Abstract

This talk presents a Bayesian reformulation of covariate-assisted principal (CAP) regression for covariance matrix outcomes, which aims to identify low-dimensional components in the covariance that are associated with covariates. We introduce a geometric formulation that respects the nonlinear geometry of the dimension reduced covariances, upon which we leverage Euclidean geometry to estimate the dimension reduction parameters and model heteroscedasticity with covariates. This method enables joint estimation and uncertainty quantification of relevant model parameters associated with heteroscedasticity. An important challenge in covariance regression, generally, is the high dimensionality, as the number of covariance elements increase quadratically in the response variable’s dimension. CAP regression developed in Zhao et al. (2021), and its extension developed here, is useful if there is no need to model the generation of the entire observations, and one is only interested in isolating the data into a potentially low-dimensional representation in which they exhibit certain desired characteristics, such as maximizing the model likelihood associated with subject-level covariates. We apply the proposed method to analyze associations between covariates and brain functional connectivity, utilizing resting-state fMRI data from the Human Connectome Project.