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

MARS: A second-order reduction algorithm for high-dimensional sparse precision matrices estimation