Nonconvex ADMM for Rank-Constrained Matrix Sensing Problem

TL;DR

This paper introduces a new ADMM-based algorithm for rank-constrained matrix sensing that guarantees global convergence and outperforms existing methods in low sampling rate noisy matrix completion scenarios.

Contribution

It proposes the first ADMM algorithm with convergence guarantees specifically for the rank-constrained matrix sensing problem, including matrix completion.

Findings

The algorithm converges globally under the Kurdyka-Lojasiewicz property.

It performs better than existing methods at low sampling rates in noisy matrix completion.

Numerical experiments validate the effectiveness of the proposed approach.

Abstract

Low-rank matrix approximation (LRMA) has been arisen in many applications, such as dynamic MRI, recommendation system and so on. The alternating direction method of multipliers (ADMM) has been designed for the nuclear norm regularized least squares problem and shows a good performance. However, due to the lack of guarantees for the convergence, there are few ADMM algorithms designed directly for the rank-constrained matrix sensing problem (RCMS). Therefore, in this paper, we propose an ADMM-based algorithm for the RCMS. Based on the Kurdyka-Lojasiewicz (KL) property, we prove that the proposed algorithm globally converges. And we discuss a specific case: the rank-constrained matrix completion problem (RCMC). Numerical experiments show that specialized for the matrix completion, the proposed algorithm performs better when the sampling rate is really low in noisy case, which is the key for the matrix completion.