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

02/16/2021

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by   Cosme Louart, et al.

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Starting from concentration of measure hypotheses on m random vectors Z_1,…, Z_m, this article provides an expression of the concentration of functionals ϕ(Z_1,…, Z_m) where the variations of ϕ on each variable depend on the product of the norms (or semi-norms) of the other variables (as if ϕ 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 - 1/nXDX^T)^-1, where X and D are random, which have fundamental interest in statistical machine learning applications.