Translator Disclaimer
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.


Download Citation

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

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 5 • September 2013
Back to Top