Intrinsic Gaussian Process Regression Modeling for Manifold-valued Response Variable

Extrinsic Gaussian process regression methods, such as the 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 a manifold-valued response variable. In this paper, we first propose a novel intrinsic Gaussian process regression model for the response variable measured on Riemannian manifold. We apply the parallel transport operator to define an intrinsic covariance structure that addresses a critical aspect of constructing a well-defined Gaussian process regression model. We 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. This method can be applied to data situated not only on Euclidean submanifolds but also on manifolds without a natural ambient space. The asymptotic properties for estimating the posterior distribution are established. Numerical studies, including simulation and real examples, indicate that the proposed method works well.