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

الأصل · EN

Let K be an isotropic convex body in ⁿ. Given >0, how many independent points Xᵢ uniformly distributed on K are needed for the empirical covariance matrix to approximate the identity up to with overwhelming probability? Our paper answers this question posed by Kannan, Lovasz and Simonovits. More precisely, let X∈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 >0, there exists C()>0, such that if N C() n and (Xᵢ)ᵢ≤ ₙ are i.i.d. copies of X, then 1/N∑ᵢ₌₁ⁿ Xᵢ⊗ Xᵢ - ≤ ε, with probability larger than 1-(-c√ n).