Let persistence ShePHerD your sheaves

Persistence-augmented graph neural networks enrich message passing with global topological information via persistent homology (PH). While some existing PH-based topological descriptors make the underlying graph filtration learnable, they fix a coefficient system. This imposes an unnecessary constraint that may limit the expressive power of PH-based descriptors and potentially hinder downstream predictive performance. To overcome this, we introduce ShePHerD as a model that learns the coefficient system with cellular sheaves, where we persist the resulting sheaf cohomology and enrich persistence events with the spectra of the sheaf Laplacian. Notably, we theoretically prove that integrating persistence sheaf cohomology into PH diagrams increases expressivity.