Perturbation analysis for t-product based tensor inverse, Moore-Penrose inverse and tensor system

TL;DR

This paper develops perturbation analysis for tensor inverses and tensor systems based on the t-product, extending classical formulas like Sherman-Morrison-Woodbury to tensor scenarios for sensitivity analysis.

Contribution

It generalizes the Sherman-Morrison-Woodbury formula to t-product tensors and applies it to analyze the sensitivity of tensor inverse and tensor systems.

Findings

Extended SMW formula for t-product tensors

Provided perturbation bounds for tensor inverses

Enabled sensitivity analysis of multilinear tensor systems

Abstract

This paper establishes some perturbation analysis for the tensor inverse, the tensor Moore-Penrose inverse and the tensor system based on the t-product. In the settings of structured perturbations, we generalize the Sherman-Morrison-Woodbury (SMW) formula to the t-product tensor scenarios. The SMW formula can be used to perform sensitivity analysis for a multilinear system of equations.