A non-convex approach to low-rank and sparse matrix decomposition

TL;DR

This paper introduces a non-convex method for low-rank and sparse matrix decomposition using a fraction function penalty, demonstrating improved recovery of corrupted matrices.

Contribution

It proposes a novel non-convex approach replacing traditional convex functions with a fraction function for better matrix decomposition.

Findings

Effective in recovering heavily corrupted low-rank matrices

Outperforms traditional convex methods in experiments

Uses an ADMM algorithm for optimization

Abstract

In this paper, we develop a nonconvex approach to the problem of low-rank and sparse matrix decomposition. In our nonconvex method, we replace the rank function and the $l_{0}$-norm of a given matrix with a non-convex fraction function on the singular values and the elements of the matrix respectively. An alternative direction method of multipliers algorithm is utilized to solve our proposed nonconvex problem with the nonconvex fraction function penalty. Numerical experiments on some low-rank and sparse matrix decomposition problems show that our method performs very well in recovering low-rank matrices which are heavily corrupted by large sparse errors.