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We provide sharp lower and upper bounds for the Gelfand widths of p-balls in the N-dimensional 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.

Let L be a lattice in IR n and K a convex body disjoint from L. The classical Flatness Theorem asserts that then w(K, L), the L-width of K, doesn't exceed 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.

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)

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.

- Alexander Litvak, Alain Pajor, Mark Rudelson, Nicole Tomczak-Jaegermann, Roman Vershynin

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

- Djalil Chafa¨i, Djalil Chafa¨ı, Olivier Guédon, Guillaume Lecué, Alain Pajor, Shahar Mendelson
- 2011

These notes are an expanded version of short courses given at the occasion of a school held in Université Paris-Est Marne-la-Vallée, The central motivation is compressed sensing, involving interactions between empirical processes, high dimensional geometry, and random matrices.

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)

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