Extrinsic Gaussian processes for regression and classification on manifolds

Lin, L., Niu, M., Cheung, P. and Dunson, D. (2019) Extrinsic Gaussian processes for regression and classification on manifolds. Bayesian Analysis, 14(3), pp. 887-906. (doi: 10.1214/18-BA1135)

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Gaussian processes (GPs) are very widely used for modeling of unknown functions or surfaces in applications ranging from regression to classification to spatial processes. Although there is an increasingly vast literature on applications, methods, theory and algorithms related to GPs, the overwhelming majority of this literature focuses on the case in which the input domain corresponds to a Euclidean space. However, particularly in recent years with the increasing collection of complex data, it is commonly the case that the input domain does not have such a simple form. For example, it is common for the inputs to be restricted to a non-Euclidean manifold, a case which forms the motivation for this article. In particular, we propose a general extrinsic framework for GP modeling on manifolds, which relies on embedding of the manifold into a Euclidean space and then constructing extrinsic kernels for GPs on their images. These extrinsic Gaussian processes (eGPs) are used as prior distributions for unknown functions in Bayesian inferences. Our approach is simple and general, and we show that the eGPs inherit fine theoretical properties from GP models in Euclidean spaces. We consider applications of our models to regression and classification problems with predictors lying in a large class of manifolds, including spheres, planar shape spaces, a space of positive definite matrices, and Grassmannians. Our models can be readily used by practitioners in biological sciences for various regression and classification problems, such as disease diagnosis or detection. Our work is also likely to have impact in spatial statistics when spatial locations are on the sphere or other geometric spaces.

Item Type:Articles
Additional Information:Lizhen Lin acknowledges the support of NSF grants IIS1663870 and DMS CAREER 1654579, and a DARPA grant N66001-17-1-4041.
Glasgow Author(s) Enlighten ID:Niu, Dr Mu
Authors: Lin, L., Niu, M., Cheung, P., and Dunson, D.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Statistics
Journal Name:Bayesian Analysis
Publisher:International Society for Bayesian Analysis
ISSN (Online):1931-6690
Published Online:11 June 2019
Copyright Holders:Copyright © 2019 International Society for Bayesian Analysis
First Published:First published in Bayesian Analysis 14(3): 887-906
Publisher Policy:Reproduced under a Creative Commons License

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