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

arXiv: https://arxiv.org/abs/2307.12438 · HTML: https://mlsanity.com/paper/2307.12438/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 64 references that are in the map.
- First posted on arXiv 23 Jul 2023.

## Builds on (most cited first, 9 of 9)

- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [A Survey on Metric Learning for Feature Vectors and Structured Data](https://mlsanity.com/paper/1306.6709/) (2013)
- [The Why and How of Nonnegative Matrix Factorization](https://mlsanity.com/paper/1401.5226/) (2014)
- [Geometric Mean Metric Learning](https://mlsanity.com/paper/1607.05002/) (2016)
- [Lognormal Distributions and Geometric Averages of Positive Definite Matrices](https://mlsanity.com/paper/1407.6383/) (2014)
- [Exponential-wrapped distributions on symmetric spaces](https://mlsanity.com/paper/2009.01983/) (2020)
- [Multivariate extensions of the Multilevel Best Linear Unbiased Estimator for ensemble-variational data assimilation](https://mlsanity.com/paper/2306.07017/) (2023)
- [Multi-Fidelity Covariance Estimation in the Log-Euclidean Geometry](https://mlsanity.com/paper/2301.13749/) (2023)

## Built on by (most cited first, 1 of 1)

- [Multifidelity Simulation-based Inference for Computationally Expensive Simulators](https://mlsanity.com/paper/2502.08416/) (2025)

## Abstract (arXiv)

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.