On the Necessity of Learnable Sheaf Laplacians
This work questions the necessity of a core design component in Sheaf Neural Networks, potentially simplifying their architecture for researchers and practitioners working with heterophilous graphs.
This paper investigates whether learnable restriction maps are necessary for Sheaf Neural Networks (SNNs) to mitigate oversmoothing on heterophilous graphs. They introduce an Identity Sheaf Network baseline, where restriction maps are fixed to identity, and find it achieves comparable performance to SNN variants across five heterophilous benchmarks. They also show that the empirical oversmoothing behavior in trained networks does not align with theoretical predictions for SNNs.
Sheaf Neural Networks (SNNs) were introduced as an extension of Graph Convolutional Networks to address oversmoothing on heterophilous graphs by attaching a sheaf to the input graph and replacing the adjacency-based operator with a sheaf Laplacian defined by (learnable) restriction maps. Prior work motivates this design through theoretical properties of sheaf diffusion and the kernel of the sheaf Laplacian, suggesting that suitable non-identity restriction maps can avoid representations converging to constants across connected components. Since oversmoothing can also be mitigated through residual connections and normalization, we revisit a trivial sheaf construction to ask whether the additional complexity of learning restriction maps is necessary. We introduce an Identity Sheaf Network baseline, where all restriction maps are fixed to the identity, and use it to ablate the empirical improvements reported by sheaf-learning architectures. Across five popular heterophilic benchmarks, the identity baseline achieves comparable performance to a range of SNN variants. Finally, we introduce the Rayleigh quotient as a normalized measure for comparing oversmoothing across models and show that, in trained networks, the behavior predicted by the diffusion-based analysis of SNNs is not reflected empirically. In particular, Identity Sheaf Networks do not appear to suffer more significant oversmoothing than their SNN counterparts.