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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles
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by Radosław Adamczak, Alexander E. Litvak, Alain Pajor and Nicole Tomczak-Jaegermann PDF
J. Amer. Math. Soc. 23 (2010), 535-561 Request permission

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 $\Big \|\frac {1}{N}\sum _{i=1}^N X_i\otimes X_i - \operatorname {Id}\Big \| \le \varepsilon ,$ 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
  • MR Author ID: 367520
  • 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
  • MR Author ID: 173265
  • Email: nicole.tomczak@ualberta.ca
  • 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
  • © Copyright 2009 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 23 (2010), 535-561
  • MSC (2000): Primary 52A20, 46B09, 52A21; Secondary 15A52, 60E15
  • DOI: https://doi.org/10.1090/S0894-0347-09-00650-X
  • MathSciNet review: 2601042