The Hanson-Wright Inequality for Random Tensors

TL;DR

This paper extends the Hanson-Wright inequality to random tensors formed by Kronecker products of subgaussian vectors, providing tight moment bounds, a decoupling inequality, and improved concentration results.

Contribution

It introduces a Hanson-Wright type inequality for tensor products of subgaussian vectors, including a decoupling inequality and enhanced concentration bounds.

Findings

Tight moment bounds for tensor quadratic forms.

A new decoupling inequality for tensor expressions.

Improved concentration inequalities for tensor-based linear transformations.

Abstract

We provide moment bounds for expressions of the type $(X^{(1)} \otimes \dots \otimes X^{(d)})^T A (X^{(1)} \otimes \dots \otimes X^{(d)})$ where $\otimes$ denotes the Kronecker product and $X^{(1)}, \dots, X^{(d)}$ are random vectors with independent, mean 0, variance 1, subgaussian entries. The bounds are tight up to constants depending on $d$ for the case of Gaussian random vectors. Our proof also provides a decoupling inequality for expressions of this type. Using these bounds, we obtain new, improved concentration inequalities for expressions of the form $\|B (X^{(1)} \otimes \dots \otimes X^{(d)})\|_2$.