Adaptive Estimation of Graphical Models under Total Positivity
This work addresses the challenge of estimating graphical models with specific structural constraints, which is incremental as it builds on prior methods for M-matrices.
The paper tackles the problem of estimating precision matrices in Gaussian graphical models under total positivity constraints, proposing an adaptive multiple-stage method that outperforms state-of-the-art methods in precision matrix estimation and graph edge identification on synthetic and financial data.
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.