Covariance Estimation for Distributions with 2 + Ε Moments

We study the minimal sample size N = N(n) that suffices to estimate the covariance matrix of an n-dimensional distribution by the sample covariance matrix in the operator norm, with an arbitrary fixed accuracy. We establish the optimal bound N = O(n) for every distribution whose k-dimensional marginals have uniformly bounded 2+ε moments outside the sphere of radius O( √ k). In the specific case of log-concave distributions, this result provides an alternative approach to the Kannan-Lovasz-Simonovits problem, which was recently solved by Adamczak, Litvak, Pajor and Tomczak-Jaegermann [1]. Moreover, a lower estimate on the covariance matrix holds under a weaker assumption – uniformly bounded 2 + ε moments of one-dimensional marginals. Our argument consists of randomizing the deterministic spectral sparsification technique of Batson, Spielman and Srivastava [4]. The new randomized method allows one to control the spectral edges of the sample covariance matrix via the Stieltjes transform evaluated at carefully chosen random points.