Hypergraph Neural Sheaf Diffusion: A Symmetric Simplicial Set Framework for Higher-Order Learning

TL;DR

This paper introduces a novel symmetric simplicial set framework for hypergraphs, enabling higher-order neural diffusion that preserves relational details and improves learning performance.

Contribution

It develops the symmetric simplicial lifting for hypergraphs and extends neural sheaf diffusion to this setting, addressing orientation and adjacency challenges.

Findings

HNSD performs competitively on benchmark datasets.

The framework preserves all structural information from hypergraphs.

Mathematically consistent with traditional graph sheaf Laplacians.

Abstract

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.