Block Gauss-Seidel methods for t-product tensor regression

Abstract: Randomized iterative algorithms, such as the randomized Kaczmarz method and the randomized Gauss-Seidel method, have gained considerable popularity due to their efficacy in solving matrix-vector and matrix-matrix regression problems. Our present work leverages the insights gained from studying such algorithms to develop regression methods for tensors, which are the natural setting for many application problems, e.g., image deblurring. In particular, we extend two variants of the block-randomized Gauss-Seidel method to solve a t-product tensor regression problem. We additionally develop methods for the special case where the measurement tensor is given in factorized form. We provide theoretical guarantees of the exponential convergence rate of our algorithms, accompanied by illustrative numerical simulations.