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Robust PCA Based on Adaptive Weighted Least Squares and Low-Rank Matrix Factorization

Kexin Li, You-wei Wen, Xu Xiao, Mingchao Zhao

TL;DR

RPCA seeks to decompose data as $\boldsymbol{Y}=\boldsymbol{X}+\boldsymbol{S}$ into a low-rank $\boldsymbol{X}$ and a sparse $\boldsymbol{S}$, but nuclear-norm/$\ell_1$ relaxations can bias estimates and incur expensive SVDs. The authors propose an adaptive weighted least squares RPCA with low-rank matrix factorization (AWLS-LRMF) and a self-attention–inspired weight update that uses a weighted Frobenius penalty $\|\boldsymbol{W}\circ \boldsymbol{S}\|_F^2$, solved via an alternating minimization with explicit updates. They prove convergence of the weight sequence and that the objective decreases and converges, yielding a scalable and robust decomposition. Empirically, the method demonstrates superior accuracy and stability over non-convex regularization on synthetic data, background subtraction, and face-shadow removal tasks, offering a practical and efficient tool for robust data analysis in imaging and video applications.

Abstract

Robust Principal Component Analysis (RPCA) is a fundamental technique for decomposing data into low-rank and sparse components, which plays a critical role for applications such as image processing and anomaly detection. Traditional RPCA methods commonly use $\ell_1$ norm regularization to enforce sparsity, but this approach can introduce bias and result in suboptimal estimates, particularly in the presence of significant noise or outliers. Non-convex regularization methods have been proposed to mitigate these challenges, but they tend to be complex to optimize and sensitive to initial conditions, leading to potential instability in solutions. To overcome these challenges, in this paper, we propose a novel RPCA model that integrates adaptive weighted least squares (AWLS) and low-rank matrix factorization (LRMF). The model employs a {self-attention-inspired} mechanism in its weight update process, allowing the weight matrix to dynamically adjust and emphasize significant components during each iteration. By employing a weighted F-norm for the sparse component, our method effectively reduces bias while simplifying the computational process compared to traditional $\ell_1$-norm-based methods. We use an alternating minimization algorithm, where each subproblem has an explicit solution, thereby improving computational efficiency. Despite its simplicity, numerical experiments demonstrate that our method outperforms existing non-convex regularization approaches, offering superior performance and stability, as well as enhanced accuracy and robustness in practical applications.

Robust PCA Based on Adaptive Weighted Least Squares and Low-Rank Matrix Factorization

TL;DR

RPCA seeks to decompose data as into a low-rank and a sparse , but nuclear-norm/ relaxations can bias estimates and incur expensive SVDs. The authors propose an adaptive weighted least squares RPCA with low-rank matrix factorization (AWLS-LRMF) and a self-attention–inspired weight update that uses a weighted Frobenius penalty , solved via an alternating minimization with explicit updates. They prove convergence of the weight sequence and that the objective decreases and converges, yielding a scalable and robust decomposition. Empirically, the method demonstrates superior accuracy and stability over non-convex regularization on synthetic data, background subtraction, and face-shadow removal tasks, offering a practical and efficient tool for robust data analysis in imaging and video applications.

Abstract

Robust Principal Component Analysis (RPCA) is a fundamental technique for decomposing data into low-rank and sparse components, which plays a critical role for applications such as image processing and anomaly detection. Traditional RPCA methods commonly use norm regularization to enforce sparsity, but this approach can introduce bias and result in suboptimal estimates, particularly in the presence of significant noise or outliers. Non-convex regularization methods have been proposed to mitigate these challenges, but they tend to be complex to optimize and sensitive to initial conditions, leading to potential instability in solutions. To overcome these challenges, in this paper, we propose a novel RPCA model that integrates adaptive weighted least squares (AWLS) and low-rank matrix factorization (LRMF). The model employs a {self-attention-inspired} mechanism in its weight update process, allowing the weight matrix to dynamically adjust and emphasize significant components during each iteration. By employing a weighted F-norm for the sparse component, our method effectively reduces bias while simplifying the computational process compared to traditional -norm-based methods. We use an alternating minimization algorithm, where each subproblem has an explicit solution, thereby improving computational efficiency. Despite its simplicity, numerical experiments demonstrate that our method outperforms existing non-convex regularization approaches, offering superior performance and stability, as well as enhanced accuracy and robustness in practical applications.

Paper Structure

This paper contains 15 sections, 3 theorems, 42 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Related work
  5. Proposed Model and Algorithm
  6. Alternating Minimization Method
  7. Self-attention mechanism for weight W
  8. Convergence Analysis
  9. Numerical experiments
  10. The weight W.
  11. Comparison with the state-of-art methods
  12. Synthetic experiments
  13. Background subtraction from video
  14. Face shadow removal
  15. Conclusion

Key Result

Lemma 1

If $\boldsymbol{W}^{k}$ is generated by W1-W3, then there exists a diagonal matrix $\boldsymbol{W}^{*}$ such that

Figures (4)

  • Figure 1: Results of the proposed method tested on synthetic data. Rank is 10, sparsity is $26.4\%$, the RMSE of the low-rank matrix is 7.2e-09, and the RMSE of the sparse matrix is 1.1e-08. (a) Low-rank component $\boldsymbol{X}_{\text{true}}$; (b) Sparse component $\boldsymbol{S}_{\text{true}}$; (c) $\boldsymbol{X}_{\text{true}}+ \boldsymbol{S}_{\text{true}}$; (d) Separated $\hat{\boldsymbol{X}}$; (e) Separated $\hat{\boldsymbol{S}}$; (f) Estimated $\boldsymbol{W}_k$.
  • Figure 2: Evolution of the Weight Matrix During Iterations in the proposed model.
  • Figure 3: Background subtraction for five 200-frame surveillance video sequences. From top to bottom: Pedestrians, PETS2006, Blizzard, Highway, and WaterSurface, respectively.
  • Figure 4: Face shadow removal for Yale B dataset.

Theorems & Definitions (5)

  • Lemma 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof