Wrapped Gaussian Process Functional Regression Model for Batch Data on Riemannian Manifolds

TL;DR

This paper introduces a novel regression model for manifold-valued data that captures nonlinear relationships using wrapped Gaussian processes, applicable to both simulated and real datasets on Riemannian manifolds.

Contribution

It proposes a concurrent functional regression model for batch data on Riemannian manifolds that estimates mean and covariance structures simultaneously, handling nonlinear relationships.

Findings

Effective on simulated data

Efficient on real data

Captures nonlinear manifold relationships

Abstract

Regression is an essential and fundamental methodology in statistical analysis. The majority of the literature focuses on linear and nonlinear regression in the context of the Euclidean space. However, regression models in non-Euclidean spaces deserve more attention due to collection of increasing volumes of manifold-valued data. In this context, this paper proposes a concurrent functional regression model for batch data on Riemannian manifolds by estimating both mean structure and covariance structure simultaneously. The response variable is assumed to follow a wrapped Gaussian process distribution. Nonlinear relationships between manifold-valued response variables and multiple Euclidean covariates can be captured by this model in which the covariates can be functional and/or scalar. The performance of our model has been tested on both simulated data and real data, showing it is an effective and efficient tool in conducting functional data regression on Riemannian manifolds.