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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)
In [CT1] Candes and Tao studied problems of approximate and exact reconstruction of sparse signals from incomplete random measurements and related them to the eigenvalue behavior of submatrices of matrices of random measurements. In particular they introduced the notion they called the uniform uncertainty principle (UUP, defined below) and studied it for(More)
We present a randomized method to approximate any vector v from some set T ⊂ R. The data one is given is the set T , vectors (Xi)i=1 of R and k scalar products (〈Xi, v〉)i=1, where (Xi)i=1 are i.i.d. isotropic subgaussian random vectors in R, and k n. We show that with high probability, any y ∈ T for which (〈Xi, y〉)i=1 is close to the data vector (〈Xi,(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)