Research map / Graph learning / Graph algorithms and clustering
Gaussian Processes on Cellular Complexes
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Defines Gaussian processes on cellular complexes via random cochains, proposing a cellular Matérn kernel and a reaction-diffusion kernel that mixes information across cell types, evaluated on synthetic and ocean-current/electromagnetic field data.
Problem
Graph GPs model only dyadic (vertex-pair) relations and cannot represent polyadic interactions, oriented signals on edges, or jointly modelled signals across vertices, edges and higher-order cells with uncertainty.
Approach
Extends GPs from graphs to cellular complexes by defining GPs as direct sums of Gaussian random cochains, deriving a Matérn kernel from the Hodge Laplacian and a reaction-diffusion kernel from the Dirac operator that mixes cochains of different orders.
Key results
- On ocean current magnitude, the edge Matérn GP achieves NLL -703.42 vs -684.54 for the graph Matérn baseline, with comparable MSE (0.029 vs 0.030).
- On synthetic cellular data, the reaction-diffusion GP reduces vertex MSE from 0.165 to 0.076 relative to the CC-Matérn kernel.
- On electromagnetic fields, RD outperforms independent Matérn on 8/10 seeds for E and B fields, but gains are small and the MSE for scalar potential is slightly worse.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the cellular-complex Matérn kernel on edges instead of a graph Matérn kernel on the line graph | Ocean current magnitude prediction · Ocean current magnitude | MSE: 0.030 ± 0.000 -> 0.029 ± 0.001 | No clear effect |
| Use the cellular-complex Matérn kernel on edges instead of a graph Matérn kernel on the line graph | Ocean current magnitude prediction · Ocean current magnitude | NLL: -684.54 ± 4.20 -> -703.42 ± 5.10 | Helped |
| Mix information across vertices, edges, and triangles with the reaction-diffusion kernel | CC-GP · Synthetic vertices | MSE: 0.165 ± 0.005 -> 0.076 ± 0.004 | Helped |
| Mix information across vertices, edges, and triangles with the reaction-diffusion kernel | CC-GP · Synthetic edges | MSE: 0.335 ± 0.014 -> 0.200 ± 0.010 | Helped |
| Mix information across vertices, edges, and triangles with the reaction-diffusion kernel | CC-GP · Synthetic triangles | MSE: 0.272 ± 0.005 -> 0.166 ± 0.005 | Helped |
| Mix information across vertices, edges, and triangles with the reaction-diffusion kernel | CC-GP · Synthetic vertices | NLL: 28.81 ± 1.75 -> -9.31 ± 1.61 | Helped |
| Mix information across vertices, edges, and triangles with the reaction-diffusion kernel | CC-GP · Synthetic edges | NLL: 136.77 ± 4.32 -> 71.07 ± 6.29 | Helped |
| Mix information across vertices, edges, and triangles with the reaction-diffusion kernel | CC-GP · Synthetic triangles | NLL: 82.84 ± 1.56 -> 39.78 ± 2.47 | Helped |
| Jointly model field types with the reaction-diffusion kernel instead of independent Matérn GPs | Electromagnetic field prediction · Electromagnetic scalar potential (V) | MSE: 0.102 ± 0.008 -> 0.11 ± 0.011 | Hurt |
| Jointly model field types with the reaction-diffusion kernel instead of independent Matérn GPs | Electromagnetic field prediction · Electromagnetic electric field (E) | MSE: 0.130 ± 0.008 -> 0.128 ± 0.009 | Helped |
| Jointly model field types with the reaction-diffusion kernel instead of independent Matérn GPs | Electromagnetic field prediction · Electromagnetic magnetic field (B) | MSE: 0.190 ± 0.026 -> 0.178 ± 0.023 | Helped |
| Jointly model field types with the reaction-diffusion kernel instead of independent Matérn GPs | Electromagnetic field prediction · Electromagnetic scalar potential (V) | NLL: 45.75 ± 7.20 -> 45.41 ± 7.35 | Helped |
Surprising findings
- On the electromagnetic experiment the RD kernel does not improve scalar potential MSE, and gains are much smaller than in the imposed-correlation experiment.
Limitations
- Eigendecomposition cost grows quickly with network size.
- Reaction-diffusion kernel controlled by only three hyperparameters (r, c, d), possibly too restrictive.
- Experiments are on simple 2D domains; no boundary conditions or complex manifolds explored.
Experimental setup
Datasets: geostrophic current data (ocean current database) synthetic simplicial mesh signals PyCharge simulated electromagnetic fields
Benchmarks: geostrophic ocean current (Southern Africa) synthetic 2D simplicial mesh signals synthetic electromagnetic fields (V, E, B)
Metrics: MSE NLL
Baselines: Graph Matérn GP on line graph Independent Matérn GPs on cellular complex
Builds on: graph Matérn GP Hodge Laplacian Dirac operator Gaussian Markov random fields
Closest papers by meaning
- Graph and Simplicial Complex Prediction Gaussian Process via Hodgelet Representations (NeurIPS 2026 · similarity 0.85)
- Deep Gaussian Processes on Directed Acyclic Graphs (NeurIPS 2026 · similarity 0.84)
- Higher-Order Graph Convolutional Network with Flower-Petals Laplacians on Simplicial Complexes (AAAI 2024 · similarity 0.84)
- From Latent Graph to Latent Topology Inference: Differentiable Cell Complex Module (ICLR 2024 · similarity 0.84)
- Uncertainty Estimation on Graphs with Structure Informed Stochastic Partial Differential Equations (NeurIPS 2025 · similarity 0.84)
- Rethinking Cancer Gene Identification Through Graph Anomaly Analysis (AAAI 2025 · similarity 0.83)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.