# Wasserstein F-tests for Fr'echet regression on Bures-Wasserstein manifolds

arXiv: https://arxiv.org/abs/2404.03878 · HTML: https://mlsanity.com/paper/2404.03878/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 82 references that are in the map.
- First posted on arXiv 5 Apr 2024.

## Builds on (most cited first, 3 of 3)

- [A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm](https://mlsanity.com/paper/1902.03736/) (2019)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)

## Built on by (most cited first, 3 of 3)

- [DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responses](https://mlsanity.com/paper/2510.17072/) (2025)
- [Fr'echet Regression on the Bures-Wasserstein Manifold](https://mlsanity.com/paper/2604.03566/) (2026)
- [Distance Profile Embedding for Independence and Conditional Independence Testing of Random Objects](https://mlsanity.com/paper/2607.28981/) (2026)

## Abstract (arXiv)

This paper considers the problem of regression analysis with random covariance matrix as outcome and Euclidean covariates in the framework of Fr'echet regression on the Bures-Wasserstein manifold. Such regression problems have many applications in single cell genomics and neuroscience, where we have covariance matrix measured over a large set of samples. Fr'echet regression on the Bures-Wasserstein manifold is formulated as estimating the conditional Fr'echet mean given covariates x. A non-asymptotic √n-rate of convergence (up to log n factors) is obtained for our estimator hatQn(x) uniformly for left|xright| lesssim √log n, which is crucial for deriving the asymptotic null distribution and power of our proposed statistical test for the null hypothesis of no association. In addition, a central limit theorem for the point estimate hatQn(x) is obtained, giving insights to a test for covariate effects. The null distribution of the test statistic is shown to converge to a weighted sum of independent chi-squares, which implies that the proposed test has the desired significance level asymptotically. Also, the power performance of the test is demonstrated against a sequence of contiguous alternatives. Simulation results show the accuracy of the asymptotic distributions. The proposed methods are applied to a single cell gene expression data set that shows the change of gene co-expression network as people age.