Test of partial effects for Frechet regression on Bures-Wasserstein manifolds

In many modern applications, positive definite matrices are often used to summarize the marginal covariance structure among sets of variables. Examples include medical imaging (Dryden et al., 2009; Fillard et al., 2007), neuroscience (Friston, 2011; Kong et al., 2020; Hu et al., 2021) and gene coexpression analysis in single cell genomics. A central challenge in these fields is how to perform regression analysis where the covariance matrix serves as the outcome variable in relation to a set of Euclidean covariates and how to test for the association between these matrix and covariates. Several regression approaches for covariance matrix outcomes have been proposed. Chiu et al. (1996) developed a method that models the elements of the logarithm of the covariance matrix as a linear function of the covariates, but this approach requires estimating a large number of parameters. Hoff & Niu (2012) proposed a regression model where the covariance matrix is expressed as a quadratic function of the explanatory variables. Zou et al. (2017) linked the matrix outcome to a linear combination of similarity matrices derived from the covariates and examined the asymptotic properties of different estimators under this framework. Xu & Li (2025) introduced Fr echet regression with covariate matrix as the outcome.