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.

翻译：随机迭代算法，如随机Kaczmarz方法和随机高斯-赛德尔方法，因其在解决矩阵-向量和矩阵-矩阵回归问题中的高效性而广受欢迎。本研究借鉴了分析此类算法所获得的见解，以开发适用于张量的回归方法，张量是许多应用问题（如图像去模糊）的自然表示框架。具体而言，我们扩展了块随机高斯-赛德尔方法的两种变体，以解决t积张量回归问题。此外，我们还针对测量张量以因子化形式给出的特殊情况开发了相应方法。我们为所提算法的指数收敛速率提供了理论保证，并辅以说明性的数值模拟。