Estimation of the precision matrix (or inverse covariance matrix) is of great
importance in statistical data analysis. 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 β1β 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
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