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

Abstract

Let K K be an isotropic convex body in R n \mathbb {R}^n . Given ε > 0 \varepsilon >0 , how many independent points X i X_i uniformly distributed on K K are needed for the empirical covariance matrix to approximate the identity up to ε \varepsilon with overwhelming probability? Our paper answers this question posed by Kannan, Lovász, and Simonovits. More precisely, let X ∈ R n 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 X is a random point in an isotropic convex body. We show that for any ε > 0 \varepsilon >0 , there exists C ( ε ) > 0 C(\varepsilon )>0 , such that if N ∼ C ( ε ) n N\sim C(\varepsilon )\, n and ( X i ) i ≤ N (X_i)_{i\le N} are i.i.d. copies of X X , then ‖ 1 N ∑ i = 1 N X i ⊗ X i − Id ⁡ ‖ ≤ ε , \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 n ) 1-\exp (-c\sqrt n) .