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Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles

Abstract : Let K be an isotropic convex body in Rn. Given ε > 0, how many independent points Xi uniformly distributed on K are neededfor the empirical covariance matrix to approximate the identity up to ε with overwhelming probability? Our paper answers this question from [12]. More precisely, let X ∈ Rn 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 ε > 0, there existsC(ε) > 0, suﰁch that if N ∼ C(ε)n and (Xi)i≤N are i.i.d. copies of ﱞﱞ1 N ﱞﱞ X, then ﱞN i=1 Xi ⊗ Xi − Idﱞ ≤ ε, with probability larger than 1 − exp(−c√n).
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https://hal.archives-ouvertes.fr/hal-00793769
Contributor : Alain Pajor Connect in order to contact the contributor
Submitted on : Friday, February 22, 2013 - 11:11:40 PM
Last modification on : Tuesday, October 19, 2021 - 4:07:09 PM

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  • HAL Id : hal-00793769, version 1

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Radosław Adamczak, Alexander Litvak, Alain Pajor, Nicole Tomczak-Jaegermann. Quantitative estimates of the convergence of the empirical covariance matrix in Log-concave Ensembles. Journal of the American Mathematical Society, American Mathematical Society, 2010, 23, pp.535-561. ⟨hal-00793769⟩

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