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.

翻译：本文考虑在Bures-Wasserstein流形上的Fr´echet回归框架下，以随机协方差矩阵为响应变量、欧几里得协变量进行回归分析的问题。此类回归问题在单细胞基因组学和神经科学中具有广泛的应用，其中我们在大量样本上测量了协方差矩阵。Bures-Wasserstein流形上的Fr´echet回归被表述为估计给定协变量$x$的条件Fr´echet均值。对于我们的估计量$\hat{Q}_n(x)$，在$\left\|x\right\| \lesssim \sqrt{\log n}$的条件下，一致地获得了非渐近的$\sqrt{n}$收敛速率（至多相差$\log n$因子），这对于推导所提出的无关联原假设检验的渐近零分布和检验功效至关重要。此外，我们得到了点估计量$\hat{Q}_n(x)$的中心极限定理，为协变量效应的检验提供了见解。研究表明，检验统计量的零分布收敛于独立卡方变量的加权和，这意味着所提出的检验在渐近意义上具有所需的显著性水平。同时，针对一系列邻近备择假设，验证了检验的功效性能。仿真结果显示了渐近分布的准确性。所提出的方法被应用于一个单细胞基因表达数据集，该数据集展示了基因共表达网络随年龄增长的变化。