Adaptive Estimation of Graphical Models under Total Positivity

Jiaxi Ying, José Vinı́cius De Miranda Cardoso, Daniel P. Palomar
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:40054-40074, 2023.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-ying23a, title = {Adaptive Estimation of Graphical Models under Total Positivity}, author = {Ying, Jiaxi and De Miranda Cardoso, Jos\'{e} Vin\'{\i}cius and Palomar, Daniel P.}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {40054--40074}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/ying23a/ying23a.pdf}, url = {https://proceedings.mlr.press/v202/ying23a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Adaptive Estimation of Graphical Models under Total Positivity %A Jiaxi Ying %A José Vinı́cius De Miranda Cardoso %A Daniel P. Palomar %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-ying23a %I PMLR %P 40054--40074 %U https://proceedings.mlr.press/v202/ying23a.html %V 202 %X 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.
APA
Ying, J., De Miranda Cardoso, J.V. & Palomar, D.P.. (2023). Adaptive Estimation of Graphical Models under Total Positivity. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:40054-40074 Available from https://proceedings.mlr.press/v202/ying23a.html.

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