Probabilistic geodesic models for regression and dimensionality reduction on riemannian manifolds

Abstract

We present a probabilistic formulation for two closely related statistical models for Riemannian manifold data: geodesic regression and principal geodesic analysis. These models generalize linear regression and principal component analysis to the manifold setting. The foundation of the approach is the particular choice of a Riemannian normal distribution law as the likelihood model. Under this distributional assumption, we show that least-squares fitting of geodesic models is equivalent to maximum-likelihood estimation when the manifold is a homogeneous space. We also provide a method for maximum-likelihood estimation of the dispersion of the noise, as well as a novel method for Monte Carlo sampling from the Riemannian normal distribution. We demonstrate the inference procedures on synthetic sphere data, as well as in a shape analysis of the corpus callosum, represented in Kendall’s shape space.