  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.
