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Manifold-valued Dirichlet ProcessesHyunwoo J. Kim, Jia Xu, Baba C. Vemuri, Vikas Singh, Manifold-valued Dirichlet Processes , In International Conference on Machine Learning (ICML), July 2015.AbstractStatistical 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 defined locally for most cases - this makes it hard to design parametric models globally on 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 covariates X and manifold-valued responses Y, 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.AcknowledgmentsThis work was supported in part by NIH grants AG040396 (VS), NS066340 (BCV), NSF CAREER award 1252725 (VS). Partial support was also provided by the Center for Predictive Computational Phenotyping (CPCP) at UW-Madison (AI117924). We are grateful to Michael A. Newton, Vamsi K. Ithapu and WonHwa Kim for various discussions related to the content presented in this paper. |