We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. These models exhibit intriguing properties, such as the existence of the maximum likelihood estimator with merely two observations for M-matrices \citep{lauritzen2019maximum,slawski2015estimation} and even one observation for diagonally dominant M-matrices \citep{truell2021maximum}. We propose an adaptive multiple-stage estimation method that refines the estimate by solving a weighted $\ell_1$-regularized problem at each stage. Furthermore, we develop a unified framework based on the gradient projection method to solve the regularized problem, incorporating distinct projections to handle the constraints of M-matrices and diagonally dominant M-matrices. A theoretical analysis of the estimation error is provided. Our method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification, as evidenced by synthetic and financial time-series data sets.

翻译：我们考虑在高斯图模型中估计（对角占优）M矩阵作为精度矩阵的问题。这些模型展现出引人注目的性质，例如：M矩阵的最大似然估计量仅需两个观测值即可存在 \citep{lauritzen2019maximum,slawski2015estimation} ，而对角占优M矩阵甚至仅需一个观测值 \citep{truell2021maximum} 。我们提出一种自适应多阶段估计方法，通过在每个阶段求解加权$\ell_1$正则化问题来优化估计量。此外，我们基于梯度投影法发展了一个统一框架以求解该正则化问题，通过引入不同的投影算子分别处理M矩阵与对角占优M矩阵的约束条件。本文还提供了估计误差的理论分析。在合成数据集与金融时间序列数据集上的实验表明，我们的方法在精度矩阵估计和图边识别任务中均优于现有最先进方法。