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Order statistics and Mallat--Zeitouni problem

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ASCW01 - Challenges in optimal recovery and hyperbolic cross approximation

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 $$\mathbb{E} \sum \limits_{j=1}k j\mbox{}\min {X{i}}2 \leq C \mathbb{E} \sum\limits_{j=1}k j\mbox{}\min {Y{i}}^2$$ 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.


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