Doubly Robust Principal Component Analysis through Robust Latent Subspace with Bounded Noise

Principal component analysis (PCA) is a fundamental technique for dimensionality reduction and the recovery of underlying data structure, with broad applicability across real-world scenarios. To enhance its robustness against noise and outliers, numerous robust PCA (RPCA) methods have been proposed. However, most existing approaches focus predominantly on the input space, overlooking the potential presence of noise in the latent space. Moreover, their optimization typically relies on alternating updates across multiple matrix variables, such as the alternating direction method of multipliers (ADMM), which is prone to premature convergence and suboptimal solutions. To overcome these limitations, we introduce a novel doubly robust PCA approach, termed DRPCA, which jointly addresses robustness in both the input and latent spaces. By deriving upper bounds on the DRPCA objective, we further establish a theoretical connection between model robustness and sparsity-inducing regularization. To efficiently solve the resulting models, we propose a hybrid alternating optimization algorithm grounded in parameter-free exact solvers, which decomposes the original DRPCA problem into tractable subproblems, integrating rapid matrix-wise updates with element-wise updates that admit closed-form, globally optimal solutions. A prescreening-based acceleration strategy is additionally incorporated to expedite the optimization process without compromising solution quality. Extensive experiments on benchmark datasets of varying scales demonstrate the superior performance of DRPCA over state-of-the-art RPCA methods. Notably, the proposed hybrid optimization algorithm consistently surpasses its matrix-wise-only or element-wise-only counterparts, yielding better solutions and higher-quality reconstruction.