  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.
