General inertial smoothing proximal gradient algorithm for the relaxation of matrix rank minimization problem

TL;DR

This paper introduces a novel inertial smoothing proximal gradient algorithm for matrix rank minimization, demonstrating convergence properties and efficiency through numerical experiments on various data types.

Contribution

It proposes the GIMSPG algorithm, incorporating inertial techniques for the first time in this context, with proven convergence and finite support recovery of singular values.

Findings

Convergence to a lifted stationary point is established.

Finite iteration support recovery of singular values is achieved.

Numerical experiments show the algorithm's efficiency on real data.

Abstract

We consider the exact continuous relaxation model of matrix rank minimization problem proposed by Yu and Zhang (Comput.Optim.Appl. 1-20, 2022). Motivated by the inertial techinique, we propose a general inertial smoothing proximal gradient algorithm(GIMSPG) for this kind of problems. It is shown that the singular values of any accumulation point have a common support set and the nonzero singular values have a unified lower bound. Besides, the zero singular values of the accumulation point can be achieved within finite iterations. Moreover, we prove that any accumulation point of the sequence generated by the GIMSPG algorithm is a lifted stationary point of the continuous relaxation model under the flexible parameter constraint. Finally, we carry out numerical experiments on random data and image data respectively to illustrate the efficiency of the GIMSPG algorithm.