Comptes Rendus
no. 3-4
Functional Analysis/Probability Theory
Sharp bounds on the rate of convergence of the empirical covariance matrix
[Ordre asymptotique des valeurs singulières extrêmes de la matrice de covariance empirique]

Présenté par : Gilles Pisier

Radosław Adamczak1 ; 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 Équipe d'analyse 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. 3-4, pp. 195-200.

Soient X1,…,XN∈Rn des vecteurs aléatoires indépendants centrés, de matrice de covariance l'identité et à densité log-concave. On démontre qu'avec une grande probabilité, on a

supx∈Sn−1|1N∑i=1N(|〈Xi,x〉|2−E|〈Xi,x〉|2)|⩽CnN,
où C>0 est une constante numérique. Ce résultat reste vrai dans le cadre beaucoup plus général où les formes linéaires (〈Xi,x〉)i⩽N,x∈Sn−1 et les normes euclidiennes (|Xi|/n)i⩽N vérifient des inégalités de type sous-exponentiel. Il en résulte que si A désigne la matrice dont les colonnes sont (Xi), alors avec grande probabilité, les valeurs singulières extrêmes λmin et λmax de AA⊤ vérifient 1−CnN⩽λminN⩽λmaxN⩽1+CnN, ce qui est une version quantitative du théorème de Bai–Yin (Z.D. Bai, Y.Q. Yin, 1993 [4]) bien connu pour les matrices aléatoires à coefficients i.i.d.

Let X1,…,XN∈Rn be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability one has

supx∈Sn−1|1N∑i=1N(|〈Xi,x〉|2−E|〈Xi,x〉|2)|⩽CnN,
where C is an absolute positive constant. This result is valid in a more general framework when the linear forms (〈Xi,x〉)i⩽N,x∈Sn−1 and the Euclidean norms (|Xi|/n)i⩽N exhibit uniformly a sub-exponential decay. As a consequence, if A denotes the random matrix with columns (Xi), then with overwhelming probability, the extremal singular values λmin and λmax of AA⊤ satisfy the inequalities 1−CnN⩽λminN⩽λmaxN⩽1+CnN which is a quantitative version of Bai–Yin theorem (Z.D. Bai, Y.Q. Yin, 1993 [4]) known for random matrices with i.i.d. entries.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.12.014
Radosław Adamczak 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 Équipe d'analyse et mathématiques appliquées, université Paris Est, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallee cedex 2, France 
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     author = {Rados{\l}aw Adamczak and Alexander E. Litvak and Alain Pajor and Nicole Tomczak-Jaegermann},
     title = {Sharp bounds on the rate of convergence of the empirical covariance matrix},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {195--200},
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Radosław Adamczak; Alexander E. Litvak; Alain Pajor; Nicole Tomczak-Jaegermann. Sharp bounds on the rate of convergence of the empirical covariance matrix. Comptes Rendus. Mathématique, Volume 349 (2011) no. 3-4, pp. 195-200. doi : 10.1016/j.crma.2010.12.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.12.014/

[1] R. Adamczak; A.E. Litvak; A. Pajor; N. Tomczak-Jaegermann Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles, J. Amer. Math. Soc., Volume 234 (2010), pp. 535-561

[2] G. Aubrun Random points in the unit ball of ℓpn, Positivity, Volume 10 (2006), pp. 755-759

[3] G. Aubrun Sampling convex bodies: a random matrix approach, Proc. Amer. Math. Soc., Volume 135 (2007), pp. 1293-1303

[4] Z.D. Bai; Y.Q. Yin Limit of the smallest eigenvalue of a large dimensional sample covariance matrix, Ann. Probab., Volume 21 (1993), pp. 1275-1294

[5] R. Kannan; L. Lovász; M. Simonovits Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom., Volume 13 (1995) no. 3–4, pp. 541-559

[6] R. Kannan; L. Lovász; M. Simonovits Random walks and O(n5) volume algorithm for convex bodies, Random Structures Algorithms, Volume 2 (1997) no. 1, pp. 1-50

[7] G. Paouris Concentration of mass on convex bodies, Geom. Funct. Anal., Volume 16 (2006) no. 5, pp. 1021-1049

Cité par Sources :

☆ The research was conducted while the authors participated in the Thematic Program on Asymptotic Geometric Analysis at the Fields Institute in Toronto in Fall 2010.

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