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Invertibility via distance for random matrices with continuous distributions

Geometric functional analysis and applications November 13, 2017 - November 17, 2017

November 17, 2017 (02:00 PM PST - 03:00 PM PST)
Speaker(s): Konstantin Tikhomirov (Princeton University)
Location: MSRI: Simons Auditorium
Tags/Keywords
  • random matrices

  • condition number

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

19-Tikhomirov

Abstract

Let A be an n by n random matrix with independent centered rows, so that each row has real-valued components, is isotropic and log-concave. Further, let M be any fixed n by n real matrix. We derive small ball probability estimates for the smallest singular value of the non-centered random matrix A+M. Our method is free from any use of covering arguments, and is principally different from a standard (by now) approach involving a decomposition of the unit sphere and coverings, as well as from an approach of Sankar-Spielman-Teng for non-centered Gaussian matrices. Our method gives an estimate for the condition number of A+M which essentially matches the known bounds for a non-centered Gaussian matrix.

Supplements
30107?type=thumb Tikhomirov Notes 902 KB application/pdf Download
Video/Audio Files

19-Tikhomirov

H.264 Video 19-Tikhomirov.mp4 340 MB video/mp4 rtsp://videos.msri.org/data/000/030/032/original/19-Tikhomirov.mp4 Download
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