title: Manifold-valued Dirichlet Processes.
abstract: Statistical models for manifold-valued data permit 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 definedlocallyfor most cases - this makes it hard to design parametric modelsgloballyon smooth manifolds. Thus, most (manifold specific) 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 unifies multivariate general linear models (MGLMs) using multiple tangent spaces. Our framework generalizes existing work on (both Euclidean and non-Euclidean) general linear models providing 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 structure (geodesic relationships) in the data. This yields a framework to group observations and discover geodesic relationships between covariatesXand manifold-valued responsesY, which we call Dirichlet process mixtures of multivariate general linear models (DP-MGLM) on Riemannian manifolds. Finally, we present proof of concept experiments to validate our model.
authors.affiliation: University of Wisconsin-Madison, Madison, WI 53706, USA.
authors.forename: Hyunwoo J
authors.initials: HJ
authors.lastname: Kim
authors.affiliation: University of Wisconsin-Madison, Madison, WI 53706, USA.
authors.forename: Jia
authors.initials: J
authors.lastname: Xu
authors.affiliation: University of Florida, Gainesville, FL 32611, USA.
authors.forename: Baba C
authors.initials: BC
authors.lastname: Vemuri
authors.affiliation: University of Wisconsin-Madison, Madison, WI 53706, USA.
authors.forename: Vikas
authors.initials: V
authors.lastname: Singh