Guaranteed Rank Minimization via Singular Value Projection

TL;DR

This paper introduces SVP, a simple, fast algorithm for rank minimization that guarantees recovery under weaker conditions, is robust to noise, and outperforms existing methods in matrix completion tasks.

Contribution

The paper presents SVP, a novel singular value projection algorithm that improves recovery guarantees, simplicity, speed, and robustness over previous approaches for rank minimization and matrix completion.

Findings

SVP recovers minimum rank solutions under weaker isometry conditions.

SVP is significantly faster and more robust to noise than existing methods.

Empirical results show SVP outperforms prior algorithms in real-world and synthetic problems.

Abstract

Minimizing the rank of a matrix subject to affine constraints is a fundamental problem with many important applications in machine learning and statistics. In this paper we propose a simple and fast algorithm SVP (Singular Value Projection) for rank minimization with affine constraints (ARMP) and show that SVP recovers the minimum rank solution for affine constraints that satisfy the "restricted isometry property" and show robustness of our method to noise. Our results improve upon a recent breakthrough by Recht, Fazel and Parillo (RFP07) and Lee and Bresler (LB09) in three significant ways: 1) our method (SVP) is significantly simpler to analyze and easier to implement, 2) we give recovery guarantees under strictly weaker isometry assumptions 3) we give geometric convergence guarantees for SVP even in presense of noise and, as demonstrated empirically, SVP is significantly faster on real-world and synthetic problems. In addition, we address the practically important problem of low-rank matrix completion (MCP), which can be seen as a special case of ARMP. We empirically demonstrate that our algorithm recovers low-rank incoherent matrices from an almost optimal number of uniformly sampled entries. We make partial progress towards proving exact recovery and provide some intuition for the strong performance of SVP applied to matrix completion by showing a more restricted isometry property. Our algorithm outperforms existing methods, such as those of \cite{RFP07,CR08,CT09,CCS08,KOM09,LB09}, for ARMP and the matrix-completion problem by an order of magnitude and is also significantly more robust to noise.