Randomized Kaczmarz methods for t-product tensor linear systems with factorized operators

TL;DR

This paper extends the randomized Kaczmarz method to solve tensor linear systems with factorized solutions using t-product operations, providing theoretical convergence guarantees and demonstrating effectiveness through numerical simulations.

Contribution

It introduces a novel randomized Kaczmarz algorithm for tensor systems with factorized solutions, expanding the applicability of such methods to tensor data.

Findings

The algorithms converge exponentially under certain conditions.

Numerical simulations confirm the effectiveness of the methods.

The approach generalizes previous Kaczmarz methods to tensor settings.

Abstract

Randomized iterative algorithms, such as the randomized Kaczmarz method, have gained considerable popularity due to their efficacy in solving matrix-vector and matrix-matrix regression problems. Our present work leverages the insights gained from studying such algorithms to develop regression methods for tensors, which are the natural setting for many application problems, e.g., image deblurring. In particular, we extend the randomized Kaczmarz method to solve a tensor system of the form $\mathbf{\mathcal{A}}\mathcal{X} = \mathcal{B}$, where $\mathcal{X}$ can be factorized as $\mathcal{X} = \mathcal{U}\mathcal{V}$, and all products are calculated using the t-product. We develop variants of the randomized factorized Kaczmarz method for matrices that approximately solve tensor systems in both the consistent and inconsistent regimes. We provide theoretical guarantees of the exponential convergence rate of our algorithms, accompanied by illustrative numerical simulations. Furthermore, we situate our method within a broader context by linking our novel approaches to earlier randomized Kaczmarz methods.