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

@article{Adamczak2009QuantitativeEO,
title={Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles},
author={Radosław Adamczak and Alexander E. Litvak and Alain Pajor and Nicole Tomczak-Jaegermann},
journal={Journal of the American Mathematical Society},
year={2009},
volume={23},
pages={535-561}
}
• Published 13 March 2009
• Mathematics
• Journal of the American Mathematical Society
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 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 with the identity as covariance matrix. An example of such a vector X is a random point in an isotropic convex body. We show that for any e…
174 Citations

### On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence

• Mathematics
• 2015
Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of

### Invertibility via distance for noncentered random matrices with continuous distributions

The method is principally different from a standard approach involving a decomposition of the unit sphere and coverings, as well as an approach of Sankar-Spielman-Teng for non-centered Gaussian matrices.

### Covariance estimation for distributions with 2+ε moments

• Mathematics
• 2013
We study the minimal sample size N=N(n) that suffices to estimate the covariance matrix of an n-dimensional distribution by the sample covariance matrix in the operator norm, with an arbitrary fixed

### How Close is the Sample Covariance Matrix to the Actual Covariance Matrix?

Given a probability distribution in ℝn with general (nonwhite) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent

### On a multi-integral norm defined by weighted sums of log-concave random vectors

Let C and K be centrally symmetric convex bodies in R n . We show that if C is isotropic then for every s > 1 and t = ( t 1 , . . . , t s ) ∈ R s , where L C is the isotropic constant of C and M ( K

### COVARIANCE ESTIMATION FOR DISTRIBUTIONS WITH 2+epsilon MOMENTS

We study the minimal sample size N = N ( n ) that suﬃces to estimate the covariance matrix of an n -dimensional distribution by the sample covariance matrix in the operator norm, with an arbitrary

### A pr 2 01 8 Approximating the covariance ellipsoid

We explore ways in which the covariance ellipsoid B = {v ∈ R : E 〈X, v〉2 ≤ 1} of a centred random vector X in R can be approximated by a simple set. The data one is given for constructing the

## References

SHOWING 1-10 OF 36 REFERENCES

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

• Mathematics
• 2009
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

### Reconstruction and subgaussian operators

• Mathematics
• 2005
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 and k scalar products (h Xi, vi ) k=1 , where (Xi) k=1 are i.i.d. isotropic

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

• Mathematics
• 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

### Random Vectors in the Isotropic Position

Abstract Letybe a random vector in R n, satisfying E y⊗y=id. LetMbe a natural number and lety1, …, yMbe independent copies ofy. We study the question of approximation of the identity operator by

### Random Points in Isotropic Convex Sets

Let K be a symmetric convex body of volume 1 whose inertia tensor is isotropic, i.e., for some constant L we have R K〈x, y〉2 dx = L2|y|2 for all y. It is shown that if m is about n(log n)3 then with

### On the Limiting Empirical Measure of the sum of rank one matrices with log-concave distribution

• Mathematics
• 2007
We consider $n\times n$ real symmetric and hermitian random matrices $H_{n,m}$ equals the sum of a non-random matrix $H_{n}^{(0)}$ matrix and the sum of $m$ rank-one matrices determined by $m$ i.i.d.

### On weakly bounded empirical processes

Let F be a class of functions on a probability space (Ω, μ) and let X1,...,Xk be independent random variables distributed according to μ. We establish an upper bound that holds with high probability

### Sampling convex bodies: a random matrix approach

We prove the following result: for any e > 0, only C(e)n sample points are enough to obtain (1 + e)-approximation of the inertia ellipsoid of an unconditional convex body in R". Moreover, for any p >

### On singular values of matrices with independent rows

• Mathematics
• 2006
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