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We present a randomized method to approximate any vector v from some set T ⊂ R n. The data one is given is the set T , vectors (X i) k i=1 of R n and k scalar products (X i , v) k i=1 , where (X i) k i=1 are i.i.d. isotropic subgaussian random vectors in R n , and k n. We show that with high probability, any y ∈ T for which (X i , y) k i=1 is close to the… (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)

1 Introduction 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… (More)

- Rados law Adamczak, Alexander E. Litvak, Alain Pajor, Nicole Tomczak-Jaegermann
- 2009

Let K be an isotropic convex body in R n. Given ε > 0, how many independent points X i 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 ∈ R n be a centered random vector with a log-concave distribution… (More)

- Simon Foucart, Alain Pajor, Holger Rauhut, Tino Ullrich
- J. Complexity
- 2010

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.

- S. Mendelson, A. Pajor
- 2005

We present deviation inequalities of random operators of the form 1 N N i=1 X i ⊗ X i from the average operator E(X ⊗ X), where X i are independent random vectors distributed as X, which is a random vector in R n or in 2. We use these inequalities to estimate the singular values of random matrices with independent rows (without assuming that the entries are… (More)

In this article we prove that for any orthonormal system (ϕ j) n j=1 ⊂ L 2 that is bounded in L ∞ , and any 1 < k < n, there exists a subset I of cardinality greater than n − k such that on span{ϕ i } i∈I , the L 1 norm and the L 2 norm are equivalent up to a factor µ(log µ) 5/2 , where µ = n/k √ log k. The proof is based on a new estimate of the supremum… (More)

We investigate properties of subspaces of L 2 spanned by subsets of a finite orthonormal system bounded in the L ∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L 1 and the L 2 norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two… (More)

- Shahar Mendelson, Alain Pajor, Mark Rudelson
- Discrete & Computational Geometry
- 2005

We study the Restricted Isometry Property of a random matrix Γ with independent isotropic log-concave rows. To this end, we introduce a parameter Γ k,m that controls uniformly the operator norm of sub-matrices with k rows and m columns. This parameter is estimated by means of new tail estimates of order statistics and deviation inequalities for norms of… (More)