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.

翻译：精确矩阵(或逆差矩阵)的估测在统计数据分析和机器学习中非常重要。然而,由于参数数量与维度成正比,当美元数额巨大时,计算就变得非常困难。在本文中,我们建议采用适应性测分算算法,以产生一种解决方案路径,用于估算受美元1美元惩罚的D-跟踪损失下的精确矩阵,每个子问题都由二级算法解决。在每次算法的反复演算中,我们能够根据Karush-Kuhn-Tucker(KKKTTT)的条件和先前迭代中估计精确矩阵的稀疏结构,大大减少{问题。因此,我们的算法能够处理超出现有方法能力的非常高的数据集。除了直接解决原始问题之外,我们每次的解算法中的子问题,我们开发了一个半mothon Newgrangian 算法的半增缩数, 与我们所建的精准的轨算法的精准趋同率相比, 也就是我们所建的双轨化的精度分析, 我们的精度算法的精度算法的精度的精度的精度算法的精度的精度的精度的精度的精度的精度分析, 是要提高的精度的精度的精度的精度的精度的精度的精度的精度的精度的精度的精度分析。