Hypergraph Neural Sheaf Diffusion: A Symmetric Simplicial Set Framework for Higher-Order Learning
This addresses a fundamental problem in hypergraph learning for researchers in graph theory and machine learning, though it appears incremental as it extends neural sheaf diffusion from graphs to hypergraphs.
The paper tackles the challenge of constructing sheaf Laplacians for hypergraphs due to missing adjacency relations and orientation systems, by introducing symmetric simplicial sets that encode oriented subrelations and preserve hyperedge provenance, resulting in the Hypergraph Neural Sheaf Diffusion (HNSD) method that shows competitive performance on benchmarks.
The absence of intrinsic adjacency relations and orientation systems in hypergraphs creates fundamental challenges for constructing sheaf Laplacians of arbitrary degrees. We resolve these limitations through symmetric simplicial sets derived directly from hypergraphs, called symmetric simplicial lifting, which encode all possible oriented subrelations within each hyperedge as ordered tuples. This construction canonically defines adjacency via facet maps while inherently preserving hyperedge provenance. We establish that the normalized degree zero sheaf Laplacian on our symmetric simplicial lifting reduces exactly to the traditional graph normalized sheaf Laplacian when restricted to graphs, validating its mathematical consistency with prior graph-based sheaf theory. Furthermore, the induced structure preserves all structural information from the original hypergraph, ensuring that every multi-way relational detail is faithfully retained. Leveraging this framework, we introduce Hypergraph Neural Sheaf Diffusion (HNSD), the first principled extension of neural sheaf diffusion to hypergraphs. HNSD operates via normalized degree zero sheaf Laplacian over symmetric simplicial lifting, resolving orientation ambiguity and adjacency sparsity inherent to hypergraph learning. Experimental evaluations demonstrate HNSDs competitive performance across established benchmarks.