Estimating sparse precision matrix: optimal rates of convergence and adaptive estimation.

Content Provider	CiteSeerX
Author	Cai, T. Tony Liu, Weidong Zhou, Harrison H.
Abstract	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 ℓ1 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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Subject Keyword	Optimal Rate Sparse Precision Matrix Adaptive Estimation Parameter Space Matrix Norm Loss Bound Technique Significant Importance Multivariate Analysis Major Step Aclime Estimator Adaptive Minimax Estimation Sparse Precision Matrix Estimation Minimax Rate Optimal Minimax Rate Wide Range Data Driven High Dimensional Setting Precision Matrix
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Resource Type	Article