Multi-Fidelity Covariance Estimation in the Log-Euclidean Geometry
This addresses the challenge of tractable covariance estimation in expensive simulation or data collection scenarios, such as physical applications like heat conduction and fluid dynamics, though it is incremental as it builds on existing multi-fidelity and geometric approaches.
The paper tackles the problem of estimating covariance matrices when data collection is expensive by introducing a multi-fidelity estimator that fuses samples from different data sources, guaranteeing definiteness and achieving speedups of over one order of magnitude compared to benchmarks.
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.