# The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphs

arXiv: https://arxiv.org/abs/2407.08086 · HTML: https://mlsanity.com/paper/2407.08086/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 73 references that are in the map.
- First posted on arXiv 10 Jul 2024.

## Builds on (most cited first, 30 of 32)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [A Survey on Graph Kernels](https://mlsanity.com/paper/1903.11835/) (2019)
- [GPflow: A Gaussian process library using TensorFlow](https://mlsanity.com/paper/1610.08733/) (2016)
- [Efficiently Sampling Functions from Gaussian Process Posteriors](https://mlsanity.com/paper/2002.09309/) (2020)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [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)
- [Graph Kernels: State-of-the-Art and Future Challenges](https://mlsanity.com/paper/2011.03854/) (2020)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [Sparse Gaussian Processes with Spherical Harmonic Features](https://mlsanity.com/paper/2006.16649/) (2020)
- [The Multiscale Laplacian Graph Kernel](https://mlsanity.com/paper/1603.06186/) (2016)
- [Practical Hilbert space approximate Bayesian Gaussian processes for probabilistic programming](https://mlsanity.com/paper/2004.11408/) (2020)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [Geometry-aware Bayesian Optimization in Robotics using Riemannian Mat'ern Kernels](https://mlsanity.com/paper/2111.01460/) (2021)
- [Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels](https://mlsanity.com/paper/2110.14423/) (2021)
- [Hodge-Compositional Edge Gaussian Processes](https://mlsanity.com/paper/2310.19450/) (2023)
- [The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running](https://mlsanity.com/paper/2111.01084/) (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)
- [Gaussian Processes on Cellular Complexes](https://mlsanity.com/paper/2311.01198/) (2023)
- [Graph Based Gaussian Processes on Restricted Domains](https://mlsanity.com/paper/2010.07242/) (2020)
- [Bringing motion taxonomies to continuous domains via GPLVM on hyperbolic manifolds](https://mlsanity.com/paper/2210.01672/) (2022)
- [Gaussian Process Landmarking on Manifolds](https://mlsanity.com/paper/1802.03479/) (2018)
- [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)
- [Implicit Gaussian process representation of vector fields over arbitrary latent manifolds](https://mlsanity.com/paper/2309.16746/) (2023)
- [Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach](https://mlsanity.com/paper/2310.19270/) (2023)

## Built on by (most cited first, 11 of 11)

- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Mesh-Informed Neural Operator : A Transformer Generative Approach](https://mlsanity.com/paper/2506.16656/) (2025)
- [Kernel-Gradient Drifting Models](https://mlsanity.com/paper/2605.10727/) (2026)
- [Deep Random Features for Scalable Interpolation of Spatiotemporal Data](https://mlsanity.com/paper/2412.11350/) (2024)
- [Riemann2: Learning Riemannian Submanifolds from Riemannian Data](https://mlsanity.com/paper/2503.05540/) (2025)
- [Do Deep Ensembles Actually Capture Uncertainty in Graph Neural Networks?](https://mlsanity.com/paper/2605.22593/) (2026)
- [Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later](https://mlsanity.com/paper/2509.19929/) (2025)
- [Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves](https://mlsanity.com/paper/2605.06395/) (2026)
- [IIKL: Isometric Immersion Kernel Learning with Riemannian Manifold for Geometric Preservation](https://mlsanity.com/paper/2505.06288/) (2025)
- [Heat and Mat'ern Kernels on Matchings](https://mlsanity.com/paper/2604.14331/) (2026)
- [A General Kernel Framework for Non-CND Distance Measures Using |D|-Dimensional Sparse Landmark Embeddings](https://mlsanity.com/paper/2609.19083/) (2026)

## Abstract (arXiv)

Kernels are a fundamental technical primitive in machine learning. In recent years, kernel-based methods such as Gaussian processes are becoming increasingly important in applications where quantifying uncertainty is of key interest. In settings that involve structured data defined on graphs, meshes, manifolds, or other related spaces, defining kernels with good uncertainty-quantification behavior, and computing their value numerically, is less straightforward than in the Euclidean setting. To address this difficulty, we present GeometricKernels, a Python software package which implements the geometric analogs of classical Euclidean squared exponential - also known as heat - and Mat'ern kernels, which are widely-used in settings where uncertainty is of key interest. As a byproduct, we obtain the ability to compute Fourier-feature-type expansions, which are widely used in their own right, on a wide set of geometric spaces. Our implementation supports automatic differentiation in every major current framework simultaneously via a backend-agnostic design. In this companion paper to the package and its documentation, we outline the capabilities of the package and present an illustrated example of its interface. We also include a brief overview of the theory the package is built upon and provide some historic context in the appendix.