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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.… (More)

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

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… (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… (More)

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

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… (More)

We study the 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… (More)

- Rados law ADAMCZAK, Alexander E. Litvak, Alain Pajor, Nicole Tomczak-Jaegermann
- 2010

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… (More)

— This book is based on a series of lectures given at Université ParisEst Marne-la-Vallée in fall 2009, by Djalil Chafäı, Olivier Guédon, Guillaume Lecué, Shahar Mendelson, and Alain Pajor.

- Y. Gordon, Alexander E. Litvak, Shahar Mendelson, Alain Pajor
- Journal of Approximation Theory
- 2007

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… (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… (More)

- Mendelson, Alain Pajor
- 2005

We present deviation inequalities of random operators of the form 1 N ∑N i=1 Xi ⊗ Xi from the average operator E(X ⊗ X), where Xi are independent random vectors distributed as X, which is a random… (More)