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Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles


Authors: Radosław Adamczak, Alexander E. Litvak, Alain Pajor and Nicole Tomczak-Jaegermann
Journal: J. Amer. Math. Soc. 23 (2010), 535-561
MSC (2000): Primary 52A20, 46B09, 52A21; Secondary 15A52, 60E15
Published electronically: October 9, 2009
MathSciNet review: 2601042
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Abstract: Let $ K$ be an isotropic convex body in $ \mathbb{R}^n$. Given $ \varepsilon>0$, how many independent points $ X_i$ uniformly distributed on $ K$ are needed for the empirical covariance matrix to approximate the identity up to $ \varepsilon$ with overwhelming probability? Our paper answers this question posed by Kannan, Lovász, and Simonovits. More precisely, let $ X\in\mathbb{R}^n$ be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector $ X$ is a random point in an isotropic convex body. We show that for any $ \varepsilon>0$, there exists $ C(\varepsilon)>0$, such that if $ N\sim C(\varepsilon) n$ and $ (X_i)_{i\le N}$ are i.i.d. copies of $ X$, then $ \Big\Vert\frac{1}{N}\sum_{i=1}^N X_i\otimes X_i - \operatorname{Id}\Big\Vert \le \varepsilon, $ with probability larger than $ 1-\exp(-c\sqrt n)$.


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Additional Information

Radosław Adamczak
Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland
Email: radamcz@mimuw.edu.pl

Alexander E. Litvak
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: alexandr@math.ualberta.ca

Alain Pajor
Affiliation: Université Paris-Est, Équipe d’Analyse et Mathématiques Appliquées, 5, boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée, Cedex 2, France
Email: Alain.Pajor@univ-mlv.fr

Nicole Tomczak-Jaegermann
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: nicole.tomczak@ualberta.ca

DOI: http://dx.doi.org/10.1090/S0894-0347-09-00650-X
Keywords: Convex bodies, log-concave measures, isotropic measures, random matrices, norm of random matrices, uniform laws of large numbers, approximation of covariance matrices
Received by editor(s): December 4, 2008
Published electronically: October 9, 2009
Additional Notes: Work on this paper began when the first author held a postdoctoral position at the Department of Mathematical and Statistical Sciences, University of Alberta in Edmonton, Alberta. The position was partially sponsored by the Pacific Institute for the Mathematical Sciences.
The fourth author holds the Canada Research Chair in Geometric Analysis
Article copyright: © Copyright 2009 American Mathematical Society