Robust Principal Component Completion

Robust tensor completion recovers multidimensional data from partial observations corrupted by sparse gross errors. Existing convolutional low-rank models typically construct translated copies using circular continuation, which introduces artificial wrap-around neighborhoods for finite nonperiodic data. We propose reflective convolution nuclear norm minimization (RCNNM), which replaces circular shifts with endpoint-nonrepeating reflection. The resulting lifting has nonuniform entry multiplicities and satisfies a weighted Gram identity that supports both the recovery analysis and the optimization method. Under random sampling and sparse corruption, we establish high-probability exact recovery of the underlying tensor and sparse errors, together with stability under bounded dense perturbations. We further develop a two-block ADMM with a closed-form entrywise tensor update, while singular-value thresholding is implemented through the smaller right Gram matrix. Experiments on synthetic tensors, BSDS color images, and CAVE multispectral images show that RCNNM consistently improves over its circular-lifting counterpart, with the clearest gains near image boundaries. In particular, average boundary-PSNR improvements reach 3.26 dB while global reconstruction quality remains competitive.

Low-rank tensor factorization provides a flexible framework for completing multidimensional data from incomplete and corrupted observations. However, unweighted spectral regularizers impose a common shrinkage profile across singular components, which may excessively attenuate dominant low-rank components, and factorized variants either lack component-specific weighting or require costly singular value decompositions (SVDs). This paper proposes two weighted Schatten-$p$ tensor factorization models, termed \WSpTFI{} and \WSpTFII{}, under the tensor-tensor product (t-product) framework to address these limitations. \WSpTFI{} is motivated by a factorized weighted tensor Schatten-$p$ norm identity and permits flexible, possibly asymmetric factor exponents. \WSpTFII{} constructs a regularizer from transform-domain column-pair energies, yielding SVD-free main factor updates and a column-pruning mechanism for reducing redundant rank components. This paper further develops an iteratively reweighted alternating direction method of multipliers (ADMM)-type scheme for \WSpTFI{} and an iteratively reweighted least squares (IRLS)--block successive upper-bound minimization (BSUM) scheme for \WSpTFII{}. Theoretical analysis establishes the weighted factorization relation and provides a conditional limiting Karush--Kuhn--Tucker (KKT) characterization for \WSpTFI{}. For \WSpTFII{}, the actual damped quadratic block updates yield a quantitative sufficient-decrease mechanism for the fixed-$δ$ smoothed factor objective. This implies asymptotic regularity, and every accumulation point of the fixed-dimensional tail is stationary. Experiments on synthetic tensor completion, color-image restoration, hyperspectral inpainting, and printed-circuit-board defect detection demonstrate competitive reconstruction quality and robustness under various degradation conditions.

While principal component analysis (PCA) is a fundamental tool for dimensionality reduction, its dense representations make it ill-suited for high-dimensional data. Existing methods address this by promoting sparsity through explicit $\ell_1$-penalties, but these are not obvious to tune due to the unsupervised nature of the task. In contrast, we propose Adversarial PCA (AdvPCA), which leverages robust optimization to achieve sparsity by optimizing the reconstruction objective against bounded, worst-case latent space perturbations. We show that this formulation admits a closed-form reduction, leading to a practical iterative algorithm that alternates between adversarial linear regression-style updates for the sparse encoder and orthogonal updates for the decoder. By theoretically characterizing the solution, we derive a data-adaptive parameterization that allows the algorithm to perform effectively out of the box. We validate these claims through numerical experiments on synthetic and real-world genomics data.