Multi-Fidelity Covariance Estimation in the Log-Euclidean Geometry

• URL: http://arxiv.org/abs/2301.13749v2
• Date: Fri, 26 May 2023 18:34:07 GMT
• Title: Multi-Fidelity Covariance Estimation in the Log-Euclidean Geometry
• Authors: Aimee Maurais and Terrence Alsup and Benjamin Peherstorfer and Youssef Marzouk
• Abstract summary: We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. We develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Evaluations of our approach using data from physical applications demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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