Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Random sections of ellipsoids and the power of random information   We study the circumradius of the intersection of an $m$-dimensional ellipsoid
$\mathcal E$ with semi-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, i.e., $\mathbb E[\mathcal{R}_n] =
\sigma_1$. 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
important case $\sigma_k \asymp k^{-\alpha} \ln^{-\beta}(k+1)$, where $\alpha >
0$ and $\beta\in\mathbb R$, we obtain that $$ \mathbb E [\mathcal{R}_n(\sigma)]
\asymp \begin{cases} \sigma_1 & : \alpha<1/2 \,\text{ or }\,
\beta\leq\alpha=1/2 \\ \sigma_n \, \sqrt{\ln(n+1)} & : \beta>\alpha=1/2 \\
\sigma_{n+1} & : \alpha>1/2. \end{cases} $$ 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. The upper bound is constructive. It is
proven for the worst case error of a least squares estimator.

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The categories of this paper are: astro-ph
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-th
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The categories of this paper are: math-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
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The categories of this paper are: eess