Alexander E. Litvak

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Let K be an isotropic convex body in Rn. Given ε > 0, how many independent points Xi 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 ∈ Rn be a centered random vector with a log-concave distribution and(More)
We study behaviour of the smallest singular value of a rectangular random matrix, i.e., matrix whose entries are independent random variables satisfying some additional conditions. We prove a deviation inequality and show that such a matrix is a “good” isomorphism on its image. Then we obtain asymptotically sharp estimates for volumes and other geometric(More)
This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors ±X1, . . . ,±XN ∈ Rn, (N ≥ n). We introduce a class of random sampling matrices and show that they satisfy a restricted isometry property (RIP) with overwhelming probability. In particular, we prove that matrices with i.i.d.(More)
Let X1, . . . , XN ∈ R be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability one has sup x∈Sn−1 ∣∣∣∣ 1 N N ∑ i=1 ( |〈Xi, x〉| − E|〈Xi, x〉| ) ∣∣∣∣ ≤ C√ n N , where C is an absolute positive constant. This result is valid in a more general framework when the(More)
Let L be a lattice in IRn and K a convex body disjoint from L. The classical Flatness Theorem asserts that w(K, L), the L-width of K, doesn’t exceed then some bound, depending only on the dimension n; this fact was later found relevant to questions in integer programming. Kannan and Lovász (1988) showed that under the above assumptions w(K, L) ≤ C n2, where(More)
Let (R , ‖ · ‖) be the space R 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(More)
In this paper we study Euclidean projections of a p-convex body in IR. Precisely, we prove that for any integer k satisfying lnn ≤ k ≤ n/2, there exists a projection of rank k with the distance to the Euclidean ball not exceeding Cp(k/ln(1 + nk )) 1/p−1/2, where Cp is an absolute positive constant depending only on p. Moreover, we obtain precise estimates(More)
We prove sharp bounds for the expectation of the supremum of the Gaussian process indexed by the intersection of Bn p with ρB n q for 1 ≤ p, q ≤ ∞ and ρ > 0, and by the intersection of Bn p∞ with ρBn 2 for 0 < p ≤ 1 and ρ > 0. We present an application of this result to a statistical problem known as the approximate reconstruction problem.
Let (RN , ‖ · ‖) be the space RN 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(More)