# Matrix Completion Via Reweighted Logarithmic Norm Minimization

arXiv: https://arxiv.org/abs/2512.21050 · HTML: https://mlsanity.com/paper/2512.21050/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 30 references that are in the map.
- First posted on arXiv 24 Dec 2025.

## Builds on (most cited first, 7 of 7)

- [Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview](https://mlsanity.com/paper/1809.09573/) (2018)
- [Weighted Schatten p-Norm Minimization for Image Denoising and Background Subtraction](https://mlsanity.com/paper/1512.01003/) (2015)
- [Bilinear Factor Matrix Norm Minimization for Robust PCA: Algorithms and Applications](https://mlsanity.com/paper/1810.05186/) (2018)
- [Generalized Nonconvex Nonsmooth Low-Rank Minimization](https://mlsanity.com/paper/1404.7306/) (2014)
- [Robust PCA via Nonconvex Rank Approximation](https://mlsanity.com/paper/1511.05261/) (2015)
- [Top-N Recommender System via Matrix Completion](https://mlsanity.com/paper/1601.04800/) (2016)
- [Multi-channel Nuclear Norm Minus Frobenius Norm Minimization for Color Image Denoising](https://mlsanity.com/paper/2209.08094/) (2022)

## Abstract (arXiv)

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.