Gaussian Sheaf Neural Networks

TL;DR

Gaussian Sheaf Neural Networks (GSNNs) extend graph neural networks to effectively handle Gaussian-distributed node features by leveraging sheaf theory, preserving geometric structure and improving learning on relational data.

Contribution

The paper introduces GSNNs, a novel framework based on sheaf theory, for graph learning with Gaussian node features, including a new Laplacian operator that maintains key properties.

Findings

GSNNs outperform standard GNNs on synthetic and real-world datasets.

The proposed Laplacian preserves geometric and algebraic structures of Gaussian features.

Theoretical analysis confirms the properties of the sheaf Laplacian.

Abstract

Graph Neural Networks (GNNs) have become the de facto standard for learning on relational data. While traditional GNNs' message passing is well suited for vector-valued node features, there are cases in which node features are better represented by probability distributions than real vectors. Concretely, when node features are Gaussians, characterized by a mean and a covariance matrix, naively concatenating their parameters into a single vector and applying standard message passing discards the geometric and algebraic structure that governs means and covariances. We propose Gaussian Sheaf Neural Networks (GSNNs), a principled framework that incorporates these inductive biases into graph-based learning. Building on the theory of cellular sheaves, we derive a new Laplacian operator that generalizes the sheaf Laplacian to this setting and preserves its key properties. We complement our theoretical contributions with experiments on synthetic and real-world data that illustrate the practical relevance of GSNNs.