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The Power of Convex Relaxation: NearOptimal Matrix Completion
, 2009
"... This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In ..."
Abstract

Cited by 146 (6 self)
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This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In general, accurate recovery of a matrix from a small number of entries is impossible; but the knowledge that the unknown matrix has low rank radically changes this premise, making the search for solutions meaningful. This paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever (the information theoretic limit). More importantly, the paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit (up to logarithmic factors). This convex program simply finds, among all matrices consistent with the observed entries, that with minimum nuclear norm. As an example, we show that on the order of nr log(n) samples are needed to recover a random n × n matrix of rank r by any method, and to be sure, nuclear norm minimization succeeds as soon as the number of entries is of the form nrpolylog(n).
Matrix Completion with Noise
"... On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest ..."
Abstract

Cited by 91 (5 self)
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On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries, and comes up in many areas of science and engineering including collaborative filtering, machine learning, control, remote sensing, and computer vision to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown lowrank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclearnorm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown n × n matrix of low rank r from just about nr log 2 n noisy samples with an error which is proportional to the noise level. We present numerical results which complement our quantitative analysis and show that, in practice, nuclear norm minimization accurately fills in the many missing entries of large lowrank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout.