  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 $\ell_1$ 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 ofconvergence 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.
