Date of this Version
The Annals of Statistics
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.
The original and published work is available at: https://projecteuclid.org/euclid.aos/1458245724#abstract
Constrained ℓ1-minimization, covariance matrix, graphical model, minimax lower bound, optimal rate of convergence, precision matrix, sparsity, spectral norm.
Cai, T., Liu, W., & Zhou, H. H. (2016). Estimating Sparse Precision Matrix: Optimal Rates of Convergence and Adaptive Estimation. The Annals of Statistics, 44 (2), 455-488. http://dx.doi.org/10.1214/13-AOS1171
Date Posted: 27 November 2017
This document has been peer reviewed.