Applied Math And Analysis Seminar
Wednesday, April 10, 2019, 12:00pm, 119 Physics
Alexander Litvak (University of Alberta)
Order statistics and Mallat--Zeitouni problem
Abstract:- 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. [video]
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