Organizations: Department of Mathematics, Faculty of Science National University of Singapore Singapore · School of Physical and Mathematical Sciences Nanyang Technological University Singapore
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 L. Existing analyses connect sheaf diffusion to oversmoothing via the harmonic space (kerL), taking its absolute dimension as an indicator of anti-oversmoothing capacity. However, absolute dimension alone is not a reliable measure: certain sheaf configurations inflate dimkerL 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 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.
Figures & tables
Figure 1: Normalized Dirichlet energy (top) and MAD (bottom) on Cora, Citeseer, and Texas in the untrained regime. Curves and shaded bands show per-seed mean and min–max range over five seeds.
Figure 2: Normalized Dirichlet energy (left), MAD (middle), and test accuracy (right) on Cora in trained regime. Legend and shading as in Figure 1 .
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: Asymptotic decay of R(F) in trivial sheaves ( dim(H0)=1) .
Figure 4: Mitigation of representation collapse through intermediate harmonic enrichment.
Figure 5: Robust information preservation in high-capacity geometric sheaf frameworks.
Hyperparameter
Value
Learning rate
0.004
Weight decay
10−4
Epochs
220
Patience
50
Label smoothing
0.05
LR scheduler
ReduceLROnPlateau
Appendix
Table 1: Training hyperparameters (shared across all model families).
Figure 6: PCA projections of input features (left), GyroSheaf (middle), and SPDSheaf (right) at the clean ( η=0 , top) and strongly-perturbed ( η=0.8 , bottom) endpoints; ellipses show per-class covariance. Under strong perturbation, SPDSheaf collapses into a narrow band while GyroSheaf retains separable class structure. Right: test accuracy versus AffErr(X) . SPDSheaf degrades monotonically as AffErr grows; GyroSheaf remains flat.