# Multi-Fidelity Covariance Estimation in the Log-Euclidean Geometry

arXiv: https://arxiv.org/abs/2301.13749 · HTML: https://mlsanity.com/paper/2301.13749/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 55 references that are in the map.
- First posted on arXiv 31 Jan 2023.

## Builds on (most cited first, 12 of 12)

- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [Practical Multi-fidelity Bayesian Optimization for Hyperparameter Tuning](https://mlsanity.com/paper/1903.04703/) (2019)
- [High-dimensional learning of linear causal networks via inverse covariance estimation](https://mlsanity.com/paper/1311.3492/) (2013)
- [Multi-Information Source Optimization](https://mlsanity.com/paper/1603.00389/) (2016)
- [Multi-Fidelity Bayesian Optimization via Deep Neural Networks](https://mlsanity.com/paper/2007.03117/) (2020)
- [Geometric Mean Metric Learning](https://mlsanity.com/paper/1607.05002/) (2016)
- [Covariance Estimation in High Dimensions via Kronecker Product Expansions](https://mlsanity.com/paper/1302.2686/) (2013)
- [Contour location via entropy reduction leveraging multiple information sources](https://mlsanity.com/paper/1805.07489/) (2018)
- [A Multifidelity Ensemble Kalman Filter with Reduced Order Control Variates](https://mlsanity.com/paper/2007.00793/) (2020)
- [Active multi-fidelity Bayesian online changepoint detection](https://mlsanity.com/paper/2103.14224/) (2021)
- [Multilevel Stein variational gradient descent with applications to Bayesian inverse problems](https://mlsanity.com/paper/2104.01945/) (2021)
- [Multi-fidelity Monte Carlo: a pseudo-marginal approach](https://mlsanity.com/paper/2210.01534/) (2022)

## Built on by (most cited first, 2 of 2)

- [Multifidelity Covariance Estimation via Regression on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2307.12438/) (2023)
- [Balancing Accuracy and Speed: A Multi-Fidelity Ensemble Kalman Filter with a Machine Learning Surrogate Model](https://mlsanity.com/paper/2512.12276/) (2025)

## Abstract (arXiv)

We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.