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

Researchers Biswarup Karmakar and Ratikanta Behera have introduced a breakthrough method for robust low-rank tensor completion that addresses fundamental limitations in existing approaches. Traditional methods using tensor nuclear norm and ℓ₁ norm regularization apply uniform shrinkage to all components, often destroying critical structural information in high-dimensional data. Their novel Tensor Weighted Correlated Total Variation (TWCTV) regularizer operates within an M-product framework, combining a weighted Schatten-p norm on gradient tensors with adaptive thresholding that preserves dominant singular values and sparse components differently.

The approach specifically targets three common real-world data problems simultaneously: missing entries, outliers, and sparse noise. Through systematic algorithmic development, the researchers created an enhanced alternating direction method of multipliers (ADMM) that offers both computational efficiency and theoretical guarantees, with comprehensive convergence analysis within the M-product framework. Numerical evaluations across image completion, denoising, and background subtraction tasks demonstrate superior performance, with the method outperforming established benchmarks by significant margins in preserving both structural elements and nuanced details.

What makes this approach particularly innovative is its adaptive weighting scheme, which reduces thresholding levels for important components while maintaining strong regularization for noise. This allows the method to recover corrupted tensor data with unprecedented accuracy while maintaining computational tractability. The 32-page paper provides extensive validation across multiple domains, showing consistent improvements over state-of-the-art methods in both reconstruction quality and preservation of critical data structures.