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.Comment: 30 pages + 15-page supplemen