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 $Z_1,\dots ,Z_m$, this article provides an expression of the concentration of functionals $\varphi (Z_1,\dots ,Z_m)$ where the variations of $\varphi$ on each variable depend on the product of the norms (or semi-norms) of the other variables (as if $\varphi$ 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 $XD{X}^T$ and its resolvent $Q=(I_p-\frac{1}{n}XD{X}^T{)}^{-1}$, where $X$ and $D$ are random, which have fundamental interest in statistical machine learning applications.

48 pages

References in corpus (1)

• On the Limiting Empirical Measure of the sum of rank one matrices with log-concave distribution

Cited by in corpus (1)

• Resolvent convergence under heterogeneous second-moment profiles and quadratic-form control