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.

我们介绍了一种作为对对称正定矩阵流形上的回归问题的解的协方差矩阵的多保真度估计器。该估计器由于构造方式是正定的，通过最小化得到它的马哈拉诺比斯距离具有可实用的计算性质。我们证明了我们的流形回归多保真度 (MRMF) 协方差估计器是在特定误差模型下的最大似然估计器。更广泛地，我们证明了我们的黎曼回归框架包含了从控制变量构造的现有多保真度协方差估计器。通过数值实例，我们证明了我们的估计器可以相对于单保真度和其他多保真度协方差估计器显著减少，可达一个数量级的平方估计误差。同时，保持正定性确保了我们的估计器与下游任务的兼容性，如数据同化和度量学习，其中这个属性是必要的。