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July 2016 Sharp nonasymptotic bounds on the norm of random matrices with independent entries
Afonso S. Bandeira, Ramon van Handel
Ann. Probab. 44(4): 2479-2506 (July 2016). DOI: 10.1214/15-AOP1025

Abstract

We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results. If $X$ is the $n\times n$ symmetric matrix with $X_{ij}\sim N(0,b_{ij}^{2})$, we show that

\[\mathbf{E}\Vert X\Vert \lesssim\max_{i}\sqrt{\sum_{j}b_{ij}^{2}}+\max_{ij}\vert b_{ij}\vert \sqrt{\log n}.\] This bound is optimal in the sense that a matching lower bound holds under mild assumptions, and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum. Analogous results are obtained for rectangular matrices and for more general sub-Gaussian or heavy-tailed distributions of the entries, and we derive tail bounds in addition to bounds on the expected norm. The proofs are based on a combination of the moment method and geometric functional analysis techniques. As an application, we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse Wigner matrices. We also recover a result of Seginer on the norm of Rademacher matrices.

Citation

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Afonso S. Bandeira. Ramon van Handel. "Sharp nonasymptotic bounds on the norm of random matrices with independent entries." Ann. Probab. 44 (4) 2479 - 2506, July 2016. https://doi.org/10.1214/15-AOP1025

Information

Received: 1 August 2014; Revised: 1 March 2015; Published: July 2016
First available in Project Euclid: 2 August 2016

zbMATH: 1372.60004
MathSciNet: MR3531673
Digital Object Identifier: 10.1214/15-AOP1025

Subjects:
Primary: 60B20
Secondary: 46B09, 60F10

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 4 • July 2016
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