Adaptive Estimation of Graphical Models under Total Positivity

• URL: http://arxiv.org/abs/2210.15471v2
• Date: Thu, 8 Jun 2023 22:23:46 GMT
• Title: Adaptive Estimation of Graphical Models under Total Positivity
• Authors: Jiaxi Ying, Jos\'e Vin\'icius de M. Cardoso, Daniel P. Palomar
• Abstract summary: We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. We propose an adaptive multiple-stage estimation method that refines the estimate. We develop a unified framework based on the gradient projection method to solve the regularized problem.
• Score: 13.47131471222723
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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