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SUMMARY:Random sections of ellipsoids and  the power of random information
  - Aicke Hinrichs (Johannes Kepler Universität)
DTSTART:20190218T110000Z
DTEND:20190218T113500Z
UID:TALK119944@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:We study the circumradius of the intersection of an $m$-dimens
 ional ellipsoid~$mathcal E$ with half axes $sigma_1geqdotsgeq sigma_m$ wit
 h random subspaces of codimension $n$. We find that\, under certain assump
 tions on $sigma$\, this random radius $mathcal{R}_n=mathcal{R}_n(sigma)$ i
 s of the same order as the minimal such radius $sigma_{n+1}$ with high pro
 bability. In other situations $mathcal{R}_n$ is close to the maximum~$sigm
 a_1$. The random variable $mathcal{R}_n$ naturally corresponds to the wors
 t-case error of the best algorithm based on random information for $L_2$-a
 pproximation of functions from a compactly embedded Hilbert space $H$ with
  unit ball $mathcal E$.<br> <br>In particular\, $sigma_k$ is the $k$th lar
 gest singular value of the embedding $Hhookrightarrow L_2$. In this formul
 ation\, one can also consider the case $m=infty$\, and we prove that rando
 m information behaves very differently depending on whether $sigma in ell_
 2$ or not. For $sigma otin ell_2$ 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 $n	oinfty$. <br><br>In the proofs
  we use a comparison result for Gaussian processes a la Gordon\, exponenti
 al estimates for sums of chi-squared random variables\, and estimates for 
 the extreme singular values of (structured) Gaussian random matrices.<br><
 br>This is joint work with David Krieg\, Erich Novak\, Joscha Prochno and 
 Mario Ullrich.
LOCATION:Seminar Room 1\, Newton Institute
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