Generalized Nonconvex Nonsmooth Low-Rank Minimization

TL;DR

This paper introduces IRNN, an iterative algorithm leveraging nonconvex penalties for improved low-rank matrix recovery, with theoretical guarantees and superior experimental performance over convex methods.

Contribution

It proposes a novel IRNN algorithm that effectively solves nonconvex low-rank minimization problems by exploiting the properties of concave penalty functions.

Findings

IRNN monotonically decreases the objective function.

IRNN achieves better low-rank recovery than convex algorithms.

Theoretical proof of convergence to stationary points.

Abstract

As surrogate functions of $L_0$-norm, many nonconvex penalty functions have been proposed to enhance the sparse vector recovery. It is easy to extend these nonconvex penalty functions on singular values of a matrix to enhance low-rank matrix recovery. However, different from convex optimization, solving the nonconvex low-rank minimization problem is much more challenging than the nonconvex sparse minimization problem. We observe that all the existing nonconvex penalty functions are concave and monotonically increasing on $[0,\infty)$. Thus their gradients are decreasing functions. Based on this property, we propose an Iteratively Reweighted Nuclear Norm (IRNN) algorithm to solve the nonconvex nonsmooth low-rank minimization problem. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem. By setting the weight vector as the gradient of the concave penalty function, the WSVT problem has a closed form solution. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthetic data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms.