Estimating Sparse Precision Matrix: Optimal Rates of Convergence and Adaptive Estimation
Tony Cai, Weidong Liu, and Harrison Zhou
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.
Cai, T., Zhang, C.-H. & Zhou, H. (2010).
Optimal rates of convergence for covariance matrix estimation.
The Annals of Statistics 38, 2118-2144.
Cai, T., Wang, L. & Xu, G. (2010).
Shifting inequality and recovery of sparse signals.
IEEE Transactions on Signal Processing 58, 1300-1308.
Cai, T., Liu, W. & Luo, X. (2011).
A constrained l_{1} minimization approach to sparse precision matrix estimation.
J. American Statistical Association 106, 594-607.
Cai, T. & Zhou, H. (2012).
Minimax estimation of large covariance matrices under l_{1} norm (with discussion).
Statistica Sinica 22, 1319-1378.
Cai, T. & Zhou, H. (2012).
Optimal rates of convergence for sparse covariance matrix estimation.
The Annals of Statistics 40, 2389-2420.
Cai, T. & Yuan, M. (2012).
Adaptive covariance matrix estimation through block thresholding.
The Annals of Statistics 40, 2014-2042.