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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
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
Uniform Uncertainty Principle for Bernoulli and Subgaussian Ensembles
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
Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis
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
Subspaces of small codimension of finite-dimensional Banach spaces
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
Restricted Isometry Property of Matrices with Independent Columns and Neighborly Polytopes by Random Sampling
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
Interactions between compressed sensing, random matrices, and high dimensional geometry
Short courses given at the occasion of a school held in Universite Paris-Est Marne-la-Vallee, 16–20 November 2009, by Djalil Chafai, Olivier Guedon, Guillaume Lecue, Alain Pajor, and Shahar Mendelson are expanded.
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
Sharp bounds on the rate of convergence of the empirical covariance matrix
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