# Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case

arXiv: https://arxiv.org/abs/2208.14960 · HTML: https://mlsanity.com/paper/2208.14960/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 45 references that are in the map.
- First posted on arXiv 31 Aug 2022.

## Builds on (most cited first, 14 of 14)

- [Efficiently Sampling Functions from Gaussian Process Posteriors](https://mlsanity.com/paper/2002.09309/) (2020)
- [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)
- [Pathwise Conditioning of Gaussian Processes](https://mlsanity.com/paper/2011.04026/) (2020)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels](https://mlsanity.com/paper/2111.01460/) (2021)
- [Bayesian Optimization Meets Riemannian Manifolds in Robot Learning](https://mlsanity.com/paper/1910.04998/) (2019)
- [Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels](https://mlsanity.com/paper/2110.14423/) (2021)
- [Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces](https://mlsanity.com/paper/1905.10913/) (2019)
- [Manifold GPLVMs for discovering non-Euclidean latent structure in neural data](https://mlsanity.com/paper/2006.07429/) (2020)
- [Gaussian Process Manifold Interpolation for Probabilistic Atrial Activation Maps and Uncertain Conduction Velocity](https://mlsanity.com/paper/2004.10586/) (2020)
- [Intrinsic Gaussian processes on complex constrained domains](https://mlsanity.com/paper/1801.01061/) (2018)
- [Exact heat kernel on a hypersphere and its applications in kernel SVM](https://mlsanity.com/paper/1702.01373/) (2017)
- [Intrinsic Gaussian Processes on Manifolds and Their Accelerations by Symmetry](https://mlsanity.com/paper/2006.14266/) (2020)

## Built on by (most cited first, 25 of 25)

- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Learning Layer-wise Equivariances Automatically using Gradients](https://mlsanity.com/paper/2310.06131/) (2023)
- [Hodge-Compositional Edge Gaussian Processes](https://mlsanity.com/paper/2310.19450/) (2023)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spaces](https://mlsanity.com/paper/2301.13088/) (2023)
- [The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphs](https://mlsanity.com/paper/2407.08086/) (2024)
- [Geometric Neural Diffusion Processes](https://mlsanity.com/paper/2307.05431/) (2023)
- [Intrinsic Gaussian Vector Fields on Manifolds](https://mlsanity.com/paper/2310.18824/) (2023)
- [Isotropic Gaussian Processes on Finite Spaces of Graphs](https://mlsanity.com/paper/2211.01689/) (2022)
- [Intrinsic Gaussian Process on Unknown Manifolds with Probabilistic Metrics](https://mlsanity.com/paper/2301.06533/) (2023)
- [Posterior Contraction Rates for Mat'ern Gaussian Processes on Riemannian Manifolds](https://mlsanity.com/paper/2309.10918/) (2023)
- [Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach](https://mlsanity.com/paper/2310.19270/) (2023)
- [Stiefel Flow Matching for Moment-Constrained Structure Elucidation](https://mlsanity.com/paper/2412.12540/) (2024)
- [Implicit Manifold Gaussian Process Regression](https://mlsanity.com/paper/2310.19390/) (2023)
- [Geometric Learning with Positively Decomposable Kernels](https://mlsanity.com/paper/2310.13821/) (2023)
- [Residual Deep Gaussian Processes on Manifolds](https://mlsanity.com/paper/2411.00161/) (2024)
- [Stochastic Poisson Surface Reconstruction with One Solve using Geometric Gaussian Processes](https://mlsanity.com/paper/2503.19136/) (2025)
- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)
- [A Bayesian Approach for Task-Specific Next-Best-View Selection with Uncertain Geometry](https://mlsanity.com/paper/2605.05095/) (2026)
- [On power sum kernels on symmetric groups](https://mlsanity.com/paper/2211.05650/) (2022)
- [Hyperboloid GPLVM for Discovering Continuous Hierarchies via Nonparametric Estimation](https://mlsanity.com/paper/2410.16698/) (2024)
- [Intrinsic Gaussian Process Regression Modeling for Manifold-valued Response Variable](https://mlsanity.com/paper/2411.18989/) (2024)
- [Information Theoretic Bayesian Optimization over the Probability Simplex](https://mlsanity.com/paper/2603.09793/) (2026)
- [Heat and Mat'ern Kernels on Matchings](https://mlsanity.com/paper/2604.14331/) (2026)
- [What is Smoothness?](https://mlsanity.com/paper/2609.03246/) (2026)
- [RW-Flow: One-Step Generation on Compact Manifolds via Wasserstein Gradient Flows](https://mlsanity.com/paper/2609.39271/) (2026)

## Abstract (arXiv)

Gaussian processes are arguably the most important class of spatiotemporal models within machine learning. They encode prior information about the modeled function and can be used for exact or approximate Bayesian learning. In many applications, particularly in physical sciences and engineering, but also in areas such as geostatistics and neuroscience, invariance to symmetries is one of the most fundamental forms of prior information one can consider. The invariance of a Gaussian process' covariance to such symmetries gives rise to the most natural generalization of the concept of stationarity to such spaces. In this work, we develop constructive and practical techniques for building stationary Gaussian processes on a very large class of non-Euclidean spaces arising in the context of symmetries. Our techniques make it possible to (i) calculate covariance kernels and (ii) sample from prior and posterior Gaussian processes defined on such spaces, both in a practical manner. This work is split into two parts, each involving different technical considerations: part I studies compact spaces, while part II studies non-compact spaces possessing certain structure. Our contributions make the non-Euclidean Gaussian process models we study compatible with well-understood computational techniques available in standard Gaussian process software packages, thereby making them accessible to practitioners.