Robust High-Order Tensor Compressive Sensing Based on M-Estimators

Tensor Compressive Sensing (TCS) has gained significant attention recently due to its strong ability to preserve the multidimensional structure of data. However, existing TCS methods face three critical challenges: 1) Biased approximation of tensor rank imposed by the convex surrogate Tensor Nuclear Norm (TNN) may interfere with the original low-rank structure of tensor data. 2) Vulnerability to non-Gaussian noise and outliers makes TCS methods highly susceptible to complex noise environments ubiquitous in real-world applications. 3) Most of them are confined to third-order tensors and cannot handle high-order tensor data effectively. Being aware of these, we propose Robust Tensor Compressive Sensing (RTCS) based on M-estimators with three key innovations: 1) We design a novel M-estimator-based low-rank regularizer for high-order tensors, which provides a superior approximation of tensor rank and better preserves the original data structure. 2) RTCS incorporates a robust Welsch estimator that adaptively mitigates the influence of complex noises and outliers in tensor recovery. 3) RTCS is developed to handle high-order tensors, thereby allowing for broader applicability beyond conventional third-order tensors. We further design an efficient algorithm based on the Alternating Direction Method of Multipliers (ADMM) to handle the complex optimization problem. Experiments show that RTCS consistently outperforms existing approaches across various noises.