In this paper, we present a Bayesian formulation of polynomial regression on a Riemannian manifold. Previous methods for fitting a curve to manifold-valued data have been formulated as geometric, least-squares estimation problems. We show that least-squares estimation on manifolds, much like the familiar Euclidean case, suffers from overfitting when using higher-order polynomials. Our Bayesian model mitigates this overfitting by placing a prior on the polynomial coefficients that shrinks their magnitude, analogous to Bayesian Euclidean regression with a Gaussian prior on the coefficients. We develop an algorithm for computing maximum a posteriori estimates of polynomial coefficients and the noise variance. Experiments on synthetically generated sphere data and a real shape regression problem demonstrate the advantages of our approach.

Longitudinal data arises in many applications in which the goal is to understand changes in individual entities over time. In this paper, we present a method for analyzing longitudinal data that take values in a Riemannian manifold. A driving application is to characterize anatomical shape changes and to distinguish between trends in anatomy that are healthy versus those that are due to disease. We present a generative hierarchical model in which each individual is modeled by a geodesic trend, which in turn is considered as a perturbation of the mean geodesic trend for the population. Each geodesic in the model can be uniquely parameterized by a starting point and velocity, i.e., a point in the tangent bundle. Comparison between these parameters is achieved through the Sasaki metric, which provides a natural distance metric on the tangent bundle. We develop a statistical hypothesis test for differences between two groups of longitudinal data by generalizing the Hotelling T2statistic to manifolds. We demonstrate the ability of these methods to distinguish differences in shape changes in a comparison of longitudinal corpus callosum data in subjects with dementia versus healthily aging controls.