On the geometry of polytopes generated by heavy-tailed random vectors

Stochastics Seminar
Friday, September 1, 2023 - 1:00pm for 1 hour (actually 50 minutes)
Skiles 006
Felix Krahmer – Technical University of Munich
Cheng Mao

In this talk, we present recent results on the geometry of centrally-symmetric random polytopes generated by N independent copies of a random vector X. We show that under minimal assumptions on X, for N>Cn, and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector - namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on X we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when X is q-stable or when X has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing - noise blind sparse recovery. This is joint work with Olivier Guédon (University of Paris-Est Marne La Vallée), Christian Kümmerle (UNC Charlotte), Shahar Mendelson (Sorbonne University Paris), and Holger Rauhut (LMU Munich).

Bio: Felix Krahmer received his PhD in Mathematics in 2009 from New York University under the supervision of Percy Deift and Sinan Güntürk. He was a Hausdorff postdoctoral fellow in the group of Holger Rauhut at the University of Bonn, Germany from 2009-2012. In 2012 he joined the University of Göttingen as an assistant professor for mathematical data analysis, where he has been awarded an Emmy Noether Junior Research Group. From 2015-2021 he was assistant professor for optimization and data analysis in the department of mathematics at the Technical University of Munich, before he was tenured and promoted to associate professor in 2021. His research interests span various areas at the interface of probability, analysis, machine learning, and signal processing including randomized sensing and reconstruction, fast random embeddings, quantization, and the computational sensing paradigm.