Low-Rank Matrix Completion: A Contemporary Survey
Luong Trung Nguyen Junhan Kim Byonghyo Shim
cs.DS cs.IT math.IT math.OC











































































































































  
  
  
  
  

  
  
  
  
  
  
  
  





































=2500










































































































































































shapes.symbols,patterns
shapes,shapes.multipart,shapes.geometric
arrows.meta


































	
	







shapes,shapes.geometric
calc,backgrounds

pic shift/.store in=,
	pic shift = (0,0),
	pics/mySensor/.style args=scale #1 RFrange #2 Number #3 Opacity #4
		code=		
		cyl/.style=aspect=2,minimum height=0.1cm,minimum width=1.5cm,cylinder,draw=black,line width=2,shape border rotate=90,cylinder uses custom fill, cylinder body fill=blue!10			
			
			[shift=,scale=#1, every node/.append style=scale=#1]
			[cyl] (-cyl) ;
			#3=0
			at ([yshift=-1]-cyl) 1.2 #3;
			[blue!70!black,very thick] (0,0.4) – (0.75,0.4);
			[fill=blue!70!black,draw=blue] (-0.1,0.5) – (0.3,0.5) – (0.1,0.3) – (-0.3,0.3) – cycle ;
			ıin 1,...,3
				[blue!70!black,very thick] (-0.05+ı*0.1,0.55) – (-0.35+ı*0.1,0.25);
			
			 
			[blue!70!black,line width=4,line cap=round] (0.75,0.4) – (0.75,2.5-1);
			

			[fill=blue!70!black,blue!70!black,line width=4] (0.75,1.5) – (0.75+0.29,1.5+0.5) – (0.75-0.29,1.5+0.5) – cycle;
			





			#2=0
			[on background layer]
			[very thick,red!40!green,fill=red!40!green,opacity=#4] (0,0) ellipse (#2*2 and #2);
			[line width=1mm,dashed,black] (0,0) ellipse (#2*2 and #2);
			
				
			
			
			
	


pic shift/.store in=,
	pic shift = (0,0),
	pics/myManifold/.style args=scale #1
		code=	
			[shift=,scale=#1, every node/.append style=scale=#1]
			
			
			-0.0682
			-0.2079
			0.9758








        
        	[hide axis]
			3[samples=21,
            	domain=-20:20,
            	y domain=70:110, 
            	colormap/greenyellow,           	
				scatter/use mapped color=
					draw=mapped color,
					fill=mapped color
				,				
				surf,
				shader=faceted interp,opacity = 1, 
				z buffer=sort,draw=green!20](sin(x)*sin(y),cos(y), cos(x)*sin(y));	















			3[samples=2,
            	domain=-0.2:0.1,
            	y domain=-0.4:0.2,
            	colormap/blackwhite,           	
				scatter/use mapped color=
					draw=black,
					fill=none
				,
				surf,
				opacity = 0, 
				z buffer=sort,fill=white,draw=black,very thick](x,y, 1-(/)*x-(/)*y);
				
			
			
			
			
			
	


pic shift/.store in=,
	pic shift = (0,0),
	pics/myManifoldv2/.style args=scale #1 retraction #2 number #3
		code=	
			[shift=,scale=#1, every node/.append style=scale=#1]
			
			
			[on background layer]
				(0,0) pic myManifold=scale 1.6;
			
			[black!80!green, very thick] (1.6,1.8) – (4.8,1.8) – (8.2,7.4) – (5,7.4) – cycle;
			
			#2=0
			
			[black!50!red,thick,dashed]  (5.8,6.4) – (5.8,5.2);
			[anchor=north,xshift=0.2cm] at (5.8,5.2) R_𝐘(𝐁);
			[black!50!red] (5.8,5.2) circle [radius=2pt];
			[rectangle,draw=black!50!red,thick,inner sep=0.1cm] at (5.7,5.3) ;
			
			[black!50!red,very thick] (3.4,3.4) .. controls (3.65,4.8) and (5.4,5.2) .. (5.8,5.2);
			[black!50!red] at (3.8,3.4) γ(t);
			[black!80!green,very thick,->] (3.8,4.25) – (5.8,6.4);	
			#2=0
			
			[anchor=south west,xshift=-0.6cm,yshift=-0.2cm] at (5.8,6.4) 𝐁=γ^' (t)|_t=0;
			[black!80!green,very thick,->] (0.2,3.6) – (0.2,7);
			[anchor=east,xshift=-0.1cm] at (0.2,7) t∈ℝ;

			[black!80!green,very thick] (0.3,5.4) – (0.1,5.4);
			[anchor=east] at (0.1,5.4) 0;
	
			[black!80!green,thick,dotted] (0.2,5.4) – (3.8,4.25);
			[black!80!green,thick,dotted] (0.2,3.8) – (3.5,3.8);
			[black!80!green,thick,dotted] (0.2,6.4) – (4.2,4.6);
			
			
			[anchor=south west,xshift=-0.6cm,yshift=-0.2cm] at (5.8,6.4) 𝐁;
			
			[black] at (4.1,4.1) 𝐘;
			[black!50!red] (3.8,4.25) circle [radius=2pt];
			

			at (8.2,1) 𝒴;
			at (2.4,2.2) T_𝐘𝒴;
			
			at (4.8,0.4) (#3);
			
			
			
	





theoremTheorem
remarkRemark
lemma[theorem]Lemma
definitionDefinition
proposition[theorem]Proposition
corollary[theorem]Corollary
exampleExample


