Subsampling for tensor least squares: Optimization and statistical perspectives

Abstract

In this paper, we propose the random subsampling method for tensor least squares problem with respect to the popular t-product. From the optimization perspective, we give the error bounds in the sense of probability for the solution and residual obtained by the proposed method. This perspective only considers the randomness of sampling, and the results indicate that leverage score sampling is superior to uniform sampling. From the statistical perspective, we derive the expressions of the conditional and unconditional expectations and variances for the solution. This perspective takes into account the randomness of both sampling and model noises simultaneously, and the results show that the unconditional variances for uniform sampling and leverage score sampling are both large and neither of them is dominant. In view of this, an optimal subsampling probability distribution is obtained by minimizing the trace of the unconditional variance. Finally, the feasibility and effectiveness of the proposed method and the correctness of the theoretical results are verified by numerical experiments.