Frontal Slice Approaches for Tensor Linear Systems

TL;DR

This paper introduces frontal slice methods for solving tensor linear systems using the t-product, including cyclic, block, and randomized variants, with theoretical analysis and practical applications in image and video processing.

Contribution

It presents a novel frontal slice approach for tensor systems, along with convergence analysis and multiple variations tailored for different operational needs.

Findings

Methods effectively solve tensor systems in synthetic and real data.

Approaches outperform traditional methods in image/video deblurring tasks.

Theoretical convergence guarantees support practical efficiency.

Abstract

Inspired by the row and column action methods for solving large-scale linear systems, in this work, we explore the use of frontal slices for solving tensor linear systems. In particular, this paper presents a novel approach for using frontal slices of a tensor $\mathcal{A}$ to solve tensor linear systems $\mathcal{A} * \mathcal{X} = \mathcal{B}$ where $*$ denotes the t-product. In addition, we consider variations of this method, including cyclic, block, and randomized approaches, each designed to optimize performance in different operational contexts. Our primary contribution lies in the development and convergence analysis of these methods. Experimental results on synthetically generated and real-world data, including applications such as image and video deblurring, demonstrate the efficacy of our proposed approaches and validate our theoretical findings.