A note on perturbation analysis for T-product based tensor singular values

TL;DR

This paper extends classical matrix perturbation results to T-product based tensor singular values, providing interlacing theorems and inequalities for tensor singular value differences.

Contribution

It introduces perturbation analysis for tensor singular values, including interlacing theorems and inequalities, extending matrix results to tensor cases.

Findings

Established Cauchy's interlacing-type theorem for tensor singular values.

Extended Mirsky's inequalities to tensors.

Derived inequalities for tensor products and sums.

Abstract

In this note, we present perturbation analysis for the T-product based tensor singular values defined by Lu et al. First, the Cauchy's interlacing-type theorem for tensor singular values is given. Then, the inequalities about the difference between the singular values of two matrices proposed by Mirsky are extended to tensor cases. Finally, we introduce some useful inequalities for the singular values of tensor products and sums.