Robust low-rank tensor completion via factorized weighted tensor schatten-p norm minimization

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.