Adaptive Estimation of Graphical Models under Total Positivity

TL;DR

This paper introduces an adaptive multi-stage estimation approach for precision matrices in Gaussian graphical models with total positivity, improving accuracy over existing methods through a novel gradient projection framework.

Contribution

It proposes a new adaptive estimation method with a unified gradient projection framework for M-matrices and diagonally dominant M-matrices, with theoretical error analysis.

Findings

Outperforms state-of-the-art methods in synthetic data

Achieves better graph edge identification in financial data

Provides theoretical bounds on estimation error

Abstract

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.