Robust Low-Rank Tensor Completion based on M-product with Weighted Correlated Total Variation and Sparse Regularization

TL;DR

This paper introduces a novel tensor completion method using weighted correlated total variation and sparse regularization within an M-product framework, improving the recovery of corrupted high-dimensional data.

Contribution

It proposes a new regularizer and weighting scheme that better preserves tensor structures and details, with an efficient ADMM algorithm and convergence analysis.

Findings

Outperforms benchmark methods in image completion and denoising.

Effectively preserves critical tensor structures and details.

Demonstrates superior robustness to noise and outliers.

Abstract

The robust low-rank tensor completion problem addresses the challenge of recovering corrupted high-dimensional tensor data with missing entries, outliers, and sparse noise commonly found in real-world applications. Existing methodologies have encountered fundamental limitations due to their reliance on uniform regularization schemes, particularly the tensor nuclear norm and $\ell_1$ norm regularization approaches, which indiscriminately apply equal shrinkage to all singular values and sparse components, thereby compromising the preservation of critical tensor structures. The proposed tensor weighted correlated total variation (TWCTV) regularizer addresses these shortcomings through an $M$-product framework that combines a weighted Schatten-$p$ norm on gradient tensors for low-rankness with smoothness enforcement and weighted sparse components for noise suppression. The proposed weighting scheme adaptively reduces the thresholding level to preserve both dominant singular values and sparse components, thus improving the reconstruction of critical structural elements and nuanced details in the recovered signal. Through a systematic algorithmic approach, we introduce an enhanced alternating direction method of multipliers (ADMM) that offers both computational efficiency and theoretical substantiation, with convergence properties comprehensively analyzed within the $M$-product framework.Comprehensive numerical evaluations across image completion, denoising, and background subtraction tasks validate the superior performance of this approach relative to established benchmark methods.