Woodruff Yahoo Labs IBM Research New York Almaden brPage 2br Singular Value Decomposition matrix A rank Lowrank matrix approximation problem min rank Singular Value Decomposition SVD z z z and V Solution via EckartYoung Theorem AV mn min time br ID: 73194
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CURMatrixDecompositionCURreplacestheleftandrightsingularvectorsintheSVDwithactualcolumnsandrowsfromthematrix,respectively0BBBB@A1CCCCA=0BBBB@C1CCCCA�U�R+0BBBB@E1CCCCA0BBBB@A1CCCCA=0BBBB@Uk1CCCCA�k�Vk+0BBBB@E1CCCCA Optimizationproblem Denition(TheCURProblem) Given A2Rmn krank(A) "�0construct C2Rmc R2Rrn U2Rcrsuchthat:kA�CURk2F(1+")kA�Akk2F:withc;r;andrank(U)beingassmallaspossible. Openproblems 1OptimalCUR:Canwendrelative-errorCURalgorithmsselectingtheoptimalnumberofcolumnsandrows,togetherwithamatrixUwithoptimalrank? 2Input-sparsity-timeCUR:Canwendrelative-errorCURalgorithmsrunningininput-sparsity-time(nnz(A)time)? 3DeterministicCUR:Canwendrelative-errorCURalgorithmsthataredeterministicandruninpolytime? Lowerbound Theorem FixappropriatematrixA2Rnn.ConsiderafactorizationCUR,kA�CURk2F(1+")kA�Akk2F:Then,foranyk1andforany"1=3:c= (k=");andr= (k=");andrank(U)k=2: Extendedlowerboundin[DeshpandeandVempala,2006],[Boutsidisetal,2011],[SinopandGuruswami,2011] Input-sparsity-timeCUR Theorem ThereexistsarandomizedalgorithmtoconstructaCURwithc=O(k=")andr=O(k=")andrank(U)=ksuchthat,withconstantprobabilityofsuccess,kA�CURk2F(1+")kA�Akk2F:Runningtime:O(nnz(A)logn+(m+n)poly(logn;k;1=")): DeterministicCUR Theorem ThereexistsadeterministicalgorithmtoconstructaCURwithc=O(k=")andr=O(k=")andrank(U)=ksuchthatkA�CURk2F(1+")kA�Akk2F:Runningtime:O(mn3k="): Step2 2ConstructRwithO(k=")rows: 1FindZ22RmkinthespanofCsuchthat:kA�Z2ZT2Ak2F(1+")kA�Akk2F: 2Howtodothisefciently? InsteadofprojectingcolumnsofAontoC,weprojectthecolumnsofAW,whereWisarandomsubspaceembedding Findbestrank-kapproximationofthecolumnsofAWinC 3SampleO(klogk)rowswithleveragescores(fromZ2).Down-samplethoserowstor1=O(k)rowswithBatson/Spielman/Srivastava(BSS)sampling.(R12Rr1n)kA�ARy1R1k2FO(1)kA�Z2ZT2Ak2F 4Sampler2=O(k=")rowswithadaptivesampling++kA�Z2ZT2ARyRk2FkA�Z2ZT2Ak2F+rank(Z2ZT2A) r2kA�AR1yR1k2F DeterministicCUREverythingshouldruninpolynomialtimeandbedeterministic. 1Existingtools: StandardSVDalgorithm. Standardmethodtondthe“best”rankkapproximationtoamatrixinagivensubspace. Batson/Spielman/Srivastava(BSS)samplingasin[Boutsidisetall,FOCS2011]. 2Newtools: Derandomizationoftheadaptivesamplingof[Desphandeetal,RANDOM2006]and[WangandZhang,JMLR2013].