Chevet type inequality and norms of submatrices

Tom 210 / 2012

Radosław Adamczak, Rafał Latała, Alexander E. Litvak, Alain Pajor, Nicole Tomczak-Jaegermann 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

  • Radosław AdamczakInstitute of Mathematics
    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
    Polish Academy of Sciences
    Śniadeckich 8
    00-956 Warszawa, Poland
    e-mail
  • Alexander E. LitvakDepartment of Mathematical and Statistical Sciences
    University of Alberta
    Edmonton, Alberta, Canada, T6G 2G1
    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
    Edmonton, Alberta, Canada, T6G 2G1
    e-mail

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