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，该估计器被构建为对称正定矩阵流形上回归问题的解。该估计器本质上具有正定性，且为获取该估计器而最小化的马氏距离具有支持实际计算的特性。我们证明，在流形切空间的特定误差模型下，我们的流形回归多保真度（MRMF）协方差估计器是最大似然估计器。更广泛而言，我们表明，所提出的黎曼回归框架涵盖了基于控制变量法的现有多保真度协方差估计器。通过数值算例，我们证明该估计器相较于单保真度及其他多保真度协方差估计器，可将平方估计误差显著降低一个数量级以上。此外，正定性的保持确保了该估计器与后续任务（如数据同化和度量学习）的兼容性，而这一特性在其中至关重要。