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

arXiv: https://arxiv.org/abs/2604.13525 · HTML: https://mlsanity.com/paper/2604.13525/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 47 references that are in the map.
- First posted on arXiv 15 Apr 2026.

## Builds on (most cited first, 11 of 11)

- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination](https://mlsanity.com/paper/1401.6497/) (2014)
- [Hyperspectral Image Restoration via Total Variation Regularized Low-rank Tensor Decomposition](https://mlsanity.com/paper/1707.02477/) (2017)
- [Tensor Completion Algorithms in Big Data Analytics](https://mlsanity.com/paper/1711.10105/) (2017)
- [Guaranteed Tensor Recovery Fused Low-rankness and Smoothness](https://mlsanity.com/paper/2302.02155/) (2023)
- [Smooth PARAFAC Decomposition for Tensor Completion](https://mlsanity.com/paper/1505.06611/) (2015)
- [Efficient Nonnegative Tucker Decompositions: Algorithms and Uniqueness](https://mlsanity.com/paper/1404.4412/) (2014)
- [A Benchmark for Sparse Coding: When Group Sparsity Meets Rank Minimization](https://mlsanity.com/paper/1709.03979/) (2017)
- [Nonconvex Robust High-Order Tensor Completion Using Randomized Low-Rank Approximation](https://mlsanity.com/paper/2305.11495/) (2023)
- [Non-Convex Weighted Lp Minimization based Group Sparse Representation Framework for Image Denoising](https://mlsanity.com/paper/1704.01429/) (2017)
- [Multi-Tubal Rank of Third Order Tensor and Related Low Rank Tensor Completion Problem](https://mlsanity.com/paper/2012.05065/) (2020)

## Built on by (most cited first, 1 of 1)

- [Robust Tensor Completion via Reflective Convolution Nuclear Norm Minimization](https://mlsanity.com/paper/2610.04979/) (2026)

## Abstract (arXiv)

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 ℓ1 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.