OmniSheaf: Unifying Sheaf Hypergraph Networks via Efficient Universal Approximators

Higher-order relations among multiple entities frequently appear in fields such as neuroscience, radio allocation and computer networks. These relations can be modeled via hypergraphs, i.e., topological structures that extend traditional graphs via hyperedges connecting multiple nodes at once. Recent work has introduced Sheaf Hypergraph Networks (SHNs), which further augment this topological structure via a cellular sheaf. The latter attaches d-dimensional vector spaces named stalks to each node and hyperedge, as well as learnable linear transformations called restriction maps between them. The key limitation of current sheaf methods for hypergraphs is that they compute the graph clique, which scales quadratically with the number of elements of each hyperedge. Ultimately, this causes loss of valuable topological information and makes computation prohibitive. In this paper, we introduce OmniSheaf (OS), a new Sheaf Neural Network (SNN) that for the first time works directly over hypergraphs via multiset functions over stalk spaces. We mathematically prove that OS generalizes SNNs, SHNs, AllSet, DHNN, and Dir-GNN via its universal approximators and that it can be made directionality-aware over hyperedges via both a complex phase and tail-head indices. Extensive experiments across 25 benchmarks against hypergraph networks, set-based architectures, and sheaf models demonstrate that OS achieves up to +14.37 higher accuracy on undirected node classification benchmarks (on Senate, while reaching 95.58% on Congress) and +4.11 on directed ones, while on Roman-Empire it trains 7.1× faster than DSHNLight, uses 3.6× less peak GPU memory and 14.9× fewer training FLOPs at equal stalk dimension.