Computation of tensors generalized inverses under $M$-product and applications

TL;DR

This paper develops new tensor generalized inverses using M-product, proposes algorithms for their computation, and applies these methods to solve multilinear systems and image deblurring tasks.

Contribution

It introduces tensor Drazin and core-EP inverses via M-product, along with algorithms and applications for solving multilinear equations and image processing.

Findings

Effective algorithms for tensor Drazin and core-EP inverses.

Application of tensor inverses to solve multilinear systems.

Tensor M-product-based regularization improves image deblurring.

Abstract

This paper introduces notions of the Drazin and the core-EP inverses on tensors via M-product. We propose a few properties of the Drazin and core-EP inverses of tensors, as well as effective tensor-based algorithms for calculating these inverses. In addition, definitions of composite generalized inverses are presented in the framework of the M-product, including CMP, DMP, and MPD inverse of tensors. Tensor-based higher-order Gauss-Seidel and Gauss-Jacobi iterative methods are designed. Algorithms for these two iterative methods to solve multilinear equations are developed. Certain multilinear systems are solved using the Drazin inverse, core-EP inverses, and composite generalized inverses such as CMP, DMP, and MPD inverse. A tensor M-product-based regularization technique is applied to solve the color image deblurring.