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159
Robust principal component analysis?
 Journal of the ACM,
, 2011
"... Abstract This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a lowrank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the lowrank and the ..."
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Cited by 569 (26 self)
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Abstract This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a lowrank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the lowrank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the 1 norm. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. We discuss an algorithm for solving this optimization problem, and present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces.
A unified framework for highdimensional analysis of Mestimators with decomposable regularizers
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Nuclear norm penalization and optimal rates for noisy low rank matrix completion.
 Annals of Statistics,
, 2011
"... AbstractThis paper deals with the trace regression model where n entries or linear combinations of entries of an unknown m1 × m2 matrix A0 corrupted by noise are observed. We propose a new nuclear norm penalized estimator of A0 and establish a general sharp oracle inequality for this estimator for ..."
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Cited by 107 (7 self)
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AbstractThis paper deals with the trace regression model where n entries or linear combinations of entries of an unknown m1 × m2 matrix A0 corrupted by noise are observed. We propose a new nuclear norm penalized estimator of A0 and establish a general sharp oracle inequality for this estimator for arbitrary values of n, m1, m2 under the condition of isometry in expectation. Then this method is applied to the matrix completion problem. In this case, the estimator admits a simple explicit form and we prove that it satisfies oracle inequalities with faster rates of convergence than in the previous works. They are valid, in particular, in the highdimensional setting m1m2 n. We show that the obtained rates are optimal up to logarithmic factors in a minimax sense and also derive, for any fixed matrix A0, a nonminimax lower bound on the rate of convergence of our estimator, which coincides with the upper bound up to a constant factor. Finally, we show that our procedure provides an exact recovery of the rank of A0 with probability close to 1. We also discuss the statistical learning setting where there is no underlying model determined by A0 and the aim is to find the best trace regression model approximating the data.
Stable principal component pursuit
 In Proc. of International Symposium on Information Theory
, 2010
"... We consider the problem of recovering a target matrix that is a superposition of lowrank and sparse components, from a small set of linear measurements. This problem arises in compressed sensing of structured highdimensional signals such as videos and hyperspectral images, as well as in the analys ..."
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Cited by 94 (3 self)
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We consider the problem of recovering a target matrix that is a superposition of lowrank and sparse components, from a small set of linear measurements. This problem arises in compressed sensing of structured highdimensional signals such as videos and hyperspectral images, as well as in the analysis of transformation invariant lowrank structure recovery. We analyze the performance of the natural convex heuristic for solving this problem, under the assumption that measurements are chosen uniformly at random. We prove that this heuristic exactly recovers lowrank and sparse terms, provided the number of observations exceeds the number of intrinsic degrees of freedom of the component signals by a polylogarithmic factor. Our analysis introduces several ideas that may be of independent interest for the more general problem of compressed sensing and decomposing superpositions of multiple structured signals. 1
SOLVING A LOWRANK FACTORIZATION MODEL FOR MATRIX COMPLETION BY A NONLINEAR SUCCESSIVE OVERRELAXATION ALGORITHM
"... Abstract. The matrix completion problem is to recover a lowrank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclearnorm minimization which requires computing singular value decompositions – a task that is increasingly costly as matrix sizes an ..."
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Cited by 91 (10 self)
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Abstract. The matrix completion problem is to recover a lowrank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclearnorm minimization which requires computing singular value decompositions – a task that is increasingly costly as matrix sizes and ranks increase. To improve the capacity of solving largescale problems, we propose a lowrank factorization model and construct a nonlinear successive overrelaxation (SOR) algorithm that only requires solving a linear least squares problem per iteration. Convergence of this nonlinear SOR algorithm is analyzed. Numerical results show that the algorithm can reliably solve a wide range of problems at a speed at least several times faster than many nuclearnorm minimization algorithms. Key words. Matrix Completion, alternating minimization, nonlinear GS method, nonlinear SOR method AMS subject classifications. 65K05, 90C06, 93C41, 68Q32
Restricted strong convexity and weighted matrix completion: Optimal bounds with noise
, 2012
"... We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with high probability, it satisfies a form of restricted strong ..."
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Cited by 84 (10 self)
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We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with high probability, it satisfies a form of restricted strong convexity with respect to weighted Frobenius norm. Using this property, we obtain as corollaries a number of error bounds on matrix completion in the weighted Frobenius norm under noisy sampling and for both exact and near lowrank matrices. Our results are based on measures of the “spikiness” and “lowrankness” of matrices that are less restrictive than the incoherence conditions imposed in previous work. Our technique involves an Mestimator that includes controls on both the rank and spikiness of the solution, and we establish nonasymptotic error bounds in weighted Frobenius norm for recovering matrices lying with ℓq“balls ” of bounded spikiness. Using informationtheoretic methods, we show that no algorithm can achieve better estimates (up to a logarithmic factor) over these same sets, showing that our conditions on matrices and associated rates are essentially optimal.
Parallel stochastic gradient algorithms for largescale matrix completion
 MATHEMATICAL PROGRAMMING COMPUTATION
, 2013
"... This paper develops Jellyfish, an algorithm for solving dataprocessing problems with matrixvalued decision variables regularized to have low rank. Particular examples of problems solvable by Jellyfish include matrix completion problems and leastsquares problems regularized by the nuclear norm or ..."
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Cited by 74 (8 self)
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This paper develops Jellyfish, an algorithm for solving dataprocessing problems with matrixvalued decision variables regularized to have low rank. Particular examples of problems solvable by Jellyfish include matrix completion problems and leastsquares problems regularized by the nuclear norm or γ2norm. Jellyfish implements a projected incremental gradient method with a biased, random ordering of the increments. This biased ordering allows for a parallel implementation that admits a speedup nearly proportional to the number of processors. On largescale matrix completion tasks, Jellyfish is orders of magnitude more efficient than existing codes. For example, on the Netflix Prize data set, prior art computes rating predictions in approximately 4 hours, while Jellyfish solves the same problem in under 3 minutes on a 12 core workstation.
SpaRCS: Recovering lowrank and sparse matrices from compressive measurements
, 2011
"... We consider the problem of recovering a matrix M that is the sum of a lowrank matrix L and a sparse matrix S from a small set of linear measurements of the form y = A(M) =A(L + S). This model subsumes three important classes of signal recovery problems: compressive sensing, affine rank minimization ..."
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Cited by 46 (4 self)
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We consider the problem of recovering a matrix M that is the sum of a lowrank matrix L and a sparse matrix S from a small set of linear measurements of the form y = A(M) =A(L + S). This model subsumes three important classes of signal recovery problems: compressive sensing, affine rank minimization, and robust principal component analysis. We propose a natural optimization problem for signal recovery under this model and develop a new greedy algorithm called SpaRCS to solve it. Empirically, SpaRCS inherits a number of desirable properties from the stateoftheart CoSaMP and ADMiRA algorithms, including exponential convergence and efficient implementation. Simulation results with video compressive sensing, hyperspectral imaging, and robust matrix completion data sets demonstrate both the accuracy and efficacy of the algorithm. 1
Collaborative filtering in a nonuniform world: Learning with the weighted trace norm
, 2010
"... We show that matrix completion with tracenorm regularization can be significantly hurt when entries of the matrix are sampled nonuniformly, but that a properly weighted version of the tracenorm regularizer works well with nonuniform sampling. We show that the weighted tracenorm regularization i ..."
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Cited by 41 (5 self)
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We show that matrix completion with tracenorm regularization can be significantly hurt when entries of the matrix are sampled nonuniformly, but that a properly weighted version of the tracenorm regularizer works well with nonuniform sampling. We show that the weighted tracenorm regularization indeed yields significant gains on the highly nonuniformly sampled Netflix dataset.