🤖 AI Summary
To address the limited expressive power of conventional hypergraphs in modeling higher-order relationships, this paper introduces the sheaf structure—previously unexplored in hypergraph learning—yielding the sheaf hypergraph model. We define both linear and nonlinear sheaf hypergraph Laplacians to explicitly encode local higher-order dependencies between nodes and hyperedges, thereby endowing the model with stronger inductive biases. Building upon this, we propose the Sheaf Hypergraph Neural Network (SheafHGNN), which integrates sheaf-based convolution with message-passing mechanisms. Extensive experiments on multiple hypergraph node classification benchmarks demonstrate that SheafHGNN consistently outperforms state-of-the-art methods, validating the effectiveness and generalization advantage of sheaf structures for capturing higher-order topological patterns.
📝 Abstract
Higher-order relations are widespread in nature, with numerous phenomena involving complex interactions that extend beyond simple pairwise connections. As a result, advancements in higher-order processing can accelerate the growth of various fields requiring structured data. Current approaches typically represent these interactions using hypergraphs. We enhance this representation by introducing cellular sheaves for hypergraphs, a mathematical construction that adds extra structure to the conventional hypergraph while maintaining their local, higherorder connectivity. Drawing inspiration from existing Laplacians in the literature, we develop two unique formulations of sheaf hypergraph Laplacians: linear and non-linear. Our theoretical analysis demonstrates that incorporating sheaves into the hypergraph Laplacian provides a more expressive inductive bias than standard hypergraph diffusion, creating a powerful instrument for effectively modelling complex data structures. We employ these sheaf hypergraph Laplacians to design two categories of models: Sheaf Hypergraph Neural Networks and Sheaf Hypergraph Convolutional Networks. These models generalize classical Hypergraph Networks often found in the literature. Through extensive experimentation, we show that this generalization significantly improves performance, achieving top results on multiple benchmark datasets for hypergraph node classification.
