The Hanson-Wright Inequality for Random Tensors

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$\otimes$denotestheKroneckerproductand$X^{(1)}, \dots, X^{(d)}$ 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 $\mathrm{\parallel }B(X^{(1)}\otimes \cdots \otimes X^{(d)}){\mathrm{\parallel }}_2$.