Hierarchical Pooling for Sheaf Neural Networks

Sheaf neural networks (SNNs) extend graph neural networks (GNNs) by operating on cellular sheaves, which assign to each node and edge its own vector space, along with linear restriction maps from node spaces to the spaces associated with their incident edges. These maps determine the sheaf Laplacian and can be learned at each layer, enabling layer-specific diffusion dynamics. Existing SNN architectures, architectures, however, can only operate at a fixed graph resolution. We introduce Hierarchical Sheaf Pooling (HiSP), a pooling layer that enables hierarchical representation learning in SNNs. Following the Select–Reduce–Connect framework, HiSP groups fine-scale nodes, reduces their features using the fine-scale sheaf structure, and constructs the corresponding coarse graph. A new sheaf is then learned at the coarse resolution. By interleaving sheaf diffusion and pooling, HiSP enables hierarchical SNNs to learn distinct diffusion dynamics across layers and graph resolutions. This construction supports multiscale problems in which aggregation changes both the interacting entities and their relations.