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