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Lower bounds for the smallest singular value of structured random matrices. (English) Zbl 1426.60006

Summary: 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 M. Rudelson and O. Zeitouni [Random Struct. Algorithms 48, No. 1, 183–212 (2016; Zbl 1362.15028)] 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 D. A. Spielman and N. Srivastava [Isr. J. Math. 190, 83–91 (2012; Zbl 1261.46007)].

MSC:

60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
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References:

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