Concentration of measure and generalized product of random vectors with an application to Hanson-Wright-like inequalities

Abstract

Starting from concentration of measure hypotheses on $m$ random vectors $Z_1,\ldots, Z_m$, this article provides an expression of the concentration of functionals $\phi(Z_1,\ldots, Z_m)$ where the variations of $\phi$ on each variable depend on the product of the norms (or semi-norms) of the other variables (as if $\phi$ were a product). We illustrate the importance of this result through various generalizations of the Hanson-Wright concentration inequality as well as through a study of the random matrix $XDX^T$ and its resolvent $Q = (I_p - \frac{1}{n}XDX^T)^{-1}$, where $X$ and $D$ are random, which have fundamental interest in statistical machine learning applications.