The Power of Convex Relaxation: Near-Optimal Matrix Completion

Abstract

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 x 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 nr polylog(n).

arXiv comment51 pages, 12 figures

References in corpus (5)

• A Singular Value Thresholding Algorithm for Matrix Completion
• Matrix Completion from a Few Entries
• Low-rank matrix factorization with attributes
• Fixed Point and Bregman Iterative Methods for Matrix Rank Minimization
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• Statistical Inferences of Linear Forms for Noisy Matrix Completion
• Matrix Coherence and the Nystrom Method
• SET: an algorithm for consistent matrix completion
• Provable Near-Optimal Low-Multilinear-Rank Tensor Recovery
• Low-rank Matrix Completion with Noisy Observations: a Quantitative Comparison
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• Recommendation on a Budget: Column Space Recovery from Partially Observed Entries with Random or Active Sampling
• On the Estimation of Coherence