Random Euclidean embeddings in spaces of bounded volume ratio

Alexander Litvak; Alain Pajor; Mark Rudelson; Nicole Tomczak-Jaegermann; Roman Vershynin

doi:10.1016/j.crma.2004.04.019

Functional Analysis/Probability Theory

Random Euclidean embeddings in spaces of bounded volume ratio
[Plongements aléatoires de l'espace euclidien dans un espace à volume ratio borné]

Présenté par : Gilles Pisier

Alexander Litvak ¹ ; Alain Pajor ² ; Mark Rudelson ³ ; Nicole Tomczak-Jaegermann ¹ ; Roman Vershynin ⁴

¹ Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
² Équipe d'analyse et mathématiques appliquées, université de Marne-la-Vallée, 5, boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée cedex 2, France
³ Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
⁴ Department of Mathematics, University of California, Davis, CA 95616, USA

Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 33-38.

Résumé
Abstract

Soit $(ℝ^{N}, ∥ \cdot ∥)$ l'espace $ℝ^{N}$ muni d'une norme ‖·‖ dont la boule unité est à volume ratio borné par rapport à la boule unité euclidienne. On montre qu'une matrice aléatoire Γ, de taille N×n (N>n), dont les coefficients sont des variables aléatoires indépendantes, vérifiant certaines hypothèses de moments, réalise avec une grande probabilité, un bon isomorphisme de l'espace euclidien de dimension n, de norme |·|, sur son image dans $(ℝ^{N}, ∥ \cdot ∥)$ : il existe α,β>0 tels que pour tout $x \in ℝ^{n}$ , $α \sqrt{N} | x | ⩽ ∥ Γ x ∥ ⩽ β \sqrt{N} | x |$ ; ce qui démontre une conjecture de Schechtman sur les plongements aléatoires de ℓ₂ⁿ dans ℓ₁^N.

Let $(ℝ^{N}, ∥ \cdot ∥)$ be the space $ℝ^{N}$ equipped with a norm ‖·‖ whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N×n matrix with N>n, whose entries are independent random variables satisfying some moment assumptions. We show that with high probability Γ is a good isomorphism from the n-dimensional Euclidean space $(ℝ^{n}, | \cdot |)$ onto its image in $(ℝ^{N}, ∥ \cdot ∥)$ : there exist α,β>0 such that for all $x \in ℝ^{n}$ , $α \sqrt{N} | x | ⩽ ∥ Γ x ∥ ⩽ β \sqrt{N} | x |$ . This solves a conjecture of Schechtman on random embeddings of ℓ₂ⁿ into ℓ₁^N.

Reçu le : 2004-04-01
Accepté le : 2004-04-27
Publié le : 2004-05-28

DOI : 10.1016/j.crma.2004.04.019

Affiliations des auteurs :

Alexander Litvak ¹ ; Alain Pajor ² ; Mark Rudelson ³ ; Nicole Tomczak-Jaegermann ¹ ; Roman Vershynin ⁴

¹ Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
² Équipe d'analyse et mathématiques appliquées, université de Marne-la-Vallée, 5, boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée cedex 2, France
³ Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
⁴ Department of Mathematics, University of California, Davis, CA 95616, USA

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     author = {Alexander Litvak and Alain Pajor and Mark Rudelson and Nicole Tomczak-Jaegermann and Roman Vershynin},
     title = {Random {Euclidean} embeddings in spaces of bounded volume ratio},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {33--38},
     publisher = {Elsevier},
     volume = {339},
     number = {1},
     year = {2004},
     doi = {10.1016/j.crma.2004.04.019},
     language = {en},
}

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Alexander Litvak; Alain Pajor; Mark Rudelson; Nicole Tomczak-Jaegermann; Roman Vershynin. Random Euclidean embeddings in spaces of bounded volume ratio. Comptes Rendus. Mathématique, Volume 339 (2004) no. 1, pp. 33-38. doi : 10.1016/j.crma.2004.04.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.04.019/

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