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：我们提出一种基于对称正定流形对数欧几里得几何的多保真度协方差矩阵估计器。该估计器融合来自不同保真度与成本层级数据源的样本，在保证矩阵正定性的同时实现方差缩减，这区别于以往方法。当模拟或数据采集成本高昂时，新估计器使协方差估计在应用中变得可行；为此，我们开发了最优样本分配方案，在固定预算下最小化估计器的均方误差。保证正定性对度量学习、数据同化及其他下游任务至关重要。在物理应用（热传导、流体动力学）数据上的评估表明，与基准方法相比，该方法实现了更精确的度量学习和一个数量级以上的加速比。