BEGIN:VCALENDAR VERSION:2.0 PRODID:-//talks.cam.ac.uk//v3//EN BEGIN:VTIMEZONE TZID:Europe/London BEGIN:DAYLIGHT TZOFFSETFROM:+0000 TZOFFSETTO:+0100 TZNAME:BST DTSTART:19700329T010000 RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU END:DAYLIGHT BEGIN:STANDARD TZOFFSETFROM:+0100 TZOFFSETTO:+0000 TZNAME:GMT DTSTART:19701025T020000 RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU END:STANDARD END:VTIMEZONE BEGIN:VEVENT CATEGORIES:Isaac Newton Institute Seminar Series SUMMARY:Random sections of ellipsoids and the power of ra ndom information - Aicke Hinrichs (Johannes Kepler Universität) DTSTART;TZID=Europe/London:20190218T110000 DTEND;TZID=Europe/London:20190218T113500 UID:TALK119944AThttp://talks.cam.ac.uk URL:http://talks.cam.ac.uk/talk/index/119944 DESCRIPTION:We study the circumradius of the intersection of a n \$m\$-dimensional ellipsoid~\$mathcal E\$ with half axes \$sigma_1geqdotsgeq sigma_m\$ with random subsp aces of codimension \$n\$. We find that\, under cert ain assumptions on \$sigma\$\, this random radius \$m athcal{R}_n=mathcal{R}_n(sigma)\$ is of the same or der 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 v ariable \$mathcal{R}_n\$ naturally corresponds to th e worst-case error of the best algorithm based on random information for \$L_2\$-approximation of func tions 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 \$Hhookrightarrow L_2\$. In this form ulation\, one can also consider the case \$m=infty\$ \, and we prove that random information behaves ve ry differently depending on whether \$sigma in ell_ 2\$ or not. For \$sigma otin ell_2\$ random informati on is completely useless. For \$sigma in ell_2\$ the expected radius of random information tends to ze ro at least at rate \$o(1/sqrt{n})\$ as \$n oinfty\$.

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

Thi s is joint work with David Krieg\, Erich Novak\, J oscha Prochno and Mario Ullrich. LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR