Adaptive Estimation of Graphical Models under Total Positivity

We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. Such models have shown interesting properties, e.g., the maximum likelihood estimator exists with as little as two observations in the case of M-matrices, and exists even with one observation in the case of diagonally dominant M-matrices. We propose an adaptive multiple-stage estimation method, which refines the estimate by solving a weighted $\ell_1$-regularized problem in each stage. We further design a unified framework based on gradient projection method to solve the regularized problem, equipped with different projections to handle the constraints of M-matrices and diagonally dominant M-matrices. Theoretical analysis of the estimation error is established. The proposed method outperforms state-of-the-art methods in estimating precision matrices and identifying graph edges, as evidenced by synthetic and financial time-series data sets.