Fast and Provable Nonconvex Robust Matrix Completion

TL;DR

This paper introduces ARMC, a non-convex algorithm for robust matrix completion that improves efficiency and guarantees convergence, outperforming existing methods in theory and practice.

Contribution

The paper proposes ARMC, a novel accelerated non-convex method with provable guarantees and improved sample complexity for robust matrix completion.

Findings

ARMC achieves faster convergence than prior non-convex methods.

Theoretical bounds on sample complexity and outlier sparsity are improved.

Numerical experiments demonstrate ARMC's superior performance on synthetic and real data.

Abstract

We study the robust matrix completion (RMC) problem subject to both sparse outliers and stochastic noise. A non-convex method termed Accelerated Robust Matrix Completion (ARMC) is proposed, which accelerates a prior non-convex approach by incorporating an explicit subspace projection step into the low-rank update, leading to significantly improved computational efficiency. Through a delicate analysis based on the leave-one-out technique, the entrywise linear convergence guarantee of ARMC has been established. Notably, the derived bounds for sample complexity and outlier sparsity improve upon existing guarantees of the convex relaxation approach that also accounts for both sparse outliers and stochastic noise. Moreover, numerical experiments on synthetic and real data show that ARMC is superior to existing non-convex RMC methods.