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Efficient Low Rank Matrix Recovery With Flexible Group Sparse Regularization

Quan Yu, Minru Bai, Xinzhen Zhang

TL;DR

This work introduces a flexible group sparse regularizer (FLGSR) to tackle low-rank matrix recovery (LRMR) by allowing arbitrary grouping of matrix columns in the factorization $C=XY^T$. The authors establish strong theoretical links: FLGSR can exactly represent matrix rank under mild conditions and admits equivalence with relaxed and penalty formulations, enabling a spectrum of reformulations with the same global minimizers. They propose an efficient IRAL-ELAM algorithm that linearizes each group, exploits information from earlier groups, and uses restarts to accelerate convergence, with rigorous convergence guarantees. Empirical results on grayscale and high-altitude aerial image inpainting show FLGSR consistently delivers higher PSNR/SSIM and faster runtimes than state-of-the-art group-sparse and nuclear-norm based methods. Overall, FLGSR provides a scalable, effective, and theoretically grounded approach to LRMR with tangible practical impact in image reconstruction tasks.

Abstract

In this paper, we present a novel approach to the low rank matrix recovery (LRMR) problem by casting it as a group sparsity problem. Specifically, we propose a flexible group sparse regularizer (FLGSR) that can group any number of matrix columns as a unit, whereas existing methods group each column as a unit. We prove the equivalence between the matrix rank and the FLGSR under some mild conditions, and show that the LRMR problem with either of them has the same global minimizers. We also establish the equivalence between the relaxed and the penalty formulations of the LRMR problem with FLGSR. We then propose an inexact restarted augmented Lagrangian method, which solves each subproblem by an extrapolated linearized alternating minimization method. We analyze the convergence of our method. Remarkably, our method linearizes each group of the variable separately and uses the information of the previous groups to solve the current group within the same iteration step. This strategy enables our algorithm to achieve fast convergence and high performance, which are further improved by the restart technique. Finally, we conduct numerical experiments on both grayscale images and high altitude aerial images to confirm the superiority of the proposed FLGSR and algorithm.

Efficient Low Rank Matrix Recovery With Flexible Group Sparse Regularization

TL;DR

This work introduces a flexible group sparse regularizer (FLGSR) to tackle low-rank matrix recovery (LRMR) by allowing arbitrary grouping of matrix columns in the factorization . The authors establish strong theoretical links: FLGSR can exactly represent matrix rank under mild conditions and admits equivalence with relaxed and penalty formulations, enabling a spectrum of reformulations with the same global minimizers. They propose an efficient IRAL-ELAM algorithm that linearizes each group, exploits information from earlier groups, and uses restarts to accelerate convergence, with rigorous convergence guarantees. Empirical results on grayscale and high-altitude aerial image inpainting show FLGSR consistently delivers higher PSNR/SSIM and faster runtimes than state-of-the-art group-sparse and nuclear-norm based methods. Overall, FLGSR provides a scalable, effective, and theoretically grounded approach to LRMR with tangible practical impact in image reconstruction tasks.

Abstract

In this paper, we present a novel approach to the low rank matrix recovery (LRMR) problem by casting it as a group sparsity problem. Specifically, we propose a flexible group sparse regularizer (FLGSR) that can group any number of matrix columns as a unit, whereas existing methods group each column as a unit. We prove the equivalence between the matrix rank and the FLGSR under some mild conditions, and show that the LRMR problem with either of them has the same global minimizers. We also establish the equivalence between the relaxed and the penalty formulations of the LRMR problem with FLGSR. We then propose an inexact restarted augmented Lagrangian method, which solves each subproblem by an extrapolated linearized alternating minimization method. We analyze the convergence of our method. Remarkably, our method linearizes each group of the variable separately and uses the information of the previous groups to solve the current group within the same iteration step. This strategy enables our algorithm to achieve fast convergence and high performance, which are further improved by the restart technique. Finally, we conduct numerical experiments on both grayscale images and high altitude aerial images to confirm the superiority of the proposed FLGSR and algorithm.
Paper Structure (23 sections, 14 theorems, 82 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 23 sections, 14 theorems, 82 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Flexible group sparse regularizer
  3. Efficient LRMR with FLGSR: Equivalence analysis
  4. Link between problems \ref{['Q:p']} and \ref{['Q:p0']}
  5. Link between problems \ref{['Q:p0']} and \ref{['Q:p_phi']}
  6. Link between problems \ref{['Q:p_phi']} and \ref{['Q:p_reg']}
  7. Link between problems \ref{['Q:p0']} and \ref{['Q:p_reg']}
  8. stationary point of \ref{['Q:p_reg']}.
  9. Characterizations of lifted stationary points of \ref{['Q:p_reg']}
  10. An Inexact Restarted Augmented Lagrangian Method with the Extrapolated Linearized Alternating Minimization
  11. The proposed IRAL
  12. The proposed ELAM
  13. Convergence analysis of Algorithm \ref{['alg1']}
  14. Convergence analysis of Algorithm \ref{['alg2']}
  15. Experimental Results
  16. ...and 8 more sections

Key Result

Theorem 2.3

Let $C=[C_1,\ldots, C_s] \in \mathbb{R}^{m\times n}$ be a matrix of rank $r$, where $C_i \in\mathbb{R}^{m\times n_i}$ with $\sum_{i=1}^sn_i=n$. If there are $n_{i_1}, \ldots, n_{i_p} \in \left\lbrace n_1,\ldots, n_s \right\rbrace$ such that $\sum_{j=1}^{p} n_{i_j}=r$, then $G_p^{\left|\cdot\right|^0

Figures (5)

  • Figure 2: The PSNR, SSIM and running time with different number of groups.
  • Figure 3: The PSNR, SSIM and running time with different iteration methods for $S$.
  • Figure 4: Examples of grayscale image inpainting. From top to bottom are respectively corresponding to "Peppers", "Sailboat", "Bridge" and "Mandrill". For better visualization, we show the zoom-in region and the corresponding partial residuals of the region. Under each image, we show enlargements of a demarcated patch and the corresponding error map (difference from the Original). Error maps with less color information indicate better restoration performance.
  • Figure 5: Comparison of the PSNR, SSIM and the running time on the randomly selected 20 images.
  • Figure 6: Examples of HAA image inpainting. From top to bottom are respectively corresponding to "San Diego" and "Woodland Hills". For better visualization, we show the zoom-in region and the corresponding partial residuals of the region. Under each image, we show enlargements of a demarcated patch and the corresponding error map (difference from the Original). Error maps with less color information indicate better restoration performance.

Theorems & Definitions (37)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Theorem 3.1
  • proof
  • ...and 27 more