Multifidelity Covariance Estimation via Regression on the Manifold of Symmetric Positive Definite Matrices
This work addresses covariance estimation for applications like data assimilation and metric learning, offering a novel method that ensures positive definiteness, though it appears incremental by extending existing multifidelity frameworks.
The authors tackled the problem of estimating covariance matrices by introducing a multifidelity estimator formulated as regression on the manifold of symmetric positive definite matrices, achieving up to an order of magnitude reduction in squared estimation error compared to existing methods.
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.