Exploring Algorithmic Limits of Matrix Rank Minimization under Affine Constraints

TL;DR

This paper introduces a simple probabilistic PCA-like algorithm capable of recovering low-rank matrices under affine constraints at the theoretical limit, outperforming traditional convex and non-convex methods in challenging scenarios.

Contribution

The paper proposes a novel, parameter-free probabilistic PCA-like algorithm for matrix rank minimization that succeeds at the theoretical measurement limit and handles ill-conditioned constraints.

Findings

Algorithm successfully recovers matrices at the measurement limit.

Performs well even with highly ill-conditioned affine constraints.

Shows promising results in computer vision and collaborative filtering tasks.

Abstract

Many applications require recovering a matrix of minimal rank within an affine constraint set, with matrix completion a notable special case. Because the problem is NP-hard in general, it is common to replace the matrix rank with the nuclear norm, which acts as a convenient convex surrogate. While elegant theoretical conditions elucidate when this replacement is likely to be successful, they are highly restrictive and convex algorithms fail when the ambient rank is too high or when the constraint set is poorly structured. Non-convex alternatives fare somewhat better when carefully tuned; however, convergence to locally optimal solutions remains a continuing source of failure. Against this backdrop we derive a deceptively simple and parameter-free probabilistic PCA-like algorithm that is capable, over a wide battery of empirical tests, of successful recovery even at the theoretical limit where the number of measurements equal the degrees of freedom in the unknown low-rank matrix. Somewhat surprisingly, this is possible even when the affine constraint set is highly ill-conditioned. While proving general recovery guarantees remains evasive for non-convex algorithms, Bayesian-inspired or otherwise, we nonetheless show conditions whereby the underlying cost function has a unique stationary point located at the global optimum; no existing cost function we are aware of satisfies this same property. We conclude with a simple computer vision application involving image rectification and a standard collaborative filtering benchmark.