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 
@article{CRMATH_2011__349_13-14_783_0,
     author = {Rados{\l}aw Adamczak and Rafa{\l} Lata{\l}a and Alexander E. Litvak and Alain Pajor and Nicole Tomczak-Jaegermann},
     title = {Geometry of log-concave ensembles of random matrices and approximate reconstruction},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {783--786},
     publisher = {Elsevier},
     volume = {349},
     number = {13-14},
     year = {2011},
     doi = {10.1016/j.crma.2011.06.025},
     language = {en},
}
TY  - JOUR
AU  - Radosław Adamczak
AU  - Rafał Latała
AU  - Alexander E. Litvak
AU  - Alain Pajor
AU  - Nicole Tomczak-Jaegermann
TI  - Geometry of log-concave ensembles of random matrices and approximate reconstruction
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 783
EP  - 786
VL  - 349
IS  - 13-14
PB  - Elsevier
DO  - 10.1016/j.crma.2011.06.025
LA  - en
ID  - CRMATH_2011__349_13-14_783_0
ER  - 
%0 Journal Article
%A Radosław Adamczak
%A Rafał Latała
%A Alexander E. Litvak
%A Alain Pajor
%A Nicole Tomczak-Jaegermann
%T Geometry of log-concave ensembles of random matrices and approximate reconstruction
%J Comptes Rendus. Mathématique
%D 2011
%P 783-786
%V 349
%N 13-14
%I Elsevier
%R 10.1016/j.crma.2011.06.025
%G en
%F CRMATH_2011__349_13-14_783_0
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/

[1] R. Adamczak, O. Guédon, A.E. Litvak, A. Pajor, N. Tomczak-Jaegermann, Condition number of a square matrix with i.i.d. columns drawn from a convex body, Proc. Amer. Math. Soc., , in press. | DOI

[2] R. Adamczak; O. Guédon; A.E. Litvak; A. Pajor; N. Tomczak-Jaegermann Smallest singular value of random matrices with independent columns, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008), pp. 853-856

[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

[5] R. Adamczak; A.E. Litvak; A. Pajor; N. Tomczak-Jaegermann Sharp bounds on the rate of convergence of empirical covariance matrix, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011), pp. 195-200

[6] R. Adamczak, R. Latała, A.E. Litvak, A. Pajor, N. Tomczak-Jaegermann, Tail estimates for norms of sums of log-concave random vectors, preprint, available at . | arXiv

[7] R. Adamczak, R. Latała, A.E. Litvak, A. Pajor, N. Tomczak-Jaegermann, Chevet type inequality and norms of submatrices, preprint, available at . | arXiv

[8] J. Bourgain Random points in isotropic convex sets, Berkeley, CA, 1996 (Math. Sci. Res. Inst. Publ.), Volume vol. 34, Cambridge Univ. Press, Cambridge (1999), pp. 53-58

[9] E.J. Candés; T. Tao Decoding by linear programming, IEEE Trans. Inform. Theory, Volume 51 (2005), pp. 4203-4215

[10] D.L. Donoho Neighborly Polytopes and Sparse Solutions of Underdetermined Linear Equations, Department of Statistics, Stanford University, 2005

[11] R. Kannan; L. Lovász; M. Simonovits Random walks and O⁎(n5) volume algorithm for convex bodies, Random Structures and Algorithms, Volume 2 (1997), pp. 1-50

[12] R. Latała Order statistics and concentration of lr norms for log-concave vectors, J. Funct. Anal., Volume 261 (2011), pp. 681-696

[13] S. Mendelson Empirical processes with a bounded ψ1 diameter, Geom. Funct. Anal., Volume 20 (2010), pp. 988-1027

[14] S. Mendelson; A. Pajor; N. Tomczak-Jaegermann Reconstruction and subgaussian operators in asymptotic geometric analysis, Geom. Funct. Anal., Volume 17 (2007), pp. 1248-1282

[15] G. Paouris Concentration of mass on convex bodies, Geom. Funct. Anal., Volume 16 (2006), pp. 1021-1049

  • Zakhar Kabluchko; Joscha Prochno Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 60 (2024) no. 2 | DOI:10.1214/22-aihp1340
  • Niklas Koep; Arash Behboodi; Rudolf Mathar An Introduction to Compressed Sensing, Compressed Sensing and Its Applications (2019), p. 1 | DOI:10.1007/978-3-319-73074-5_1
  • Sjoerd Dirksen; Guillaume Lecue; Holger Rauhut On the Gap Between Restricted Isometry Properties and Sparse Recovery Conditions, IEEE Transactions on Information Theory, Volume 64 (2018) no. 8, p. 5478 | DOI:10.1109/tit.2016.2570244
  • Sjoerd Dirksen Dimensionality Reduction with Subgaussian Matrices: A Unified Theory, Foundations of Computational Mathematics, Volume 16 (2016) no. 5, p. 1367 | DOI:10.1007/s10208-015-9280-x
  • David Alonso-Gutiérrez; Alexander E. Litvak; Nicole Tomczak-Jaegermann On the Isotropic Constant of Random Polytopes, The Journal of Geometric Analysis, Volume 26 (2016) no. 1, p. 645 | DOI:10.1007/s12220-015-9567-9
  • Radosław Adamczak; Rafał Latała; Alexander E. Litvak; Krzysztof Oleszkiewicz; Alain Pajor; Nicole Tomczak-Jaegermann A Short Proof of Paouris' Inequality, Canadian Mathematical Bulletin, Volume 57 (2014) no. 1, p. 3 | DOI:10.4153/cmb-2012-014-5
  • Radosław Adamczak; Olivier Guédon; Rafał Latała; Alexander Litvak; Krzysztof Oleszkiewicz; Alain Pajor; Nicole Tomczak-Jaegermann Moment estimates for convex measures, Electronic Journal of Probability, Volume 17 (2012) no. none | DOI:10.1214/ejp.v17-2150

Cité par 7 documents. Sources : Crossref

Commentaires - Politique

Il n'y a aucun commentaire pour cet article. Soyez le premier à écrire un commentaire !

Publier un nouveau commentaire:

Publier une nouvelle réponse:

Ces articles pourraient vous intéresser

Smallest singular value of random matrices with independent columns

Radosław Adamczak; Olivier Guédon; Alexander Litvak; ...

C. R. Math (2008)

Sharp bounds on the rate of convergence of the empirical covariance matrix

Radosław Adamczak; Alexander E. Litvak; Alain Pajor; ...

C. R. Math (2011)