Graph and Simplicial Complex Prediction Gaussian Process via the Hodgelet Representations

TL;DR

This work extends Gaussian processes to simplicial complexes by embedding edge and higher-order simplex attributes through SC wavelet transforms and then applying the Hodge decomposition to separate signals into exact, co-exact, and harmonic components, yielding the Hodgelet representations. By pooling these components and constructing additive or product kernels over the Hodge subspaces, the approach achieves data-efficient, topology-aware predictions for SCs and graphs, with joint optimization of wavelet and kernel hyperparameters via marginal likelihood. Empirical results across synthetic vector-field tasks and real benchmarks (TUDataset, MoleculeNet, PowerGraph) show that edge information and the Hodge decomposition consistently improve predictive performance over vertex-only baselines and standard GNNs. The framework is extensible to higher-dimensional simplicial structures and offers a principled path toward geometry- and topology-aware GP models for complex non-Euclidean data.

Abstract

Predicting the labels of graph-structured data is crucial in scientific applications and is often achieved using graph neural networks (GNNs). However, when data is scarce, GNNs suffer from overfitting, leading to poor performance. Recently, Gaussian processes (GPs) with graph-level inputs have been proposed as an alternative. In this work, we extend the Gaussian process framework to simplicial complexes (SCs), enabling the handling of edge-level attributes and attributes supported on higher-order simplices. We further augment the resulting SC representations by considering their Hodge decompositions, allowing us to account for homological information, such as the number of holes, in the SC. We demonstrate that our framework enhances the predictions across various applications, paving the way for GPs to be more widely used for graph and SC-level predictions.

Figures (6)

• Figure 1: Orange vertices are the $0$-simplices, black edges are the $1$-simplices, and blue triangles are the $2$-simplices.
• Figure 2: Edge signals in the exact subspace are curl-free, while they are divergence-free in the co-exact subspace. In the harmonic subspace, they are both divergence-free and curl-free.
• Figure 3: Comparison of the WTGP/HTGP models on the vector field classification benchmarks. We consider ablation in mesh resolution. Shaded regions indicate standard error from the mean accuracy.
• Figure 4: Projection of a continuous vector field onto a regular triangular mesh by applying the de Rham map.
• Figure 5: Illustration of the random vector field data generating process.

Theorems & Definitions (2)

• Remark 1
• Remark 2