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September 2013 Covariance estimation for distributions with ${2+\varepsilon}$ moments
Nikhil Srivastava, Roman Vershynin
Ann. Probab. 41(5): 3081-3111 (September 2013). DOI: 10.1214/12-AOP760


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 accuracy. We establish the optimal bound $N=O(n)$ for every distribution whose $k$-dimensional marginals have uniformly bounded $2+\varepsilon$ moments outside the sphere of radius $O(\sqrt{k})$. In the specific case of log-concave distributions, this result provides an alternative approach to the Kannan–Lovasz–Simonovits problem, which was recently solved by Adamczak et al. [J. Amer. Math. Soc. 23 (2010) 535–561]. Moreover, a lower estimate on the covariance matrix holds under a weaker assumption—uniformly bounded $2+\varepsilon$ moments of one-dimensional marginals. Our argument consists of randomizing the spectral sparsifier, a deterministic tool developed recently by Batson, Spielman and Srivastava [SIAM J. Comput. 41 (2012) 1704–1721]. The new randomized method allows one to control the spectral edges of the sample covariance matrix via the Stieltjes transform evaluated at carefully chosen random points.


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Nikhil Srivastava. Roman Vershynin. "Covariance estimation for distributions with ${2+\varepsilon}$ moments." Ann. Probab. 41 (5) 3081 - 3111, September 2013.


Published: September 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1293.62121
MathSciNet: MR3127875
Digital Object Identifier: 10.1214/12-AOP760

Primary: 62H12
Secondary: 60B20

Keywords: Covariance matrices , High-dimensional distributions , log-concave distributions , random matrices , Stieltjes transform

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 5 • September 2013
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