Manifold-valued Dirichlet Processes

Statistical models for manifold-valued data per-mit capturing the intrinsic nature of the curved spaces in which the data lie and have been a topic of research for several decades. Typically, these formulations use geodesic curves and distances defined locally for most cases — this makes it hard to design parametric models globally on smooth manifolds. Thus, most (manifold spe-cific) parametric models available today assume that the data lie in a small neighborhood on the manifold. To address this ‘locality ’ problem, we propose a novel nonparametric model which uni-fies multivariate general linear models (MGLMs) using multiple tangent spaces. Our framework generalizes existing work on (both Euclidean and non-Euclidean) general linear models pro-viding a recipe to globally extend the locally-defined parametric models (using a mixture of local models). By grouping observations into sub-populations at multiple tangent spaces, our method provides insights into the hidden struc-ture (geodesic relationships) in the data. This yields a framework to group observations and discover geodesic relationships between covari-ates X and manifold-valued responses Y, which we call Dirichlet process mixtures of multivari-ate general linear models (DP-MGLM) on Rie-mannian manifolds. Finally, we present proof of concept experiments to validate our model. 1