🤖 AI Summary
To address the suboptimal solutions in low-rank matrix completion (LRMC) caused by nuclear norm’s excessive shrinkage of singular values, this paper proposes a compact nonconvex rank surrogate—the reweighted logarithmic norm (RLogN)—and introduces it to LRMC modeling for the first time. RLogN more accurately approximates matrix rank and mitigates singular value bias. We further develop an alternating direction method of multipliers (ADMM)-based optimization framework with theoretically guaranteed convergence, balancing computational efficiency and numerical stability. On image inpainting tasks, our method significantly outperforms state-of-the-art LRMC algorithms: PSNR improves by 2.1 dB and SSIM by 0.032, yielding more natural visual results and sharper edge recovery. The core contributions are: (1) introducing RLogN as a novel nonconvex rank approximation; (2) designing an efficient, provably convergent solver; and (3) empirically validating its superiority on real-world low-rank prior tasks.
📝 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.