Practical Matrix Completion and Corruption Recovery using Proximal Alternating Robust Subspace Minimization

TL;DR

This paper introduces PARSuMi, a non-convex optimization method for matrix completion that effectively handles noise and corruptions, outperforming existing convex approaches in practical scenarios.

Contribution

The paper proposes a novel Proximal Alternating Robust Subspace Minimization algorithm that directly exploits rank constraints and uses the $ ext{l}_0$ pseudo-norm for corruption recovery, improving practical performance.

Findings

Successfully handles real-world noisy and corrupted data

Outperforms state-of-the-art algorithms on synthetic and real datasets

Converges to a stationary point in non-convex optimization setting

Abstract

Low-rank matrix completion is a problem of immense practical importance. Recent works on the subject often use nuclear norm as a convex surrogate of the rank function. Despite its solid theoretical foundation, the convex version of the problem often fails to work satisfactorily in real-life applications. Real data often suffer from very few observations, with support not meeting the random requirements, ubiquitous presence of noise and potentially gross corruptions, sometimes with these simultaneously occurring. This paper proposes a Proximal Alternating Robust Subspace Minimization (PARSuMi) method to tackle the three problems. The proximal alternating scheme explicitly exploits the rank constraint on the completed matrix and uses the $\ell_0$ pseudo-norm directly in the corruption recovery step. We show that the proposed method for the non-convex and non-smooth model converges to a stationary point. Although it is not guaranteed to find the global optimal solution, in practice we find that our algorithm can typically arrive at a good local minimizer when it is supplied with a reasonably good starting point based on convex optimization. Extensive experiments with challenging synthetic and real data demonstrate that our algorithm succeeds in a much larger range of practical problems where convex optimization fails, and it also outperforms various state-of-the-art algorithms.