T product Tensors Part II: Tail Bounds for Sums of Random T product Tensors

TL;DR

This paper develops new probabilistic bounds for sums of random T product tensors, extending classical scalar inequalities to the tensor setting and analyzing their eigenvalues and norms.

Contribution

It generalizes Chernoff and Bernstein bounds to T product tensors and introduces tail bounds for sums, eigenvalues, and tensor martingales.

Findings

Derived tail bounds for eigenvalues of sums of T product tensors.

Extended classical scalar inequalities to the tensor setting.

Established bounds for tensor martingales and related inequalities.

Abstract

This paper is the Part II of a serious work about T product tensors focusing at establishing new probability bounds for sums of random, independent, T product tensors. These probability bounds characterize large deviation behavior of the extreme eigenvalue of the sums of random T product tensors. We apply Lapalace transform method and Lieb concavity theorem for T product tensors obtained from our Part I paper, and apply these tools to generalize the classical bounds associated with the names Chernoff, and Bernstein from the scalar to the T product tensor setting. Tail bounds for the norm of a sum of random rectangular T product tensors are also derived from corollaries of random Hermitian T product tensors cases. The proof mechanism is also applied to T product tensor valued martingales and T product tensor based Azuma, Hoeffding and McDiarmid inequalities are derived.