TY - GEN
T1 - Manifold-valued Dirichlet processes
AU - Kim, Hyunwoo J.
AU - Xu, Jia
AU - Vemuri, Baba C.
AU - Singh, Vikas
N1 - Funding Information:
This 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 (All 17924). We are grateful to Michael A. Newton, Vamsi K. Ithapu and WonHwa Kim for various discussions related to the content presented in this paper.
Publisher Copyright:
Copyright © 2015 by the author(s).
PY - 2015
Y1 - 2015
N2 - 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 covari-ates X and manifold-valued responses Y, which we call Dirichlet process mixtures of multivariate general linear models (DP-MGLM) on Rie-mannian manifolds. Finally, we present proof of concept experiments to validate our model.
AB - 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 covari-ates X and manifold-valued responses Y, which we call Dirichlet process mixtures of multivariate general linear models (DP-MGLM) on Rie-mannian manifolds. Finally, we present proof of concept experiments to validate our model.
UR - http://www.scopus.com/inward/record.url?scp=84969835201&partnerID=8YFLogxK
M3 - Conference contribution
AN - SCOPUS:84969835201
T3 - 32nd International Conference on Machine Learning, ICML 2015
SP - 1199
EP - 1208
BT - 32nd International Conference on Machine Learning, ICML 2015
A2 - Blei, David
A2 - Bach, Francis
PB - International Machine Learning Society (IMLS)
T2 - 32nd International Conference on Machine Learning, ICML 2015
Y2 - 6 July 2015 through 11 July 2015
ER -