Open Access
April 2016 Estimating sparse precision matrix: Optimal rates of convergence and adaptive estimation
T. Tony Cai, Weidong Liu, Harrison H. Zhou
Ann. Statist. 44(2): 455-488 (April 2016). DOI: 10.1214/13-AOS1171


Precision matrix is of significant importance in a wide range of applications in multivariate analysis. This paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting. Optimal rates of convergence are established for a range of matrix norm losses. A fully data driven estimator based on adaptive constrained $\ell_{1}$ minimization is proposed and its rate of convergence is obtained over a collection of parameter spaces. The estimator, called ACLIME, is easy to implement and performs well numerically.

A major step in establishing the minimax rate of convergence is the derivation of a rate-sharp lower bound. A “two-directional” lower bound technique is applied to obtain the minimax lower bound. The upper and lower bounds together yield the optimal rates of convergence for sparse precision matrix estimation and show that the ACLIME estimator is adaptively minimax rate optimal for a collection of parameter spaces and a range of matrix norm losses simultaneously.


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T. Tony Cai. Weidong Liu. Harrison H. Zhou. "Estimating sparse precision matrix: Optimal rates of convergence and adaptive estimation." Ann. Statist. 44 (2) 455 - 488, April 2016.


Received: 1 January 2013; Revised: 1 June 2013; Published: April 2016
First available in Project Euclid: 17 March 2016

zbMATH: 1341.62115
MathSciNet: MR3476606
Digital Object Identifier: 10.1214/13-AOS1171

Primary: 62H12
Secondary: 62F12 , 62G09

Keywords: Constrained $\ell_{1}$-minimization , Covariance matrix , Graphical model , minimax lower bound , Optimal rate of convergence , precision matrix , Sparsity , spectral norm

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 2 • April 2016
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