# Intrinsic Gaussian Process Regression Modeling for Manifold-valued Response Variable

arXiv: https://arxiv.org/abs/2411.18989 · HTML: https://mlsanity.com/paper/2411.18989/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 46 references that are in the map.
- First posted on arXiv 28 Nov 2024.

## Builds on (most cited first, 16 of 16)

- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [Gaussians on Riemannian Manifolds: Applications for Robot Learning and Adaptive Control](https://mlsanity.com/paper/1909.05946/) (2019)
- [Graph Based Gaussian Processes on Restricted Domains](https://mlsanity.com/paper/2010.07242/) (2020)
- [Single Index Fr'echet Regression](https://mlsanity.com/paper/2108.05437/) (2021)
- [Intrinsic Gaussian processes on complex constrained domains](https://mlsanity.com/paper/1801.01061/) (2018)
- [Geodesic analysis in Kendall's shape space with epidemiological applications](https://mlsanity.com/paper/1906.11950/) (2019)
- [Intrinsic Riemannian Functional Data Analysis](https://mlsanity.com/paper/1812.01831/) (2018)
- [Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics](https://mlsanity.com/paper/2301.06533/) (2023)
- [Wrapped Gaussian Process Functional Regression Model for Batch Data on Riemannian Manifolds](https://mlsanity.com/paper/2409.03181/) (2024)
- [Generalized Gaussian Process Regression Model for Non-Gaussian Functional Data](https://mlsanity.com/paper/1401.8189/) (2014)
- [Gaussian Process on the Product of Directional Manifolds](https://mlsanity.com/paper/2303.06799/) (2023)

## Abstract (arXiv)

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.