Intrinsic Gaussian Process Regression Modeling for Manifold-valued Response Variable

TL;DR

This paper introduces an intrinsic Gaussian process regression model for manifold-valued data, utilizing parallel transport on Riemannian manifolds to define covariance, with proven asymptotic properties and demonstrated effectiveness through simulations and real data.

Contribution

It develops the first intrinsic Gaussian process regression model for manifold data, addressing limitations of extrinsic methods and ensuring invariance and consistency.

Findings

Model performs well in simulations and real data examples.

Proven asymptotic and posterior consistency of the model.

Invariance to orthonormal frame choices in the covariance structure.

Abstract

Extrinsic Gaussian process regression methods, such as wrapped Gaussian process, have been developed to analyze manifold data. However, there is a lack of intrinsic Gaussian process methods for studying complex data with manifold-valued response variables. In this paper, we first apply the parallel transport operator on Riemannian manifold to propose an intrinsic covariance structure that addresses a critical aspect of constructing a well-defined Gaussian process regression model. We then propose a novel intrinsic Gaussian process regression model for manifold-valued data, which can be applied to data situated not only on Euclidean submanifolds but also on manifolds without a natural ambient space. We establish the asymptotic properties of the proposed models, including information consistency and posterior consistency, and we also show that the posterior distribution of the regression function is invariant to the choice of orthonormal frames for the coordinate representations of the covariance function. Numerical studies, including simulation and real examples, indicate that the proposed methods work well.