  Estimation of the precision matrix (or inverse covariance matrix) is of great
importance in statistical data analysis and machine learning. However, as the
number of parameters scales quadratically with the dimension $p$, computation
becomes very challenging when $p$ is large. In this paper, we propose an
adaptive sieving reduction algorithm to generate a solution path for the
estimation of precision matrices under the $\ell_1$ penalized D-trace loss,
with each subproblem being solved by a second-order algorithm. In each
iteration of our algorithm, we are able to greatly reduce the number of
variables in the {problem} based on the Karush-Kuhn-Tucker (KKT) conditions and
the sparse structure of the estimated precision matrix in the previous
iteration. As a result, our algorithm is capable of handling datasets with very
high dimensions that may go beyond the capacity of the existing methods.
Moreover, for the sub-problem in each iteration, other than solving the primal
problem directly, we develop a semismooth Newton augmented Lagrangian algorithm
with global linear convergence rate on the dual problem to improve the
efficiency. Theoretical properties of our proposed algorithm have been
established. In particular, we show that the convergence rate of our algorithm
is asymptotically superlinear. The high efficiency and promising performance of
our algorithm are illustrated via extensive simulation studies and real data
applications, with comparison to several state-of-the-art solvers.
