# Hodge-Compositional Edge Gaussian Processes

arXiv: https://arxiv.org/abs/2310.19450 · HTML: https://mlsanity.com/paper/2310.19450/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 62 references that are in the map.
- First posted on arXiv 30 Oct 2023.

## Builds on (most cited first, 24 of 24)

- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Principled Simplicial Neural Networks for Trajectory Prediction](https://mlsanity.com/paper/2102.10058/) (2021)
- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [Simplicial Convolutional Neural Networks](https://mlsanity.com/paper/2110.02585/) (2021)
- [The Vector Heat Method](https://mlsanity.com/paper/1805.09170/) (2018)
- [Graph-based Semi-Supervised & Active Learning for Edge Flows](https://mlsanity.com/paper/1905.07451/) (2019)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [Gaussian Processes on Graphs via Spectral Kernel Learning](https://mlsanity.com/paper/2006.07361/) (2020)
- [Finite Impulse Response Filters for Simplicial Complexes](https://mlsanity.com/paper/2103.12587/) (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 Over Graphs](https://mlsanity.com/paper/1803.05776/) (2018)
- [Gaussian processes at the Helm(holtz): A more fluid model for ocean currents](https://mlsanity.com/paper/2302.10364/) (2023)
- [Non-separable Spatio-temporal Graph Kernels via SPDEs](https://mlsanity.com/paper/2111.08524/) (2021)
- [Edges in Brain Networks: Contributions to Models of Structure and Function](https://mlsanity.com/paper/2105.07069/) (2021)
- [The decomposition of the higher-order homology embedding constructed from the k-Laplacian](https://mlsanity.com/paper/2107.10970/) (2021)
- [Intrinsic Gaussian Vector Fields on Manifolds](https://mlsanity.com/paper/2310.18824/) (2023)
- [Helmholtzian Eigenmap: Topological feature discovery & edge flow learning from point cloud data](https://mlsanity.com/paper/2103.07626/) (2021)
- [Isotropic Gaussian Processes on Finite Spaces of Graphs](https://mlsanity.com/paper/2211.01689/) (2022)
- [Structure of complex networks: Quantifying edge-to-edge relations by failure-induced flow redistribution](https://mlsanity.com/paper/1303.6241/) (2013)
- [Hodge-Decomposition of Functional Brain Networks](https://mlsanity.com/paper/2211.10542/) (2022)

## Built on by (most cited first, 16 of 16)

- [Gaussian Processes on Cellular Complexes](https://mlsanity.com/paper/2311.01198/) (2023)
- [MANTRA: The Manifold Triangulations Assemblage](https://mlsanity.com/paper/2410.02392/) (2024)
- [The GeometricKernels Package: Heat and Mat'ern Kernels for Geometric Learning on Manifolds, Meshes, and Graphs](https://mlsanity.com/paper/2407.08086/) (2024)
- [Topological Flow Matching](https://mlsanity.com/paper/2606.15897/) (2026)
- [Topological Schr"odinger Bridge Matching](https://mlsanity.com/paper/2504.04799/) (2025)
- [Graph Classification Gaussian Processes via Hodgelet Spectral Features](https://mlsanity.com/paper/2410.10546/) (2024)
- [Residual Deep Gaussian Processes on Manifolds](https://mlsanity.com/paper/2411.00161/) (2024)
- [Simplicial Gaussian Models: Representation and Inference](https://mlsanity.com/paper/2510.12983/) (2025)
- [Graph and Simplicial Complex Prediction Gaussian Process via the Hodgelet Representations](https://mlsanity.com/paper/2505.10877/) (2025)
- [Discrete Gaussian Vector Fields On Meshes](https://mlsanity.com/paper/2507.20024/) (2025)
- [Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes](https://mlsanity.com/paper/2602.03055/) (2026)
- [Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding](https://mlsanity.com/paper/2505.17939/) (2025)
- [Families of Optimal Transport Kernels for Cell Complexes](https://mlsanity.com/paper/2507.16569/) (2025)
- [Colored Markov Random Fields for Probabilistic Topological Modeling](https://mlsanity.com/paper/2512.03727/) (2025)
- [Topological Signal Processing: An Application-Oriented Tutorial](https://mlsanity.com/paper/2605.22853/) (2026)
- [Kernel Regression with Tensor Trains and Hadamard Overparameterization](https://mlsanity.com/paper/2607.17390/) (2026)

## Abstract (arXiv)

We propose principled Gaussian processes (GPs) for modeling functions defined over the edge set of a simplicial 2-complex, a structure similar to a graph in which edges may form triangular faces. This approach is intended for learning flow-type data on networks where edge flows can be characterized by the discrete divergence and curl. Drawing upon the Hodge decomposition, we first develop classes of divergence-free and curl-free edge GPs, suitable for various applications. We then combine them to create Hodge-compositional edge GPs that are expressive enough to represent any edge function. These GPs facilitate direct and independent learning for the different Hodge components of edge functions, enabling us to capture their relevance during hyperparameter optimization. To highlight their practical potential, we apply them for flow data inference in currency exchange, ocean currents and water supply networks, comparing them to alternative models.