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Precision matrix is of significant importance in a wide range of applicationsin multivariate analysis. This paper considers adaptive minimax estimation ofsparse precision matrices in the high dimensional setting. Optimal rates ofconvergence are established for a range of matrix norm losses. A fully datadriven estimator based on adaptive constrained $\ell_1$ minimization isproposed and its rate of convergence is obtained over a collection of parameterspaces. The estimator, called ACLIME, is easy to implement and performs wellnumerically.  A major step in establishing the minimax rate of convergence is thederivation of a rate-sharp lower bound. A "two-directional" lower boundtechnique is applied to obtain the minimax lower bound. The upper and lowerbounds together yield the optimal rates ofconvergence for sparse precisionmatrix estimation and show that the ACLIME estimator is adaptively minimax rateoptimal for a collection of parameter spaces and a range of matrix norm lossessimultaneously.

Estimating Sparse Precision Matrix: Optimal Rates of Convergence and Adaptive Estimation