The Annals of Probability

Lower bounds for the smallest singular value of structured random matrices

Nicholas Cook

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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.

Article information

Ann. Probab., Volume 46, Number 6 (2018), 3442-3500.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 15B52: Random matrices

Random matrices condition number regularity lemma metric entropy


Cook, Nicholas. Lower bounds for the smallest singular value of structured random matrices. Ann. Probab. 46 (2018), no. 6, 3442--3500. doi:10.1214/17-AOP1251.

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Supplemental materials

  • Supplement to “Lower bounds for the smallest singular value of structured random matrices”. This supplement contains the proofs of Corollary 1.16 and Lemmas 2.5, 2.7 and 2.8.