Subsampling for tensor least squares: Optimization and statistical perspectives

TL;DR

This paper explores random subsampling techniques for tensor least squares problems using the t-product, providing theoretical error bounds, statistical analysis, and an optimal sampling distribution, validated through numerical experiments.

Contribution

It introduces a comprehensive analysis of subsampling methods for tensor least squares, including error bounds, statistical expectations, variances, and an optimal sampling strategy.

Findings

Error bounds for residuals and solutions established

Explicit expressions for expectations and variances derived

Optimal subsampling probability distribution identified

Abstract

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.