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November 2018 Lower bounds for the smallest singular value of structured random matrices
Nicholas Cook
Ann. Probab. 46(6): 3442-3500 (November 2018). DOI: 10.1214/17-AOP1251

Abstract

We obtain lower tail estimates for the smallest singular value of random matrices with independent but nonidentically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form

\[M=A\circ X+B=(a_{ij}\xi_{ij}+b_{ij}),\] where $X=(\xi_{ij})$ has i.i.d. centered entries of unit variance and $A$ and $B$ are fixed matrices. In our main result, we obtain polynomial bounds on the smallest singular value of $M$ for the case that $A$ has bounded (possibly zero) entries, and $B=Z\sqrt{n}$ where $Z$ is a diagonal matrix with entries bounded away from zero. As a byproduct of our methods we can also handle general perturbations $B$ under additional hypotheses on $A$, which translate to connectivity hypotheses on an associated graph. In particular, we extend a result of Rudelson and Zeitouni for Gaussian matrices to allow for general entry distributions satisfying some moment hypotheses. Our proofs make use of tools which (to our knowledge) were previously unexploited in random matrix theory, in particular Szemerédi’s regularity lemma, and a version of the restricted invertibility theorem due to Spielman and Srivastava.

Citation

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Nicholas Cook. "Lower bounds for the smallest singular value of structured random matrices." Ann. Probab. 46 (6) 3442 - 3500, November 2018. https://doi.org/10.1214/17-AOP1251

Information

Received: 1 December 2016; Revised: 1 November 2017; Published: November 2018
First available in Project Euclid: 25 September 2018

zbMATH: 06975491
MathSciNet: MR3857860
Digital Object Identifier: 10.1214/17-AOP1251

Subjects:
Primary: 60B20
Secondary: 15B52

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 6 • November 2018
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