On the Necessity of Learnable Sheaf Laplacians

TL;DR

This paper questions the necessity of learnable sheaf Laplacians in Sheaf Neural Networks by demonstrating that a simple identity restriction map baseline performs comparably to more complex models across multiple benchmarks.

Contribution

The study introduces an Identity Sheaf Network baseline and shows that learnable restriction maps are not essential for performance or oversmoothing mitigation in SNNs.

Findings

Identity baseline achieves similar accuracy to complex SNN variants.

Learnable restriction maps do not significantly reduce oversmoothing.

Diffusion-based analysis does not match empirical oversmoothing behavior.

Abstract

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.