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

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.