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Matrix Completion Via Reweighted Logarithmic Norm Minimization

Zhijie Wang, Liangtian He, Qinghua Zhang, Jifei Miao, Liang-Jian Deng, Jun Liu

TL;DR

This work tackles LRMC by introducing a reweighted matrix logarithmic norm (RMLN) as a nonconvex surrogate that more accurately approximates the rank than the nuclear norm. An ADMM-based solver accommodates any $0<p\le 1$, with a singular-value shrinkage step derived via a DC decomposition (Theorem 1) and reweighted proximal updates. Empirical results on image inpainting across Set12 and BSD68 demonstrate superior PSNR/SSIM to state-of-the-art LRMC methods, including under random and block masks, and show the reweighting strategy and $p$-parameter choice (notably $p=0.8$) yield robust performance. The proposed framework offers a more faithful rank surrogate with practical recovery benefits and potential extensions to noisy completion and robust PCA.

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.

Matrix Completion Via Reweighted Logarithmic Norm Minimization

TL;DR

This work tackles LRMC by introducing a reweighted matrix logarithmic norm (RMLN) as a nonconvex surrogate that more accurately approximates the rank than the nuclear norm. An ADMM-based solver accommodates any , with a singular-value shrinkage step derived via a DC decomposition (Theorem 1) and reweighted proximal updates. Empirical results on image inpainting across Set12 and BSD68 demonstrate superior PSNR/SSIM to state-of-the-art LRMC methods, including under random and block masks, and show the reweighting strategy and -parameter choice (notably ) yield robust performance. The proposed framework offers a more faithful rank surrogate with practical recovery benefits and potential extensions to noisy completion and robust PCA.

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.
Paper Structure (15 sections, 18 equations, 6 figures, 2 tables, 1 algorithm)

This paper contains 15 sections, 18 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. LRMC Problem
  5. Matrix Logarithmic Norm
  6. The Proposed Method
  7. Reweighted Matrix Logarithmic Norm
  8. RMLN for Matrix Completion
  9. Numerical experiments
  10. Experimental Setup
  11. Random Mask Experiments
  12. Block Mask Experiments
  13. Role of The Reweighted Strategy
  14. Analysis of Power $p$
  15. Conclusion

Figures (6)

  • Figure 1: Comparison of the rank function, the convex envelope of rank (nuclear norm) Cai2010SVT, the MLN chen2021logarithmic, and the proposed RMLN for scalar $x$.
  • Figure 2: The Set12 dataset. The test images from left-to-right and top-to-bottom are labeled as Img1 to Img12, respectively.
  • Figure 3: The visual quality of different methods on image "Img2" from the Set12 dataset with a random mask (MR = 0.50).
  • Figure 4: The visual quality of different methods on image "Img9" from the Set12 dataset with a random mask (MR = 0.65).
  • Figure 5: The visual quality and SSIM values of different methods on image "Img11" from the Set12 dataset with block masks. (a) Observed image; (b) Geman kang2015robust (0.9712); (c) TNNR Hu2013TNNM (0.9412); (d) WNNM Gu2017WNNM (0.9663); (e) SC$p$li2020matrix (0.9736); (f) NMF shan2023multi (0.9658); (g) RMLN (0.9842).
  • ...and 1 more figures