T-product Tensor Expander Chernoff Bound

TL;DR

This paper extends the Chernoff bound to dependent T-product tensors, providing a more general probabilistic tail bound applicable to higher-dimensional data arrays with broader conditions.

Contribution

It introduces a novel T-product tensor expander Chernoff bound using majorization, expanding the scope from matrices to tensors and allowing polynomial functions and Ky Fan norms.

Findings

Generalizes matrix Chernoff bounds to T-product tensors

Allows for polynomial functions of tensor sums

Removes zero-sum restriction in tensor bounds

Abstract

In probability theory, the Chernoff bound gives exponentially decreasing bounds on tail distributions for sums of independent random variables and such bound is applied at different fields in science and engineering. In this work, we generalize the conventional Chernoff bound from the summation of independent random variables to the summation of dependent random T-product tensors. Our main tool used at this work is majorization technique. We first apply majorizaton method to establish norm inequalitites for T-product tensors and these norm inequalities are used to derive T-product tensor expander Chernoff bound. Compared with the matrix expander Chernoff bound obtained by Garg et al., the T-product tensor expander Chernoff bound proved at this work contributes following aspects: (1) the random objects dimensions are increased from matrices (two-dimensional data array) to T-product tensors (three-dimensional data array); (2) this bound generalizes the identity map of the random objects summation to any polynomial function of the random objects summation; (3) Ky Fan norm, instead only the maximum or the minimum eigenvalues, for the function of the random T-product tensors summation is considered; (4) we remove the restriction about the summation of all mapped random objects is zero, which is required in the matrix expander Chernoff bound derivation.