 China-UK Low Carbon College, Shanghai Jiao Tong University, Shanghai, China 2 Materials Genome Initiative Center, School of Material Science and Engineering, Shanghai Jiao Tong University, Shanghai, China †Equal contribution Corresponding author: shenghong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='ju@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S1 Section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Downselection of polymer descriptors Polymer descriptors were downselected in three stages: removing features with low variance (Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' ), primary filtering referred to different correlation coefficients (Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' ), and final selection assisted with ML model (Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' ), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Downselection and analysis of descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (a) Process of descriptors downselection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (b), (c) and (d) Relationship between descriptors at different stages (Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=', Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' and Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=') and thermal conductivity based on various metrics or frequency of occurrence of descriptors though 100 Random Forest (RF) model selections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (e), (f), (g) and (h) Pearson, Spearman, Distance and MIC correlation matrices showing correlations among optimized descriptors and TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The inset shows the distribution statistics of the corresponding correlation coefficient values respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The description of the 20 optimized descriptors is available in Section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Initial (320) Variance (264) Correlations (53) Optimized (20)S2 Section S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' List of MD-inspired and Mordred-based fingerprints The MD-inspired descriptors include force field related (FF-related) descriptors and thermal conductivity related (TC- related) descriptors, listed in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Descriptions of all MD-inspired descriptors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Epsilon_max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Maximum value of the depth of the energy potential (Lennard–Jones parameter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Epsilon_min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Minimum value of the depth of the energy potential (Lennard–Jones parameter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Epsilon_average Average value of the depth of the energy potential (Lennard–Jones parameter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Sigma_max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Maximum value of the equilibrium distance (Lennard–Jones parameter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Sigma_min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Minimum value of the equilibrium distance (Lennard–Jones parameter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Sigma_average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Average value of the equilibrium distance (Lennard–Jones parameter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Kb_max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Maximum value of force constants of the bond ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Kb_min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Minimum value of force constants of the bond ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Kb_average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Average value of force constants of the bond ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='R0_max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Maximum value of equilibration structural parameters of the bond ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='R0_min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Minimum value of equilibration structural parameters of the bond ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='R0_average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Average value of equilibration structural parameters of the bond ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Ka_max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Maximum value of force constants of the 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of the bond angle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Theta0_min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Minimum value of equilibration structural parameters of the bond angle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Theta0_average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Average value of equilibration structural parameters of the bond angle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Kd_max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Maximum value of phase angle for the torsional angle parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Delta_min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Minimum value of phase angle for the torsional angle parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Delta_average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Average value of phase angle for the torsional angle parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Ki_max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Maximum value of force constants of the improper angle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Ki_min ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Minimum value of force constants of the improper angle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Ki_average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Average value of force constants of the improper angle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='MW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Molecular weight of monomer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='MW_ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='The ratio of the molecular weight of the main chain to that of the monomer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Vdw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Van der Waals volume of the monomer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Cross-sectional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Equivalent cross-sectional area of polymer chain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='S3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='The Mordred-based descriptors were calculated by Mordred software 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' the 286 Mordred-based descriptors in this work are listed in Table S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' and the detailed meaning could be found at https://mordred-descriptor.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='JGI9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='JGI10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='JGT10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='TopoShapeIndex ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='mZagreb1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='S5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Section S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Descriptions of optimized descriptors for polymer thermal conductivity (TC) prediction Polymer optimized descriptors were obtained by downselection, listed in Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Descriptions of all MD-inspired descriptors No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='ABC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Atom-bond connectivity index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Atom-bond connectivity index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='LogEE_A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='LogEE of adjacency matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Adjacency matrix descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Nh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Number of H atoms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Atom count descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='ATSC0dv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Centered moreau-broto autocorrelation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='of lag 0 weighted by valence electrons ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Centered Autocorrelation of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Topological Structure descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='BCUTdv-1h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='First heighest eigenvalue of Burden ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='matrix weighted by valence electrons ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='BCUT descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Xc-4d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='4-ordered Chi cluster weighted by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='sigma electrons ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Chi descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='ETA_dBeta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='ETA delta beta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='ETA delta beta descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='CIC0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='0-ordered complementary information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='content ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Complementary information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='content descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Kier2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='kappa shape index 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Kappa shape index 2 descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='SMR_VSA6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='MOE MR VSA Descriptor 6 ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='75 <= x < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05) MOE type descriptors using Wildman-Crippen MR and surface area contribution 11 SlogP_VSA3 MOE logP VSA Descriptor 3 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='20 <= x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00) MOE type descriptors using Wildman-Crippen LogP and surface area contribution 12 EState_VSA10 EState VSA Descriptor 10 ( 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='17 <= x < 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='MOE type descriptors using ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='EState indices and surface area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='contribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='TopoPSA(NO) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Topological polar surface area (use ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='only nitrogen and oxygen) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Topological polar surface area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='mZagreb1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='modified Zagreb index (version 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Zagreb index descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Kd_average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Average value of force constants of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='dihedral angle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='FF-related descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Nd_average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Average value of multiplicity for the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='torsional angle parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='FF-related descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='MW ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Molecular weight of monomer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='TC- related descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='MW_ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='The ratio of the molecular weight of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='the main chain to that of the monomer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='TC- related descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Vdw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Van der Waals volume of the monomer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='TC- related descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Cross-sectional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Equivalent cross-sectional area of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='polymer chain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='TC- related descriptor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='S6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='Section S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Construction of different ML models with optimized descriptors Figure S2 shows the RF, XGBoost and MLP models trained with the optimized descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The test set is consisted of 100 randomly selected polymer structures from the benchmark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Construction of different ML models with optimized descriptors Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='87 Training R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='95 Training R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='81 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='84 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='87 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='88 %8 Training data Training data Training data Testdata Testdata TestdataS7 Section S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Accuracy of ML models at different descriptors downselection stages Figure S3 shows the accuracy of ML models at different descriptors downselection stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The extra principal component analysis (PCA) 2 technique with 95% variance was used to compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Accuracy of ML models at different descriptors downselection stages Figure S4 represents the relationship between the number of principal components and cumulative variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Here, the PCA with 19 principal components of 95% variance was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Relationship between the number of principal components and cumulative variance Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='91 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='89 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='91 Test R?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' :0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='87 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='88 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='79 80 Training data Training data Trainingdata Testdata Test data Testdata Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='97 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='88 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='89 Test R2 :0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='89 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='89 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='78 % Training data Training data Training data Testdata Testdata Testdata Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='84 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='82 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='87 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='81 Test R2 :0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='83 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='81 Training data Training data Training data Testdata Testdata Testdata95% cut-off threshotdS8 Section S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Accuracy of ML models with various graph descriptors Figure S5 shows the training and testing accuracy of ML models using Mol2vec 3, MACCS 4, Morgan and Morgan count (cMorgan) 5 graph descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Accuracy of ML models with various graph descriptors Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='90 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='92 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='75 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='74 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='79 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='79 Training data Training data Trainingdata Testdata Testdata Testdata Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='77 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='74 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='82 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='74 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='74 Test R?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='71 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Trainingdata Trainingdata Trainingdata Testdata Testdata Test data Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='67 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='78 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='92 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='70 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='77 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='66 88 800 Training data Training data Training data Test data Testdata Test data Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='70 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='85 Training R2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='92 Test R?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='68 Test R2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='76 Test R?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='0 500 Trainingdata Training data Training data Testdata Testdata TestdataS9 Section S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Analysis of the feature importance Figure S6 exhibits the SHAP values 6 for each optimized descriptor of the training data set polymers as a function of corresponding descriptor value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The subplots are sorted by the average SHAP value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The SHAP values of the top-ranked features corresponding to the training dataset polymers show a monotonic relationship with the feature values overall, while the lower-ranked features have difficulty in capturing this rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Wherein, the relationships between two important MD-inspired descriptors and TC are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Distribution of SHAP values for each optimized descriptor Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Relationships between two important MD-inspired descriptors and TC of 1735 benchmark polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='4 value SHAP value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='4 SHAP value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='3 SHAP value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='04 0 50 100 150200250 01020 30 4050 607080 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='0 0 10 20 30 40 50 TopoPSA(NO) EStateVSA10 LogEE A ABCS10 Section S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Discovery of polymers with high thermal conductivity (PHTC) by proposed ML workflow We performed virtual screening of polymer data in PolyInfo and PI1M based on the trained RF, XGBoost and MLP machine learning models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' In this work, totally of 107 polymer structures with TC greater than 20 W/mK were identified and validated by MD simulations, which are listed in Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' In this work, a total of 107 polymer structures with thermal conductivity greater than 20 W/mK were identified and validated by molecular dynamics simulations, as listed in Table 4, and the relevant monomer structure can be viewed in Figure S8 by polymer ID (PID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The TC obtained from RF, XGBoost and MLP model predictions and MD simulations are log2TC in W/mK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The effective cross-sectional area of the polymer is given in Å2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' MD-validated polymer structures with thermal conductivity greater than 20 W/mK PID SMILES SA score Cross- sectional RF XGBoost MLP MD PHTC001 [*]c1ccc([*])cc1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='32 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='77 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='81 PHTC002 [*]CC[*] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='12 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='69 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='91 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='66 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='28 PHTC003 [*]c1cccc([*])c1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='50 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='86 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='59 PHTC004 [*]CC(=O)N[*] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='07 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='41 PHTC005 [*]c1cccc([*])n1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='15 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='72 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='98 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='65 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='06 PHTC006 [*]C=Cc1ccc([*])cc1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='17 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='13 PHTC007 [*]c1ccc2cc([*])ccc2c1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='17 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='12 PHTC008 [*]CC(=O)O[*] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='21 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='09 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='98 PHTC009 [*]c1cccc2c([*])cccc12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='22 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='86 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='81 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='35 PHTC010 [*]c1ccc([*])c(Cl)c1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='23 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='73 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='61 PHTC011 [*]c1ccc2ccc([*])cc2c1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='28 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='81 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='49 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='62 PHTC012 [*]OCC([*])=O 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='28 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='86 3.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='87 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='10 PHTC106 [*]/C=C/C=C/N=N/[*] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='76 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='71 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='11 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='59 PHTC107 [*]C([2H])=C([*])[2H] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='27 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='07 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='48 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Continued PHTC001 PHTC002 PHTC003 PHTC004 PHTC005 PHTC006 PHTC007 PHTC008 PHTC009 PHTC010 PHTC011 PHTC012 PHTC013 PHTC014 PHTC015 PHTC016 PHTC017 PHTC018 PHTC019 PHTC020 PHTC021 PHTC022 PHTC023 PHTC024 PHTC025 PHTC026 PHTC027 PHTC028 PHTC029 PHTC030 PHTC031 PHTC032 PHTC033 PHTC034 PHTC035 PHTC036S14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Continued 人 PHTC037 PHTC038 PHTC039 PHTC040 PHTC041 PHTC042 Y PHTC043 PHTC044 PHTC045 PHTC046 PHTC047 PHTC048 a X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' X PHTC049 PHTC050 PHTC051 PHTC052 PHTC053 PHTC054 PHTC055 PHTC056 PHTC057 PHTC058 PHTC059 PHTC060 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' PHTC061 PHTC062 PHTC063 PHTC064 PHTC065 PHTC066 PHTC067 PHTC068 PHTC069 PHTC070 PHTC071 PHTC072 X Y PHTC073 PHTC074 PHTC075 PHTC076 PHTC077 PHTC078 PHTC079 PHTC080 PHTC081 PHTC082 PHTC083 PHTC084S15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Polymer monomer structures with TC greater than 20 W/mK verified by MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Polymer properties can be found in Table S4 using the PID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' PHTC085 PHTC086 PHTC087 PHTC088 PHTC089 PHTC090 PHTC091 PHTC092 PHTC093 PHTC094 PHTC095 PHTC096 PHTC097 PHTC098 PHTC099 PHTC100 PHTC101 PHTC102 X PHTC103 PHTC104 PHTC105 PHTC106 PHTC107S16 Section S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Symbolic regression for characterizing the TC of promising polymers The 107 promising polymer structures (TC > 20 W/mK) with optimized descriptors were utilized for symbolic regression (SR), where 20% structures were randomly selected for testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The mathematical formulae were acquired and selected using an efficient stepwise strategy with SR based on a genetic programming (GPSR) as implemented in the gplearn code 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' At first, Pearson coefficients were used as evaluation metrics of training fitness to filter optimized descriptors and generate sub- descriptors, and the grid search strategy with the hyperparameters listed in Table S5 was applied in GPSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The hyperparametric combinations corresponding to the 22380 formulas are characterized by complexity and training fitness of Pearson coefficients (PC) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The complexity is equivalent to the formula length in gplearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Among these formulas, 4365 of them with PC values not less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='85 are statistically by complexity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Generally, the formulas with large complexity have a higher training fitness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' But the formulas are also usually quite long and do not facilitate the calculation and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Only formulas with large fitness and low complexity are appropriate, so we selected 158 formulas with the PC value not less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='85 and complexity within 10 for subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' By counting the frequency of occurrence of the 20 optimized descriptors in 158 formulas, the first 8 descriptors were finally retained, as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S9c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' It is worth emphasizing that the MD- inspired descriptors of cross-sectional area (cross-sectional) and dihedral force constants (Kd_average) appeared in each of the formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' And, we additionally estimated the frequency of occurrence of sub- descriptors created by taking the inverse (−𝑥), logarithm (ln 𝑥), and so on, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S9d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' As a result, a new ensemble containing 22 descriptors which have a strong association with TC is listed in Table S6 for retraining in gplearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Further, we reset the grid search hyperparameters listed in Table S7, and utilized the accuracy R2 as the evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The 19580 formulas with fitting accuracy greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='6 versus complexity are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The effective accuracy is the average of the training and testing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Despite the fact that the effective accuracy of some formulas exceeds 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='8, and their complexity is greater than 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The Pareto front of accuracy R2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' complexity (no more than 30) of 9073 mathematical formulas shown via density plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S10b, and the points of c, d, e and f were identified by Latin hypercube sampling approach 8,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The formulas for the four points are listed in Table S8, and their fitting results compared with MD labeled log2TC are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 10c-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The definitions of the variables 𝑥0 to 𝑥21 can be found in Table S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Table S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Setup of hyperparameters in gplearn for filter and generate new sub-descriptors Parameter Value Combination Generations 300 1 Population size in every generation 1000,2000 2 Probability of crossover (pc) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='30,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='90], step=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05 746 Subtree mutation (ps) [(1-pc)/3,(1-pc)/2] (step= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01) Hoist mutation (ph) [(1-pc)/3,(1-pc)/2] (step = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01) Point mutation (pp) 1-pc-ps-ph Function set {+, −,×,÷, √𝑥, ln 𝑥 , |𝑥|, −𝑥, 1/𝑥} 1 Parsimony coefficient auto 1 Metric Pearson coefficient 1 Stopping criterial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='900 1 Random_state 0, 1, 2, 3, 4 5 Init_depth [2, 6], [4, 8], [6, 10] 3 S17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Filtering optimized descriptors and creating sub-descriptors in gplearn through Pearson coefficient (PC) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (a) Mathematical formula complexity versus PC values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (b) Statistics of formulas with Pearson coefficient not less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='85 by complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (c) and (d) Frequency of occurrence of optimized descriptors and sub-descriptors in 158 mathematical formulas (PC values >=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='85 and complexity <=10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Table S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' New descriptors ensemble for GPSR Variable Format PC values Variable Format PC values 𝑥0 ABC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='37 𝑥11 ETA_dBeta 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='11 𝑥1 10/ABC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='44 𝑥12 Kd_average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='20 𝑥2 √ABC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='39 𝑥13 √Kd_average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='12 𝑥3 ATSC0dv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='32 𝑥14 LogEE_A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='40 𝑥4 Cross-sectional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='52 𝑥15 MW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='39 𝑥5 10/Cross-sectional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='63 𝑥16 MW/100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='39 𝑥6 1-10/Cross-sectional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='59 𝑥17 100/MW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='52 𝑥7 100/Cross-sectional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='63 𝑥18 √MW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='43 𝑥8 √100/Cross − sectional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='62 𝑥19 Nd_average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='31 𝑥9 10/Cross-sectional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='63 𝑥20 log Nd_average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='35 𝑥10 ETA_dBeta 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='11 𝑥21 − log Nd_average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='44 inwsum log_sum ne_sum Isort_sum AC OgEE APO de Kier Bet: ectiona: averag ectional averageS18 Table S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Reset of hyperparameters in gplearn for GPSR Parameter Value Combination Generations 300 1 Population size in every generation 5000 1 Probability of crossover (pc) [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='30,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='90], step=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='05 746 Subtree mutation (ps) [(1-pc)/3,(1-pc)/2] (step= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01) Hoist mutation (ph) [(1-pc)/3,(1-pc)/2] (step = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='01) Point mutation (pp) 1-pc-ps-ph Function set {+, −,×,÷, √𝑥, ln 𝑥 , |𝑥|, −𝑥, 1/𝑥} 1 Parsimony coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='005 3 Metric R2 1 Stopping criterial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='900 1 Random_state 0, 1, 2, 3, 4 5 Init_depth [2, 6], [4, 8], [6, 10], [2, 10] 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' GPSR for TC prediction of promising polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (a) 19580 formulas with fitting accuracy greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='6 versus complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' (b) Pareto front of accuracy R2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' complexity of 9073 mathematical formulas shown via density plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The four points of c, d, e and f represent the four formulas at the Pareto front, whose fitting results vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' MD labeled TC are plotted in (c), (d), (e) and (f) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 888S19 Table S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' The four mathematical formulas at the Pareto front in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S10b Point Formulas R2 Complexity c 𝑙𝑜𝑔2 𝐾 = √(𝑥12 − 𝑥8 − 𝑥16 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='49) × 𝑥9 × (𝑥6 + 𝑥20) + √𝑙𝑛\u2061(𝑥12) + 𝑥21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='71 20 d 𝑙𝑜𝑔2 𝐾 = √𝑥12 − 𝑙𝑛 [(𝑥8 − 𝑥11) × 𝑥8 𝑙𝑛(√𝑥19) ] × (𝑥6 + 𝑥20) × 𝑥7 + 𝑥21 + √𝑙𝑛 𝑥12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='74 25 e 𝑙𝑜𝑔2 𝐾 = 𝑙𝑛\u2061[ 1 𝑥10 × (𝑥1 − 𝑥2) − 𝑥12] × (𝑥5 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='824) + 𝑥17 √ 𝑥3 (𝑥12 − 𝑥11) × 𝑥5 + 𝑥21 + 𝑥13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='74 28 f 𝑙𝑜𝑔2 𝐾 = √𝑥8 + [ 𝑥10 𝑙𝑛(𝑙𝑛 𝑥17 − 𝜒12) + 𝑥8 − 𝑥12] × (𝑥6 + 𝑥20) × 𝑥8 𝑥20 + 𝑥21 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='108972 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' 9 Agarwal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=', Doan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=', Robertson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=', Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' & Assary, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Discovery of Energy Storage Molecular Materials Using Quantum Chemistry-Guided Multiobjective Bayesian Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content=' Chemistry of Materials 33, 8133-8144, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='chemmater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} +page_content='1c02040 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE1T4oBgHgl3EQfPwMX/content/2301.03030v1.pdf'} diff --git a/rtA0T4oBgHgl3EQfK_-E/vector_store/index.faiss b/rtA0T4oBgHgl3EQfK_-E/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ea7342dab352f480a2caaf2f83f6a4c8be3fd17e --- /dev/null +++ b/rtA0T4oBgHgl3EQfK_-E/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1c28254c1516f52fc60b519a010a92eb25d3df682c4d469f05fc08a947f2fc1e +size 7798829 diff --git a/rtFKT4oBgHgl3EQf0y6v/content/2301.11917v1.pdf b/rtFKT4oBgHgl3EQf0y6v/content/2301.11917v1.pdf new file mode 100644 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Let X be the 2-sphere S2 or the real projective plane RP2. We show that +if Γ is a finitely generated group acting minimally and distally on X, then Γ contains a +nonabelian free subgroup. +1. INTRODUCTION +The aim of the note is continuing the study of the following question: +Given a discrete group G and a compact metric space X, +can G act on X distally and minimally? +The answer to this question involves the discussions around the algebraic structure of G +and the topology of X. Here, we are mainly concerned on the case that X is a closed sur- +face (a compact connected 2-manifold with no boundary). There have been several related +investigations around this topic. A remarkable result due to Furstenberg says that if a non- +trivial space X admits a distal minimal action by a locally compact abelian group, then X +cannot be simply connected (see [4, Theorem 11.1] or [1, Chapter 7-Theorem 16]). In [9], +Shi proved further that if a continuum X admits a distal minimal amenable group action, +then the first ˇCech cohomology group ˇH1(X) of X with integer coefficients is nontrivial; +in particular, if X is a CW-complex, then the fundamental group of X cannot be finite; +so, the 2-sphere S2 and the real projective plane RP2 admit no distal minimal actions by +amenable groups. Bronˇsteˇın proved that if X is a connected and locally connected finitely +dimensional compact metric space which admits a distal minimal group action, then X +must be a manifold and the fundamental group π1(X) is virtually nilpotent ([3]); this im- +plies that if X is a closed surface except for the sphere S2, the real projective plane RP2, +the torus T2, and the Klein bottle K2, then it admits no distal minimal actions by any +group. Shi showed that no closed surface admits a distal minimal action by SL(n,Z) with +n ≥ 3 ([10]). +The following is the main theorem of this paper. Recall that a group G is a small group +if it contains no free nonabelian subgroups. +2010 Mathematics Subject Classification. 37B05. +Key words and phrases. distality, amenable group, free group, minimality, cohomology. +1 + +2 +E. H. Shi & Hui Xu +Theorem 1.1. Let X be the 2-sphere S2 or the real projective plane RP2. If Γ is a finitely +generated group acting minimally and distally on X, then Γ contains a nonabelian free +subgroup. Equivalently, X admits no distal minimal actions by a small group. +Here we remark that the class of small groups is strictly larger than that of amenable +groups, so this theorem is not implied by the main theorem in [9]; and it is easy to con- +struct distal minimal actions on S2 and RP2 by Z ∗ Z. In addition, the theorem does not +hold when X is either the torus T2 or the Klein bottle K2, since they admit distal minimal +actions by abelian groups (see the appendix). +Now we summarize all the known results around the existence of distal minimal group +actions on closed surfaces in the following tabular. +Closed surfaces +Existence +Non-existence +S2,RP2 +Z∗Z +small groups, SL(n,Z)(n ≥ 3) +T2,K2 +Z +SL(n,Z)(n ≥ 3) +Others +Any groups +2. PRELIMINARIES +In this section, we just list the theorems that will be used in the proof of the main +theorem without mentioning the related notions and notations already appeared in [9, 10]. +Theorem 2.1. [1, p.98] Let (X,G,φ) and (Y,G,ψ) be distal minimal actions, and let +f : X → Y be a homomorphism. Then f is open. +Theorem 2.2. [1, p.104] Suppose X is not a single point. If (X,G,φ) is distal minimal, +then it has a nontrivial equicontinuous factor. +Theorem 2.3. [3, Theorem 3.17.12] Let π : (X,G,φ) → (Y,G,ψ) be a homomorphism +between minimal systems. Suppose that π is open and G is finitely generated. If (Y,G,ψ) is +equicontinuous and there is some y ∈Y such that f −1(y) is of 0-dimension. Then (X,G,φ) +is also equicontinuous. +Theorem 2.4. [1, p.52] Let (X,G,φ) be equicontinuous. Then the closure φ(G) inC(X,X) +with respect to the uniform convergence topology is a compact topological group. +Theorem 2.5. [8] Let (X,G,φ) and (Y,G,ψ) be distal minimal actions, and let f : X →Y +be a homomorphism. Then for every y ∈ Y, we have dim(Y)+dim( f −1(y)) = dim(X). +Theorem 2.6. [3, Theorem 3.17.10] Let (X,G,φ) be a distal minimal system with X being +a connected and locally connected compact metric space of finite dimension, then X is a +manifold. +Theorem 2.7. [5, Corollary 4.25] Let G be a compact Lie group and let g be the Lie +algebra of G. Then g = Z(g)�[g,g], where Z(g) is the center of g and [g,g] is semisimple. + +Distal minimal group actions +3 +Theorem 2.8. [5, Corollary 1.103] Let G be a compact connected commutative Lie group +of dimension n. Then G is isomorphic to the n-torus Tn. +Theorem 2.9. [11, Theorem 3.50] Let G be a connected Lie group with Lie algebra g. +Then the center of G is a closed Lie subgroup of G with Lie algebra the center of g. +Theorem 2.10. [6, p.65] Let X be a compact metric space and let (X,G) be an action of +group G on X. Suppose G is compact. Then for every x ∈ X, G/Gx is homeomorphic to +Gx, where Gx = {g ∈ G : gx = x}. +Theorem 2.11. [6, p.99] Let G be a compact group and let U be an open neighborhood +of the identity e. Then U contains a closed normal subgroup H of G such that G/H is +isomorphic to a Lie group. +Theorem 2.12. [6, p.61] Let X be a compact metric space and let (X,G) be an action of +group G on X. Suppose G is compact and H is a closed normal subgroup of G. Then G/H +can act on X/H by letting gH ·H(x) = H(gx) for gH ∈ G/H and H(x) ∈ X/H. +We use ˇH1(X) to denote the first ˇCech cohomology group of X with integer coefficients. +Theorem 2.13. [9, Corollary 2.15] Let f : X → Y be an open map from a continuum X +onto a continuum Y. Then f ∗ : ˇH1(Y) → ˇH1(X) is injective. +Theorem 2.14. [2, Theorem 1.3] Let G be a connected non-solvable real Lie group of +dimension d. Then any finitely generated dense subgroup of G contains a dense free sub- +group of rank 2d. +It is well known that a compact connected Hausdorff space is locally connected metriz- +able if and only if it is a continuous image of the closed interval [0,1] ([7, Theorem 8.18]). +Thus the following result is direct. +Theorem 2.15. Let X be a compact metric space which is connected and locally con- +nected and Y be a Hausdorff space. If there is a continuous surjection f : X → Y, then Y +is locally connected and metrizable. +3. PROOF OF THE MAIN THEOREM +Lemma 3.1. Let X be the 2-sphere S2 or the real projective plane RP2. Let Γ be a finitely +generated group and φ : Γ → Homeo(X) be a distal minimal action on X. Then (X,Γ,φ) +is equicontinuous. +Proof. Assume to the contrary that φ is not equicontinuous. From Theorem 2.2, we let +(Y,Γ,ψ) be the maximal equicontinuous factor of (X,Γ,φ) with a factor map π. Then Y +is a compact manifold of dimension ≤ 2 by Theorem 2.5, Theorem 2.6, and Theorem 2.15. +If dim(Y) = 2, then it follows from Theorem 2.5 that for every x ∈ X, dim(π−1(x)) = 0. + +4 +E. H. Shi & Hui Xu +This together with Theorem 2.3 and Theorem 2.1 implies that (X,Γ,φ) is equicontinuous, +which contradicts the assumption. So, we may assume that dim(Y) = 1; this means that +Y is the circle S1. Since π is open and ˇH1(S1) is the integer group, from Theorem 2.1 and +Theorem 2.13, we have that ˇH1(X) is infinite, which is a contradiction. +□ +Lemma 3.2. Let G be a connected compact Lie group acting faithfully and transitively +on a closed surface X with finite fundamental group. Then G is semisimple. +Proof. Assume to the contrary that G is not semisimple. Then by Theorem 2.7, Theorem +2.8, and Theorem 2.9, the connected component Z(G)0 of the center of G is isomorphic +to some torus Tn with n > 0. Set K = Z(G)0. For x ∈ X, let Stab(x) := {k ∈ K : kx = x} +be the stabilizer of x in K. Then from Theorem 2.10, Kx is homeomorphic to K/Stab(x) +which is also a torus. Thus Kx is either a point or a circle. If for every x ∈ X, Kx is a circle, +then similar to the arguments in Lemma 3.1, X/K is a circle and the ˇCech cohomology +group ˇH1(X) with integer coefficients is infinite. This is a contradiction. So, there is some +x0 ∈ X with Kx0 = x0. Since K is in the center of G, we have Kgx0 = gKx0 = gx0 for +every g ∈ G. Noting that Gx0 = X, the action of K on X is trivial, which contradicts the +faithfulness of the action. So, G is semisimple. +□ +Proof of the main theorem. Let φ : Γ → Homeo(X) be the distal minimal action. From +Lemma 3.1, we see that the (X,Γ,φ) is equicontinuous. Let K be the closure of φ(Γ) with +respect to the uniform topology on Homeo(X). It follows from Theorem 2.4 that K is a +compact metric group acting transitively on X. By Theorem 2.11, we can take a closed +normal subgroup N of K such that K/N is a Lie group and X/N is not a single point. Then +it canonically induces an action of K on X/N and the natural quotient map X → X/N is a +equicontinuous extension. Thus it follows from Theorem 2.5, Theorem 2.6, and Theorem +2.15 that X/N is a manifold of dimension ≤ 2. Similar to the arguments in Lemma 3.1, +we have dim(X/N) = 2. +Set p : S2 → X be a covering and q : X → X/N be the quotient map. Let π : Y → X/N +be the universal covering and � +qp : S2 → Y be the lifting of qp. Since π is open and p and +q are local homeomorphisms, we see that � +qp is open. Thus � +qp(S2) is open and closed in +Y. Thus Y = � +qp(S2) by the connectedness. So Y is compact (homeomorphic to S2). Thus +π is a finite cover and then the fundamental group of X/N is finite. +Now consider the natural action of K/N on X/N (see Theorem 2.12). Since the con- +nected component (K/N)0 has finite index in K/N, the (K/N)0 action on X/N is still +transitive and φ(Γ)N ∩ (K/N)0 is dense in (K/N)0. Applying Lemma 3.2, we see that +a quotient group of (K/N)0 is semisimple. This together with Theorem 2.14 implies the +existence of free nonabelian subgroups in Γ. +□ + +Distal minimal group actions +5 +4. APPENDIX +In this appendix, we show that there is a minimal distal homeomorphism on the Klein +bottle. +Let the torus T2 be R2/Z2. Then there is a Z2 action on T2 by +h(x,y) = (x+ 1 +2,1−y) +mod Z2, +where h is the nonidentity element in Z2. It is easy to see that this action is free and +properly discontinuous and the quotient space T2/Z2 is the Klein bottle K2. +Now for a homeomorphism T of T2, if it commutes with h, i.e., Th = hT, then it +induces a homeomorphism �T of K2. +Let α be an irrational number and φ : T → T be a continuous mapping. Further, define +a homeomorphism T : T2 → T2 by +T(x,y) = (x+α,y+φ(x)) +mod Z2. +The system (T2,T) is a skew product system and it is well known that this system is +distal, since it is a group extension of a minimal equicontinuous system. The following +theorem characterizes the minimality of such skew product system. +Theorem 4.1. [1, Chapter 5, Theorem 10] The above defined system (T2,T) is minimal +if and only if for each k ∈ Z \ {0}, there is no continuous function f : T → T such that +f(x+α) = f(x)+kφ(x) for each x ∈ T. +Now we take φ : T → T to be φ(x) = 1 − |1 − 2x| for x ∈ [0,1). By comparing the +Fourier coefficients, there is no continuous function f : T → T such that f(x + α) = +f(x) +kφ(x) for each k ∈ Z\ {0}. Thus the system (T2,T) is minimal by Theorem 4.1. +It is straightforward to calculate that for each (x,y) ∈ T2, Th(x,y) = (x +α + 1 +2,1 −y+ +φ(x+ 1 +2)) and hT(x,y) = (x+α + 1 +2,1−y−φ(x)). Note that +φ(x+ 1 +2) = +� +1−2x, +x ∈ [0,1/2) +2x−1, +x ∈ [1/2,1) and −φ(x) = +� +−2x, +x ∈ [0,1/2) +2x−2, +x ∈ [1/2,1) . +Therefore, it follows that T commutes with h and thus the induced homeomorphism �T on +K2 is also minimal and distal. +REFERENCES +[1] J. Auslander, Miniml flows and their extensions. North-Holland Mathematics Stud- +ies, 153. Notas de Matem´atica [Mathematical Notes], 122. North-Holland Publish- +ing Co., Amsterdam, 1988. +[2] E. Breuillard, T. Gelander, On dense free subgroups of Lie groups. J. Algebra 261 +(2003), 448-467. + +6 +E. H. Shi & Hui Xu +[3] I. Bronˇsteˇın, Extensions of minimal transformation groups, Suthoff & Noordhoff, +1979. +[4] H. Furstenberg, The Structure of distal flows. Amer. J. Math. 85 (1963), 477-515. +[5] A. Knapp, Lie groups beyaond an introduction (Second Edition). PM 140, +Birkhauser, 2002. +[6] D. Montgomery, L. Zippin, Topological transformation groups. Interscience Pub- +lishers, New York-London, 1955. +[7] S. Nadler, Continuum theory, volume 158 of Monographs and Textbooks in Pure +and Applied Mathematics. Marcel Dekker, Inc., New York, 1992. An introduction. +[8] M. Rees, On the structure of minimal distal transformation groups with manifolds +as phase spaces, thesis, University of Warwick, Coventry, England. +[9] E. +Shi, +Continua +having +distal +minimal +actions +by +amenable +groups. +arXiv:2001.03755. +[10] E. Shi, Distal higher rank lattice actions on surfaces. arXiv:2001.01183. +[11] F. Warner, Foundations of differentiable manifolds and Lie groups. Springer-Verlag, +GTM 94, 1983. +(E.H. Shi) SCHOOL OF MATHEMATICAL SCIENCES, SOOCHOW UNIVERSITY, SUZHOU 215006, P. +R. CHINA +Email address: ehshi@suda.edu.cn +(H. Xu) CAS WU WEN-TSUN KEY LABORATORY OF MATHEMATICS, UNIVERSITY OF SCIENCE +AND TECHNOLOGY OF CHINA, HEFEI, ANHUI 230026, CHINA +Email address: huixu2734@ustc.edu.cn + diff --git a/tNE5T4oBgHgl3EQfKw5g/content/tmp_files/load_file.txt b/tNE5T4oBgHgl3EQfKw5g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..830af93c5950600f4261c14b408b8aac55f8c3d4 --- /dev/null +++ b/tNE5T4oBgHgl3EQfKw5g/content/tmp_files/load_file.txt @@ -0,0 +1,276 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf,len=275 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='05468v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='DS] 13 Jan 2023 GROUPS ACTING DISTALLY AND MINIMALLY ON S2 AND RP2 ENHUI SHI AND HUI XU ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Let X be the 2-sphere S2 or the real projective plane RP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' We show that if Γ is a finitely generated group acting minimally and distally on X, then Γ contains a nonabelian free subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' INTRODUCTION The aim of the note is continuing the study of the following question: Given a discrete group G and a compact metric space X, can G act on X distally and minimally?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' The answer to this question involves the discussions around the algebraic structure of G and the topology of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Here, we are mainly concerned on the case that X is a closed sur- face (a compact connected 2-manifold with no boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' There have been several related investigations around this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' A remarkable result due to Furstenberg says that if a non- trivial space X admits a distal minimal action by a locally compact abelian group, then X cannot be simply connected (see [4, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='1] or [1, Chapter 7-Theorem 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' In [9], Shi proved further that if a continuum X admits a distal minimal amenable group action, then the first ˇCech cohomology group ˇH1(X) of X with integer coefficients is nontrivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' in particular, if X is a CW-complex, then the fundamental group of X cannot be finite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' so, the 2-sphere S2 and the real projective plane RP2 admit no distal minimal actions by amenable groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Bronˇsteˇın proved that if X is a connected and locally connected finitely dimensional compact metric space which admits a distal minimal group action, then X must be a manifold and the fundamental group π1(X) is virtually nilpotent ([3]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' this im- plies that if X is a closed surface except for the sphere S2, the real projective plane RP2, the torus T2, and the Klein bottle K2, then it admits no distal minimal actions by any group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Shi showed that no closed surface admits a distal minimal action by SL(n,Z) with n ≥ 3 ([10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' The following is the main theorem of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Recall that a group G is a small group if it contains no free nonabelian subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' 37B05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' distality, amenable group, free group, minimality, cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' 1 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Shi & Hui Xu Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Let X be the 2-sphere S2 or the real projective plane RP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' If Γ is a finitely generated group acting minimally and distally on X, then Γ contains a nonabelian free subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Equivalently, X admits no distal minimal actions by a small group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Here we remark that the class of small groups is strictly larger than that of amenable groups, so this theorem is not implied by the main theorem in [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' and it is easy to con- struct distal minimal actions on S2 and RP2 by Z ∗ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' In addition, the theorem does not hold when X is either the torus T2 or the Klein bottle K2, since they admit distal minimal actions by abelian groups (see the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Now we summarize all the known results around the existence of distal minimal group actions on closed surfaces in the following tabular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Closed surfaces Existence Non-existence S2,RP2 Z∗Z small groups, SL(n,Z)(n ≥ 3) T2,K2 Z SL(n,Z)(n ≥ 3) Others Any groups 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' PRELIMINARIES In this section, we just list the theorems that will be used in the proof of the main theorem without mentioning the related notions and notations already appeared in [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='98] Let (X,G,φ) and (Y,G,ψ) be distal minimal actions, and let f : X → Y be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then f is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='104] Suppose X is not a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' If (X,G,φ) is distal minimal, then it has a nontrivial equicontinuous factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='12] Let π : (X,G,φ) → (Y,G,ψ) be a homomorphism between minimal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Suppose that π is open and G is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' If (Y,G,ψ) is equicontinuous and there is some y ∈Y such that f −1(y) is of 0-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then (X,G,φ) is also equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='52] Let (X,G,φ) be equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then the closure φ(G) inC(X,X) with respect to the uniform convergence topology is a compact topological group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [8] Let (X,G,φ) and (Y,G,ψ) be distal minimal actions, and let f : X →Y be a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then for every y ∈ Y, we have dim(Y)+dim( f −1(y)) = dim(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='10] Let (X,G,φ) be a distal minimal system with X being a connected and locally connected compact metric space of finite dimension, then X is a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [5, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='25] Let G be a compact Lie group and let g be the Lie algebra of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then g = Z(g)�[g,g], where Z(g) is the center of g and [g,g] is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Distal minimal group actions 3 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [5, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='103] Let G be a compact connected commutative Lie group of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then G is isomorphic to the n-torus Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='50] Let G be a connected Lie group with Lie algebra g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then the center of G is a closed Lie subgroup of G with Lie algebra the center of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='65] Let X be a compact metric space and let (X,G) be an action of group G on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Suppose G is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then for every x ∈ X, G/Gx is homeomorphic to Gx, where Gx = {g ∈ G : gx = x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='99] Let G be a compact group and let U be an open neighborhood of the identity e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then U contains a closed normal subgroup H of G such that G/H is isomorphic to a Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='61] Let X be a compact metric space and let (X,G) be an action of group G on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Suppose G is compact and H is a closed normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then G/H can act on X/H by letting gH ·H(x) = H(gx) for gH ∈ G/H and H(x) ∈ X/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' We use ˇH1(X) to denote the first ˇCech cohomology group of X with integer coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [9, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='15] Let f : X → Y be an open map from a continuum X onto a continuum Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then f ∗ : ˇH1(Y) → ˇH1(X) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='3] Let G be a connected non-solvable real Lie group of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then any finitely generated dense subgroup of G contains a dense free sub- group of rank 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' It is well known that a compact connected Hausdorff space is locally connected metriz- able if and only if it is a continuous image of the closed interval [0,1] ([7, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Thus the following result is direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Let X be a compact metric space which is connected and locally con- nected and Y be a Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' If there is a continuous surjection f : X → Y, then Y is locally connected and metrizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' PROOF OF THE MAIN THEOREM Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Let X be the 2-sphere S2 or the real projective plane RP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Let Γ be a finitely generated group and φ : Γ → Homeo(X) be a distal minimal action on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then (X,Γ,φ) is equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Assume to the contrary that φ is not equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='2, we let (Y,Γ,ψ) be the maximal equicontinuous factor of (X,Γ,φ) with a factor map π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then Y is a compact manifold of dimension ≤ 2 by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='6, and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' If dim(Y) = 2, then it follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='5 that for every x ∈ X, dim(π−1(x)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Shi & Hui Xu This together with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='3 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='1 implies that (X,Γ,φ) is equicontinuous, which contradicts the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' So, we may assume that dim(Y) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' this means that Y is the circle S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Since π is open and ˇH1(S1) is the integer group, from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='13, we have that ˇH1(X) is infinite, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Let G be a connected compact Lie group acting faithfully and transitively on a closed surface X with finite fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then G is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Assume to the contrary that G is not semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='7, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='8, and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='9, the connected component Z(G)0 of the center of G is isomorphic to some torus Tn with n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Set K = Z(G)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' For x ∈ X, let Stab(x) := {k ∈ K : kx = x} be the stabilizer of x in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='10, Kx is homeomorphic to K/Stab(x) which is also a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Thus Kx is either a point or a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' If for every x ∈ X, Kx is a circle, then similar to the arguments in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='1, X/K is a circle and the ˇCech cohomology group ˇH1(X) with integer coefficients is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' So, there is some x0 ∈ X with Kx0 = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Since K is in the center of G, we have Kgx0 = gKx0 = gx0 for every g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Noting that Gx0 = X, the action of K on X is trivial, which contradicts the faithfulness of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' So, G is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' □ Proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Let φ : Γ → Homeo(X) be the distal minimal action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='1, we see that the (X,Γ,φ) is equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Let K be the closure of φ(Γ) with respect to the uniform topology on Homeo(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' It follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='4 that K is a compact metric group acting transitively on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='11, we can take a closed normal subgroup N of K such that K/N is a Lie group and X/N is not a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then it canonically induces an action of K on X/N and the natural quotient map X → X/N is a equicontinuous extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Thus it follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='6, and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='15 that X/N is a manifold of dimension ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Similar to the arguments in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='1, we have dim(X/N) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Set p : S2 → X be a covering and q : X → X/N be the quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Let π : Y → X/N be the universal covering and � qp : S2 → Y be the lifting of qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Since π is open and p and q are local homeomorphisms, we see that � qp is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Thus � qp(S2) is open and closed in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Thus Y = � qp(S2) by the connectedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' So Y is compact (homeomorphic to S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Thus π is a finite cover and then the fundamental group of X/N is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Now consider the natural action of K/N on X/N (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Since the con- nected component (K/N)0 has finite index in K/N, the (K/N)0 action on X/N is still transitive and φ(Γ)N ∩ (K/N)0 is dense in (K/N)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='2, we see that a quotient group of (K/N)0 is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' This together with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='14 implies the existence of free nonabelian subgroups in Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' □ Distal minimal group actions 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' APPENDIX In this appendix, we show that there is a minimal distal homeomorphism on the Klein bottle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Let the torus T2 be R2/Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Then there is a Z2 action on T2 by h(x,y) = (x+ 1 2,1−y) mod Z2, where h is the nonidentity element in Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' It is easy to see that this action is free and properly discontinuous and the quotient space T2/Z2 is the Klein bottle K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Now for a homeomorphism T of T2, if it commutes with h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=', Th = hT, then it induces a homeomorphism �T of K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Let α be an irrational number and φ : T → T be a continuous mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Further, define a homeomorphism T : T2 → T2 by T(x,y) = (x+α,y+φ(x)) mod Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' The system (T2,T) is a skew product system and it is well known that this system is distal, since it is a group extension of a minimal equicontinuous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' The following theorem characterizes the minimality of such skew product system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' [1, Chapter 5, Theorem 10] The above defined system (T2,T) is minimal if and only if for each k ∈ Z \\ {0}, there is no continuous function f : T → T such that f(x+α) = f(x)+kφ(x) for each x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Now we take φ : T → T to be φ(x) = 1 − |1 − 2x| for x ∈ [0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' By comparing the Fourier coefficients, there is no continuous function f : T → T such that f(x + α) = f(x) +kφ(x) for each k ∈ Z\\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Thus the system (T2,T) is minimal by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' It is straightforward to calculate that for each (x,y) ∈ T2, Th(x,y) = (x +α + 1 2,1 −y+ φ(x+ 1 2)) and hT(x,y) = (x+α + 1 2,1−y−φ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Note that φ(x+ 1 2) = � 1−2x, x ∈ [0,1/2) 2x−1, x ∈ [1/2,1) and −φ(x) = � −2x, x ∈ [0,1/2) 2x−2, x ∈ [1/2,1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Therefore, it follows that T commutes with h and thus the induced homeomorphism �T on K2 is also minimal and distal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Auslander, Miniml flows and their extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' North-Holland Mathematics Stud- ies, 153.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Springer-Verlag, GTM 94, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Shi) SCHOOL OF MATHEMATICAL SCIENCES, SOOCHOW UNIVERSITY, SUZHOU 215006, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' CHINA Email address: ehshi@suda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='cn (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content=' Xu) CAS WU WEN-TSUN KEY LABORATORY OF MATHEMATICS, UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA, HEFEI, ANHUI 230026, CHINA Email address: huixu2734@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfKw5g/content/2301.05468v1.pdf'} diff --git a/utFLT4oBgHgl3EQfjS8b/content/tmp_files/2301.12110v1.pdf.txt b/utFLT4oBgHgl3EQfjS8b/content/tmp_files/2301.12110v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cef04968b4e203e83e111f0280759e4975fa5e22 --- /dev/null +++ b/utFLT4oBgHgl3EQfjS8b/content/tmp_files/2301.12110v1.pdf.txt @@ -0,0 +1,2084 @@ +arXiv:2301.12110v1 [math.CO] 28 Jan 2023 +FREE FERMIONIC SCHUR FUNCTIONS +SLAVA NAPRIENKO +Abstract. In this paper, we introduce a new family of Schur functions that depend on two +sets of variables and two doubly infinite sequences of parameters. These functions general- +ize and unify various existing Schur functions, including classical Schur functions, factorial +Schur functions, supersymmetric Schur functions, Frobenius-Schur functions, factorial su- +persymmetric Schur functions, and dual Schur functions. We prove that the new family +of functions satisfies several well-known properties, such as the combinatorial description, +Jacobi-Trudi identity, N¨agelsbach-Kostka formula, Giambelli formula, Ribbon formula, Weyl +formula, Berele-Regev factorization, and Cauchy identity. +Our approach is based on the integrable six vertex model with free fermionic weights. +We show that these weights satisfy the refined Yang-Baxter equation, which results in su- +persymmetry for the Schur functions. Furthermore, we derive refined operator relations for +the row transfer operators and use them to find partition functions with various boundary +conditions. Our results provide new proofs for known results as well as new identities for +the Schur functions. +Contents +1. +Introduction +1 +2. +Six Vertex Model +4 +2.1. +Combinatorial description +5 +2.2. +Non-intersecting lattice paths +6 +2.3. +Row transfer operators +9 +2.4. +The Yang-Baxter equation and operator relations +12 +2.5. +Partition functions +16 +2.6. +The six vertex model on infinite strip +18 +3. +Partitions, Maya diagrams, and ribbons +22 +4. +Free fermionic Schur functions +23 +4.1. +Further properties +28 +References +29 +1. Introduction +In this paper, we introduce a new family of Schur functions sλ/µ;a,b(x, y), which depend +on two sets of variables x = (x1, . . . , xn) and y = (y1, . . . , yn), as well as two doubly infinite +sequences of parameters (ai)i∈Z and (bi)i∈Z. We provide a hands-on definition in the style of +Olshanski, Regev, and Vershik [ORV03]. +1 + +First, we define the complete symmetric free fermionic functions hk;a,b by the generating +series +1 + +∞ +� +k=1 +hk;a,b(x, y)1 − akbk +1 − a0b0 +(z − b0) +(1 − a1z) +(z − b1) +(1 − a2z) . . . (z − bk−1) +(1 − akz) = +n +� +i=1 +1 + yiz +1 − xiz +1 − b0xi +1 + b0yi +. +As usual, we set h0;a,b = 1 and hk;a,b = 0 for k < 0. +For any two partitions λ and µ, the free fermionic Schur functions sλ/µ;a,b are given by +sλ/µ;a,b = det(hλi−µj−i+j;τ 1−ja,τ 1−jb)1≤i,j≤l(λ), +where τ s((ai)i∈Z) = (ai+s)i∈Z is the shift operator. +We show that this new family of Schur functions unifies and generalizes existing families +of Schur functions from literature that we discuss now. +The classical Schur functions sλ(x) are a type of symmetric function that arise in various +areas of mathematics, such as representation theory, algebraic combinatorics, and the theory +of special functions. They can be used to provide characters for polynomial representations +of the general linear group or the symmetric group, and also appear in algebraic geometry +as representatives of cycles in flag varieties. There are numerous generalizations of Schur +functions that have been studied in different contexts. +One such generalization is the factorial Schur function sλ(x|a), which depends on a se- +quence of parameters (ai)i∈Z. When ai = −i + 1, these functions are known as shifted Schur +functions s∗ +λ(x) and were introduced by Olshanski and Okounkov in [OO97]. They form a +natural basis for the center of the universal enveloping algebra U(gln). Another variant, the +double Schur function sλ(x || a), was introduced by Molev in [Mol09] and differs from the +factorial Schur functions only by a reparametrization. Both the factorial and double Schur +functions have the property of stability, which allows for their definition in infinitely many +variables. The factorial Schur functions have also applications in the combinatorics of flag +varieties, where they appear as the equivariant Schubert classes. +Another generalization is the supersymmetric Schur function sλ(x/y), which depends on +two sets of variables x and y and satisfies the property of supersymmetry. This property +relates the Schur functions to the representation theory of the superalgebra gl(m|n) [BR87]. +Molev [Mol98] further generalized the factorial and supersymmetric Schur functions by +introducing the factorial supersymmetric Schur functions sλ(x/y || a). However, Olshanski, +Regev, and Vershik [ORV03] pointed out that Molev’s generalization does not have the +stability property. +They introduced the Frobenius-Schur functions sλ;a(x, y) as a shifted +version of Molev’s functions, which do possess the stability property. When the number of +variables x and y are equal, the Frobenius-Schur functions differ from Molev’s functions only +by a shift in the parameters. +One of the most fundamental results in the study of classical Schur functions sλ(x) is the +Cauchy identity, which provides a closed product formula for the sum of Schur functions +over all partitions: +� +λ +sλ(x)sλ(y) = +� +i,j +1 +1 − xiyj +. +2 + +Berele and Regev [BR87] demonstrated that the supersymmetric Schur functions sλ(x/y) +also satisfy a Cauchy identity of the form: +� +λ +sλ(x/y)sλ(z/w) = +� +i,j +1 + yizj +1 − xizj +1 + xiwj +1 − yiwj +. +However, it remained unclear whether a similar result held for the factorial Schur functions +sλ(x||a). In Theorem 3.1 of [Mol09], Molev introduced a new family of dual Schur functions +�sλ(x || a) and demonstrated that they satisfy a Cauchy identity with the factorial Schur +functions: +� +λ +sλ(x||a)�sλ(z||a) = +� +i,j +1 − aizj +1 − xizj +. +Furthermore, Molev proved in Corollary 3.2 of [Mol09] that the factorial supersymmetric +Schur functions sλ(x/y||a) (also known as Frobenius-Schur functions sλ;a(x, y)) and the dual +Schur functions satisfy the following Cauchy identity: +� +λ +sλ(x/y||a)�sλ(y||a) = +� +i,j +1 + yizj +1 − xiyj +. +However, it was not known if a Cauchy identity in the style of Berele-Regev involving two +factorial supersymmetric Schur functions existed. +In this article, we present a new family of free fermionic Schur functions that unifies and +generalizes previously mentioned Schur functions. Specifically, we show that the new family +of functions sλ/µ;a,b(x, y) encompasses the following cases: +(1) sλ/µ;0,0(x, 0) = sλ/µ(x): classical Schur functions, +(2) sλ/µ;0,0(x, y) = sλ/µ(x/y): supersymmetric Schur functions, +(3) sλ/µ;a,0(x, a′) = sλ/µ(x || a): factorial Schur functions, +(4) sλ/µ;a′,0(x, y) = sλ/µ(x/y || a): factorial supersymmetric Schur functions, +(5) sλ/µ;a,0(x, y) = sλ/µ;a(x, y): Frobenius-Schur functions, +(6) sλ/µ;0,b(x, 0) = �sλ/µ(x || b): dual Schur functions. +Furthermore, we demonstrate that the free fermionic Schur functions satisfy a Berele- +Regev Cauchy identity, which relates the free fermionic Schur functions to their dual coun- +terparts: +Theorem (See Theorem 4.7 in text). +(1.1) +� +λ +sλ;a,b(x, y)�sλ;a,b(z, w) = +� +i,j +1 + yizj +1 − xiyj +1 + xiwj +1 − yiwj +. +It is noteworthy that the right-hand side of the identity does not depend on the doubly +infinite sequences of parameters a = (ai)i∈Z and b = (bi)i∈Z. Naturally, our version of the +Cauchy identity degenerates to all Cauchy identities mentioned above. +In addition to the Cauchy identity, we prove that the free fermionic Schur functions possess +various properties commonly associated with Schur functions, such as the combinatorial +description, Jacobi-Trudi identity, N¨agelsbach-Kostka formula, Giambelli formula, Ribbon +formula, Weyl determinant formula, Berele-Regev factorization, and others. +Our approach is based on the integrable six vertex model with free fermionic weights. +Specifically, we define the free fermionic Schur functions as the partition function of the six +3 + +vertex model, where each vertex is assigned two spectral parameters x, y associated with the +row and two spectral parameters a, b associated with the column. The weights assigned to +each vertex are defined as follows: +a1(x, y; a, b) = 1 − bx, +a2(x, y; a, b) = y + a, +b1(x, y; a, b) = 1 + by, +b2(x, y; a, b) = x − a, +c1(x, y; a, b) = 1 − ab, +c2(x, y; a, b) = x + y. +The central tool in the theory of integrable lattice models is the Yang-Baxter equation. +It relates the weights of two vertices by exchanging their row spectral parameters simulta- +neously. In particular, if T(x, y; a, b) represents a vertex with labels x, y and a, b, then the +classical Yang-Baxter equation is given by +R(x1, y1; x2, y2)T(x1, y1; a, b)T(x2, y2; a, b) = T(x2, y2; a, b)T(x1, y1; a, b)R(x1, y1; x2, y2). +One of the main novelties of our work is the introduction of new refined Yang-Baxter +equations, which allow us to exchange the spectral parameters x or y separately: +Theorem (See Theorem 2.8 in text). +Rx(x1, x2; y1)T(x1, y1; a, b)T(x2, y2; a, b) = T(x2, y1; a, b)T(x1, y2; a, b)Rx(x1, x2; y1), +Ry(y1, y2; x2)T(x1, y2; a, b)T(x2, y2; a, b) = T(x1, y2; a, b)T(x2, y1; a, b)Ry(y1, y2; x2). +The refined Yang-Baxter equations lead to the refined Yang-Baxter algebra of the row +transfer operators which allows us to demonstrate that the free fermionic Schur functions +defined in our model possess supersymmetry. Specifically, they are symmetric separately in +x and y, and satisfy the cancellation property (Proposition 4.1). +The free fermionic six vertex model has been previously used to define generalizations +of Schur functions. In [Mot17b], a family of Schur functions was defined using a specific +parametrization of weights. In [ABPW21], Aggarwal, Borodin, Petrov, and Wheeler studied +the partition functions of the free fermionic six vertex models with a different parametrization +of the weights and showed that the partition functions specialize to the factorial Schur +functions and supersymmetric Schur functions. +It is possible to relate the weights from +[ABPW21] to our weights through a sequence of reparametrizations, which means that many +results obtained using different parametrizations are analogous to each other. +However, our choice of parametrization, normalization, and shift of parameters enables us +to define a new family of Schur functions that are both supersymmetric and stable under +specialization. These properties are important as they allow for the definition of symmetric +Schur functions in infinitely many variables as elements of a graded ring. +Furthermore, +our choice of parametrization unifies the factorial supersymmetric Schur functions and the +dual Schur functions from Molev’s work [Mol09]. By unifying these various types of Schur +functions and establishing their stability and supersymmetry, we provide a uniform approach +to studying Schur functions and their generalizations using the free fermionic six vertex +model. +Acknowledgements. I am sincerely grateful to Daniel Bump for his invaluable support +and guidance and mentorship throughout the course of this project. Thank you! +2. Six Vertex Model +In this section, we review the six vertex model from statistical mechanics. We show that +the six vertex model has a combinatorial description in terms of admissible states on a lattice +4 + +I4 +I3 +I2 +I1 +J1 +J2 +J3 +J4 +J5 +J6 +J7 +J8 +I4 +I3 +I2 +I1 +J1 +J2 +J3 +J4 +J5 +J6 +J7 +J8 +Figure 1. The six vertex model and a typical state. +i +j +i +j +i +j +i +j +i +j +i +j +a1(i, j) +a2(i, j) +b1(i, j) +b2(i, j) +c1(i, j) +c2(i, j) +Figure 2. The six admissible types of vertices. +The types of vertices are +traditionally called a1, a2, b1, b2, c1, c2, following Baxter [Bax82] +as well as an operator description in terms of row transfer operators. A good overview of +the six vertex model is given in Section 1 of [BBF11] and Section 1-4 of [ABPW21]. +2.1. Combinatorial description. The six vertex model is a model on a rectangular lattice +that is determined by the following data: +(1) Row labels I = (I1, I2, . . . , IN), where N is the number of rows, +(2) Column labels J = (J1, J2, . . . , JM), where M is the number of columns, +(3) Boundaries β = (βl, βt, βr, βb) with βl, βt, βr, βb ∈ 2N. +Given the data, the six vertex model is the configurations of paths on a rectangular lattice +with N rows and M columns. The paths enter the lattice on the left at rows βl and on +the top at columns βt, and leave the lattice on the right at rows βr, and on the bottom at +columns βb. Paths travel from NW to SE, and they can intersect, but they can only move +right and down. Due to these restrictions, there are only six possible configurations for each +vertex, hence the name of the model. +An admissible state in a six vertex model is any configuration of paths that respects the +boundaries. It is important to note that the six vertex model satisfies a preservation law: +each vertex has the same number of incoming and outgoing paths. Therefore, for there to +be any admissible states at all, the number of paths entering and leaving the model must +be equal: l(βl) + l(βt) = l(βr) + l(βb). We denote the set of all admissible states of the six +vertex model with the given data as S(I, J; βl, βt, βr, βb). If some of the boundaries are out +of range of the model, they are ignored. +Let a1, a2, b1, b2, c1, c2: I × J → C be the weight functions. The vertex weight wt(v) of a +vertex v in a state s ∈ S is the weight function of the vertex type. For example, the vertex +5 + +weight of a vertex of type b2 on row with label i and column with label j is equal to b2(i, j). +The state weight of a state s ∈ S is the product of the weights of all vertices in s. Finally, +the partition function Z(S) of a six vertex model S is the sum of the weights of all states +in S: +Z(S) = +� +s∈S +wt(s) = +� +s∈S +� +v∈s +wt(v). +For brevity, we write Z(I, J; βl, βt, βr, βb) for Z(S(I, J; βl, βt, βr, βb)). +One of the main objectives in the study of integrable lattice models is to identify ap- +propriate weights that result in meaningful and useful partition functions. The six vertex +model, for example, has been shown to produce a variety of special functions depending +on the choice of weights used. With one set of weights, the model generates the number +of alternating sign matrices [Kup96]. With another, it produces Schur functions, and more +generally, spherical Whittaker functions for the general linear group over a non-archimedean +local field by means of the Casselman-Shalika formula [BBF11]. Additionally, using yet an- +other set of weights, the six vertex model generates supersymmetric Schur functions [Har21]. +Recently, by using more general weights, it has been demonstrated that the six vertex model +can produce various generalizations of Schur functions [Mot17b, Mot17a, ABPW21]. +In this paper, the vertices will have the horizontal labels of the form (xi, yj) and the +vertical labels of the form (ai, bj). Then we use the following weights: +(2.1) +a1(x, y; a, b) = 1 − bx, +a2(x, y; a, b) = y + a, +b1(x, y; a, b) = 1 + by, +b2(x, y; a, b) = x − a, +c1(x, y; a, b) = 1 − ab, +c2(x, y; a, b) = x + y. +We note that similar weights were used in Section 3 of [BBF11] and Figure 4 of [ABPW21]. +By change of variables and rescaling, it is possible to relate different choices of weights to +each other. However, we have chosen our parametrization because it is most suitable for +studying the resulting symmetric functions. In particular, our choice of weights leads to the +refined Yang-Baxter equation, which gives an easy proof of the supersymmetry. +2.2. Non-intersecting lattice paths. We revisit the theory of non-intersecting lattice +paths and recall the powerful Lindstr¨om–Gessel–Viennot lemma (LGV lemma). For a com- +prehensive treatment of the topic, we refer the reader to [Lin73, GV85]. +Consider a directed acyclic graph G, in which each directed edge e ∈ G is assigned a +weight wt(e). For a directed path P between two vertices, we define the weight of the path, +wt(P), as the product of the weights of the edges in the path. For any two vertices a, b ∈ G, +we define the sum e(a, b) = � +P : a→b wt(P) over all directed paths from a to b. +Let A = (a1, . . . , an) and B = (b1, . . . , bn) be two n-tuples of vertices. We consider an +n-tuple of non-intersecting paths (P1, . . . , Pn): A → B, where Pi : ai → bi. +The weight +wt(P1, . . . , Pn) of the n-tuple is defined as the product wt(P1, . . . , Pn) = �n +i=1 wt(Pi) of the +weights of the involved paths. Additionally, we impose the restriction that if we fix the +starting points (a1, . . . , an), then each path Pi in an n-tuple (P1, . . . , Pn) of non-intersecting +paths must end exactly at bi. In other words, there is no n-tuple of non-intersecting paths +P1, . . . , Pn such that Pi : ai → bσ(i) for some permutation σ ∈ Sn. +With these conditions in place, we can state the Lindstr¨om–Gessel–Viennot lemma: +6 + +Lemma 2.1. The weighted sum of all n-tuples (P1, . . . , Pn): A → B is equal to the deter- +minant of the weights of one path traveling from ai to bj, i.e., +� +(P1,...,Pn): A→B +wt(P1, . . . , Pn) = det(e(ai, bj))1≤i,j≤n. +Remark 2.2. The LGV lemma is applicable in a vastly more general context, including the +possibility of permutations of the paths and more general graphs. However, in this paper, +we focus on the most basic and special case. +The six vertex model resembles a model of non-intersecting lattice paths, with the ex- +ception that the paths do intersect. However, under certain conditions on the weights, it +is possible to adjust these intersections to obtain a model of non-intersecting paths without +altering the normalized partition function. +We say that the weights of a six vertex model are free fermionic if they satisfy the following +condition: +(2.2) +a1a2 + b1b2 = c1c2. +The weights given by (2.1) are free fermionic as +(1 − bx)(y + a) + (1 + by)(x − a) = (1 − ab)(x + y). +Let �a1 = a1/a1 = 1, �a2 = a2/a1, �b1 = b1/a1, �b2 = b2/a1, �c1 = c1/a1, and �c2 = c2/a1 be the +normalized weight functions. We also define the normalized partition function +�Z(I; J; β) = +Z(I; J; β) +�N +i=1 +�M +j=1 a1(Ii, Jj) +. +By dividing the weight of vertices at each site by the weight of a1, we obtain a new partition +function which is equivalent to the original partition function with the normalized weights. +This normalization eliminates the contribution of vertices of type a1 to the partition function, +thus making it possible to relate the six-vertex model to a system of non-intersecting lattice +paths. +Given a six vertex model S(I; J; βl, βt, βr, βb), we can associate a corresponding directed +acyclic graph G(I; J; βl, βt, βr, βb) in which each horizontal edge is directed from left to right, +and each vertical edge is directed from top to bottom. Additionally, for each vertex, we add +a new diagonal edge connecting the midpoint of the top edge to the midpoint of the right +edge and directed from northwest to southeast. The weights of the edges in this graph are +as specified in the table in Figure 3. +edge +weight +�c1(v) +1 +�b1(v) +�b2(v)/�c1(v) +�a2(v)/�c1(v) +Figure 3. The weights of edges around the vertex v in the associated graph. +With this construction, we have the following proposition: +7 + +Figure 4. A typical state in the corresponding graph. +Proposition 2.3. Let a1, a2, b1, b2, c1, c2 be free fermionic weights. Let A1, A2, . . . , Ad be the +positions where paths enter the model counting from the left bottom corner to top left corner +to top right corner, and B1, B2, . . . , Bd be the positions where paths leave the model counting +from bottom left corner to bottom right corner to top right corner. The partition function of +a free fermionic six vertex model is given by the determinant of one-path partition functions: +�Z(S(I1, . . . , IN; J1, . . . , JM; β)) = det +� +�ZAi→Bj +� +, +where �ZAi→Bj is the normalized partition function of the system with one path entering at +the position Ai and leaving at the position Bj. +Proof. We show that the partition function equals to the weighted sum of non-intersecting +lattice paths of the corresponding graph G(I; J; βl, βt, βr, βb). We show that the equality +locally on the level of one vertex. Then the result follows globally. It is enough to show +that for each type of the six vertex vertex, the associated weights in the graph give the +same contribution. The weights of types a1, b1, b2, c1 are just mapped to the same weights. +Consider a vertex of type a2 in the six vertex model which corresponds to the intersection +of two paths. In the associated graph, the paths do not intersect: +→ += +× +. +Hence, the six vertex weight �a2(v) splits into the product two edge weights �c1 and �a2/�c1. +Next consider a vertex of type c2 in the six vertex model. In the associated graph, there are +two possibilities for the path: +→ ++ +. +8 + +Then the six vertex weight �c2 splits into the sum of �b1�b2/�c1 and �a2/�c1. But by the free- +fermionic condition for the normalized weights, we have +�c1�c2 = �b1�b2 + �a2. +Since the weight preserves at each vertex, it preserves globally, and we showed that the +partition function equals to the weighted sum of non-intersecting lattice paths. Now we +apply the LGV lemma to get the result. +□ +2.3. Row transfer operators. In this subsection, we introduce an alternative definition of +the six vertex model, in terms of row transfer operators, which will be useful for our analysis. +The six vertex model can be viewed as a composition of operators acting on parametrized +vector spaces. Specifically, we consider two families of vector spaces: +(1) V (x, y) = C e0 ⊕ C e1 ∼= C2, which correspond to the horizontal edges, +(2) W(a, b) = C e0 ⊕ C e1 ∼= C2, which correspond to the vertical edges. +We define the vertex operator +T(x, y; a, b) : V (x, y) ⊗ W(a, b) → W(a, b) ⊗ V (a, b), +which is given by its matrix in the standard basis e0 ⊗ e0, e0 ⊗ e1, e1 ⊗ e0, e1 ⊗ e1: +T(x, y; a, b) = + + + + +a1(x, y; a, b) +c1(x, y; a, b) +b1(x, y; a, b) +b2(x, y; a, b) +c2(x, y; a, b) +a2(x, y; a, b) + + + + += + + + + +1 − bx +1 − ab +1 + by +x − a +x + y +y + a + + + + . +Despite not all matrix elements depend on all variables x, y, a, b, we write the dependence +for the uniform notation. +In our convention, the operators act on the left. For vectors v, w, we write ⟨v|T|w⟩ for the +corresponding matrix coefficient. Hence, we have +a1(x, y; a, b) = ⟨e0 ⊗ e0|T(x, y; a, b)|e0 ⊗ e0⟩ = 1 − bx, +a2(x, y; a, b) = ⟨e1 ⊗ e1|T(x, y; a, b)|e1 ⊗ e1⟩ = y + a, +b1(x, y; a, b) = ⟨e0 ⊗ e1|T(x, y; a, b)|e1 ⊗ e0⟩ = 1 + by, +b2(x, y; a, b) = ⟨e1 ⊗ e0|T(x, y; a, b)|e0 ⊗ e1⟩ = x − a, +c1(x, y; a, b) = ⟨e0 ⊗ e1|T(x, y; a, b)|e0 ⊗ e1⟩ = 1 − ab, +c2(x, y; a, b) = ⟨e1 ⊗ e0|T(x, y; a, b)|e1 ⊗ e0⟩ = x + y. +Graphically, we represent the vertex operator as a vertex in a rectangular lattice. We label +the horizontal edge by x, y to represent V (x, y) and the vertical edge by a, b to represent +W(a, b). Then we draw empty edges for the basis elements e0 of the corresponding spaces, +9 + +and shaded edges for the basis elements e1. For example, we have +x, y +a, b +x, y +a, b +x, y +a, b +x, y +a, b +x, y +a, b +x, y +a, b +a1(x, y; a, b) +a2(x, y; a, b) +b1(x, y; a, b) +b2(x, y; a, b) +c1(x, y; a, b) +c2(x, y; a, b) +1 − bx +y + a +1 + by +x − a +1 − ab +x + y +Let x = (x1, . . . , xn) and y = (y1, . . . , yn). Let a = (a1, . . . , am) and b = (b1, . . . , bm). We +define the vector spaces +V (x, y) = V (xn, yn) ⊗ V (xn−1, yn−1) ⊗ · · · ⊗ V (x1, y1), +W(a, b) = W(a1, b1) ⊗ W(a2, b2) ⊗ · · · ⊗ W(am, bm) +which represents the space of rows/columns in the six vertex model. Note that when x and +y (or a and b) are single variables, the definition reduces to the single row/column space. +We define the row operator +T(x, y; a1, . . . , am; b1, . . . , bm) = T(x, y; a1, b1)T(x, y; a2, b2) . . . T(x, y; am, bm). +from V (x, y) ⊗ W(a1, . . . , am; b1, . . . , bm) to W(a1, . . . , am; b1, . . . , bm) ⊗ V (x, y). +Here we +mean that every operator T(x, y; ai, bi) transposes V (x, y) with W(ai, bi) and acts by identity +elsewhere. +Graphically, we can represent the row operator as a row in a rectangular lattice consisting +of the vertex operators: +x, y +a1, b1 +a2, b2 +a3, b3 +a4, b4 +a5, b5 +a6, b6 +a7, b7 +a8, b8 +Naturally, several rows stacked together form an operator +T(x1, . . . , xn; y1, . . . , yn; a1, . . . , am; b1, . . . , bm) +from V (x, y) ⊗ W(a, b) to W(a, b) ⊗ V (x, y) which is the six vertex operator. +Graphically, we represent the six vertex operator as the rectangular lattice with rows +labeled by (x1, y1), . . . , (xn, yn) and columns labeled by (a1, b1), . . . , (am, bm). +10 + +x3, y3 +x2, y2 +x1, y1 +a1, b1 +a2, b2 +a3, b3 +a4, b4 +a5, b5 +a6, b6 +a7, b7 +a8, b8 +Now, if we apply the six vertex operator to the basis vectors of the vector space V (x, y) ⊗ +W(a, b), we can represent the action of each state graphically as a system of paths traveling +from the top left corner to the bottom right corner of the rectangular lattice. In this way, +we have recovered the combinatorial definition of the six vertex model. In particular, the +partition functions of the model are given by the matrix coefficients of the six vertex operator, +where the boundary conditions correspond to the basis elements of the vector space. +One way to study the six vertex operator is to decompose it into a composition of so-called +row transfer operators acting on the column space. There are four different types of row +transfer operators, depending on the boundary conditions of a given row. Since the row +operators depend on the parameters x and y, while a and b are constant, we usually omit +the parameters a and b from the notation. +We define the row transfer operators +A(x, y), B(x, y), C(x, y), D(x, y): W(a; b) → W(a; b). +We define these operators by their matrix coefficients: +⟨v|A(x, y; a, b)|w⟩ = ⟨e0 ⊗ v|T(x, y; a, b)|w ⊗ e0⟩, +⟨v|B(x, y; a, b)|w⟩ = ⟨e1 ⊗ v|T(x, y; a, b)|w ⊗ e0⟩, +⟨v|C(x, y; a, b)|w⟩ = ⟨e1 ⊗ v|T(x, y; a, b)|w ⊗ e0⟩, +⟨v|D(x, y; a, b)|w⟩ = ⟨e1 ⊗ v|T(x, y; a, b)|w ⊗ e1⟩. +Graphically, the four row transfer operators are shown in Figure 5. +Lemma 2.4. We have +⟨v ⊗ u|A(x, y; a, b) = ⟨v|A(x, y; a, b)| ⊗ ⟨u|A(x, y; a, b)| + ⟨v|B(x, y; a, b)| ⊗ ⟨u|C(x, y; a, b)|, +⟨v ⊗ u|B(x, y; a, b) = ⟨v|A(x, y; a, b)| ⊗ ⟨u|B(x, y; a, b)| + ⟨v|B(x, y; a, b)| ⊗ ⟨u|D(x, y; a, b)|, +⟨v ⊗ u|C(x, y; a, b) = ⟨v|C(x, y; a, b)| ⊗ ⟨u|A(x, y; a, b)| + ⟨v|D(x, y; a, b)| ⊗ ⟨u|C(x, y; a, b)|, +⟨v ⊗ u|D(x, y; a, b) = ⟨v|C(x, y; a, b)| ⊗ ⟨u|B(x, y; a, b)| + ⟨v|D(x, y; a, b)| ⊗ ⟨u|D(x, y; a, b)|, +where each operator acts on the corresponding column space of vectors v or u. +Proof. By definition. +□ +Let us connect the combinatorial definition and the row transfer operators definition on +the following example. +11 + +A(x, y; a, b) +x, y +a1, b1 +a2, b2 +a3, b3 +a4, b4 +a5, b5 +a6, b6 +a7, b7 +a8, b8 +B(x, y; a, b) +x, y +a1, b1 +a2, b2 +a3, b3 +a4, b4 +a5, b5 +a6, b6 +a7, b7 +a8, b8 +C(x, y; a, b) +x, y +a1, b1 +a2, b2 +a3, b3 +a4, b4 +a5, b5 +a6, b6 +a7, b7 +a8, b8 +D(x, y; a, b) +x, y +a1, b1 +a2, b2 +a3, b3 +a4, b4 +a5, b5 +a6, b6 +a7, b7 +a8, b8 +Figure 5. Row transfer operators for a = (a1, . . . , am), b = (b1, . . . , bm), and +m = 8. +Example 2.5. The operator +A(x1, y1)A(x2, y2) . . . A(xn, yn) +corresponds to the six vertex model +S((x1, y1), . . . , (xn, yn); (a1, b1), . . . , (am, bm); ∅, βt, ∅, βb). +Indeed, the composition of row transfer operators of type A graphically represents the six +vertex model with left and right boundary being empty. Then the matrix coefficient +⟨ei1 ⊗ · · · ⊗ eim|A(x1, y1)A(x2, y2) . . . A(xn, yn)|ej1 ⊗ · · · ⊗ ejm⟩ +corresponds to the partition function +Z((x1, y1), . . . , (xn, yn); (a1, b1), . . . , (am, bm); ∅, (1ik=1)m +k=1, ∅, (1jk=1)m +k=1). +2.4. The Yang-Baxter equation and operator relations. The central tool in the theory +of integrable lattice models is the Yang-Baxter equation which sets the relation for the +weight functions. Thanks to these relations, the partition function satisfies various functional +equations which makes it possible both to compute it and to recognize its connection to the +special functions. We formulate the Yang-Baxter equation for the row transfer operators. +For combinatorial description in terms of cross vertices, see Lemma 1 in [BBF11] or Figure +6 in [ABPW21]. +12 + +Consider the following operators: +Rx(x1, x2; y): V (x2, y1) ⊗ V (x1, y2) → V (x1, y1) ⊗ V (x2, y2), +Ry(y1, y2; x): V (x1, y2) ⊗ V (x2, y1) → V (x1, y1) ⊗ V (x2, y2), +Rx,y(x1, x2; y): V (x2, y2) ⊗ V (x1, y1) → V (x1, y1) ⊗ V (x2, y2). +given in the standard bases by matrices +Rx,y(x1, y1; x2, y2) = + + + + +x1 + y2 +x2 + y2 +y2 − y1 +x1 − x2 +x1 + y1 +x2 + y1 + + + + , +Rx(x1, x2; y) = + + + + +x1 + y +x2 + y +0 +x1 − x2 +x1 + y +x2 + y + + + + , +Ry(y1, y2; x) = + + + + +x + y2 +x + y2 +y2 − y1 +0 +x + y1 +x + y1 + + + + . +We note that the six vertex Rx,y factors into the product of two five vertex matrices: +Rx,y(x1, y1; x2, y2) = (x2 + y2)−1Rx(x1, x2; y)Ry(y1, y2; x). +See Section 4.7 of [WZJ18] for another instance of similar factorization. +Remark 2.6. We note that by setting yi = −q2xi, the R-matrix Rx,y(x1, y1; x2, y2) trans- +forms into the well-known R-matrix for the standard representations of the affine quantum +supergroup Uq( � +sl(1|1)): +Rq(x1, x2) = + + + + +x1 − q2x2 +x2 − q2x2 +q2(x1 − x2) +x1 − x2 +x1 − q2x1 +x2 − qx1 + + + + . +More generally, by setting yi = qixi, we obtain the R-matrix from [BBF11] up to a change +of variables and a rescaling. Furthermore, it is possible to establish a connection between the +R-matrix Rx,y and the R-matrix given by weights (2.6) from [ABPW21] through a similar +transformation. +13 + +Remark 2.7. The five vertex R-matrices can be written in a simple form: +Rx(x1, x2; y) = (x2 + y)I4 + (x1 − x2) + + + + +1 +0 +0 +1 +1 +0 + + + + , +Ry(y1, y2; x) = (x + y2)I4 + (y1 − y2) + + + + +1 +0 +−1 +0 +1 +1 + + + + . +The algebra generated by Rx +i and Ry +i acting on a tensor product V (x1, y1)⊗· · ·⊗V (xn, yn) +is spanned by the corresponding elements Pi and P ′ +i, where P x +i and P y +i act on the i-th and +(i + 1)-th sites by the matrices given in the above equations. This algebra could be seen as +distant relative of the Temperley Lieb algebra for the free fermionic matrices. +Theorem 2.8 (The refined Yang-Baxter equations). +Rx(x1, x2; y1)T(x1, y1; a, b)T(x2, y2; a, b) = T(x2, y1; a, b)T(x1, y2; a, b)Rx(x1, x2; y1), +Ry(y1, y2; x2)T(x1, y2; a, b)T(x2, y2; a, b) = T(x1, y2; a, b)T(x2, y1; a, b)Ry(y1, y2; x2), +Rx,y(x1, y1; x2, y2)T(x1, y1; a, b)T(x2, y2; a, b) = T(x2, y2; a, b)T(x1, y1; a, b)Rx,y(x1, y1; x2, y2). +Proof. Direct calculation. +□ +We note that we provided the refined Yang-Baxter equation, which is a generalization of +the standard Yang-Baxter equation. The standard Yang-Baxter equation, which is repre- +sented by the matrix Rx,y, simultaneously exchanges both spectral parameters x and y. In +contrast, the matrices Rx and Ry exchange only one of the spectral parameters at a time. +By composing these operators, we recover the standard Yang-Baxter equation. +The refined Yang-Baxter equation gives rise to refined relations for the row transfer oper- +ators, which involve only the exchange of x or y parameters. These refined relations allow +us to demonstrate that the partition functions are symmetric separately in x variables and +in y variables. This is a novel result that cannot be shown using the standard technique, as +the standard Yang-Baxter equation simultaneously exchanges both parameters. +Furthermore, we note that the R-matrices do not depend on the parameters a and b. By +repeatedly applying the Yang-Baxter equation, we can derive the Yang-Baxter equation for +the entire row operators, a process known as the train argument. This technique involves +moving the R-matrix through pairs of operators in a consecutive manner. +Corollary 2.9 (Train argument). Let a = (a1, . . . , am) and b = (b1, . . . , bm). Then +Rx(x1, x2; y1)T(x1, y2; a, b)T(x2, y2; a, b) = T(x2, y1; a, b)T(x1, y2; a, b)Rx(x1, x2; y1), +Ry(y1, y2; x2)T(x1, y2; a, b)T(x2, y2; a, b) = T(x1, y2; a, b)T(x2, y1; a, b)Ry(y1, y2; x2), +Rx,y(x1, y1; x2, y2)T(x1, y1; a, b)T(x2, y2; a, b) = T(x2, y2; a, b)T(x1, y1; a, b)Rx,y(x1, y1; x2, y2). +The refined Yang-Baxter equation, in conjunction with the train argument, provides rela- +tions for the row transfer operators that can be deduced by reading off the matrix coefficients +of the Yang-Baxter equation for the row operators. +Proposition 2.10 (Operator relations). The operators A, B, C, D satisfy relations. +14 + +(1) The cancellation property: +A(t, −t) = 1, +D(t, −t) = 1. +(2) The separate symmetry in x’s and y’s for operators A and D: +A(x1, y1)A(x2, y2) = A(x2, y1)A(x1, y2), +A(x1, y1)A(x2, y2) = A(x1; y2)A(x2, y1), +D(x1, y1)D(x2, y2) = D(x2, y1)D(x1, y2), +D(x1, y1)D(x2, y2) = D(x1; y2)D(x2, y1), +(3) The partial symmetry in x’s and y’s for operators B and C: +(x1 + y1)B(x1, y1)B(x2, y2) = (x2 + y1)B(x2; y1)B(x1, y2), +(x2 + y2)B(x1, y1)B(x2, y2) = (x2 + y1)B(x1; y2)B(x2, y1), +(x2 + y1)C(x1, y1)C(x2, y2) = (x1 + y1)C(x2; y1)C(x1, y2), +(x2 + y1)C(x1, y1)C(x2, y2) = (x2 + y2)C(x1; y2)C(x2, y1). +(4) The partial symmetry in x’s and y’s between operators B, C: +(x2 + y1)B(x1, y1)C(x2, y2) = (x2 + y2)B(x1, y2)C(x2, y1), +(x2 + y1)C(x1, y1)B(x2, y2) = (x1 + y1)C(x2, y1)B(x1, y2). +(5) The relations between A, B, A, C, D, B, and D, C: +(x2 − x1)A(x1, y1)B(x2, y2) = (x2 + y1)B(x2, y1)A(x1, y2) − (x1 + y1)B(x1, y1)A(x2, y2), +(y1 − y2)B(x1, y1)A(x2, y2) = (x2 + y1)A(x1, y2)B(x2, y1) − (x2 + y2)A(x1, y1)B(x2, y2), +(x1 − x2)C(x1, y1)A(x2, y2) = (x1 + y1) (A(x2, y1)C(x1, y2) − A(x1, y1)C(x2, y2)) , +(y1 − y2)A(x1, y1)C(x2, y2) = (x2 + y2) (C(x1, y2)A(x2, y1) − C(x1, y1)A(x2, y2)) . +(6) The relations between A, D, D, A, B, C, and C, B: +(x1 + y1)A(x1, y1)D(x2, y2) + (x1 − x2)C(x1, y1)B(x2, y2) = (x1 + y1)A(x2, y1)D(x1, y2), +(x2 + y1)A(x1, y1)D(x2, y2) = (x2 + y1)A(x1, y2)D(x2, y1) + (y2 − y1)B(x1, y2)C(x2, y1), +(x2 + y1)D(x1, y1)A(x2, y2) = (x2 + y1)D(x2, y1)A(x1, y2) + (x1 − x2)C(x2, y1)B(x1, y2), +(x2 + y2)D(x1, y1)A(x2, y2) + (y2 − y1)B(x1, y1)C(x2, y2) = (x2 + y2)D(x1, y2)A(x2, y1). +(7) Finally, the relations between A, B, C, D: +(x1 + y1)B(x1, y1)C(x2, y2) + (x1 − x2)D(x1, y1)A(x2, y2) = += (x2 + y1)B(x2, y1)C(x1, y2) + (x1 − x2)A(x2, y1)D(x1, y2). +(x2 + y2)C(x1, y1)B(x2, y2) + (y2 − y1)A(x1, y1)D(x2, y2) = += (x2 + y1)C(x1, y2)B(x2, y1) + (y2 − y1)D(x1, y2)A(x2, y1). +Proof. Direct calculation using the Yang-Baxter equations and the train argument. +□ +15 + +Remark 2.11. We note that these operator relations refine the relations given in Proposition +2.4 of [ABPW21] or Corollary 4.3 of [Kor21]. Specifically, since the R-matrix Rx,y is the +product of two five vertex matrices Rx and Ry, these relations imply all relations that arise +from the classical Yang-Baxter equation. +The refined Yang-Baxter equation also gives rise to new operator relations that cannot be +identified using the standard Yang-Baxter equation alone. For example, the fourth group of +operator relations involving the partial symmetry for operators B and C is a new relation +that cannot be derived from the standard R-matrix, as each side would yield two terms, +resulting in a relation involving four terms in total. +The row transfer operators A, B, C, D and the relations between them form the Yang- +Baxter algebra on W(a1, . . . , am, b1, . . . , bm). +This algebra has a rich structure that can +be exploited to solve the six vertex model. +In particular, the algebraic Bethe ansatz is +a powerful technique for finding explicit expressions for the matrix coefficients of the row +transfer operators in terms of solutions to the Bethe equations. These solutions can then be +used to compute the partition function and correlation functions of the six vertex model. +2.5. Partition functions. In this section, we show how to apply the combinatorial defini- +tion, the LGV lemma, the refined Yang-Baxter equation, and the resulting operator relations +to compute exactly the partition functions with various boundary conditions. +In the following result, we consider the row transfer operators, and say that they satisfy +certain properties if all of their matrix coefficients also satisfy those properties. We proceed +to establish further properties of these operators. +Lemma 2.12. We have the following properties. +(1) The operators B and C have a pre-factor: +(a) B(x1, y1)B(x2, y2) . . . B(xn, yn) is divisible by � +i α2 > · · · > αn > 0. Then +⟨e(α)|B(x1, y1)B(x2, y2) . . . B(xn, yn)|e(∅)⟩ = +� +i n′. In other words, ik = 1 for all large enough negative +integers, and ik = 0 for all large enough positive integers. The corresponding basis element +for a given Maya diagram is given by the expression (eσk)k∈Z. +Let λ = (λ1, λ2, . . . , λn) be a partition, that is, an n-tuple of non-negative integers such +that λ1 ≥ λ2 ≥ · · · ≥ λn. For an integer c ∈ Z and a partition λ, we define the corresponding +Maya diagram by (ik)k∈Z with ik = 1 if k ∈ {λi − i + 1 + c}∞ +i=1, and zero otherwise. We +denote the corresponding basis element by σc(λ) or ⟨λ; c|. We also denote σ(λ) = σ0(λ) and +⟨λ| = ⟨λ; c|. It is not hard to see that these elements parametrize all basis vectors, and thus +we can decompose the space W = � +c∈Z Wc, where each Wc is spanned on vectors σc(λ. The +integer c is sometimes called the “charge” or “level”. +Let ν be a skew diagram containing no 2 × 2 block of squares. Such a diagram is called a +”ribbon” or a ”skew hook” if it is connected. A skew diagram with no 2×2 blocks of squares +is a disjoint union of ribbons. We describe the ribbon in terms of the corresponding Maya +diagrams, with charge zero. +Let λ/µ be the skew partition. Let (i0; i1, i2, . . . , ik; ik+1) with ij ∈ Z be such that i0 ∈ +σ(µ) \ σ(λ), i1, . . . , ik ∈ σ(λ) ∩ σ(µ), and ik+1 ∈ σ(λ) \ σ(µ). It is not hard to see that such a +tuple corresponds exactly to a ribbon in λ/µ. We denote by Rib(λ/µ) the set of all ribbons +in λ/µ. +In the previous section, we defined operators on W. They can be seen as A: Wc → Wc, +B: Wc → Wc+1, C : Wc → Wc−1, and D: Wc → Wc. We will focus on the operators A and +D, and for convenience, we will work in charge zero. +We now show that the row operators A and D have a natural description in terms of +ribbons. +More generally, the entire Yang-Baxter algebra generated by the row transfer +operators could be described in terms of ribbons. See Section 4.3 of [Kor21] for reference. +We compute explicitly the following values which describe the action of the operators on +the basis elements: +fλ/µ;a,b(x, y) := ⟨µ|A(x, y)|λ⟩, +�fλ/µ;a,b(x, y) := ⟨λ|D(x, y)|µ⟩. +22 + +Lemma 3.1. We have +fλ/µ;a,b(x, y) = γµ;b(x, y) +� +r∈R(λ/µ) +wtr;a,b(x, y), +�fλ/µ;a,b(x, y) = γµ;b(x, y) +� +r∈R(λ/µ) +wtr;a,b(x, y), +where +γλ;b(x, y) = +∞ +� +i=1 +1 + bλ−i+1y +1 + b−i+1y +1 − bi−λ′ +ix +1 − bix , +wtr;a,b(x, y) = 1 − ai0bi0 +1 + bi0y +x + y +1 − bik+1x +ik+1−1 +� +k=i0+1 +� +(y + aj)/(1 + bjy), +if j ∈ {i1, i2, . . . , ik}, +(x − aj)/(1 − bjx), +otherwise. +� +wtr;a,b(x, y) = 1 − aik+1bik+1 +1 + bi0y +x + y +1 − bik+1x +ik+1−1 +� +k=i0+1 +� +(1 + bjy)/(y + aj), +if j ∈ {i1, i2, . . . , ik}, +(1 − bjx)/(x − aj), +otherwise. +. +Proof. By inspection, the row transfer operator of type A consists of the ribbons as illustrated +in Figure 8. Then function γλ;b renormalizes the weights such that vertices of type b1 do not +contribute. Then the ribbon weight functions wtr;a,b and � +wtr;a,b give the explicit weight of +a ribbon. Since the row trasnfer operator is the product over all such ribbons, the result +follows. The operator D is treated in a similar manner. +□ +The formula become particularly simple in the case of the simplest ribbon. Recall that a +hook is a partition of the form (p + 1, 1q) for p, q ≥ 0. In Frobenius notation, a hook has the +form (p|q). +Corollary 3.2. We have fλ;a,b = �fλ;a,b = 0 unless λ is a hook. Moreover, +f(p|q);a,b(x, y) = (1 − a(q+1)′b(q+1)′)(x + y) +(1 + b(q+1)′y)(1 − b(p+1)x) +(x|a)p +(x; b)p +(y|a′)q +(y; b′)q , +�f(p|q);a,b(z, w) = (1 − a(p+1)b(p+1))(z + w) +(1 + a(q+1)′w)(1 − a(p+1)z) +(z; b)p +(z|a)p +(w; b′)q +(w|a′)q . +4. Free fermionic Schur functions +In this section, we introduce a new family of Schur functions, defined as the partition +functions of the six vertex model with weights as specified in equation (2.1). These functions +are defined using the row transfer operator approach and will be shown to generalize and +unify existing families of Schur functions. +Let x = (x1, . . . , xn), y = (y1, . . . , yn), a = (ai)i∈Z, and b = (bi)i∈Z. +We define the +free fermionic Schur functions sλ/µ;a,b(x, y) and the dual free fermionic Schur functions +�sλ/µ;a,b(x, y) as follows: +sλ/µ;a,b(x, y) = ⟨µ|A(x1, y1) . . . A(xn, yn)|λ⟩, +�sλ/µ;a,b(x, y) = ⟨λ|D(x−1 +1 , y−1 +1 ) . . . D(x−1 +n , y−1 +n )|µ⟩. +23 + +We will now demonstrate various familiar properties for the new Schur functions sλ/µ;a,b +that generalize the well-known properties of Schur functions. +We remark the the Schur +functions are related to Gλ/µ functions from [ABPW21] by reparametrization of the weights. +A function f(x1, . . . , xn, y1, . . . , yn) is said to be supersymmetric if it is symmetric in +variables x and y, and satisfies the cancellation property: +f(x1, . . . , xn−1, t; y1, . . . , yn−1, −t) = f(x1, . . . , xn−1; y1, . . . , xn−1). +Proposition 4.1. The functions sλ/µ;a,b(x, y) and �sλ/µ;a,b(x, y) are supersymmetric. +Proof. It follows from Lemma 2.19. +□ +Remark 4.2. It is important to note that the above proof relies on the refined Yang-Baxter +equation. The standard Yang-Baxter equation would only allow for the demonstration that +the partition functions are symmetric under simultaneous permutation of variables x and y, +as seen in Lemma 4 of [BBF11] or Proposition 3.5 of [ABPW21]. +Thanks to the cancellation property, we can define the free fermionic Schur functions in +∞ + ∞ variables x = (x1, x2, . . . ) and y = (y1, y2, . . . ) such that if yk = −xk for all k > n, +then +sλ/µ;a,b(x, y) = sλ/µ;a,b(x1, . . . , xn; y1, . . . , yn), +�sλ/µ;a,b(x, y) = �sλ/µ;a,b(x1, . . . , xn; y1, . . . , yn). +Therefore, in the following discussion, we will not specify the number of variables unless it +is necessary for the argument at hand. +Proposition 4.3 (Branching rules). The following relations hold: +sλ/µ;a,b(x, y) = +� +µ⊆ν⊆λ +sν/µ;a,b(x1, . . . , xn−1, y1, . . . , yn−1)fλ/ν;a,b(xn, yn), +�sλ/µ;a,b(x, y) = +� +µ⊆ν⊆λ +�sν/µ;a,b(x1, . . . , xn−1, y1, . . . , yn−1) �fλ/ν;a,b(xn, yn), +where the branching weights fλ/µ;a,b and �fλ/µ;a,b are defined in Lemma 3.1. +Proof. It follows from the definition and Lemma 3.1. +□ +We give a combinatorial formula in terms of ribbons, generalizing (4.5) in [ORV03]. +Corollary 4.4 (Combinatorial ribbon formula). The following relations hold: +sλ/µ;a,b(x, y) = +� +µ=ν0⊂ν1⊂...νn=λ +fν1/ν0(x1, y1) . . . fνn/νn−1(xn, yn), +�sλ/µ;a,b(x, y) = +� +µ=ν0⊂ν1⊂...νn=λ +�fν1/ν0(x1, y1) . . . �fνn/νn−1(xn, yn). +where νk/νk−1 contains no 2 × 2 block of squares for each k = 1, 2, . . . , n. +Proof. The above relations follow from the successive use of the branching rules, which +separates one variable at a time. Additionally, by Lemma 3.1, we know that fλ/µ;a,b(xi, yi) +and �fλ/µ;a,b(xi, yi) are zero unless λ/µ is a ribbon. +□ +24 + +In the following results, we will use the notation from Assumption 2.17 and assume that +Θa,b(xi, zj) and Θa′,b′(yi, wj) for all i, j = 1, 2, . . . . We give a generalization of the skew +Cauchy identity. In terms of partition functions of the six vertex model, this result is the +generalization of Proposition 3.7 in [ABPW21]. +Proposition 4.5 (Skew Cauchy Identity). We have the following identity: +� +λ +sλ/µ;a,b(x, y)�sλ/ν;a,b(z, w) = +� +i,j +1 + yizj +1 − xizj +1 + xiwj +1 − yiwj +� +ρ +�sµ/ρ;a,b(z, w)sν/ρ;a,b(x, y). +Proof. This identity can be derived by repeated application of Lemma 2.20. +□ +The following special case generalizes Theorem 3.4 in [Mol09]. +Corollary 4.6. +� +i,j +1 + yizj +1 − xizj +1 + xiwj +1 − yiwj +sν;a,b(x, y) = +� +ν⊂λ +sλ;a,b(x, y)�sλ/ν;a,b(z, w), +� +i,j +1 + yizj +1 − xizj +1 + xiwj +1 − yiwj +�sµ;a,b(z, w) = +� +µ⊂λ +�sλ;a,b(z, w)sλ/µ;a,b(x, y). +Proof. Set µ = ∅ or ν = ∅ in the skew Cauchy identity. +□ +We now prove one of our main results, the Cauchy identity in the form of Berele-Regev +[BR87]. We remark that the right-hand side of the identity is independent of the parameters +(ai)i∈Z and (bi)i∈Z. This identity degenerates to Theorem 3.1 and Corollary 3.2 from [Mol09]. +Theorem 4.7 (Cauchy Identity). We have the following identity: +� +λ +sλ;a,b(x, y)�sλ;a,b(z, w) = +� +i,j +1 + yizj +1 − xizj +1 + xiwj +1 − yiwj +. +Proof. Setting µ = ν = ∅ in Proposition 4.5 yields the desired identity. +□ +We use the Cauchy identity to give the generating series for the hook Schur functions +s(p|q);a,b. This result generalizes Proposition 7.1 from [ORV03]. +Recall that we defined +(x|a)r = (x − a1)(x − a2) . . . (x − ar), +(x; a)r = (1 − a1x)(1 − a2x) . . . (1 − arx). +For a sequence a = (ai)i∈Z, let a′ = (−a−i+1)i∈Z be the dual sequence. +Corollary 4.8 (Generating series for hook functions). +1 + (z + w) +∞ +� +p,q=0 +s(p|q);a,b(x, y)(1 − ap+1bp+1) (z|b)p +(z; a)p+1 +(w|b′)q +(w; a′)q+1 = +� +i +1 + yiz +1 − xiz +1 + xiw +1 − yiw , +1 + (x + y) +∞ +� +p,q=0 +�s(p|q);a,b(z, w)(1 − a(q+1)′b(q+1)′) (x|a)p +(x; b)p+1 +(y|a′)q +(y; b′)q+1 = +� +i +1 + zjy +1 − zjx +1 + wjx +1 − wjy. +Proof. Let z, w or x, y be single variables in the Cauchy identity, then use the explicit formula +for the hook Schur functions from Corollary 3.2. +□ +25 + +Let (x, z|a, b)r be defined by +(x, z|a, b)r = (1 − ar+1br+1) (x|a)r +(x; b)r+1 +(z|b)r +(z; a)p+1. +As a special case, we get the following identity: +Corollary 4.9. +1 + (x + y)(z + w) +∞ +� +p,q=0 +(x, z|a, b)p(y, w|a′, b′)q = 1 + yz +1 − xz +1 + xw +1 − yw. +Proof. Set x, y to be single variables in the generating series for hook functions. +□ +Let hp+1;a,b = s(p|0);a,b = s(p+1);a,b and �hp+1;a,b = �s(p|0);a,b = �s(p+1);a,b be the complete +symmetric functions, and eq+1;a,b = s(0|q);a,b = s(1q+1);a,b and �eq+1;a,b = �s(0|q);a,b = �s(1q+1);a,b be +the elementary symmetric functions. We define h0;a,b = �h0;a,b = e0;a,b = �e0;a,b = 1. +Let τ r be an operator that acts on sequences (ai)i ∈ Z by shifting the indices by τ r(ai) = +(ai+r)i∈Z. We also write τ rsλ/µ;a,b for sλ/µ;τ ra,τ rb. +Corollary 4.10 (Generating series for complete and elementary functions). +1 + +∞ +� +k=1 +hk;a,b(x, y)1 − akbk +1 − a0b0 +(z|τ −1b)k +(z; a)k += +� +i +1 + yiz +1 − xiz +1 − b0xi +1 + b0yi +, +(4.1) +1 + +∞ +� +k=1 +ek;a,b(x, y)1 − akbk +1 − a1b1 +(w|(τ −1b)′)k +(w; a′)k += +� +i +1 + xiw +1 − yiw +1 + yib1 +1 − xib1 +, +(4.2) +1 + +∞ +� +k=0 +�hk;a,b(z, w)1 − ak′bk′ +1 − a0b0 +(x|τ −1a)k +(x; b)k += +� +i +1 − zja0 +1 − zjx +1 + wjx +1 + a0wj +, +(4.3) +1 + +∞ +� +k=0 +�ek;a,b(z, w)1 − ak′bk′ +1 − a1b1 +(y|(τ −1a)′)k +(y; b′)k += +� +i +1 + zjy +1 − wjy +1 + a1wj +1 − a1zj +. +(4.4) +Proof. We prove the first identity by setting w = −b0 in the generating series for complete +homogeneous symmetric functions, as given by the first identity in Corollary 4.8. Using the +property (b−1 +0 ; b′)q = 0 for q > 0, we can simplify the sum to only include p = 0, 1, 2, . . . . +Similarly, we can obtain the other identities by setting z = b1, y = −a0, and x = a1 in +the generating series for elementary homogeneous symmetric functions and complete homo- +geneous symmetric functions, respectively. Then, we simplify the expressions to obtain the +desired identities. +□ +Proposition 4.11 (Jacobi-Trudi Formula). +sλ/µ;a,b = det +� +τ µj−j+1hλi−µj−i+j;a,b +� +. +Proof. By the LGV lemma for the six vertex model (see Proposition 2.3, we have +⟨µ|A(x1, y1) . . . A(xn, yn)|λ⟩ = det (⟨e(µi − i + 1)|A(x1, y1) . . . A(xn, yn)|e(λj − j + 1)) , +where e(k1, k2, . . . , kr) denote ei1 ⊗ · · · ⊗ eim with ik = 1 only when k ∈ {k1, . . . , kr}. Using +the fact that +⟨e(µj − j + 1)|A(x1, y1) . . . A(xn, yn)|e(λi − i + 1)⟩ = τ µj−j+1hλi−µj−i+j(x, y), +26 + +we can deduce the proposition. We have absorbed the normalization factors into the defini- +tion of sλ/µ;a,b. +□ +As a consequence of the Jacobi-Trudi formula, the functions sλ/µ;a,b can be identified with +a special case of the ninth variation introduced by Macdonald in [Mac92]. Therefore, we +have the following corollary. We refer to [Mac92] for notation. +Corollary 4.12 (Determinant identities). The functions sλ/µ;a,b satisfy the following deter- +minant identities: +(1) (N¨agelsbach-Kostka formula) +sλ/µ;a,b = det(τ µj+j−1eλ′ +i−µ′ +j−i+j;a,b). +(2) (Giambelli formula) +sλ;a,b = det(s(αi|βj);a,b)1≤i,j≤d(λ). +(3) (Ribbon formula) +sλ;a,b = det(s[αi|βj]la,b)1≤i,j≤r(λ). +Proof. These are formal consequences of the Macdonald. These are formulas (9.6), (9.6’), +(9.7), and (9.9) in [Mac92]. +□ +Finally, we show that the new Schur functions generalize and unify the existing Schur +functions from literature. +Corollary 4.13 (Degenerations). +(1) sλ/µ;0,0(x, 0) = sλ/µ(x): classical Schur functions, +(2) sλ/µ;0,0(x, y) = sλ/µ(x/y): supersymmetric Schur functions, +(3) sλ/µ;a,0(x, a′) = sλ/µ(x || a): factorial Schur functions, +(4) sλ/µ;a′,0(x, y) = sλ/µ(x/y || a): factorial supersymmetric Schur functions, +(5) sλ/µ;a,0(x, y) = sλ/µ;a(x, y): Frobenius-Schur functions, +(6) �sλ/µ;0,b(x/0) = �sλ/µ(x || b): dual Schur functions. +Proof. The Frobenius-Schur functions and dual Schur functions satisfy the Jacobi-Trudi iden- +tity by Theorem A.6 in [ORV03] and Proposition 3.9 in [Mol09], respectively. Hence, it is +enough to show that the complete homogeneous functions hk;a,b degenerate to the Frobenius- +Schur’s hk;a and the dual Schur’s �hk;a. We do so by showing that the generating series de- +generates to the generating series in both cases. By Corollary 4.10, when z �→ z−1 and b = 0, +the generating series for complete homogeneous functions hk;a,b becomes +1 + +∞ +� +k=0 +hk;a,0 +(z − a1)(z − a2) . . . (z − ak) = z + yi +z − xi +, +which is exactly the generating series from (3.1) of [ORV03]. +Similarly, when we set a = 0 and w = 0 in the generating series for complete homogeneous +functions, we get +1 + +∞ +� +k=0 +�hk;a,0(1 − a0z)(1 − a1z) . . . (1 − ak−1z) = +∞ +� +i=1 +1 − a0z +1 − xiz , +27 + +which is the generating function for the dual complete symmetric functions �hk(x||b′) from +Proposition 3.8 in [Mol09]. +Since the Schur functions, supersymmetric Schur functions, factorial Schur functions, and +factorial supersymmetric Schur functions are all special cases of the Frobenius-Schur func- +tions and the dual Schur functions, the result follows. +Alternatively, one can show that the complete homogeneous functions hk;a,b degenerate +to the Frobenius-Schur functions hk;a and the dual Schur functions �hk;a by noting that they +satisfy the correct branching rules and Cauchy identities, respectively. The branching weight +from Lemma 3.3 with b = 0 degenerates to the correct branching rules from Proposition 4.3 +in [ORV03], and our Cauchy identity in Theorem 4.7 degenerates to Corollary 3.2 in [Mol09] +when b = 0 and w = 0. This implies that sλ/µ;a,0 = sλ/µ;a and �sλ;a,0(z) are equal to the dual +Schur functions �sλ(z||a). +□ +4.1. Further properties. In this section, we use the properties of the six vertex model to +derive additional properties of the Schur functions. +Let x = (x1, . . . , xn), y = (y1, . . . , yn), a = (ai)i∈Z, and b = (bi)i∈Z. We define the one +variable Schur functions Gλ/µ;a,b(x, y) and the dual Schur functions �Gλ/µ(x) as follows: +Gλ/µ;a,b(x, y) = ⟨µ|B(x1, y1) . . . B(xn, yn)|λ; n⟩, +�Gλ/µ;a,b(x, y) = ⟨λ; n|C(x1, y1) . . . C(xn, yn)|µ⟩, +G′ +λ/µ;a,b(x, y) = ⟨µc; n|B(x1, y1) . . . B(xn, yn)|λc⟩, +�G′ +λ/µ;a,b(x, y) = ⟨λc; n|C(x1, y1) . . . C(xn, yn)|µc⟩. +The non-skew shape case allows for an explicit formula for certain functions, as stated in +Proposition 2.13. Specifically, we have: +Gλ;a,b(x, y) = +� +i 0 +and ∆t > 0]. +We now try to do something similar for the traversable wormhole, using general relativity +with normal matter but allowing for a degenerate metric. +III. +SIMPLE EXAMPLE +A. +Nondegenerate metric – special case +Our basic idea can be tested by starting from the simple example discussed in Box 2 of +Ref. [1]. There, the metric is given by (recall c = 1) +ds2 ��� +(nondegen-metric-special) += −dt2 + dl2 + +� +b2 +0 + l2� � +dθ2 + sin2 θ dφ2� +, +(3.1) +with a nonzero real constant b0 (taken to be positive, for definiteness). The coordinates t +and l in (3.1) range over (−∞, ∞) and θ and φ are the standard spherical coordinates. +2 + +The Einstein equation, Rµν − 1 +2 gµν R = (8πG)−1 Tµν, then requires the following compo- +nents of the energy-momentum tensor [1]: +T t +t +��� +(nondegen-metric-special) += +1 +8πG +b2 +0 +(b2 +0 + l2)2 , +(3.2a) +T l +l +��� +(nondegen-metric-special) += − 1 +8πG +b2 +0 +(b2 +0 + l2)2 , +(3.2b) +T θ +θ +��� +(nondegen-metric-special) += +1 +8πG +b2 +0 +(b2 +0 + l2)2 , +(3.2c) +T φ +φ +��� +(nondegen-metric-special) += +1 +8πG +b2 +0 +(b2 +0 + l2)2 , +(3.2d) +with all other components vanishing. The energy density is given by ρ = T tt = −T t +t and +we have ρ < 0 from (3.2a), which definitely corresponds to exotic matter. +As shown by items (d) and (e) of Box 2 in Ref. [1], this wormhole is traversable. +B. +Degenerate metric – special case +Now consider the following metric Ansatz: +ds2 ��� +(degen-metric-special) += −dt2 + +ξ2 +ξ2 + λ2 dξ2 + +� +b2 +0 + ξ2� � +dθ2 + sin2 θ dφ2� +, +(3.3) +with nonzero real constants λ and b0 (both taken to be positive, for definiteness) and coor- +dinates t and ξ ranging over (−∞, ∞). The metric from (3.3) gives the following Ricci and +Kretschmann curvature scalars: +R +��� +(degen-metric-special) += −2 +b2 +0 − λ2 +(b2 +0 + ξ2)2 , +(3.4a) +K +��� +(degen-metric-special) += 12 (b2 +0 − λ2)2 +(b2 +0 + ξ2)4 , +(3.4b) +both of which are seen to vanish as ξ → ±∞. +Three remarks are in order. First, the metric (3.3) is degenerate with a vanishing deter- +minant at ξ = 0. In physical terms, this 3-dimensional hypersurface at ξ = 0 corresponds +to a “spacetime defect” (see references [6, 8–10] in Ref. [3] here; some further references will +be given shortly). +Second, the change of the ξ coordinate to +�l = ξ +� +1 + λ2/ξ2 +(3.5) +3 + +gives a metric similar to (3.1), +ds2 = −dt2 + d�l 2 + +� +b2 +0 + �l 2 − λ2� � +dθ2 + sin2 θ dφ2� +, +(3.6) +but this coordinate transformation ξ → �l is discontinuous and, therefore, not a diffeomor- +phism. We remark that the coordinate �l ∈ (−∞, −λ] ∪[λ, ∞) is unsatisfactory for a proper +description of the whole spacetime manifold, as, for given values of {t, θ, φ}, both �l = −λ +and �l = λ correspond to a single point of the manifold (with the single coordinate ξ = 0). For +further discussion of some of the mathematics and physics issues of such spacetime defects, +see Sec. 2 of Ref. [7] and Sec. III of Ref. [8]. +Third, the embedding diagram of the spacetime (3.3) for (t, θ) = (const, π/2) and λ2 < b2 +0 +is similar [with a 3-dimensional Euclidean embedding space] to the embedding diagram of the +spacetime (3.1) for the same values of (t, θ) and b2 +0, as given by item (b) of Box 2 in Ref. [1]. +The embedding diagram of the spacetime (3.3) for λ2 > b2 +0 is similar to the embedding +diagrams for λ2 ∈ [0, b2 +0), except that, for λ2 > b2 +0, there is a (2+1)-dimensional Minkowski +embedding space. The description of the spacetime (3.3) for λ2 = b2 +0 uses two copies of the +flat Euclidean space E3 with the interior of two balls removed and their surfaces identified, +somewhat analogous to the description of the spacetime defect of Sec. 2 in Ref. [7]. +With the metric (3.3), the Einstein equation requires +T t +t +��� +(degen-metric-special) += +1 +8πG +b2 +0 − λ2 +(b2 +0 + ξ2)2 , +(3.7a) +T ξ +ξ +��� +(degen-metric-special) += − 1 +8πG +b2 +0 − λ2 +(b2 +0 + ξ2)2 , +(3.7b) +T θ +θ +��� +(degen-metric-special) += +1 +8πG +b2 +0 − λ2 +(b2 +0 + ξ2)2 , +(3.7c) +T φ +φ +��� +(degen-metric-special) += +1 +8πG +b2 +0 − λ2 +(b2 +0 + ξ2)2 . +(3.7d) +Compared to the previous results (3.2), we see that the previous factors b2 +0 in the numerators +have been replaced by new factors (b2 +0−λ2), while the changes in the denominators are trivial. +Starting from λ2 = 0+, these new numerator factors then change sign as λ2 increases above +b2 +0 and we no longer require exotic matter. Indeed, we have from (3.7a) that ρ = −T t +t > 0 +for λ2 > b2 +0 . +There is, of course, also the special case λ2 = b2 +0, for which the energy-momentum tensor +4 + +vanishes altogether, +T µ +ν +��� +(degen-metric-special) +λ2=b2 +0 += 0 , +(3.8) +and so do the curvature scalars (3.4). In that case, we have a wormhole in the vacuum, +which will be further discussed in Sec. IV. +We can get the radial geodesics ξ(t) passing through the wormhole throat by adapting +result (3.6b) of Ref. [7] to our case: +ξ(t) +��� +(degen-metric-special) +λ2=b2 +0 += + + + + + +± +� +(B t)2 + 2 B λ t , +for t ≥ 0 , +∓ +� +(B t)2 − 2 B λ t , +for t < 0 , +(3.9) +with a dimensionless constant B ∈ (0, 1]. The same curves (3.9) can be more easily obtained +from straight lines �l(t), with radial velocity magnitude v, in the 2-dimensional Minkowski +subspace of the spacetime (3.6) and the definition (3.5) of the coordinate �l. The Minkowski- +subspace analysis identifies the constant B in (3.9) as the ratio v/c, so that the B = 1 curves +are light-like and those with B < 1 timelike. +The discussion of other geodesics for the metric (3.3) is similar to the discussion in Ref. [9], +which considers a related spacetime defect. The metric of this spacetime defect resembles +the metric of the wormhole presented here, but their global spatial structures are different. +IV. +DEGENERATE METRIC – GENERAL CASE +The special degenerate metric (3.3) can be generalized as follows: +ds2 ��� +(degen-metric-general) += −e2 �φ(ξ) dt2 + +ξ2 +ξ2 + λ2 dξ2 + �r 2(ξ) +� +dθ2 + sin2 θ dφ2� +, +(4.1) +with real functions �φ(ξ) and �r(ξ) and, again, coordinates t and ξ ranging over (−∞, ∞). +If, moreover, we assume that �φ(ξ) remains finite everywhere and that �r(ξ) is positive with +�r(ξ) ∼ |ξ| for ξ → ±∞, then the spacetime from (4.1) corresponds to a wormhole (see also +the further discussion at the beginning of Sec. 11.2 in Ref. [2]). +If the global minimum of the function �r(ξ) has the value b0 > 0 at ξ = ξ0 ≡ 0 and if +the function �φ(ξ) is essentially constant near ξ = 0, we expect interesting behavior for λ2 +of the order of b2 +0 or larger. In fact, using power series in ξ2 for �φ(ξ) and �r 2(ξ), we get +energy-momentum components without singular behavior at ξ = 0. It is clear that further +work will be cumbersome but perhaps not impossible. +5 + +Awaiting the final analysis of the metric (4.1), we recall, from Sec. III B, that we already +have one complete wormhole-type solution of the Einstein gravitational field equation: +�φ(ξ) = 0 , +�r 2(ξ) = λ2 + ξ2 , +T µ +ν(ξ) = 0 . +(4.2) +Note that, different from Minkowski spacetime, this flat vacuum-wormhole spacetime is +multiply connected (there are noncontractible loops in space, for example, loops encircling +the wormhole mouth). The solution (4.2) has one free parameter, the length scale λ, which +can be determined by considering the circumferences divided by 2π of great circles centered +on the wormhole [each circle has θ = π/2 and constant values of t and �l in the metric (3.6) +for λ2 = b2 +0]. Some further comments on this length scale λ appear in Sec. V. +V. +DISCUSSION +We have three final remarks. First, we mentioned in Sec. III B the possibility of obtaining +a vacuum wormhole if the parameters λ2 and b2 +0 of the metric (3.3) are chosen to be equal. +Vacuum wormholes also appear in certain modified-gravity theories [10], where the exotic +effects trace back to the extra terms in the gravitational action. Our vacuum-wormhole +solution does not require any change of the theory, general relativity suffices, except that we +now allow for degenerate metrics. +Second, this vacuum-wormhole solution (4.2) has the length scale λ as a free parameter +and, if there is a preferred value λ in Nature, then that value can only come from a theory +beyond general relativity. An example of such a theory would be nonperturbative superstring +theory in the formulation of the IIB matrix model [11, 12]. That matrix model could give +rise to an emergent spacetime with or without spacetime defects [13, 14] and, if defects do +appear, then the length scale λ of a remnant vacuum-wormhole defect would be related to +the matrix-model length scale. +Third, the main objective of the present paper is to reduce the hurdles to overcome in the +quest of traversable wormholes (specifically, we remove the requirement of exotic matter). +But there remains, at least, one important hurdle, namely to construct or harvest a suitable +spacetime defect. +ACKNOWLEDGMENTS +It is a pleasure to thank Z.L. Wang for useful comments on the manuscript. +6 + +[1] M.S. Morris and K.S. Thorne, “Wormholes in space-time and their use for interstellar travel: +A tool for teaching general relativity,” Am. J. Phys. 56, 395 (1988). +[2] M. Visser, Lorentzian Wormholes: From Einstein to Hawking (Springer, New York, NY, +1995). +[3] F.R. Klinkhamer, “Regularized big bang singularity,” Phys. Rev. D 100, 023536 (2019), +arXiv:1903.10450. +[4] F.R. Klinkhamer, “More on the regularized big bang singularity,” Phys. Rev. D 101, 064029 +(2020), arXiv:1907.06547. +[5] F.R. Klinkhamer and Z.L. Wang, “Nonsingular bouncing cosmology from general relativity,” +Phys. Rev. D 100, 083534 (2019), arXiv:1904.09961. +[6] F.R. Klinkhamer and Z.L. Wang, “Nonsingular bouncing cosmology from general relativity: +Scalar metric perturbations,” Phys. Rev. D 101, 064061 (2020), arXiv:1911.06173. +[7] F.R. Klinkhamer, “A new type of nonsingular black-hole solution in general relativity,” Mod. +Phys. Lett. A 29, 1430018 (2014), arXiv:1309.7011. +[8] F.R. Klinkhamer and F. Sorba, “Comparison of spacetime defects which are homeomorphic +but not diffeomorphic,” J. Math. Phys. 55, 112503 (2014), arXiv:1404.2901. +[9] F.R. Klinkhamer and Z.L. Wang, “Lensing and imaging by a stealth defect of spacetime,” +Mod. Phys. Lett. A 34, 1950026 (2019), arXiv:1808.02465. +[10] M. Calz`a, M. Rinaldi, and L. Sebastiani, “A special class of solutions in F(R)-gravity,” Eur. +Phys. J. C 78, 178 (2018), arXiv:1802.00329. +[11] N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, “A large-N reduced model as super- +string,” Nucl. Phys. B 498, 467 (1997), arXiv:hep-th/9612115. +[12] H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, A. Tsuchiya, and T. Tada, “IIB matrix model,” Prog. +Theor. Phys. Suppl. 134, 47 (1999), arXiv:hep-th/9908038. +[13] F.R. Klinkhamer, “IIB matrix model: Emergent spacetime from the master field,” Prog. +Theor. Exp. Phys. 2021, 013B04 (2021), arXiv:2007.08485. +[14] F.R. Klinkhamer, “IIB matrix model and regularized big bang,” Prog. Theor. Exp. Phys. +2021, 063B05 (2021), arXiv:2009.06525. +7 + diff --git a/vNAyT4oBgHgl3EQf0vmi/content/tmp_files/load_file.txt b/vNAyT4oBgHgl3EQf0vmi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..41e9431e911d39d45bf70ca8eb3cf39e28d588ff --- /dev/null +++ b/vNAyT4oBgHgl3EQf0vmi/content/tmp_files/load_file.txt @@ -0,0 +1,249 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf,len=248 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='00724v1 [gr-qc] 30 Dec 2022 KA–TP–31–2022 (v1) Traversable wormhole without exotic matter F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Klinkhamer∗ Institute for Theoretical Physics, Karlsruhe Institute of Technology (KIT), 76128 Karlsruhe, Germany Abstract We present a traversable-wormhole solution of the standard gravitational field equation of general relativity without need of exotic matter (exotic matter can have, for example, negative energy density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Instead of exotic matter, the solution relies on a 3-dimensional “spacetime defect” char- acterized by a locally vanishing metric determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' ∗ frans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='klinkhamer@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='edu 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' INTRODUCTION Traversable wormholes [1] appear to require “exotic” matter, for example matter violating the Null-Energy-Condition (NEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [2] for further discussion and references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' In this short paper, we look for a way around the necessity of having exotic matter, while making no essential changes in the established theories (general relativity and the standard model of elementary particle physics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Throughout, we use natural units with c = 1 and ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' BASIC IDEA The regularized-big-bang spacetime [3, 4] is a solution of general relativity with nor- mal matter and a degenerate metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' This spacetime corresponds to a traversable cosmic bounce [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' As noted briefly in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' II of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [4] and more extensively in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' II of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [6], the degeneracy of the regularized-big-bang metric gives an effective matter component which is “exotic,” specifically NEC violating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' The heuristics, then, is that the exotic effects of the metric degeneracy turn the singular (concave) big-bang behavior [a(t) ∼ √ t → 0 for t ↓ 0] into a smooth (convex) bounce behavior [a(t) ∼ aB + t2 for t ∈ (−∆t, +∆t), with aB > 0 and ∆t > 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' We now try to do something similar for the traversable wormhole, using general relativity with normal matter but allowing for a degenerate metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' SIMPLE EXAMPLE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Nondegenerate metric – special case Our basic idea can be tested by starting from the simple example discussed in Box 2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' There, the metric is given by (recall c = 1) ds2 ��� (nondegen-metric-special) = −dt2 + dl2 + � b2 0 + l2� � dθ2 + sin2 θ dφ2� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='1) with a nonzero real constant b0 (taken to be positive, for definiteness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' The coordinates t and l in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='1) range over (−∞, ∞) and θ and φ are the standard spherical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' 2 The Einstein equation, Rµν − 1 2 gµν R = (8πG)−1 Tµν, then requires the following compo- nents of the energy-momentum tensor [1]: T t t ��� (nondegen-metric-special) = 1 8πG b2 0 (b2 0 + l2)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='2a) T l l ��� (nondegen-metric-special) = − 1 8πG b2 0 (b2 0 + l2)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='2b) T θ θ ��� (nondegen-metric-special) = 1 8πG b2 0 (b2 0 + l2)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='2c) T φ φ ��� (nondegen-metric-special) = 1 8πG b2 0 (b2 0 + l2)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='2d) with all other components vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' The energy density is given by ρ = T tt = −T t t and we have ρ < 0 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='2a), which definitely corresponds to exotic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' As shown by items (d) and (e) of Box 2 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [1], this wormhole is traversable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Degenerate metric – special case Now consider the following metric Ansatz: ds2 ��� (degen-metric-special) = −dt2 + ξ2 ξ2 + λ2 dξ2 + � b2 0 + ξ2� � dθ2 + sin2 θ dφ2� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='3) with nonzero real constants λ and b0 (both taken to be positive, for definiteness) and coor- dinates t and ξ ranging over (−∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' The metric from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='3) gives the following Ricci and Kretschmann curvature scalars: R ��� (degen-metric-special) = −2 b2 0 − λ2 (b2 0 + ξ2)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='4a) K ��� (degen-metric-special) = 12 (b2 0 − λ2)2 (b2 0 + ξ2)4 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='4b) both of which are seen to vanish as ξ → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Three remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' First, the metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='3) is degenerate with a vanishing deter- minant at ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' In physical terms, this 3-dimensional hypersurface at ξ = 0 corresponds to a “spacetime defect” (see references [6, 8–10] in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [3] here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' some further references will be given shortly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Second, the change of the ξ coordinate to �l = ξ � 1 + λ2/ξ2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='5) 3 gives a metric similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='1), ds2 = −dt2 + d�l 2 + � b2 0 + �l 2 − λ2� � dθ2 + sin2 θ dφ2� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='6) but this coordinate transformation ξ → �l is discontinuous and, therefore, not a diffeomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' We remark that the coordinate �l ∈ (−∞, −λ] ∪[λ, ∞) is unsatisfactory for a proper description of the whole spacetime manifold, as, for given values of {t, θ, φ}, both �l = −λ and �l = λ correspond to a single point of the manifold (with the single coordinate ξ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' For further discussion of some of the mathematics and physics issues of such spacetime defects, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' 2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [7] and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' III of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Third, the embedding diagram of the spacetime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='3) for (t, θ) = (const, π/2) and λ2 < b2 0 is similar [with a 3-dimensional Euclidean embedding space] to the embedding diagram of the spacetime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='1) for the same values of (t, θ) and b2 0, as given by item (b) of Box 2 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' The embedding diagram of the spacetime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='3) for λ2 > b2 0 is similar to the embedding diagrams for λ2 ∈ [0, b2 0), except that, for λ2 > b2 0, there is a (2+1)-dimensional Minkowski embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' The description of the spacetime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='3) for λ2 = b2 0 uses two copies of the flat Euclidean space E3 with the interior of two balls removed and their surfaces identified, somewhat analogous to the description of the spacetime defect of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' 2 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' With the metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='3), the Einstein equation requires T t t ��� (degen-metric-special) = 1 8πG b2 0 − λ2 (b2 0 + ξ2)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='7a) T ξ ξ ��� (degen-metric-special) = − 1 8πG b2 0 − λ2 (b2 0 + ξ2)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='7b) T θ θ ��� (degen-metric-special) = 1 8πG b2 0 − λ2 (b2 0 + ξ2)2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='7c) T φ φ ��� (degen-metric-special) = 1 8πG b2 0 − λ2 (b2 0 + ξ2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='7d) Compared to the previous results (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='2), we see that the previous factors b2 0 in the numerators have been replaced by new factors (b2 0−λ2), while the changes in the denominators are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Starting from λ2 = 0+, these new numerator factors then change sign as λ2 increases above b2 0 and we no longer require exotic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Indeed, we have from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='7a) that ρ = −T t t > 0 for λ2 > b2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' There is, of course, also the special case λ2 = b2 0, for which the energy-momentum tensor 4 vanishes altogether, T µ ν ��� (degen-metric-special) λ2=b2 0 = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='8) and so do the curvature scalars (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' In that case, we have a wormhole in the vacuum, which will be further discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' We can get the radial geodesics ξ(t) passing through the wormhole throat by adapting result (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='6b) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [7] to our case: ξ(t) ��� (degen-metric-special) λ2=b2 0 = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ± � (B t)2 + 2 B λ t , for t ≥ 0 , ∓ � (B t)2 − 2 B λ t , for t < 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='9) with a dimensionless constant B ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' The same curves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='9) can be more easily obtained from straight lines �l(t), with radial velocity magnitude v, in the 2-dimensional Minkowski subspace of the spacetime (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='6) and the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='5) of the coordinate �l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' The Minkowski- subspace analysis identifies the constant B in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='9) as the ratio v/c, so that the B = 1 curves are light-like and those with B < 1 timelike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' The discussion of other geodesics for the metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='3) is similar to the discussion in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [9], which considers a related spacetime defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' The metric of this spacetime defect resembles the metric of the wormhole presented here, but their global spatial structures are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' DEGENERATE METRIC – GENERAL CASE The special degenerate metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='3) can be generalized as follows: ds2 ��� (degen-metric-general) = −e2 �φ(ξ) dt2 + ξ2 ξ2 + λ2 dξ2 + �r 2(ξ) � dθ2 + sin2 θ dφ2� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='1) with real functions �φ(ξ) and �r(ξ) and, again, coordinates t and ξ ranging over (−∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' If, moreover, we assume that �φ(ξ) remains finite everywhere and that �r(ξ) is positive with �r(ξ) ∼ |ξ| for ξ → ±∞, then the spacetime from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='1) corresponds to a wormhole (see also the further discussion at the beginning of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='2 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' If the global minimum of the function �r(ξ) has the value b0 > 0 at ξ = ξ0 ≡ 0 and if the function �φ(ξ) is essentially constant near ξ = 0, we expect interesting behavior for λ2 of the order of b2 0 or larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' In fact, using power series in ξ2 for �φ(ξ) and �r 2(ξ), we get energy-momentum components without singular behavior at ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' It is clear that further work will be cumbersome but perhaps not impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' 5 Awaiting the final analysis of the metric (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='1), we recall, from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' III B, that we already have one complete wormhole-type solution of the Einstein gravitational field equation: �φ(ξ) = 0 , �r 2(ξ) = λ2 + ξ2 , T µ ν(ξ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='2) Note that, different from Minkowski spacetime, this flat vacuum-wormhole spacetime is multiply connected (there are noncontractible loops in space, for example, loops encircling the wormhole mouth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' The solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='2) has one free parameter, the length scale λ, which can be determined by considering the circumferences divided by 2π of great circles centered on the wormhole [each circle has θ = π/2 and constant values of t and �l in the metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='6) for λ2 = b2 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Some further comments on this length scale λ appear in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' DISCUSSION We have three final remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' First, we mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' III B the possibility of obtaining a vacuum wormhole if the parameters λ2 and b2 0 of the metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='3) are chosen to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Vacuum wormholes also appear in certain modified-gravity theories [10], where the exotic effects trace back to the extra terms in the gravitational action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Our vacuum-wormhole solution does not require any change of the theory, general relativity suffices, except that we now allow for degenerate metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Second, this vacuum-wormhole solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='2) has the length scale λ as a free parameter and, if there is a preferred value λ in Nature, then that value can only come from a theory beyond general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' An example of such a theory would be nonperturbative superstring theory in the formulation of the IIB matrix model [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' That matrix model could give rise to an emergent spacetime with or without spacetime defects [13, 14] and, if defects do appear, then the length scale λ of a remnant vacuum-wormhole defect would be related to the matrix-model length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Third, the main objective of the present paper is to reduce the hurdles to overcome in the quest of traversable wormholes (specifically, we remove the requirement of exotic matter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' But there remains, at least, one important hurdle, namely to construct or harvest a suitable spacetime defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' ACKNOWLEDGMENTS It is a pleasure to thank Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' Wang for useful comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content=' 6 [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQf0vmi/content/2301.00724v1.pdf'} +page_content='S.' 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