Comptes Rendus
no. 13-14
Functional Analysis/Probability Theory
Geometry of log-concave ensembles of random matrices and approximate reconstruction
[Géométrie des ensembles log-concave des matrices aléatoires et une reconstruction approximative]

Présenté par : Gilles Pisier

Radosław Adamczak1 ; Rafał Latała1 ; Alexander E. Litvak2 ; Alain Pajor3 ; Nicole Tomczak-Jaegermann2
1 Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
3 Equipe dʼanalyse et mathématiques appliquées, université Paris Est, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallee cedex 2, France
Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 783-786.

On étudie la propriété dʼisométrie restreinte dʼune matrice aléatoire Γ dont les lignes sont des vecteurs aléatoires indépendants isotropes log-concave. Pour cela on introduit un paramètre Γk,m qui contrôle uniformément les normes dʼopérateurs des sous-matrices de k lignes et m colonnes. Ce paramètre est estimé à lʼaide de nouvelles inégalités de queue des statistiques dʼordre et dʼinégalités de déviation des normes de projections dʼun vecteur aléatoire log-concave.

We study the Restricted Isometry Property of a random matrix Γ with independent isotropic log-concave rows. To this end, we introduce a parameter Γk,m that controls uniformly the operator norm of sub-matrices with k rows and m columns. This parameter is estimated by means of new tail estimates of order statistics and deviation inequalities for norms of projections of an isotropic log-concave vector.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.06.025
Radosław Adamczak 1 ; Rafał Latała 1 ; Alexander E. Litvak 2 ; Alain Pajor 3 ; Nicole Tomczak-Jaegermann 2

1 Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
3 Equipe dʼanalyse et mathématiques appliquées, université Paris Est, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallee cedex 2, France 
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     title = {Geometry of log-concave ensembles of random matrices and approximate reconstruction},
     journal = {Comptes Rendus. Math\'ematique},
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Radosław Adamczak; Rafał Latała; Alexander E. Litvak; Alain Pajor; Nicole Tomczak-Jaegermann. Geometry of log-concave ensembles of random matrices and approximate reconstruction. Comptes Rendus. Mathématique, Volume 349 (2011) no. 13-14, pp. 783-786. doi : 10.1016/j.crma.2011.06.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.06.025/

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[3] R. Adamczak; A.E. Litvak; A. Pajor; N. Tomczak-Jaegermann Quantitative estimates of the convergence of the empirical covariance matrix in log-concave Ensembles, Journal of AMS, Volume 234 (2010), pp. 535-561

[4] R. Adamczak; A.E. Litvak; A. Pajor; N. Tomczak-Jaegermann Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling, Constructive Approximation, Volume 34 (2011), pp. 61-88

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