# Extrinsic Gaussian processes for regression and classification on manifolds

arXiv: https://arxiv.org/abs/1706.08757 · HTML: https://mlsanity.com/paper/1706.08757/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jun 2017.

## Built on by (most cited first, 9 of 9)

- [Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels](https://mlsanity.com/paper/2110.14423/) (2021)
- [Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics](https://mlsanity.com/paper/2301.06533/) (2023)
- [Intrinsic and extrinsic deep learning on manifolds](https://mlsanity.com/paper/2302.08606/) (2023)
- [Gaussian Process Subspace Regression for Model Reduction](https://mlsanity.com/paper/2107.04668/) (2021)
- [Shape Modeling with Spline Partitions](https://mlsanity.com/paper/2108.02507/) (2021)
- [Geometry-Aware Hierarchical Bayesian Learning on Manifolds](https://mlsanity.com/paper/2111.00184/) (2021)
- [Extrinsic Bayesian Optimizations on Manifolds](https://mlsanity.com/paper/2212.13886/) (2022)
- [Atlas Gaussian processes on restricted domains and point clouds](https://mlsanity.com/paper/2511.15822/) (2025)
- [Intrinsic-Hybrid Latent Diffusion Models for Generative Modeling on Unknown Manifolds](https://mlsanity.com/paper/2608.04827/) (2026)

## Abstract (arXiv)

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.