Matrix Completion Via Reweighted Logarithmic Norm Minimization

TL;DR

This paper introduces a reweighted logarithmic norm for low-rank matrix completion, offering a closer approximation to the rank function than nuclear norm, and demonstrates improved results in image inpainting tasks.

Contribution

It proposes a novel nonconvex surrogate for rank minimization and an efficient ADMM-based algorithm to solve the resulting optimization problem.

Findings

Outperforms state-of-the-art LRMC methods in image inpainting

Achieves higher visual quality and better quantitative metrics

Provides a more accurate approximation to the rank function

Abstract

Low-rank matrix completion (LRMC) has demonstrated remarkable success in a wide range of applications. To address the NP-hard nature of the rank minimization problem, the nuclear norm is commonly used as a convex and computationally tractable surrogate for the rank function. However, this approach often yields suboptimal solutions due to the excessive shrinkage of singular values. In this letter, we propose a novel reweighted logarithmic norm as a more effective nonconvex surrogate, which provides a closer approximation than many existing alternatives. We efficiently solve the resulting optimization problem by employing the alternating direction method of multipliers (ADMM). Experimental results on image inpainting demonstrate that the proposed method achieves superior performance compared to state-of-the-art LRMC approaches, both in terms of visual quality and quantitative metrics.