Alexander E. Litvak

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

Let K be an isotropic convex body in Rn. Given e > 0, how many independent points Xi uniformly distributed on K are neededfor the empirical covariance matrix to approximate the identity up to e with overwhelming probability? Our paper answers this question from [12]. More precisely, let X ∈ Rn 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 e > 0, there existsC(e) > 0, suﰁch that if N ∼ C(e)n and (Xi)i≤N are i.i.d. copies of ﱞﱞ1 N ﱞﱞ X, then ﱞN i=1 Xi ⊗ Xi − Idﱞ ≤ e, with probability larger than 1 − exp(−c√n).

The Flatness Theorem for Nonsymmetric Convex Bodies via the Local Theory of Banach Spaces

Let L be a lattice in Rn and K a convex body disjoint from L. The classical Flatness Theorem asserts that then w(K, L), the L-width of K, does not exceed some bound, depending only on the dimension n; this fact was later found relevant to questions in integer programming. Kannan and Lovasz (1988) showed that under the above assumptions w(K, L) ≤ Cn2, where C is a universal constant. Banaszczyk (1996) proved that w(K, L) ≤ Cn(1 + log n) if K has a centre of symmetry. In the present paper we show that w(K, L) ≤ Cn3/2 for an arbitrary K. It is conjectured that the exponent 3/2 may be replaced by 1, perhaps at the cost of a logarithmic factor; we prove that for some naturally arising classes of bodies.

On the minimum of several random variables

For a given sequence of real numbers a 1 ,..,a n , we denote the kth smallest one by k-min 1<i<nai . Let A be a class of random variables satisfying certain distribution conditions (the class contains N(0,1) Gaussian random variables). We show that there exist two absolute positive constants c and C such that for every sequence of real numbers 0 < x 1 ≤... ≤ x n and every k ≤ n, one has where ξ 1 ,..,1 n are independent random variables from the class A. Moreover, if k = 1, then the left-hand side estimate does not require independence of the ξ i s. We provide similar estimates for the moments of k-min 1≤i≤n |x i ξ i | as well.

The covering numbers and “low *-estimate" for quasi-convex bodies

This article gives estimates on the covering numbers and diameters of random proportional sections and projections of quasi-convex bodies in Rn. These results were known for the convex case and played an essential role in the development of the theory. Because duality relations cannot be applied in the quasi-convex setting, new ingredients were introduced that give new understanding for the convex case as well.

Euclidean embeddings in spaces of finite volume ratio via random matrices

Abstract Let (ℝ n , || ⋅ ||) be the space ℝ N equipped with a norm || ⋅ || whose unit ball has a bounded volume ratio with respect to the Euclidean unit ball. Let Γ be any random N  × n matrix with N  > n , whose entries are independent random variables satisfying some moment assumptions. We show that with high probability Γ is a good isomorphism from the n -dimensional Euclidean space (ℝ N , | ⋅ |) onto its image in (ℝ N , || ⋅ ||) , i.e. there exist α, β > 0 such that for all x ∈ ℝ N , . This solves a conjecture of Schechtman on random embeddings of ℓ 2 n into ℓ 1 N .

A Short Proof of Paouris' Inequality

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm

Tail estimates for norms of sums of log-concave random vectors

|X|

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

of an isotropic log-concave random vector

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

X\in\R^n

Geometry of spaces between polytopes and related zonotopes.

, stating that for every