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.

翻译：鲁棒低秩张量补全问题旨在解决现实应用中常见的高维张量数据存在缺失项、异常值和稀疏噪声时的恢复挑战。现有方法因依赖均匀正则化方案（特别是张量核范数和$\ell_1$范数正则化方法），对所有奇异值和稀疏分量施加同等收缩，从而破坏了关键张量结构的保持，存在根本性局限。本文提出的张量加权相关全变分（TWCTV）正则化器通过$M$乘积框架解决了这些缺陷，该框架结合了用于低秩性与光滑性约束的梯度张量加权Schatten-$p$范数，以及用于噪声抑制的加权稀疏分量。所提出的加权方案自适应地降低阈值水平，以保留主要奇异值和稀疏分量，从而改善恢复信号中关键结构元素和细微细节的重建。通过系统性的算法方法，本文引入了一种增强型交替方向乘子法（ADMM），该方法兼具计算效率和理论支撑，并在$M$乘积框架下全面分析了其收敛性。在图像补全、去噪和背景减除任务上的全面数值评估验证了该方法相较于现有基准方法的优越性能。