Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency

Eduardo Pavez
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:9711-9722, 2022.

Abstract

This paper considers the problem of estimating high dimensional Laplacian constrained precision matrices by minimizing Stein’s loss. We obtain a necessary and sufficient condition for existence of this estimator, that consists on checking whether a certain data dependent graph is connected. We also prove consistency in the high dimensional setting under the symmetrized Stein loss. We show that the error rate does not depend on the graph sparsity, or other type of structure, and that Laplacian constraints are sufficient for high dimensional consistency. Our proofs exploit properties of graph Laplacians, the matrix tree theorem, and a characterization of the proposed estimator based on effective graph resistances. We validate our theoretical claims with numerical experiments.

Cite this Paper

BibTeX
@InProceedings{pmlr-v151-pavez22a, title = { Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency }, author = {Pavez, Eduardo}, booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics}, pages = {9711--9722}, year = {2022}, editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel}, volume = {151}, series = {Proceedings of Machine Learning Research}, month = {28--30 Mar}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v151/pavez22a/pavez22a.pdf}, url = {https://proceedings.mlr.press/v151/pavez22a.html}, abstract = { This paper considers the problem of estimating high dimensional Laplacian constrained precision matrices by minimizing Stein’s loss. We obtain a necessary and sufficient condition for existence of this estimator, that consists on checking whether a certain data dependent graph is connected. We also prove consistency in the high dimensional setting under the symmetrized Stein loss. We show that the error rate does not depend on the graph sparsity, or other type of structure, and that Laplacian constraints are sufficient for high dimensional consistency. Our proofs exploit properties of graph Laplacians, the matrix tree theorem, and a characterization of the proposed estimator based on effective graph resistances. We validate our theoretical claims with numerical experiments. } }
Endnote
%0 Conference Paper %T Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency %A Eduardo Pavez %B Proceedings of The 25th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2022 %E Gustau Camps-Valls %E Francisco J. R. Ruiz %E Isabel Valera %F pmlr-v151-pavez22a %I PMLR %P 9711--9722 %U https://proceedings.mlr.press/v151/pavez22a.html %V 151 %X This paper considers the problem of estimating high dimensional Laplacian constrained precision matrices by minimizing Stein’s loss. We obtain a necessary and sufficient condition for existence of this estimator, that consists on checking whether a certain data dependent graph is connected. We also prove consistency in the high dimensional setting under the symmetrized Stein loss. We show that the error rate does not depend on the graph sparsity, or other type of structure, and that Laplacian constraints are sufficient for high dimensional consistency. Our proofs exploit properties of graph Laplacians, the matrix tree theorem, and a characterization of the proposed estimator based on effective graph resistances. We validate our theoretical claims with numerical experiments.
APA
Pavez, E.. (2022). Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency . Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 151:9711-9722 Available from https://proceedings.mlr.press/v151/pavez22a.html.

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