Manifold-valued Dirichlet Processes
Hyunwoo Kim, Jia Xu, Baba Vemuri, Vikas Singh
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 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.
BibTeX
@InProceedings{pmlr-v37-kim15,
title = {Manifold-valued Dirichlet Processes},
author = {Kim, Hyunwoo and Xu, Jia and Vemuri, Baba and Singh, Vikas},
booktitle = {Proceedings of the 32nd International Conference on Machine Learning},
pages = {1199--1208},
year = {2015},
editor = {Bach, Francis and Blei, David},
volume = {37},
series = {Proceedings of Machine Learning Research},
address = {Lille, France},
month = {07--09 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v37/kim15.pdf},
url = {https://proceedings.mlr.press/v37/kim15.html},
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 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.}
}