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

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.