Demystifying Oversmoothing in Sheaf Neural Networks: An Index-Theoretic Criterion
2026-08-17 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study Sheaf Neural Networks (SNNs), which are designed to avoid a problem called oversmoothing in Graph Convolutional Networks. They find that just counting certain mathematical features (the dimension of the harmonic space) is not enough to tell if an SNN truly avoids oversmoothing. Instead, they develop a new geometric method to better measure this ability and prove criteria to compare different sheaf structures meaningfully. They also propose a new sheaf type called GyroSheaf that works in more complex, curved settings. Experiments show their method reliably predicts when models maintain useful features as they get deeper.
Graph Convolutional NetworksOversmoothingSheaf Neural NetworksSheaf LaplacianHarmonic spaceIndex theoryStalk transportationGyroSheafGyrovector spacesLocal linearization
Authors
Junwen Dong, Yuhan Peng, Hao Li, Huitao Feng, Kelin Xia
Abstract
To combat oversmoothing in Graph Convolutional Networks, Sheaf Neural Networks (SNNs) were proposed as a generalization by equipping the graph with a sheaf structure and replacing the graph Laplacian with a sheaf Laplacian $\mathcal{L}$. Existing analyses connect sheaf diffusion to oversmoothing via the harmonic space ($\ker\mathcal{L}$), taking its absolute dimension as an indicator of anti-oversmoothing capacity. However, absolute dimension alone is not a reliable measure: certain sheaf configurations inflate $\dim \ker \mathcal{L}$ while their harmonic sections remain entirely constant, without enriching discriminative capacity. We instead introduce the first relative, geometric approach, yielding a precise characterisation of anti-oversmoothing capacity. Under natural conditions on stalk transportation and global sheaf structure, we establish an index-theoretic comparison criterion showing that one sheaf's harmonic space genuinely contains another's beyond trivial inflation. We illustrate this with a concrete instance and further introduce \textit{GyroSheaf}, a sheaf with curved gyrovector-space stalks, extending the criterion to the non-linear setting via local tangent-space linearization. Experiments across ten models confirm the theoretical criterion: sheaf models violating the criterion collapse despite possessing index jumps, while compliant models maintain depth-stable representations.