Sharp nonasymptotic bounds on the norm of random matrices with independent entries
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 \[\mathbf{E}\Vert X\Vert \lesssim\max_i\sqrt{\sum_jb_{ij}^2}+\max _{ij}\vert b_{ij}\vert \sqrt{\log n}.\] 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 subGaussian or heavytailed 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2014
 arXiv:
 arXiv:1408.6185
 Bibcode:
 2014arXiv1408.6185B
 Keywords:

 Mathematics  Probability;
 Mathematics  Functional Analysis
 EPrint:
 Published at http://dx.doi.org/10.1214/15AOP1025 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)