Robust Estimation of Tree Structured Gaussian Graphical Models

Ashish Katiyar, Jessica Hoffmann, Constantine Caramanis
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:3292-3300, 2019.

Abstract

Consider jointly Gaussian random variables whose conditional independence structure is specified by a graphical model. If we observe realizations of the variables, we can compute the covariance matrix, and it is well known that the support of the inverse covariance matrix corresponds to the edges of the graphical model. Instead, suppose we only have noisy observations. If the noise at each node is independent, we can compute the sum of the covariance matrix and an unknown diagonal. The inverse of this sum is (in general) dense. We ask: can the original independence structure be recovered? We address this question for tree structured graphical models. We prove that this problem is unidentifiable, but show that this unidentifiability is limited to a small class of candidate trees. We further present additional constraints under which the problem is identifiable. Finally, we provide an O(n^3) algorithm to find this equivalence class of trees.

Cite this Paper

BibTeX
@InProceedings{pmlr-v97-katiyar19a, title = {Robust Estimation of Tree Structured {G}aussian Graphical Models}, author = {Katiyar, Ashish and Hoffmann, Jessica and Caramanis, Constantine}, booktitle = {Proceedings of the 36th International Conference on Machine Learning}, pages = {3292--3300}, year = {2019}, editor = {Chaudhuri, Kamalika and Salakhutdinov, Ruslan}, volume = {97}, series = {Proceedings of Machine Learning Research}, month = {09--15 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v97/katiyar19a/katiyar19a.pdf}, url = {https://proceedings.mlr.press/v97/katiyar19a.html}, abstract = {Consider jointly Gaussian random variables whose conditional independence structure is specified by a graphical model. If we observe realizations of the variables, we can compute the covariance matrix, and it is well known that the support of the inverse covariance matrix corresponds to the edges of the graphical model. Instead, suppose we only have noisy observations. If the noise at each node is independent, we can compute the sum of the covariance matrix and an unknown diagonal. The inverse of this sum is (in general) dense. We ask: can the original independence structure be recovered? We address this question for tree structured graphical models. We prove that this problem is unidentifiable, but show that this unidentifiability is limited to a small class of candidate trees. We further present additional constraints under which the problem is identifiable. Finally, we provide an O(n^3) algorithm to find this equivalence class of trees.} }
Endnote
%0 Conference Paper %T Robust Estimation of Tree Structured Gaussian Graphical Models %A Ashish Katiyar %A Jessica Hoffmann %A Constantine Caramanis %B Proceedings of the 36th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Ruslan Salakhutdinov %F pmlr-v97-katiyar19a %I PMLR %P 3292--3300 %U https://proceedings.mlr.press/v97/katiyar19a.html %V 97 %X Consider jointly Gaussian random variables whose conditional independence structure is specified by a graphical model. If we observe realizations of the variables, we can compute the covariance matrix, and it is well known that the support of the inverse covariance matrix corresponds to the edges of the graphical model. Instead, suppose we only have noisy observations. If the noise at each node is independent, we can compute the sum of the covariance matrix and an unknown diagonal. The inverse of this sum is (in general) dense. We ask: can the original independence structure be recovered? We address this question for tree structured graphical models. We prove that this problem is unidentifiable, but show that this unidentifiability is limited to a small class of candidate trees. We further present additional constraints under which the problem is identifiable. Finally, we provide an O(n^3) algorithm to find this equivalence class of trees.
APA
Katiyar, A., Hoffmann, J. & Caramanis, C.. (2019). Robust Estimation of Tree Structured Gaussian Graphical Models. Proceedings of the 36th International Conference on Machine Learning, in Proceedings of Machine Learning Research 97:3292-3300 Available from https://proceedings.mlr.press/v97/katiyar19a.html.

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