Multifidelity Covariance Estimation via Regression on the Manifold of Symmetric Positive Definite Matrices

TL;DR

This paper presents a novel multifidelity covariance estimator based on Riemannian regression on the manifold of symmetric positive definite matrices, improving accuracy and ensuring positive definiteness for downstream applications.

Contribution

The authors introduce a manifold regression-based multifidelity covariance estimator that is positive definite, computationally practical, and encompasses existing methods within a unified framework.

Findings

Significant reduction in estimation error, up to tenfold, compared to existing methods.

The estimator is a maximum likelihood estimator under a specific error model.

Preserves positive definiteness, enabling reliable use in downstream tasks.

Abstract

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.