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Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space

It is a special pleasure and honor for the first named author to dedicate this paper to the 60th birthdays of two of his outstanding friends — Israel Gohberg and Ilya Piatetski-Shapiro.

Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles

- R. Adamczak, A. Litvak, A. Pajor, N. Tomczak-Jaegermann
- Mathematics
- 13 March 2009

Let K be an isotropic convex body in Rn. Given e > 0, how many independent points Xi uniformly distributed on K are neededfor the empirical covariance matrix to approximate the identity up to e with… Expand

Uniform Uncertainty Principle for Bernoulli and Subgaussian Ensembles

- S. Mendelson, A. Pajor, N. Tomczak-Jaegermann
- Mathematics
- 27 August 2006

The paper considers random matrices with independent subgaussian columns and provides a new elementary proof of the Uniform Uncertainty Principle for such matrices. The Principle was introduced by… Expand

Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis

- S. Mendelson, A. Pajor, N. Tomczak-Jaegermann
- Mathematics
- 30 August 2007

Abstract.We present a randomized method to approximate any vector
$$\upsilon$$
from a set
$$T \subset {\mathbb{R}}^n$$
. The data one is given is the set T, vectors
$$(X_i)^{k}_{i=1}$$
of … Expand

Subspaces of small codimension of finite-dimensional Banach spaces

- A. Pajor, N. Tomczak-Jaegermann
- Mathematics
- 1 April 1986

Given a finite-dimensional Banach space E and a Euclidean norm on E, we study relations between the norm and the Euclidean norm on subspaces of E of small codimension. Then for an operator taking… Expand

Restricted Isometry Property of Matrices with Independent Columns and Neighborly Polytopes by Random Sampling

- R. Adamczak, A. Litvak, A. Pajor, N. Tomczak-Jaegermann
- Mathematics
- 30 April 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 sampling matrices… Expand

Interactions between compressed sensing, random matrices, and high dimensional geometry

- Djalil CHAFAÏ, O. Guédon, Guillaume Lecué, A. Pajor
- Computer Science
- 31 December 2012

TLDR

On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

We consider n × n real symmetric and hermitian random matrices Hn that are sums of a non-random matrix H n and of mn rank-one matrices determined by i.i.d. isotropic random vectors with log-concave… Expand

Sharp bounds on the rate of convergence of the empirical covariance matrix

- R. Adamczak, A. Litvak, A. Pajor, N. Tomczak-Jaegermann
- Mathematics
- 1 December 2010

Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3… Expand

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