Let L be a lattice in IR n 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 n 2 ,… (More)
We prove sharp bounds for the expectation of the supremum of the Gaussian process indexed by the intersection of B n p with ρB n q for 1 ≤ p, q ≤ ∞ and ρ > 0, and by the intersection of B n p∞ with ρB n 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 (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 isomor-phism from the n-dimensional Euclidean… (More)
We provide sharp lower and upper bounds for the Gelfand widths of ℓ p-balls in the N-dimensional ℓ N q-space for 0 < p ≤ 1 and p < q ≤ 2. Such estimates are highly relevant to the novel theory of compressive sensing, and our proofs rely on methods from this area.
We analyze the behavior of a random matrix with independent rows, each distributed according to the same probability measure on R n or on 2. We investigate the spectrum of such a matrix and the way the ellipsoid generated by it approximates the covariance structure of the underlying measure. As an application, we provide estimates on the deviation of the… (More)
Let A be a matrix whose columns X 1 ,. .. , X N are independent random vectors in R n. Assume that the tails of the 1-dimensional marginals decay as P(| X i , a | ≥ t) ≤ t −p uniformly in a ∈ S n−1 and i ≤ N. Then for p > 4 we prove that with high probability A/ √ n has the Restricted Isometry Property (RIP) provided that Eu-clidean norms |X i | are… (More)