🤖 AI Summary
This work addresses the limitation of conventional graph neural networks (GNNs) in handling node features represented as Gaussian distributions, which often disregard their intrinsic geometric and algebraic structures, leading to information loss. To overcome this, the authors propose Gaussian Simplex Neural Networks (GSNNs), the first GNN framework that explicitly incorporates the geometry of the Gaussian manifold. Leveraging cellular sheaf theory, they design a message-passing mechanism tailored for Gaussian-distributed features and extend the graph Laplacian operator to the Gaussian space while rigorously preserving its essential mathematical properties. Experimental results on both synthetic and real-world datasets demonstrate that GSNNs significantly outperform baseline approaches that naively concatenate Gaussian parameters, thereby validating their effectiveness in capturing and leveraging the structural information inherent in Gaussian representations.
📝 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.