Exact recovery low-rank matrix via transformed affine matrix rank minimization

TL;DR

This paper introduces a novel nonconvex approximation for low-rank matrix recovery, transforming the NP-hard problem into a more tractable form and demonstrating effective image inpainting results.

Contribution

It proposes a transformed affine matrix rank minimization approach using a nonconvex fraction function, establishing equivalence and solution properties, and applies the DC algorithm for practical recovery.

Findings

Effective low-rank image recovery demonstrated

Outperforms some state-of-the-art algorithms in experiments

Provides theoretical guarantees for solution uniqueness

Abstract

The goal of affine matrix rank minimization problem is to reconstruct a low-rank or approximately low-rank matrix under linear constraints. In general, this problem is combinatorial and NP-hard. In this paper, a nonconvex fraction function is studied to approximate the rank of a matrix and translate this NP-hard problem into a transformed affine matrix rank minimization problem. The equivalence between these two problems is established, and we proved that the uniqueness of the global minimizer of transformed affine matrix rank minimization problem also solves affine matrix rank minimization problem if some conditions are satisfied. Moreover, we also proved that the optimal solution to the transformed affine matrix rank minimization problem can be approximately obtained by solving its regularization problem for some proper smaller $\lambda>0$. Lastly, the DC algorithm is utilized to solve the regularization transformed affine matrix rank minimization problem and the numerical experiments on image inpainting problems show that our method performs effectively in recovering low-rank images compared with some state-of-art algorithms.