A smoothing proximal gradient algorithm for matrix rank minimization problem

TL;DR

This paper introduces a smoothing proximal gradient algorithm for low-rank matrix minimization, providing theoretical guarantees and demonstrating its effectiveness through numerical experiments.

Contribution

It proposes a novel SPG algorithm for matrix rank minimization with convergence analysis and exact relaxation properties.

Findings

The SPG algorithm converges to a lifted stationary point.

The relaxation maintains the same minimizers as the original problem.

Numerical results confirm the algorithm's efficiency.

Abstract

In this paper, we study the low-rank matrix minimization problem, where the loss function is convex but nonsmooth and the penalty term is defined by the cardinality function. We first introduce an exact continuous relaxation, that is, both problems have the same minimzers and the same optimal value. In particular, we introduce a class of lifted stationary point of the relaxed problem and show that any local minimizer of the relaxed problem must be a lifted stationary point. In addition, we derive lower bound property for the nonzero singular values of the lifted stationary point and hence also of the local minimizers of the relaxed problem. Then the smoothing proximal gradient (SPG) algorithm is proposed to find a lifted stationary point of the continuous relaxation model. Moreover, it is shown that the whole sequence generated by SPG algorithm converges to a lifted stationary point. At last, numerical examples show the efficiency of the SPG algorithm.