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.

翻译：我们提出了一种协方差矩阵的多保真度估计器，它利用对称正定流形的对数-欧几里得几何。该估计器融合来自不同保真度和成本的数据源层次结构的样本，以实现方差减少，同时保证正定性，这与先前的方法形成对比。新估计器使得在模拟或数据采集成本高昂的应用中，协方差估计变得可行；为此，我们开发了一种最优样本分配方案，该方案在给定预算下最小化估计器的均方误差。保证正定性对于度量学习、数据同化及其他下游任务至关重要。使用来自物理应用（热传导、流体动力学）的数据对我们的方法进行评估，结果表明与基准相比，度量学习更准确，且加速效果超过一个数量级。