  Brain structural networks are often represented as discrete adjacency
matrices with elements summarizing the connectivity between pairs of regions of
interest (ROIs). These ROIs are typically determined a-priori using a brain
atlas. The choice of atlas is often arbitrary and can lead to a loss of
important connectivity information at the sub-ROI level. This work introduces
an atlas-free framework that overcomes these issues by modeling brain
connectivity using smooth random functions. In particular, we assume that the
observed pattern of white matter fiber tract endpoints is driven by a latent
random function defined over a product manifold domain. To facilitate
statistical analysis of these high dimensional functional data objects, we
develop a novel algorithm to construct a data-driven reduced-rank function
space that offers a desirable trade-off between computational complexity and
flexibility. Using real data from the Human Connectome Project, we show that
our method outperforms state-of-the-art approaches that use the traditional
atlas-based structural connectivity representation on a variety of connectivity
analysis tasks. We further demonstrate how our method can be used to detect
localized regions and connectivity patterns associated with group differences.
