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.
Abstract (translated)
低秩矩阵补全（LRMC）在广泛的应用中表现出显著的成功。为了应对秩最小化问题的NP难性质，通常使用核范数作为秩函数的有效且计算可行的凸替代品。然而，这种方法往往由于奇异值过度收缩而产生次优解。在这封信中，我们提出了一种新颖的加权对数范数作为一种更有效的非凸替代方法，它比许多现有方案提供了一个更为接近的近似值。通过采用交替方向乘子法（ADMM），我们高效地解决了由此产生的优化问题。在图像插值实验中，所提方法在视觉质量和定量指标方面均优于最先进的LRMC方法。
URL
https://arxiv.org/abs/2512.21050
