Demystifying Oversmoothing in Sheaf Neural Networks: An Index-Theoretic Criterion

2026-08-17

AI Summary

This study addresses the over-smoothing problem in deep neural networks and the unreliability of existing harmonic space dimension metrics by proposing a relative geometric criterion grounded in index theory to precisely characterize over-smoothing resistance. Pioneering a relative geometric analysis perspective, this work integrates local tangent space linearization with the stalk structure of curved gyrovector spaces to construct a nonlinear gyrosheaf diffusion model, thereby overcoming the limitations of absolute dimensionality. Extensive experiments across ten architectures validate the effectiveness of the proposed criterion. Notably, the gyrosheaf diffusion model successfully maintains representation stability and prevents collapse in deep networks, confirming the theoretical validity of this approach. These findings establish a robust framework for analyzing and mitigating over-smoothing through rigorous geometric principles rather than conventional spectral methods.

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.

Problem

• Oversmoothing
• Sheaf Neural Networks
• Harmonic Space
• Index Theory
• Anti-oversmoothing Capacity

Innovation

• Sheaf Neural Networks
• Index-Theoretic Criterion
• Oversmoothing
• GyroSheaf
• Harmonic Space