# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Chevet type inequality and norms of submatrices

### Tom 210 / 2012

Studia Mathematica 210 (2012), 35-56 MSC: Primary 52A23, 46B06, 46B09, 60B20; Secondary 15B52, 60E15, 94B75. DOI: 10.4064/sm210-1-3

#### Streszczenie

We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of the expectation of the supremum of “symmetric exponential” processes, compared to the Gaussian ones in the Chevet inequality. This is used to give a sharp upper estimate for a quantity ${\Gamma }_{k,m}$ that controls uniformly the Euclidean operator norm of the submatrices with $k$ rows and $m$ columns of an isotropic log-concave unconditional random matrix. We apply these estimates to give a sharp bound for the restricted isometry constant of a random matrix with independent log-concave unconditional rows. We also show that our Chevet type inequality does not extend to general isotropic log-concave random matrices.

#### Autorzy

University of Warsaw
Banacha 2
02-097 Warszawa, Poland
e-mail
• Rafał LatałaInstitute of Mathematics
University of Warsaw
Banacha 2
02-097 Warszawa, Poland
and
Institute of Mathematics
00-956 Warszawa, Poland
e-mail
• Alexander E. LitvakDepartment of Mathematical and Statistical Sciences
University of Alberta
e-mail
• Alain PajorÉquipe d'Analyse et Mathématiques Appliquées
Université Paris-Est
5, boulevard Descartes, Champs sur Marne
77454 Marne-la-Vallée, Cedex 2, France
e-mail
• Nicole Tomczak-JaegermannDepartment of Mathematical and Statistical Sciences
University of Alberta