Robust Low-Rank Matrix Completion via a New Sparsity-Inducing Regularizer

TL;DR

This paper introduces a new sparsity-inducing regularizer with a closed-form proximity operator, applied to robust low-rank matrix completion, leading to improved performance over existing methods.

Contribution

The paper proposes a novel regularizer with a convex Moreau envelope and closed-form proximity operator, enhancing robustness and efficiency in matrix completion tasks.

Findings

The regularizer is quasiconvex with a convex Moreau envelope.

The derived proximity operator has a closed-form solution.

Experimental results show superior restoration performance.

Abstract

This paper presents a novel loss function referred to as hybrid ordinary-Welsch (HOW) and a new sparsity-inducing regularizer associated with HOW. We theoretically show that the regularizer is quasiconvex and that the corresponding Moreau envelope is convex. Moreover, the closed-form solution to its Moreau envelope, namely, the proximity operator, is derived. Compared with nonconvex regularizers like the lp-norm with 0<p<1 that requires iterations to find the corresponding proximity operator, the developed regularizer has a closed-form proximity operator. We apply our regularizer to the robust matrix completion problem, and develop an efficient algorithm based on the alternating direction method of multipliers. The convergence of the suggested method is analyzed and we prove that any generated accumulation point is a stationary point. Finally, experimental results based on synthetic and real-world datasets demonstrate that our algorithm is superior to the state-of-the-art methods in terms of restoration performance.