期刊
ANNALS OF STATISTICS
卷 44, 期 2, 页码 455-488出版社
INST MATHEMATICAL STATISTICS
DOI: 10.1214/13-AOS1171
关键词
Constrained l(1)-minimization; covariance matrix; graphical model; minimax lower bound; optimal rate of convergence; precision matrix; sparsity; spectral norm
资金
- NSF FRG Grant [DMS-08-54973, DMS-08-54975]
- NSF [DMS-12-08982, DMS-06-45676]
- NSFC [11201298, 11322107, 11431006]
- Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning
- Shanghai Shuguang Program
- Foundation for the Author of National Excellent Doctoral Dissertation of PR China
- 973 Program [2015CB856004]
- Shanghai Pujiang Program
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1208982] Funding Source: National Science Foundation
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 l(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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