On sketch-and-project methods for solving tensor equations

TL;DR

This paper introduces new sketch-and-project algorithms for tensor equations based on the t-product, including adaptive strategies, with proven linear convergence and demonstrated effectiveness through numerical experiments.

Contribution

It proposes the first regular and adaptive sketch-and-project methods for tensor equations with convergence guarantees and explores Fourier domain variants and special cases.

Findings

All methods have linear convergence in expectation.

Adaptive strategies improve convergence performance.

Numerical experiments confirm effectiveness and feasibility.

Abstract

We first propose the regular sketch-and-project method for solving tensor equations with respect to the popular t-product. Then, three adaptive sampling strategies and three corresponding adaptive sketch-and-project methods are derived. We prove that all the proposed methods have linear convergence in expectation. Furthermore, we investigate the Fourier domain versions and some special cases of the new methods, where the latter corresponds to some famous matrix equation methods. Finally, numerical experiments are presented to demonstrate and test the feasibility and effectiveness of the proposed methods for solving tensor equations