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Life sciences · Preprint
arXiv · August 17, 2026
Raises a question worth testing. It does not answer one.
This preprint introduces an index-theoretic criterion to characterize anti-oversmoothing capacity in Sheaf Neural Networks, arguing that absolute dimension of the harmonic space is insufficient and proposing a relative geometric measure instead. Experiments across ten models show that sheaf models violating the criterion collapse despite index jumps, while compliant models maintain depth-stable representations, but the work remains unreviewed theoretical work.
Theoretical analysis with computational experiments. Sheaf Neural Network configurations and Graph Convolutional Network variants; computational rather than empirical population.. Intervention: Index-theoretic criterion for anti-oversmoothing capacity; GyroSheaf with curved gyrovector-space stalks.. Compared with: Prior analyses based on absolute dimension of harmonic space (ker ℒ); sheaf models violating vs. complying with the proposed criterion..
Absolute dimension of harmonic space (ker ℒ) alone is not a reliable measure of anti-oversmoothing capacity Index-theoretic comparison criterion established showing when one sheaf's harmonic space genuinely contains another's beyond trivial inflation GyroSheaf introduced with curved gyrovector-space stalks, extending criterion to non-linear setting
Absolute dimension of harmonic space (ker ℒ) alone is not a reliable measure of anti-oversmoothing capacity Index-theoretic comparison criterion established showing when one sheaf's harmonic space genuinely contains another's beyond trivial inflation
The source did not state who this applies to in practice.
A theoretical framework with mathematical proofs and limited experimental validation across models, establishing a criterion for understanding oversmoothing in graph neural networks rather than demonstrating clinical or practical efficacy.
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Graded across the dimensions that decide whether you should act, each from what the source actually supports. There is no single score, and where a dimension was not assessed it says so.
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.
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