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Alternating minimization for square root principal component pursuit

Shengxiang Deng, Xudong Li, Yangjing Zhang

TL;DR

This work tackles robust matrix recovery via SRPCP by recasting the problem as a distributionally robust optimization (DRO) with a square-root loss, which enables tuning-free recovery of a low-rank component $L_0$ and a sparse component $S_0$ from data $D = L_0 + S_0 + Z_0$. It then develops a two-block alternating minimization (AltMin) algorithm with closed-form updates for both $S$ and $L$, along with a convergence guarantee to coordinatewise minima under mild conditions. To further boost efficiency, the authors integrate a BM-decomposition-based acceleration that leverages variational nuclear norms to use partial SVDs and a rank-estimation scheme, yielding substantial speedups on large-scale problems. Numerical experiments on synthetic and real dark-raw-video data demonstrate that AltMin is highly efficient and robust, with the accelerated version offering significant improvements in low-rank regimes and large-scale settings. The approach provides a scalable, tuning-free framework for low-rank plus sparse decomposition with clear practical impact in tasks such as background/foreground separation in challenging imaging scenarios.

Abstract

Recently, the square root principal component pursuit (SRPCP) model has garnered significant research interest. It is shown in the literature that the SRPCP model guarantees robust matrix recovery with a universal, constant penalty parameter. While its statistical advantages are well-documented, the computational aspects from an optimization perspective remain largely unexplored. In this paper, we focus on developing efficient optimization algorithms for solving the SRPCP problem. Specifically, we propose a tuning-free alternating minimization (AltMin) algorithm, where each iteration involves subproblems enjoying closed-form optimal solutions. Additionally, we introduce techniques based on the variational formulation of the nuclear norm and Burer-Monteiro decomposition to further accelerate the AltMin method. Extensive numerical experiments confirm the efficiency and robustness of our algorithms.

Alternating minimization for square root principal component pursuit

TL;DR

This work tackles robust matrix recovery via SRPCP by recasting the problem as a distributionally robust optimization (DRO) with a square-root loss, which enables tuning-free recovery of a low-rank component and a sparse component from data . It then develops a two-block alternating minimization (AltMin) algorithm with closed-form updates for both and , along with a convergence guarantee to coordinatewise minima under mild conditions. To further boost efficiency, the authors integrate a BM-decomposition-based acceleration that leverages variational nuclear norms to use partial SVDs and a rank-estimation scheme, yielding substantial speedups on large-scale problems. Numerical experiments on synthetic and real dark-raw-video data demonstrate that AltMin is highly efficient and robust, with the accelerated version offering significant improvements in low-rank regimes and large-scale settings. The approach provides a scalable, tuning-free framework for low-rank plus sparse decomposition with clear practical impact in tasks such as background/foreground separation in challenging imaging scenarios.

Abstract

Recently, the square root principal component pursuit (SRPCP) model has garnered significant research interest. It is shown in the literature that the SRPCP model guarantees robust matrix recovery with a universal, constant penalty parameter. While its statistical advantages are well-documented, the computational aspects from an optimization perspective remain largely unexplored. In this paper, we focus on developing efficient optimization algorithms for solving the SRPCP problem. Specifically, we propose a tuning-free alternating minimization (AltMin) algorithm, where each iteration involves subproblems enjoying closed-form optimal solutions. Additionally, we introduce techniques based on the variational formulation of the nuclear norm and Burer-Monteiro decomposition to further accelerate the AltMin method. Extensive numerical experiments confirm the efficiency and robustness of our algorithms.
Paper Structure (18 sections, 15 theorems, 86 equations, 2 figures, 3 tables, 3 algorithms)

This paper contains 18 sections, 15 theorems, 86 equations, 2 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Robust squared loss regression problems
  3. An alternating minimization method for \ref{['prob:srpcp']}
  4. Explicit formula for updating $S$ in Algorithm \ref{['algo:AltMin']}
  5. Explicit formula for updating $L$ in Algorithm \ref{['algo:AltMin']}
  6. Convergence analysis
  7. Acceleration of the AltMin method via the BM-decomposition
  8. Numerical experiments
  9. Synthetic data
  10. Real data from dark raw videos
  11. Conclusion
  12. Proof of Theorem \ref{['thm:dro_formulation']}
  13. Proof of Proposition \ref{['prop:1']}
  14. Proof of Proposition \ref{['pro:unique']}
  15. Proof of Proposition \ref{['pro:diagA']}
  16. ...and 3 more sections

Key Result

Proposition 1

Let the squared loss function be $\ell (X,Y;B) = (Y - \langle X,B \rangle)^2,\,Y\in\mathbb{R},X,B\in\mathbb{R}^{2n_1\times n_2}$ and the cost function $c: \, \mathbb{R}^{2n_1\times n_2+1} \times \mathbb{R}^{2n_1\times n_2+1} \to [0, \infty]$ be defined by Let $\mathbb{P}_n:= \frac{1}{n} \sum_{i=1}^{n} {\bf 1}_{\{(X_i,Y_i)\}}$ denote the empirical distribution. Then it holds that

Figures (2)

  • Figure 1: Decomposition of the 30-th, 60-th, 90-th frames for the windmill video dataset using the approximated solution pairs returned by \ref{['algo:AltMin']}. The first row depicts the 30-th raw frames, the second row shows the 60-th raw frames, and the last row shows the 90-th raw frames. The first column presents the original raw frames $D$, the second column shows the denoised image $\widehat{L}+\widehat{S}$, the third and fourth columns display the background $\widehat{L}$ and the moving foreground $\widehat{S}$, respectively, and the last column shows the estimated noise $\widehat{Z}$, calculated as the residual $D-\widehat{L}-\widehat{S}$.
  • Figure 2: The rank identification of Algorithm \ref{['algo:AltMin']}.

Theorems & Definitions (30)

  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Theorem 2
  • proof
  • Remark 1
  • Proposition 3
  • Proposition 4
  • Theorem 3
  • proof
  • ...and 20 more