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

TL;DR

This paper establishes that for isotropic log-concave distributions in high dimensions, a number of samples proportional to the dimension suffices for the empirical covariance matrix to approximate the true covariance with high probability.

Contribution

It provides a quantitative bound on the sample size needed for the empirical covariance to approximate the identity matrix in high-dimensional log-concave ensembles.

Findings

Sample size proportional to dimension ensures covariance approximation

High probability bounds for empirical covariance convergence

Applicable to isotropic convex bodies and log-concave distributions

Abstract

Let $K$ be an isotropic convex body in $\R^n$. Given $\eps>0$, how many independent points $X_i$ uniformly distributed on $K$ are needed for the empirical covariance matrix to approximate the identity up to $\eps$ with overwhelming probability? Our paper answers this question posed by Kannan, Lovasz and Simonovits. More precisely, let $X\in\R^n$ 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 $\eps>0$, there exists $C(\eps)>0$, such that if $N\sim C(\eps) n$ 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 - \Id\Big\| \le \epsilon, $ with probability larger than $1-\exp(-c\sqrt n)$.