Concentration of measure and generalized product of random vectors with an application to Hanson-Wright-like inequalities
This work provides theoretical tools for concentration of measure in high-dimensional statistics, which is incremental but important for applications in machine learning.
The paper tackles the problem of deriving concentration inequalities for functionals of multiple random vectors, where the functional's variations depend on products of norms, and applies this to generalize Hanson-Wright inequalities and analyze random matrices like XDX^T and its resolvent, with implications for statistical machine learning.
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 $φ(Z_1,\ldots, Z_m)$ 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 $Q = (I_p - \frac{1}{n}XDX^T)^{-1}$, where $X$ and $D$ are random, which have fundamental interest in statistical machine learning applications.