We introduce a multifidelity estimator of covariance matrices formulated as the solution to a regression problem on the manifold of symmetric positive definite matrices. The estimator is positive definite by construction, and the Mahalanobis distance minimized to obtain it possesses properties which enable practical computation. We show that our manifold regression multifidelity (MRMF) covariance estimator is a maximum likelihood estimator under a certain error model on manifold tangent space. More broadly, we show that our Riemannian regression framework encompasses existing multifidelity covariance estimators constructed from control variates. We demonstrate via numerical examples that our estimator can provide significant decreases, up to one order of magnitude, in squared estimation error relative to both single-fidelity and other multifidelity covariance estimators. Furthermore, preservation of positive definiteness ensures that our estimator is compatible with downstream tasks, such as data assimilation and metric learning, in which this property is essential.

翻译：我们提出了一种协方差矩阵的多保真度估计器，其被表述为对称正定矩阵流形上回归问题的解。该估计器通过构造保证正定性，且用于求解它的马氏距离具有便于实际计算的性质。我们证明，在流形切空间上的特定误差模型下，我们的流形回归多保真度（MRMF）协方差估计器是一种最大似然估计器。更广泛地说，我们证明该黎曼回归框架囊括了现有基于控制变量构造的多保真度协方差估计器。通过数值算例，我们证明该估计器能够显著降低平方估计误差（相比单保真度及其他多保真度协方差估计器可降低一个数量级）。此外，正定性的保持确保该估计器与数据同化、度量学习等下游任务兼容，而该性质在这些任务中至关重要。