Estimates of norms of log-concave random matrices with dependent entries

Abstract

We prove estimates for $\mathbb{E}\Vert X:{\ell }_{p^{\prime }}^n\to {\ell }_q^m\Vert$ for $p,q\ge 2$ and any random matrix $X$ having the entries of the form $a_{ij}{Y}_{ij}$, where $Y=(Y_{ij}{)}_{1\le i\le m,1\le j\le n}$ has i.i.d. isotropic log-concave rows. This generalises the result of Guédon, Hinrichs, Litvak, and Prochno for Gaussian matrices with independent entries. Our estimate is optimal up to logarithmic factors. As a byproduct we provide the analogue bound for $m\times n$ random matrices, which entries form an unconditional vector in ${\mathbb{R}}^{mn}$. We also prove bounds for norms of matrices which entries are certain Gaussian mixtures.

16 pages

References in corpus (2)

• The dimension-free structure of nonhomogeneous random matrices
• Comparison of weak and strong moments for vectors with independent coordinates