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

Abstract

Let K be an isotropic convex body in R n . Given ε > 0, how many independent points X i uniformly distributed on K are needed for 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 ∈ 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 ε > 0, there exists C(ε) > 0, such that if N ∼ C(ε) n and (X i ) i≤N are i.i.d. copies of, with probability larger than 1 − exp(−c √ n).

Condition number of a square matrix with i.i.d. columns drawn from a convex body

“…When combined with estimates for the operator norm of the matrix Γ, obtained recently in [2], the above theorem also yields a corollary about the tail behavior of the so called condition number of the matrix Γ (denoted by (Γ)). The question about its behavior for random matrices was raised by Smale [22] in connection with stability of numerical algorithms for solving large systems of linear equations.…”

“…On the other hand, Proposition 2.10 with = 10 and = 10 /(2 6 ) gives for any ∈ (0, 1), Let us now turn to the proof of (2). Let be the constant in (1).…”

“…To this end we will use the following result from [2] (it is an immediate consequence of Corollary 3.8 there).…”

Geometry of log-concave ensembles of random matrices and approximate reconstruction

“…The articles [1], [2] and [3] considered random matrices with independent columns, and investigated high dimensional geometric properties of the convex hull of the columns and the RIP for various models of matrices, including the log-concave Ensemble build with independent isotropic log-concave columns. It was shown that various properties of random vectors can be efficiently studied via operator norms and the parameter Γ n,m recalled below.…”

“…It was shown that various properties of random vectors can be efficiently studied via operator norms and the parameter Γ n,m recalled below. In order to control this parameter an efficient technique of chaining was developed in [1] and [2].…”