Affine matrix rank minimization problem via non-convex fraction function penalty

TL;DR

This paper introduces a non-convex fraction function to promote low rank in affine matrix rank minimization, proposing an iterative singular value thresholding algorithm that outperforms existing methods in matrix completion and image inpainting.

Contribution

It develops a novel non-convex regularization approach and an iterative algorithm for affine matrix rank minimization, with proven convergence and improved empirical performance.

Findings

Algorithm effectively finds low-rank matrices in completion tasks.

Adjusting parameter 'a' improves results over state-of-the-art methods.

The method outperforms existing algorithms in numerical experiments.

Abstract

Affine matrix rank minimization problem is a fundamental problem with a lot of important applications in many fields. It is well known that this problem is combinatorial and NP-hard in general. In this paper, a continuous promoting low rank non-convex fraction function is studied to replace the rank function in this NP-hard problem. Inspired by our former work in compressed sensing, an iterative singular value thresholding algorithm is proposed to solve the regularization transformed affine matrix rank minimization problem. For different $a>0$, we could get a much better result by adjusting the different value of $a$, which is one of the advantages for the iterative singular value thresholding algorithm compared with some state-of-art methods. Some convergence results are established and numerical experiments show that this thresholding algorithm is feasible for solving the regularization transformed affine matrix rank minimization problem. Moreover, we proved that the value of the regularization parameter $\lambda>0$ can not be chosen too large. Indeed, there exists $\bar{\lambda}>0$ such that the optimal solution of the regularization transformed affine matrix rank minimization problem is equal to zero for any $\lambda>\bar{\lambda}$. Numerical experiments on matrix completion problems show that our method performs powerful in finding a low-rank matrix and the numerical experiments about image inpainting problems show that our algorithm has better performances than some state-of-art methods.