Wasserstein F-tests for Fréchet regression on Bures-Wasserstein manifolds

This paper addresses regression analysis for covariance matrix-valued outcomes with Euclidean covariates, motivated by applications in single-cell genomics and neuroscience where covariance matrices are observed across many samples. Our analysis leverages Fréchet regression on the Bures-Wasserstein manifold to estimate the conditional Fréchet mean given covariates $x$ . We establish a non-asymptotic uniform $\sqrt{n}$ -rate of convergence (up to logarithmic factors) over covariates with $\Vert x\Vert \lesssim \sqrt{\text{log}n}$ and derive a pointwise central limit theorem to enable statistical inference. For testing covariate effects, we devise a novel test whose null distribution converges to a weighted sum of independent chi-square distributions, with power guarantees against a sequence of contiguous alternatives. Simulations validate the accuracy of the asymptotic theory. Finally, we apply our methods to a single-cell gene expression dataset, revealing age-related changes in gene co-expression networks.