Chevet type inequality and norms of submatrices

Radosław Adamczak ¹   ; Rafał Latała ²   ; Alexander E. Litvak ³   ; Alain Pajor ⁴   ; Nicole Tomczak-Jaegermann ³

¹ Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland
² Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland and Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland
³ Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta, Canada, T6G 2G1
⁴ Équipe d'Analyse et Mathématiques Appliquées Université Paris-Est 5, boulevard Descartes, Champs sur Marne 77454 Marne-la-Vallée, Cedex 2, France

Studia Mathematica, Tome 210 (2012) no. 1, pp. 35-56

Résumé

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.

Détail

DOI : 10.4064/sm210-1-3

Keywords: prove chevet type inequality which gives upper bound norm isotropic log concave unconditional random matrix terms expectation supremum symmetric exponential processes compared gaussian chevet inequality sharp upper estimate quantity gamma controls uniformly euclidean operator norm submatrices rows columns isotropic log concave unconditional random matrix apply these estimates sharp bound restricted isometry constant random matrix independent log concave unconditional rows chevet type inequality does extend general isotropic log concave random matrices