Subsampling for tensor least squares: Optimization and statistical perspectives

In this paper, we investigate the random subsampling method for tensor least squares problem with respect to the popular t-product. From the optimization perspective, we present the error bounds in the sense of probability for the residual and solution obtained by the proposed method. From the statistical perspective, we derive the expressions of the conditional and unconditional expectations and variances for the solution, where the unconditional ones combine the model noises. Moreover, based on the unconditional variance, an optimal subsampling probability distribution is also found. Finally, the feasibility and effectiveness of the proposed method and the correctness of the theoretical results are verified by numerical experiments.