Bayesian scalar-on-network regression with applications to brain functional connectivity

This paper presents a Bayesian regression model relating scalar outcomes to brain functional connectivity represented as symmetric positive definite (SPD) matrices. Unlike many proposals that simply vectorize the matrix-valued connectivity predictors, thereby ignoring their geometric structure, the method presented here respects the Riemannian geometry of SPD matrices by using a tangent space modeling. Dimension reduction is performed in the tangent space, relating the resulting low-dimensional representations to the responses. The dimension reduction matrix is learned in a supervised manner with a sparsity-inducing prior imposed on a Stiefel manifold to prevent overfitting. Our method yields a parsimonious regression model that allows uncertainty quantification of all model parameters and identification of key brain regions that predict the outcomes. We demonstrate the performance of our approach in simulation settings and through a case study to predict Picture Vocabulary scores using data from the Human Connectome Project.