Tensor completion via Tucker decomposition with Correlated Total Variation regularization on factor matrices

Abstract

Multidimensional data, such as color images and videos, often exhibit inherent low-rank and local smoothness properties, with temporal and spatial correlations playing a crucial role in data recovery. While most existing methods focus on modeling these properties independently, they often overlook their coupled correlation in the factor space derived during tensor decomposition. In this study, we propose a novel Matrix Correlated Total Variation (MCTV) regularizer to explicitly model the coupled correlation between low-rankness and smoothness within Tucker decomposition. Unlike traditional methods, MCTV propagates the low-rankness of factor matrices to the Tucker rank, eliminating the need to predefine the core tensor size. It also preserves temporal and spatial smoothness correlations across all tensor modes through operations on smooth factor matrices. By integrating MCTV into a Tucker-based tensor completion model, we remove the dependence on hyperparameters like the Tucker rank and tradeoff parameters, creating a unified framework for capturing these coupled correlations. To optimize the proposed model, we design an efficient Alternating Direction Method of Multipliers (ADMM) algorithm. Experimental results on benchmark datasets demonstrate the superiority of our method in recovering multidimensional data by effectively modeling the synergy between low-rankness and smoothness.