# Intrinsic Gaussian Vector Fields on Manifolds

arXiv: https://arxiv.org/abs/2310.18824 · HTML: https://mlsanity.com/paper/2310.18824/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 38 references that are in the map.
- First posted on arXiv 28 Oct 2023.

## Builds on (most cited first, 12 of 12)

- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [The Vector Heat Method](https://mlsanity.com/paper/1805.09170/) (2018)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [Algorithmic Linearly Constrained Gaussian Processes](https://mlsanity.com/paper/1801.09197/) (2018)
- [Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels](https://mlsanity.com/paper/2110.14423/) (2021)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spaces](https://mlsanity.com/paper/2301.13088/) (2023)
- [Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces](https://mlsanity.com/paper/1905.10913/) (2019)
- [Gaussian Process Manifold Interpolation for Probabilistic Atrial Activation Maps and Uncertain Conduction Velocity](https://mlsanity.com/paper/2004.10586/) (2020)
- [Gaussian processes at the Helm(holtz): A more fluid model for ocean currents](https://mlsanity.com/paper/2302.10364/) (2023)
- [Posterior Contraction Rates for Mat'ern Gaussian Processes on Riemannian Manifolds](https://mlsanity.com/paper/2309.10918/) (2023)
- [Implicit Gaussian process representation of vector fields over arbitrary latent manifolds](https://mlsanity.com/paper/2309.16746/) (2023)
- [Gaussian Processes and Statistical Decision-making in Non-Euclidean Spaces](https://mlsanity.com/paper/2202.10613/) (2022)

## Built on by (most cited first, 8 of 8)

- [Hodge-Compositional Edge Gaussian Processes](https://mlsanity.com/paper/2310.19450/) (2023)
- [Gaussian Processes on Cellular Complexes](https://mlsanity.com/paper/2311.01198/) (2023)
- [The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphs](https://mlsanity.com/paper/2407.08086/) (2024)
- [Graph Random Features for Scalable Gaussian Processes](https://mlsanity.com/paper/2509.03691/) (2025)
- [Residual Deep Gaussian Processes on Manifolds](https://mlsanity.com/paper/2411.00161/) (2024)
- [Discrete Gaussian Vector Fields On Meshes](https://mlsanity.com/paper/2507.20024/) (2025)
- [Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds](https://mlsanity.com/paper/2606.13827/) (2026)
- [Riemannian Simultaneous Inference for Tangent Vector Field Regression](https://mlsanity.com/paper/2609.21910/) (2026)

## Abstract (arXiv)

Various applications ranging from robotics to climate science require modeling signals on non-Euclidean domains, such as the sphere. Gaussian process models on manifolds have recently been proposed for such tasks, in particular when uncertainty quantification is needed. In the manifold setting, vector-valued signals can behave very differently from scalar-valued ones, with much of the progress so far focused on modeling the latter. The former, however, are crucial for many applications, such as modeling wind speeds or force fields of unknown dynamical systems. In this paper, we propose novel Gaussian process models for vector-valued signals on manifolds that are intrinsically defined and account for the geometry of the space in consideration. We provide computational primitives needed to deploy the resulting Hodge-Mat'ern Gaussian vector fields on the two-dimensional sphere and the hypertori. Further, we highlight two generalization directions: discrete two-dimensional meshes and "ideal" manifolds like hyperspheres, Lie groups, and homogeneous spaces. Finally, we show that our Gaussian vector fields constitute considerably more refined inductive biases than the extrinsic fields proposed before.