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October 2012 Optimal rates of convergence for sparse covariance matrix estimation
T. Tony Cai, Harrison H. Zhou
Ann. Statist. 40(5): 2389-2420 (October 2012). DOI: 10.1214/12-AOS998

Abstract

This paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and Bregman divergence losses. A major focus is on the derivation of a rate sharp minimax lower bound. The problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems. Standard techniques fail to yield good results, and new tools are thus needed.

We first develop a lower bound technique that is particularly well suited for treating “two-directional” problems such as estimating sparse covariance matrices. The result can be viewed as a generalization of Le Cam’s method in one direction and Assouad’s Lemma in another. This lower bound technique is of independent interest and can be used for other matrix estimation problems.

We then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique. A thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm. The results are then extended to the general matrix $\ell_{w}$ operator norms for $1\le w\le\infty$. In addition, we give a unified result on the minimax rate of convergence for sparse covariance matrix estimation under a class of Bregman divergence losses.

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T. Tony Cai. Harrison H. Zhou. "Optimal rates of convergence for sparse covariance matrix estimation." Ann. Statist. 40 (5) 2389 - 2420, October 2012. https://doi.org/10.1214/12-AOS998

Information

Published: October 2012
First available in Project Euclid: 4 February 2013

zbMATH: 1373.62247
MathSciNet: MR3097607
Digital Object Identifier: 10.1214/12-AOS998

Subjects:
Primary: 62H12
Secondary: 62F12 , 62G09

Keywords: Assouad’s lemma , Bregman divergence , covariance matrix estimation , Frobenius norm , Le Cam’s method , minimax lower bound , Optimal rate of convergence , spectral norm , thresholding

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 5 • October 2012
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