  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
$\sqrt{n}$-rate of convergence (up to $\log n$ factors) is obtained for our
estimator $\hat{Q}_n(x)$ uniformly for $\left\|x\right\| \lesssim \sqrt{\log
n}$, 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 $\hat{Q}_n(x)$ 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.
