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 enabling 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 the MRMF 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：本文提出一种多保真度协方差矩阵估计器，其构建于对称正定矩阵流形上的回归问题解。该估计器在构造上保持正定性，且用于求解的Mahalanobis距离具有可实现实际计算的性质。我们证明，在流形切空间的特定误差模型下，该流形回归多保真度（MRMF）协方差估计器是极大似然估计量。更广泛而言，我们证明该黎曼回归框架包含了基于控制变量构建的现有多保真度协方差估计器。数值实验表明，相较于单保真度及其他多保真度协方差估计器，MRMF估计器能将平方估计误差显著降低达一个数量级。此外，正定性的保持确保该估计器可兼容下游任务（如数据同化与度量学习），这些任务中正定性是不可或缺的特性。