Low-rank Matrix Completion using Alternating Minimization

TL;DR

This paper provides the first theoretical analysis of alternating minimization for low-rank matrix completion, demonstrating its effectiveness and faster convergence under standard conditions, which explains its empirical success.

Contribution

It offers a novel theoretical understanding of alternating minimization's success and convergence speed in low-rank matrix completion and sensing problems.

Findings

Alternating minimization succeeds under standard conditions.

The method guarantees faster, geometric convergence.

The analysis simplifies understanding of the algorithm's performance.

Abstract

Alternating minimization represents a widely applicable and empirically successful approach for finding low-rank matrices that best fit the given data. For example, for the problem of low-rank matrix completion, this method is believed to be one of the most accurate and efficient, and formed a major component of the winning entry in the Netflix Challenge. In the alternating minimization approach, the low-rank target matrix is written in a bi-linear form, i.e. $X = UV^\dag$; the algorithm then alternates between finding the best $U$ and the best $V$. Typically, each alternating step in isolation is convex and tractable. However the overall problem becomes non-convex and there has been almost no theoretical understanding of when this approach yields a good result. In this paper we present first theoretical analysis of the performance of alternating minimization for matrix completion, and the related problem of matrix sensing. For both these problems, celebrated recent results have shown that they become well-posed and tractable once certain (now standard) conditions are imposed on the problem. We show that alternating minimization also succeeds under similar conditions. Moreover, compared to existing results, our paper shows that alternating minimization guarantees faster (in particular, geometric) convergence to the true matrix, while allowing a simpler analysis.