Comptes Rendus
Functional Analysis/Probability Theory
Lower estimates for the singular values of random matrices
Comptes Rendus. Mathématique, Volume 342 (2006) no. 4, pp. 247-252.

Let Γ be an n×n matrix, whose entries are independent identically distributed (i.i.d.) random variables satisfying the subgaussian tail estimate. We obtain polynomial type lower estimates of the singular numbers of Γ, which hold with probability close to 1. We also show that if A is an N×n matrix with N>n, whose entries are i.i.d. subgaussian random variables, then with high probability the space E=ARn satisfies the conditions of Kashin's theorem, i.e. the 2N and 1N norms are equivalent on E. Moreover the distance between these norms polynomially depends on δ=(Nn)/n.

Soit Γ une matrice n×n, ayant pour coefficients des variables aléatoires indépendantes et identiquement distribuées (i.i.d.) vérifiant une décroissance sous-gaussienne des queues. Dans ce travail, nous obtenons des minorations de type polynomial des valeurs singulières de Γ, valables avec une probabilité proche de 1. Nous montrons aussi que si A est une matrice N×n avec N>n, dont les coefficients sont des variables aléatoires sous-gaussiennes i.i.d., alors l'espace E=ARn vérifie avec une grande probablilité les conditions du théorème de Kashin, c'est à dire les normes 2N et 1N sont équivalentes sur E. De plus la distance entre ces normes dépend polynomialement de δ=(Nn)/n.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.11.013
Mark Rudelson 1

1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
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Mark Rudelson. Lower estimates for the singular values of random matrices. Comptes Rendus. Mathématique, Volume 342 (2006) no. 4, pp. 247-252. doi : 10.1016/j.crma.2005.11.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.11.013/

[1] S. Artstein, O. Friedland, V. Milman, Large sections of L1n spanned by random sign-vectors are isomorphic to Euclidean with polynomial dependence, Preprint

[2] Z.D. Bai; Y.Q. Yin Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix, Ann. Probab., Volume 21 (1993) no. 3, pp. 1275-1294

[3] K. Davidson; S.J. Szarek Local operator theory, random matrices and Banach spaces, Handbook of the Geometry of Banach Spaces, vol. I, North-Holland, Amsterdam, 2001, pp. 317-366

[4] A.Yu. Garnaev; E.D. Gluskin The widths of a Euclidean ball, Dokl. Akad. Nauk SSSR, Volume 277 (1984), pp. 1048-1052 (in Russian). English translation Soviet Math. Dokl., 30, 1984, pp. 200-204

[5] B. Kashin The widths of certain finite-dimensional sets and classes of smooth functions, Izv. Akad. Nauk SSSR Ser. Mat., Volume 41 (1977), pp. 334-351 (in Russian)

[6] A.E. Litvak; A. Pajor; M. Rudelson; N. Tomczak-Jaegermann Smallest singular value of random matrices and geometry of random polytopes, Adv. Math., Volume 195 (2005) no. 2, pp. 491-523

[7] A.E. Litvak; A. Pajor; M. Rudelson; N. Tomczak-Jaegermann; R. Vershynin Random Euclidean embeddings in spaces of bounded volume ratio, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004), pp. 33-38

[8] A.E. Litvak, A. Pajor, M. Rudelson, N. Tomczak-Jaegermann, R. Vershynin, Euclidean embeddings in spaces of finite volume ratio via random matrices, J. Reine Angew. Math., in press

[9] M. Rudelson, Invertibility of random matrices: norm of the inverse, Preprint

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