Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm

TL;DR

This paper introduces IRNN, an algorithm for low-rank matrix recovery using nonconvex surrogates of the rank function, outperforming convex methods in accuracy and convergence.

Contribution

It proposes a novel IRNN algorithm that solves nonconvex, nonsmooth low-rank minimization problems with theoretical convergence guarantees.

Findings

IRNN outperforms state-of-the-art convex algorithms in experiments.

The algorithm guarantees monotonic decrease of the objective function.

IRNN effectively handles multi-block variable problems.

Abstract

The nuclear norm is widely used as a convex surrogate of the rank function in compressive sensing for low rank matrix recovery with its applications in image recovery and signal processing. However, solving the nuclear norm based relaxed convex problem usually leads to a suboptimal solution of the original rank minimization problem. In this paper, we propose to perform a family of nonconvex surrogates of $L_0$-norm on the singular values of a matrix to approximate the rank function. This leads to a nonconvex nonsmooth minimization problem. Then we propose to solve the problem by Iteratively Reweighted Nuclear Norm (IRNN) algorithm. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem, which has a closed form solution due to the special properties of the nonconvex surrogate functions. We also extend IRNN to solve the nonconvex problem with two or more blocks of variables. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthesized data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms.