Low-rank matrix recovery problem minimizing a new ratio of two norms approximating the rank function then using an ADMM-type solver with applications

Abstract

Since the nuclear norm may lead to suboptimal solutions to rank minimization problems, many nonconvex surrogates have been proposed to provide better rank function approximations. In this paper, we use the ratio of the nuclear norm and the Frobenius norm, denoted as N/F, as a new nonconvex surrogate of the rank function. The N/F can provide a close approximation for the matrix rank. In particular, the N/F is the same as the rank for rank 1 matrices. Furthermore, compared to other popular nonconvex approximations, the N/F model has no extra parameters to tune. Theoretically, we establish the exact recovery condition and the stable recovery condition for the N/F minimization problem with or without noise, respectively. Computationally, we use the alternating direction method of multipliers with a specific variable-splitting scheme to solve the N/F minimization problem. What is more, we incorporate a box constraint in the N/F model, and propose the ADMM-box to solve it. The convergence analysis of ADMM and ADMM-box are also given. Extensive experiments on both synthetic data and color images verify the effectiveness of our proposed methods.