Regression Uncertainty on the Grassmannian

Trends in longitudinal or cross-sectional studies over time are often captured through regression models. In their simplest manifestation, these regression models are formulated in R. However, in the context of imaging studies, the objects of interest which are to be regressed are frequently best modeled as elements of a Riemannian manifold. Regression on such spaces can be accomplished through geodesic regression. This paper develops an approach to compute confidence intervals for geodesic regression models. The approach is general, but illustrated and specifically developed for the Grassmann manifold, which allows us, e.g., to regress shapes or linear dynamical systems. Extensions to other manifolds can be obtained in a similar manner. We demonstrate our approach for regression with 2D/3D shapes using synthetic and real data.