  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.
