The Hanson-Wright Inequality for Random Tensors

Abstract

We provide moment bounds for expressions of the type $(X^{(1)}\otimes \cdots \otimes X^{(d)}{)}^{T}A(X^{(1)}\otimes \cdots \otimes X^{(d)})$ 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 $\Vert B(X^{(1)}\otimes \cdots \otimes X^{(d)}){\Vert }_2$.