Alexander Litvak : Order statistics and Mallat--Zeitouni problem

Let $X$ be an $n$-dimensional random centered Gaussian vector with independent but not necessarily identically distributed coordinates and let $T$ be an orthogonal transformation of $\mathbb{R}^n$. We show that the random vector $Y=T(X)$ satisfies $$ \mathbb{E} \sum_{j=1}^k j\mbox{-}\min_{i \leq n} {X_{i}}^2 \leq C \mathbb{E} \sum_{j=1}^k j\mbox{-}\min_{i\leq n}{Y_{i}}^2 $$ for all $k \leq n$, where $ j\mbox{-}\min$ denotes the $j$-th smallest component of the corresponding vector and $C>0$ is a universal constant. This resolves (up to a multiplicative constant) an old question of S.Mallat and O.Zeitouni regarding optimality of the Karhunen--Lo`eve basis for the nonlinear reconstruction. We also show some relations for order statistics of random vectors (not only Gaussian), which are of independent interest. This is a joint work with Konstantin Tikhomirov.

• Category: Applied Math and Analysis
• Duration: 01:14:48
• Date:  April 10, 2019 at 11:55 AM
• Views: 130
• Tags:   seminar, Applied Math And Analysis Seminar