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Generalized Low-Rank Matrix Completion Model with Overlapping Group Error Representation

Wenjing Lu, Zhuang Fang, Liang Wu, Liming Tang, Hanxin Liu, Chuanjiang He

TL;DR

This work tackles the limitation that real-world data are not strictly low-rank by proposing a generalized low-rank matrix completion model that decomposes an observed matrix ${\mathbf{Y}}$ into a low-rank component ${\mathbf{X}}$ and a structured sparse error ${\mathcal{E}}$ via an overlapping group error representation (OGER). The optimization problem $\min_{\mathbf{X},\mathcal{E}} R(\mathbf{X}) + \lambda \phi(\mathcal{E})$ with ${\mathbf{Y}}_{\Omega} = {\mathbf{X}}_{\Omega} + \mathcal{E}_{\Omega}$ is solved by an ADMM-MM framework, leveraging rank-surrogate updates and MM-based sparse updates, with theoretical convergence guarantees. The method is validated on image recovery tasks under random, text, and block masks, showing superior PSNR/SNR performance compared to NNM, WNNM, SCP, and N/F, while maintaining scalable per-iteration complexity. Overall, the approach provides a more faithful priors-based reconstruction by capturing both global low-rank structure and local block sparsity, improving robustness to structured errors in matrix completion. The framework also offers potential extensions to tensor data in future work.

Abstract

The low-rank matrix completion (LRMC) technology has achieved remarkable results in low-level visual tasks. There is an underlying assumption that the real-world matrix data is low-rank in LRMC. However, the real matrix data does not satisfy the strict low-rank property, which undoubtedly present serious challenges for the above-mentioned matrix recovery methods. Fortunately, there are feasible schemes that devise appropriate and effective priori representations for describing the intrinsic information of real data. In this paper, we firstly model the matrix data ${\bf{Y}}$ as the sum of a low-rank approximation component $\bf{X}$ and an approximation error component $\cal{E}$. This finer-grained data decomposition architecture enables each component of information to be portrayed more precisely. Further, we design an overlapping group error representation (OGER) function to characterize the above error structure and propose a generalized low-rank matrix completion model based on OGER. Specifically, the low-rank component describes the global structure information of matrix data, while the OGER component not only compensates for the approximation error between the low-rank component and the real data but also better captures the local block sparsity information of matrix data. Finally, we develop an alternating direction method of multipliers (ADMM) that integrates the majorization-minimization (MM) algorithm, which enables the efficient solution of the proposed model. And we analyze the convergence of the algorithm in detail both theoretically and experimentally. In addition, the results of numerical experiments demonstrate that the proposed model outperforms existing competing models in performance.

Generalized Low-Rank Matrix Completion Model with Overlapping Group Error Representation

TL;DR

This work tackles the limitation that real-world data are not strictly low-rank by proposing a generalized low-rank matrix completion model that decomposes an observed matrix into a low-rank component and a structured sparse error via an overlapping group error representation (OGER). The optimization problem with is solved by an ADMM-MM framework, leveraging rank-surrogate updates and MM-based sparse updates, with theoretical convergence guarantees. The method is validated on image recovery tasks under random, text, and block masks, showing superior PSNR/SNR performance compared to NNM, WNNM, SCP, and N/F, while maintaining scalable per-iteration complexity. Overall, the approach provides a more faithful priors-based reconstruction by capturing both global low-rank structure and local block sparsity, improving robustness to structured errors in matrix completion. The framework also offers potential extensions to tensor data in future work.

Abstract

The low-rank matrix completion (LRMC) technology has achieved remarkable results in low-level visual tasks. There is an underlying assumption that the real-world matrix data is low-rank in LRMC. However, the real matrix data does not satisfy the strict low-rank property, which undoubtedly present serious challenges for the above-mentioned matrix recovery methods. Fortunately, there are feasible schemes that devise appropriate and effective priori representations for describing the intrinsic information of real data. In this paper, we firstly model the matrix data as the sum of a low-rank approximation component and an approximation error component . This finer-grained data decomposition architecture enables each component of information to be portrayed more precisely. Further, we design an overlapping group error representation (OGER) function to characterize the above error structure and propose a generalized low-rank matrix completion model based on OGER. Specifically, the low-rank component describes the global structure information of matrix data, while the OGER component not only compensates for the approximation error between the low-rank component and the real data but also better captures the local block sparsity information of matrix data. Finally, we develop an alternating direction method of multipliers (ADMM) that integrates the majorization-minimization (MM) algorithm, which enables the efficient solution of the proposed model. And we analyze the convergence of the algorithm in detail both theoretically and experimentally. In addition, the results of numerical experiments demonstrate that the proposed model outperforms existing competing models in performance.
Paper Structure (26 sections, 13 theorems, 106 equations, 11 figures, 4 tables, 2 algorithms)

This paper contains 26 sections, 13 theorems, 106 equations, 11 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Organization
  4. Preliminaries and problem formulation
  5. Establishment of the model
  6. Overlapping Group Error Representation
  7. The Proposed Model and Algorithm
  8. Numerical experiments
  9. Discussion of Related Parameters
  10. Random Mask Experiments
  11. Text Mask Experiments
  12. Block Mask Experiments
  13. Convergence Analysis
  14. Conclusion
  15. PROOF OF THEOREM 1
  16. ...and 11 more sections

Key Result

Theorem 1

If $f\left( u \right)$ is a smooth function, consider the optimization problem, the solution of the following iterative optimization problem converges to the solution ${u^*}$ of the above minimization problem as $n \to \infty$, where the surrogate function $g\left( {u,{u_n}} \right)$ satisfies: (i) For any $u$, $g\left( {{u_n},{u_n}} \right) = f\left( {{u_n}} \right)$ and (ii) $g\left( {u,{u_n}}

Figures (11)

  • Figure 1: An example of the decomposition result of a real image.
  • Figure 2: Illustration of the proposed method.
  • Figure 3: Decomposition results of 'Babala' and 'Butterfly' test images for different low-rank truncations.
  • Figure 4: Test images for numerical experiments.
  • Figure 5: The PSNR values of the test images ('image1', 'image3' and, 'image4') are restored by different penalty parameter $\rho$ values.
  • ...and 6 more figures

Theorems & Definitions (37)

  • Definition 1: SVD Candes2009
  • Definition 2: Rank Function Candes2009
  • Definition 3: Nuclear Norm candes2012exactcandes2010power
  • Definition 4: Schatten $p$ Norm nie2012low
  • Definition 5: Weighted Nuclear Norm gu2014weightedgu2017weighted
  • Definition 6: Weighted Schatten $p$ Norm xie2016weighted
  • Definition 7: Schatten Capped $p$ Norm li2020matrix
  • Theorem 1: MM Algorithm hunter2004tutorialpokala2021iterativelywu2022hybrid
  • Remark 1
  • Theorem 2: $k$-rank Approximation Error vershynin2018high
  • ...and 27 more