Generalized T-product Tensor Bernstein Bounds

TL;DR

This paper develops Bernstein bounds for the tail behavior of unitarily invariant norms of sums of random symmetric T-product tensors, extending tensor concentration inequalities to this class of tensors.

Contribution

It introduces Bernstein bounds for the Ky Fan norm of sums of symmetric T-product tensors, utilizing majorization, antisymmetric Kronecker products, and Laplace transform techniques.

Findings

Established Bernstein bounds for T-product tensor sums.

Extended concentration inequalities to symmetric T-product tensors.

Provided theoretical tools for analyzing random T-product tensor behavior.

Abstract

Since Kilmer et al. introduced the new multiplication method between two third-order tensors around 2008 and third-order tensors with such multiplication structure are also called as T-product tensors, T-product tensors have been applied to many fields in science and engineering, such as low-rank tensor approximation, signal processing, image feature extraction, machine learning, computer vision, and the multi-view clustering problem, etc. However, there are very few works dedicated to exploring the behavior of random T-product tensors. This work considers the problem about the tail behavior of the unitarily invariant norm for the summation of random symmetric T-product tensors. Majorization and antisymmetric Kronecker product tools are main techniques utilized to establish inequalities for unitarily norms of multivariate T-product tensors. The Laplace transform method is integrated with these inequalities for unitarily norms of multivariate T-product tensors to provide us with Bernstein Bounds estimation of Ky Fan $k$-norm for functions of the symmetric random T-product tensors summation.