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

• URL: http://arxiv.org/abs/2404.03878v1
• Date: Fri, 5 Apr 2024 04:01:51 GMT
• Title: Wasserstein F-tests for Fréchet regression on Bures-Wasserstein manifolds
• Authors: Haoshu Xu, Hongzhe Li,
• Abstract summary: Fr'echet regression on the Bures-Wasserstein manifold is developed. A test for the null hypothesis of no association is proposed. Results show that the proposed test has the desired level of significanceally.
• Score: 0.9514940899499753
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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