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Concentration of measure and generalized product of random vectors with an application to Hanson-Wright-like inequalities (2102.08020v5)

Published 16 Feb 2021 in math.PR and stat.ML

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 $XDXT$ and its resolvent $Q = (I_p - \frac{1}{n}XDXT){-1}$, where $X$ and $D$ are random, which have fundamental interest in statistical machine learning applications.

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