[Statlist] ETH/UZH Research Seminar by Shahar Mendelson, Mathematical Sciences Institute, Australian National University, 09.03.2020

[Maurer Letizia]

We are glad to announce the following talk in the ETH/UZH Research Seminar:

"On the geometry of random polytopes and the small-​ball method"   
by  Shahar Mendelson Mathematical, Sciences Institute, Australian National University

Time: Monday 09.03.2020 at 14.15 h
Place: ETH Zurich, HG G 19.1

Abstract: Let X be an isotropic random vector in R^n and let X_1,...,X_N be independent copies of X for N>cn. A well known question in Asymptotic Geometric Analysis that has been studied extensively over the last 30 years is whether (and under what conditions) the symmetric convex hull of X_1,...,X_N, absconv(X_1,...,X_N), contains a large canonical convex body. The first breakthrough was in the late 80's, when Gluskin showed that if X is the standard Gaussian vector, then with high probability, absconv(X_1,...,X_N) contains c\sqrt{log(N/n)}B_2^n. Results of a similar flavour (and what "similar flavour" means here will be explained in the talk) are known, for example, when X has iid subgaussian coordinates and when X is log-​concave. All these results rely on X exhibiting enough concentration and the arguments break down when X is no longer (very) light-​tailed. We present a general approach to the problem that is based on the small-​ball method and show that under minimal conditions on X, absconv(X_1,...,X_N) contains the dual of a natural floating body associated with X. This leads to a unified proof of all the previous results and allows one to address the problem when X is heavy-​tailed. At the heart of the proof is an idea that is used frequently in the analysis of many statistical recovery procedures: obtaining a high probability, lower bound on the infimum of a nonnegative random process - in this case, on \inf_{t \in T} \|\Gamma t\|_\infty, where T is an appropriate subset of R^n, and \Gamma is the random matrix whose rows are X_1,...,X_N. A joint work with O. Guedon, F. Krahmer, Christian Kummerle and Holger Rauhut.

Speaker invited by Afonso Bandeira

Seminar website: https://math.ethz.ch/sfs/news-and-events/research-seminar.html

Organisers: A. Bandeira, P. L. Bühlmann, L. Held, T. Hothorn, D. Kozbur, M. H. Maathuis, C. Uhler, S. van de Geer, M. Wolf