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Random sections of ellipsoids and the power of random information

Presented by: 
Aicke Hinrichs
Monday 18th February 2019 - 11:00 to 11:35
INI Seminar Room 1
We study the circumradius of the intersection of an $m$-dimensional ellipsoid~$\mathcal E$ with half axes $\sigma_1\geq\dots\geq \sigma_m$ with random subspaces of codimension $n$. We find that, under certain assumptions on $\sigma$, this random radius $\mathcal{R}_n=\mathcal{R}_n(\sigma)$ is of the same order as the minimal such radius $\sigma_{n+1}$ with high probability. In other situations $\mathcal{R}_n$ is close to the maximum~$\sigma_1$. The random variable $\mathcal{R}_n$ naturally corresponds to the worst-case error of the best algorithm based on random information for $L_2$-approximation of functions from a compactly embedded Hilbert space $H$ with unit ball $\mathcal E$.

In particular, $\sigma_k$ is the $k$th largest singular value of the embedding $H\hookrightarrow L_2$. In this formulation, one can also consider the case $m=\infty$, and we prove that random information behaves very differently depending on whether $\sigma \in \ell_2$ or not. For $\sigma \notin \ell_2$ random information is completely useless. For $\sigma \in \ell_2$ the expected radius of random information tends to zero at least at rate $o(1/\sqrt{n})$ as $n\to\infty$.

In the proofs we use a comparison result for Gaussian processes a la Gordon, exponential estimates for sums of chi-squared random variables, and estimates for the extreme singular values of (structured) Gaussian random matrices.

This is joint work with David Krieg, Erich Novak, Joscha Prochno and Mario Ullrich.
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons