Multi-Fidelity Covariance Estimation in the Log-Euclidean Geometry

Aimee Maurais, Terrence Alsup, Benjamin Peherstorfer, Youssef Marzouk
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:24214-24235, 2023.

Abstract

We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-maurais23a, title = {Multi-Fidelity Covariance Estimation in the Log-{E}uclidean Geometry}, author = {Maurais, Aimee and Alsup, Terrence and Peherstorfer, Benjamin and Marzouk, Youssef}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {24214--24235}, 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/maurais23a/maurais23a.pdf}, url = {https://proceedings.mlr.press/v202/maurais23a.html}, abstract = {We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.} }
Endnote
%0 Conference Paper %T Multi-Fidelity Covariance Estimation in the Log-Euclidean Geometry %A Aimee Maurais %A Terrence Alsup %A Benjamin Peherstorfer %A Youssef Marzouk %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-maurais23a %I PMLR %P 24214--24235 %U https://proceedings.mlr.press/v202/maurais23a.html %V 202 %X We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.
APA
Maurais, A., Alsup, T., Peherstorfer, B. & Marzouk, Y.. (2023). Multi-Fidelity Covariance Estimation in the Log-Euclidean Geometry. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:24214-24235 Available from https://proceedings.mlr.press/v202/maurais23a.html.

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