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Smoothed analysis of symmetric random matrices with continuous distributions


Authors: Brendan Farrell and Roman Vershynin
Journal: Proc. Amer. Math. Soc. 144 (2016), 2257-2261
MSC (2010): Primary 60B20
DOI: https://doi.org/10.1090/proc/12844
Published electronically: October 6, 2015
MathSciNet review: 3460183
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Abstract: We study invertibility of matrices of the form $ D+R$, where $ D$ is an arbitrary symmetric deterministic matrix and $ R$ is a symmetric random matrix whose independent entries have continuous distributions with bounded densities. We show that $ \Vert(D+R)^{-1}\Vert = O(n^2)$ with high probability. The bound is completely independent of $ D$. No moment assumptions are placed on $ R$; in particular the entries of $ R$ can be arbitrarily heavy-tailed.


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Additional Information

Brendan Farrell
Affiliation: Computing and Mathematical Sciences, California Institute of Technology, 1200 E. California Boulevard, Pasadena, California 91125
Email: farrell@cms.caltech.edu

Roman Vershynin
Affiliation: Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, Michigan 48109
Email: romanv@umich.edu

DOI: https://doi.org/10.1090/proc/12844
Received by editor(s): September 26, 2014
Received by editor(s) in revised form: May 29, 2015
Published electronically: October 6, 2015
Additional Notes: The first author was partially supported by Joel A. Tropp under ONR awards N00014-08-1-0883 and N00014-11-1002 and a Sloan Research Fellowship
The second author was partially supported by NSF grants 1001829, 1265782, and U.S. Air Force Grant FA9550-14-1-0009.
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2015 American Mathematical Society