Wasserstein F-tests for Fréchet regression on Bures-Wasserstein manifolds
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| Format: | Preprint |
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2024
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| author | Xu, Haoshu Li, Hongzhe |
| author_facet | Xu, Haoshu Li, Hongzhe |
| contents | This paper considers the problem of regression analysis with random covariance matrix as outcome and Euclidean covariates in the framework of Fréchet 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. Fréchet regression on the Bures-Wasserstein manifold is formulated as estimating the conditional Fréchet 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_03878 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wasserstein F-tests for Fréchet regression on Bures-Wasserstein manifolds Xu, Haoshu Li, Hongzhe Methodology Machine Learning This paper considers the problem of regression analysis with random covariance matrix as outcome and Euclidean covariates in the framework of Fréchet 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. Fréchet regression on the Bures-Wasserstein manifold is formulated as estimating the conditional Fréchet 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. |
| title | Wasserstein F-tests for Fréchet regression on Bures-Wasserstein manifolds |
| topic | Methodology Machine Learning |
| url | https://arxiv.org/abs/2404.03878 |