On the Necessity of Learnable Sheaf Laplacians

Ferran Hernandez Caralt; Mar Gonzàlez i Català; Adrián Bazaga; Pietro Liò

cs.LGMar 5, 2026

On the Necessity of Learnable Sheaf Laplacians

Authors: Ferran Hernandez Caralt, Mar Gonzàlez i Català, Adrián Bazaga, Pietro Liò

Organizations: Department of Computer Science and Technology, University of Cambridge, Cambridge, United Kingdom · Microsoft

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.

Figures & tables

Model	Texas	Wisconsin	Squirrel	Chameleon	Cornell
ISN	$88.01\pm 4.05$	$88.82\pm 3.83$	$53.27\pm 2.28$	$67.35\pm 1.71$	$85.95\pm 5.38$
Best-RiSNN	87.89 $\pm$ 4.28 ( -0.12 $\pm$ 4.05 ) $\approx$	88.04 $\pm$ 2.39 ( -0.78 $\pm$ 3.83 ) $\approx$	53.30 $\pm$ 3.30 ( +0.03 $\pm$ 2.28 ) $\approx$	66.58 $\pm$ 1.81 ( -0.77 $\pm$ 1.71 ) $\approx$	85.95 $\pm$ 6.14 ( 0.00 $\pm$ 5.38 ) $\approx$
Best-jDSNN	87.37 $\pm$ 5.10 ( -0.64 $\pm$ 4.05 ) $\approx$	89.22 $\pm$ 3.42 ( +0.40 $\pm$ 3.83 ) $\approx$	51.28 $\pm$ 1.80 ( -1.99 $\pm$ 2.28 ) $\approx$	66.45 $\pm$ 3.46 ( -0.90 $\pm$ 1.71 ) $\approx$	85.41 $\pm$ 4.55 ( -0.54 $\pm$ 5.38 ) $\approx$
Conn-NSD	86.16 $\pm$ 2.24 ( -1.85 $\pm$ 4.05 ) $\approx$	88.73 $\pm$ 4.47 ( -0.09 $\pm$ 3.83 ) $\approx$	45.19 $\pm$ 1.57 ( -8.08 $\pm$ 2.28 ) $\downarrow$	65.21 $\pm$ 2.04 ( -2.14 $\pm$ 1.71 ) $\downarrow$	85.95 $\pm$ 7.72 ( 0.00 $\pm$ 5.38 ) $\approx$
Best-SNN	85.95 $\pm$ 5.51 ( -2.06 $\pm$ 4.05 ) $\approx$	89.41 $\pm$ 4.74 ( +0.59 $\pm$ 3.83 ) $\approx$	56.34 $\pm$ 1.32 ( +3.07 $\pm$ 2.28 ) $\uparrow$	68.68 $\pm$ 1.73 ( +1.33 $\pm$ 1.71 ) $\approx$	86.49 $\pm$ 7.35 ( +0.54 $\pm$ 5.38 ) $\approx$
Best-NSP	87.03 $\pm$ 5.51 ( -0.98 $\pm$ 4.05 ) $\approx$	89.02 $\pm$ 3.84 ( +0.20 $\pm$ 3.83 ) $\approx$	50.11 $\pm$ 2.03 ( -3.16 $\pm$ 2.28 ) $\downarrow$	62.85 $\pm$ 1.98 ( -4.50 $\pm$ 1.71 ) $\downarrow$	76.49 $\pm$ 5.28 ( -9.46 $\pm$ 5.38 ) $\downarrow$

Table 1: Results across 5 popular heterophilic benchmarks of ISN against the results shown by Bodnar et al. (2022) ; Caralt et al. (2024) ; Suk et al. (2022) ; Barbero et al. (2022) ; Zaghen et al. (2024) ; Ribeiro et al. (2026) ; Fiorini et al. (2026) . Main numbers are

\mu\pm\sigma

. The colored annotations denote the difference vs. ISN, as

\Delta\mu\pm\sigma

( green : improvement, red : degradation, gray : no change).

Dataset	Texas	Wisconsin	Squirrel	Chameleon	Cornell
Min Gain	0.89	0.98	0.64	0.57	0.54
Max Gain	3.98	4.3	2.99	3.67	3.1
Type of Heterophily	Good	Good	Good	Good	Good

Table 2: Heterophily measures introduced by Wang et al. (2024) on the benchmarks used.

Panel of Figure 1 — Figure 1: A series of line plots, one for each dataset-neural network combination, representing the averaged (across folds) $R_{\Delta_{\mathcal{F}}}$ in red and $R_{\Delta_{\mathcal{I}}}$ in blue at each layer. These results contradict the theoretical understanding proposed by Bodnar et al. (2022) since according to Hypothesis 5.1 , the blue line should be above the red line.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 2: Rayleigh Quotients at the first and only layer of a trained SNN and ISN.

R_{\Delta_{\mathcal{F}}}

in red and

R_{\Delta_{\mathcal{I}}}

in blue at each layer.

	Texas	Wisconsin	Film	Squirrel	Chameleon	Cornell
add_hp	False	True	False	False	False	False
add_lp	False	False	False	True	True	False
d	1	1	4	1	1	1
deg_normalised	False	False	True	False	False	False
dropout	0.7	0.8	0.5	0	0	0.9
early_stopping	200	200	100	100	100	200

Appendix

Table 3: Hyperparameter configurations of the ISNs that were used in this work.

Explore similar work

Demystifying Oversmoothing in Sheaf Neural Networks: An Index-Theoretic Criterion
Aug 17, 2026Junwen Dong, Yuhan Peng, Hao Li +2Sheaf Neural Networks Graph Neural Networks
Department of Mathematics, Faculty of Science National University of Singapore Singapore · School of Physical and Mathematical Sciences Nanyang Technological University Singapore
Benchmarking Sheaf Neural Networks for Inductive Tasks
Aug 3, 2026Stefano Fiorini, Edoardo Coppola, Pietro LiòGraph Neural Networks Graph Representation Learning
1Independent Researcher · Department of Computer Science and Technology, University of Cambridge
Hierarchical Pooling for Sheaf Neural Networks
Jun 18, 2026Dionisia Naddeo, Carlo Abate, Pietro Liò +2Graph Neural Networks Graph Coarsening
UiT The Arctic University of Norway · UiT The Arctic University of Norway NORCE Norwegian Research Centre AS

Aug 17, 2026J/K move · Enter open · S save

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

Junwen Dong, Yuhan Peng, Hao Li, Huitao Feng+1

Department of Mathematics, Faculty of Science National University of Singapore Singapore · School of Physical and Mathematical Sciences Nanyang Technological University Singapore

Sheaf Neural Networks Graph Neural Networks

Open paper AskPDF arXiv

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 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.

Abstract

Explore similar work

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

Benchmarking Sheaf Neural Networks for Inductive Tasks

Hierarchical Pooling for Sheaf Neural Networks