Generalized Hanson-Wright Inequality for Random Tensors

TL;DR

This paper extends the Hanson-Wright inequality to the Ky Fan k-norm for quadratic forms of random tensors, using tensor decoupling and Chernoff bounds to derive tail probability bounds.

Contribution

It introduces a generalized Hanson-Wright inequality for random tensors under the Einstein product, incorporating decoupling and tensor Chernoff bounds.

Findings

Derived tail bounds for quadratic forms of random tensors

Extended Hanson-Wright inequality to Ky Fan k-norms

Applied decoupling method for dependent tensor sums

Abstract

The Hanson-Wright inequality is an upper bound for tails of real quadratic forms in independent random variables. In this work, we extend the Hanson-Wright inequality for the Ky Fan k-norm for the polynomial function of the quadratic sum of random tensors under Einstein product. We decompose the quadratic tensors sum into the diagonal part and the coupling part. For the diagonal part, we can apply the generalized tensor Chernoff bound directly. But, for the coupling part, we have to apply decoupling method first, i.e., decoupling inequality to bound expressions with dependent random tensors with independent random tensors, before applying generalized tensor Chernoff bound again to get the the tail probability of the Ky Fan $k$-norm of the coupling part sum of independent random tensors. At the end, the generalized Hanson-Wright inequality for the Ky Fan k-norm for the polynomial function of the quadratic sum of random tensors can be obtained by the combination of the bound from the diagonal sum part and the bound from the coupling sum part.