Deep Neural Sheaf Diffusion

Remi Bourgerie; Sarunas Girdzijauskas; Viktoria Fodor

arxiv: 2605.19021 · v1 · pith:TLZUP7LVnew · submitted 2026-05-18 · 💻 cs.LG

Deep Neural Sheaf Diffusion

Remi Bourgerie , Sarunas Girdzijauskas , Viktoria Fodor This is my paper

Pith reviewed 2026-05-20 12:05 UTC · model grok-4.3

classification 💻 cs.LG

keywords deep graph neural networksneural sheaf diffusionsheaf adjacency operatorlong-range dependenciesgraph attention mechanismsrepresentation collapse

0 comments

The pith

Replacing the sheaf Laplacian with a sheaf adjacency operator lets deep sheaf diffusion keep disagreement signals alive across layers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper identifies that Neural Sheaf Diffusion loses its useful disagreement signal as layers stack, even though theory promises no collapse. Deep Neural Sheaf Diffusion fixes this by swapping in a sheaf adjacency operator and adding normalization, odd nonlinearities, and gating so that deeper layers still contribute. This setup lets the model use many layers of aggregation on graph data. Tests show gains of up to 30 percentage points on synthetic long-range tasks and steady wins on real benchmarks. The design is also contrasted with attention models by using matrix-valued edge functions and normalizing node states rather than scores.

Core claim

Deep Neural Sheaf Diffusion replaces the sheaf Laplacian with a sheaf adjacency operator, together with normalization and gating, to preserve an informative disagreement signal across layers and support effective deep aggregation in graph tasks.

What carries the argument

Sheaf adjacency operator that replaces the Laplacian to keep edge disagreement signals from vanishing at greater depths.

If this is right

Deeper layers add meaningful information instead of causing representation collapse.
Performance improves on tasks that require information to travel far across the graph.
Sheaf-based models become practical building blocks for very deep graph networks.
Matrix-valued edge functions and node normalization distinguish the method from standard attention.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar operator changes could be tested in other diffusion-style graph models to reach greater depth.
The matrix-valued edge functions may give richer pairwise interactions than scalar attention scores.
Stable deep sheaf layers could be combined into larger architectures for graph-scale foundation models.

Load-bearing premise

The replacement of the sheaf Laplacian by a sheaf adjacency operator, together with the added normalization and gating, will preserve an informative disagreement signal at arbitrary depth without introducing new instabilities or requiring dataset-specific tuning.

What would settle it

Train both DNSD and NSD at increasing depths on the same synthetic long-range graph datasets and measure whether the disagreement signal stays away from zero and whether accuracy keeps rising rather than plateauing.

Figures

Figures reproduced from arXiv: 2605.19021 by Remi Bourgerie, Sarunas Girdzijauskas, Viktoria Fodor.

**Figure 1.** Figure 1: As depth increases (L ≫ 1), standard message passing mechanisms degrade: graph attention (a) collapses representations, while neural sheaf diffusion (b) produces vanishing signals that limits the effective depth. Deep Neural Sheaf Diffusion (c) maintains informative signal at depth. model architectures that scale in depth. The rest of the paper is organized as follows. Section 2 introduces the necessary … view at source ↗

**Figure 2.** Figure 2: Test accuracy on the synthetic community detection dataset G5 as a function of depth. Each curve corresponds to a model variant, where adj, odd, and gate denote the use of adjacency-based operator, odd nonlinearities, and gating, respectively, and diag / full refer to diagonal or full restriction maps. Shaded regions represent one standard deviation over multiple runs. izations to future work. To assess th… view at source ↗

read the original abstract

Deep Graph Neural Networks (GNNs) are essential for capturing complex dependencies in graph-structured data. However, scaling GNNs to depth remains challenging, as stacking layers leads to representation collapse and diminishing sensitivity due to repeated aggregation. While Neural Sheaf Diffusion (NSD) provides strong theoretical guarantees against such collapse, these guarantees do not translate to practice: as depth increases, the disagreement signal of the sheaf Laplacian vanishes, limiting the contribution of deeper layers. We identify mechanisms that hinder NSD effectiveness at depth and propose \emph{Deep Neural Sheaf Diffusion} (DNSD), which replaces the sheaf Laplacian with a sheaf adjacency operator to maintain informative signals across layers. This is complemented by normalization, odd nonlinearities, and gating. To provide a principled explanation of the expected performance improvement, we contrast sheaf diffusion to graph attention mechanisms, highlighting that DNSD replaces scalar attention scores with matrix-valued edge functions and normalizes node representations rather than attention scores. We demonstrate empirically that DNSD effectively utilizes deep aggregation in graph tasks, outperforming GNN and NSD baselines with up to 30pp accuracy on synthetic long-range datasets, and consistently outperforming them on real-world benchmarks. These results position sheaf-based architectures as a promising building block for graph foundation models by supporting effective deep architectures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper swaps the sheaf Laplacian for an adjacency operator plus normalization and gating to scale NSD deeper, with decent empirical lifts but no analysis to show the signal actually stays stable.

read the letter

Hi, the main point is that this work replaces the sheaf Laplacian with a sheaf adjacency operator, then layers on odd nonlinearities, gating, and node normalization to stop the disagreement signal from vanishing as depth grows in neural sheaf diffusion. That combination is the concrete new move inside the existing NSD framework, and they contrast it to scalar attention by using matrix-valued edge functions and normalizing nodes instead of scores. Empirically they report up to 30 percentage point gains on synthetic long-range tasks and consistent wins over GNN and NSD baselines on real graphs, which suggests the tweaks make deeper aggregation practical for these models. The experiments at least demonstrate that the changes let deeper layers contribute on the tasks they tested. The soft spot is the missing justification for why the new operator keeps an informative signal alive without new instabilities or heavy retuning. They note the Laplacian collapse problem in prior NSD but give only a qualitative attention comparison rather than eigenvalue bounds, contraction arguments, or any fixed-point analysis for the adjacency version. Without that, it is hard to know whether the gains generalize or just work under specific conditions. The abstract also skips error bars, ablation details, and full experimental controls, so the performance numbers are difficult to assess precisely. This paper is for people already working on sheaf diffusion or depth scaling in graph representation learning. A reader focused on long-range graph tasks or extensions of NSD would pick up the operator substitution and the reported results as useful pieces. I would send it for peer review. The idea targets a real limitation and comes with some supporting runs, even if the theory and experimental reporting need tightening.

Referee Report

3 major / 3 minor

Summary. The manuscript introduces Deep Neural Sheaf Diffusion (DNSD) to address vanishing disagreement signals in deep Neural Sheaf Diffusion (NSD). It replaces the sheaf Laplacian with a sheaf adjacency operator, adds normalization, odd nonlinearities and gating, contrasts the approach with scalar graph attention via matrix-valued edge functions and node normalization, and reports empirical outperformance of up to 30pp accuracy on synthetic long-range tasks plus consistent gains on real-world benchmarks.

Significance. If the modifications preserve an informative per-edge disagreement signal at arbitrary depth without introducing instabilities or requiring extensive retuning, the work would provide a concrete route to deeper sheaf-based GNNs capable of long-range aggregation. The dual evaluation on synthetic long-range and real-world data, together with the explicit contrast to attention mechanisms, would strengthen the case for sheaf diffusion as a building block for deeper graph architectures.

major comments (3)

[§3] §3 (Operator Definition): The central claim that the sheaf adjacency operator (together with normalization and gating) preserves an informative disagreement signal at arbitrary depth rests on a qualitative motivation from Laplacian collapse in NSD, yet no eigenvalue bounds, contraction-mapping argument, or spectral radius analysis is supplied for the new operator; this is load-bearing for the assertion that deep aggregation becomes effective.
[Experimental Results] Experimental Results (synthetic long-range tables): The reported gains of up to 30pp accuracy are presented without error bars, without ablation isolating the adjacency operator versus normalization/gating/odd nonlinearities, and without details on hyperparameter sensitivity or number of runs; these omissions directly affect confidence in the robustness of the outperformance claim.
[§5] §5 (Attention Contrast): The principled explanation contrasts matrix-valued edge functions and node normalization in DNSD against scalar attention scores, but supplies no quantitative derivation or controlled experiment showing that this structural difference accounts for the observed depth-wise gains rather than other implementation choices.

minor comments (3)

[Abstract] The abstract states 'up to 30pp accuracy' without naming the exact baseline and dataset in the summary sentence; a parenthetical clarification would improve readability.
[Method] Notation for the sheaf adjacency operator could be aligned more explicitly with the original NSD Laplacian definition to ease comparison for readers familiar with the prior work.
[Figures] Figures depicting signal propagation over depth would benefit from shaded variance bands across multiple random seeds.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive feedback and for recognizing the potential of DNSD to enable deeper sheaf-based architectures. We address each major comment below, indicating planned revisions to the manuscript.

read point-by-point responses

Referee: [§3] §3 (Operator Definition): The central claim that the sheaf adjacency operator (together with normalization and gating) preserves an informative disagreement signal at arbitrary depth rests on a qualitative motivation from Laplacian collapse in NSD, yet no eigenvalue bounds, contraction-mapping argument, or spectral radius analysis is supplied for the new operator; this is load-bearing for the assertion that deep aggregation becomes effective.

Authors: We agree that a formal spectral analysis would strengthen the central claim. The manuscript motivates the switch to the sheaf adjacency operator primarily through the observed vanishing disagreement signal under repeated Laplacian application. In the revision we will expand §3 with a discussion of the spectral radius of the normalized adjacency operator and the role of odd nonlinearities and gating in preventing contraction, including any eigenvalue bounds that follow directly from the normalization. revision: yes
Referee: [Experimental Results] Experimental Results (synthetic long-range tables): The reported gains of up to 30pp accuracy are presented without error bars, without ablation isolating the adjacency operator versus normalization/gating/odd nonlinearities, and without details on hyperparameter sensitivity or number of runs; these omissions directly affect confidence in the robustness of the outperformance claim.

Authors: We acknowledge that the current experimental presentation lacks statistical detail and component-wise ablations. The revised manuscript will report mean accuracy and standard deviation over multiple random seeds, include ablation tables that isolate the adjacency operator from normalization, odd nonlinearities and gating, and add a description of the hyperparameter search procedure together with sensitivity results for the synthetic long-range benchmarks. revision: yes
Referee: [§5] §5 (Attention Contrast): The principled explanation contrasts matrix-valued edge functions and node normalization in DNSD against scalar attention scores, but supplies no quantitative derivation or controlled experiment showing that this structural difference accounts for the observed depth-wise gains rather than other implementation choices.

Authors: Section 5 provides a conceptual contrast between matrix-valued edge functions with node normalization and scalar attention with score normalization. While a full quantitative derivation is not present, the depth-wise empirical gains are consistent with the design. We will add a controlled ablation that varies only the edge-function type (matrix versus scalar) while holding other components fixed, thereby isolating its contribution to long-range performance. revision: partial

Circularity Check

0 steps flagged

No significant circularity; core operator change and empirical claims are independent

full rationale

The paper's derivation proceeds from identifying NSD's practical signal collapse at depth, proposing the sheaf adjacency operator replacement plus normalization/gating/odd nonlinearities as a fix, and supporting this via qualitative contrast to scalar attention plus empirical gains on long-range tasks. No equations reduce a claimed prediction to a fitted input by construction, and no load-bearing step relies on a self-citation chain or imported uniqueness theorem. The modifications and results stand as independent content rather than tautological redefinitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the untested transfer of NSD theoretical guarantees to the new adjacency operator and on the empirical observation that the disagreement signal vanishes in standard NSD; no free parameters or invented entities with independent evidence are declared.

axioms (1)

domain assumption The sheaf Laplacian supplies strong theoretical guarantees against representation collapse in shallow models
Abstract states that NSD provides these guarantees yet they do not translate to practice at depth.

invented entities (1)

Sheaf adjacency operator no independent evidence
purpose: Maintain informative disagreement signals across many layers
Introduced as direct replacement for the sheaf Laplacian; no independent falsifiable prediction supplied.

pith-pipeline@v0.9.0 · 5757 in / 1337 out tokens · 41356 ms · 2026-05-20T12:05:45.965469+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

replaces the sheaf Laplacian with a sheaf adjacency operator to maintain informative signals across layers... complemented by normalization, odd nonlinearities, and gating
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

DNSD replaces scalar attention scores with matrix-valued edge functions and normalizes node representations rather than attention scores

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 7 internal anchors

[1]

Layer Normalization

Ba, J. L., Kiros, J. R., and Hinton, G. E. Layer normalization. arXiv preprint arXiv:1607.06450,

work page internal anchor Pith review Pith/arXiv arXiv
[2]

Bamberger, J., Barbero, F., Dong, X., and Bronstein, M. M. Bundle neural networks for message diffusion on graphs. arXiv preprint arXiv:2405.15540,

work page arXiv
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S., Bronstein, M., Veliˇckovi´c, P., and Li`o, P

7 Deep Neural Sheaf Diffusion Barbero, F., Bodnar, C., de Oc´ariz Borde, H. S., Bronstein, M., Veliˇckovi´c, P., and Li`o, P. Sheaf neural networks with connection Laplacians. InTopological, Algebraic and Geometric Learning Workshops 2022, pp. 28–36. PMLR, 2022a. Barbero, F., Bodnar, C., de Oc ´ariz Borde, H. S., and Lio, P. Sheaf attention networks. InNe...

work page arXiv 2022
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Billion-scale graph foundation models.arXiv preprint arXiv:2602.04768,

Bechler-Speicher, M., Gottlieb, Y ., Isakov, A., Abensur, D., Tavory, A., Haimovich, D., Guy, I., and Weinsberg, U. Billion-scale graph foundation models.arXiv preprint arXiv:2602.04768,

work page internal anchor Pith review arXiv
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Polynomial Neural Sheaf Diffusion: A Spectral Filtering Approach on Cellular Sheaves

Borgi, A., Silvestri, F., and Li `o, P. Polynomial neural sheaf diffusion: A spectral filtering approach on cellular sheaves.arXiv preprint arXiv:2512.00242,

work page internal anchor Pith review Pith/arXiv arXiv
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How Attentive are Graph Attention Networks?

Brody, S., Alon, U., and Yahav, E. How attentive are graph attention networks?arXiv preprint arXiv:2105.14491,

work page internal anchor Pith review Pith/arXiv arXiv
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Vision Transformers Need Registers

Darcet, T., Oquab, M., Mairal, J., and Bojanowski, P. Vision transformers need registers.arXiv preprint arXiv:2309.16588,

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L., Belieni, J., Souza, A

Ribeiro, A., Ten´orio, A. L., Belieni, J., Souza, A. H., and Mesquita, D. Cooperative sheaf neural networks.arXiv preprint arXiv:2507.00647,

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Graph Attention Networks

Veliˇckovi´c, P., Cucurull, G., Casanova, A., Romero, A., Lio, P., and Bengio, Y . Graph attention networks.arXiv preprint arXiv:1710.10903,

work page internal anchor Pith review Pith/arXiv arXiv
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Graph founda- tion models: A comprehensive survey.arXiv preprint arXiv:2505.15116,

8 Deep Neural Sheaf Diffusion Wang, Z., Liu, Z., Ma, T., Li, J., Zhang, Z., Fu, X., Li, Y ., Yuan, Z., Song, W., Ma, Y ., et al. Graph founda- tion models: A comprehensive survey.arXiv preprint arXiv:2505.15116,

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Efficient Streaming Language Models with Attention Sinks

Xiao, G., Tian, Y ., Chen, B., Han, S., and Lewis, M. Ef- ficient streaming language models with attention sinks. arXiv preprint arXiv:2309.17453,

work page internal anchor Pith review Pith/arXiv arXiv
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Sheaf diffusion goes nonlinear: Enhancing gnns with adaptive sheaf laplacians

Zaghen, O., Longa, A., Azzolin, S., Telyatnikov, L., Passerini, A., and Lio, P. Sheaf diffusion goes nonlinear: Enhancing gnns with adaptive sheaf laplacians. InICML 2024 Workshop on Geometry-grounded Representation Learning and Generative Modeling,

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Results are reported as mean ± std over 6 random train seeds {42,43,44,45,46,47} , evaluated on test graphs generated from 3 independent test seeds{100,101,102}

monitored on validation accuracy; the best checkpoint is restored at the end of training. Results are reported as mean ± std over 6 random train seeds {42,43,44,45,46,47} , evaluated on test graphs generated from 3 independent test seeds{100,101,102}. 11 Deep Neural Sheaf Diffusion Model complexity.Table 5 reports parameter counts at each model’s selected...

work page 2023
[14]

Training.We use the Adam optimiser

Best hyperparameters are selected per dataset–model combination based on validation accuracy. Training.We use the Adam optimiser. The learning rate is reduced on plateau (factor 0.5, patience 20 epochs). Early stopping is applied with a patience of 100 epochs monitored on validation accuracy; the best checkpoint is restored at the end of training. All res...

work page 2021

[1] [1]

Layer Normalization

Ba, J. L., Kiros, J. R., and Hinton, G. E. Layer normalization. arXiv preprint arXiv:1607.06450,

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

Bamberger, J., Barbero, F., Dong, X., and Bronstein, M. M. Bundle neural networks for message diffusion on graphs. arXiv preprint arXiv:2405.15540,

work page arXiv

[3] [3]

S., Bronstein, M., Veliˇckovi´c, P., and Li`o, P

7 Deep Neural Sheaf Diffusion Barbero, F., Bodnar, C., de Oc´ariz Borde, H. S., Bronstein, M., Veliˇckovi´c, P., and Li`o, P. Sheaf neural networks with connection Laplacians. InTopological, Algebraic and Geometric Learning Workshops 2022, pp. 28–36. PMLR, 2022a. Barbero, F., Bodnar, C., de Oc ´ariz Borde, H. S., and Lio, P. Sheaf attention networks. InNe...

work page arXiv 2022

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Billion-scale graph foundation models.arXiv preprint arXiv:2602.04768,

Bechler-Speicher, M., Gottlieb, Y ., Isakov, A., Abensur, D., Tavory, A., Haimovich, D., Guy, I., and Weinsberg, U. Billion-scale graph foundation models.arXiv preprint arXiv:2602.04768,

work page internal anchor Pith review arXiv

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Polynomial Neural Sheaf Diffusion: A Spectral Filtering Approach on Cellular Sheaves

Borgi, A., Silvestri, F., and Li `o, P. Polynomial neural sheaf diffusion: A spectral filtering approach on cellular sheaves.arXiv preprint arXiv:2512.00242,

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

How Attentive are Graph Attention Networks?

Brody, S., Alon, U., and Yahav, E. How attentive are graph attention networks?arXiv preprint arXiv:2105.14491,

work page internal anchor Pith review Pith/arXiv arXiv

[7] [7]

Vision Transformers Need Registers

Darcet, T., Oquab, M., Mairal, J., and Bojanowski, P. Vision transformers need registers.arXiv preprint arXiv:2309.16588,

work page internal anchor Pith review Pith/arXiv arXiv

[8] [8]

L., Belieni, J., Souza, A

Ribeiro, A., Ten´orio, A. L., Belieni, J., Souza, A. H., and Mesquita, D. Cooperative sheaf neural networks.arXiv preprint arXiv:2507.00647,

work page arXiv

[9] [9]

Graph Attention Networks

Veliˇckovi´c, P., Cucurull, G., Casanova, A., Romero, A., Lio, P., and Bengio, Y . Graph attention networks.arXiv preprint arXiv:1710.10903,

work page internal anchor Pith review Pith/arXiv arXiv

[10] [10]

Graph founda- tion models: A comprehensive survey.arXiv preprint arXiv:2505.15116,

8 Deep Neural Sheaf Diffusion Wang, Z., Liu, Z., Ma, T., Li, J., Zhang, Z., Fu, X., Li, Y ., Yuan, Z., Song, W., Ma, Y ., et al. Graph founda- tion models: A comprehensive survey.arXiv preprint arXiv:2505.15116,

work page arXiv

[11] [11]

Efficient Streaming Language Models with Attention Sinks

Xiao, G., Tian, Y ., Chen, B., Han, S., and Lewis, M. Ef- ficient streaming language models with attention sinks. arXiv preprint arXiv:2309.17453,

work page internal anchor Pith review Pith/arXiv arXiv

[12] [12]

Sheaf diffusion goes nonlinear: Enhancing gnns with adaptive sheaf laplacians

Zaghen, O., Longa, A., Azzolin, S., Telyatnikov, L., Passerini, A., and Lio, P. Sheaf diffusion goes nonlinear: Enhancing gnns with adaptive sheaf laplacians. InICML 2024 Workshop on Geometry-grounded Representation Learning and Generative Modeling,

work page 2024

[13] [13]

Results are reported as mean ± std over 6 random train seeds {42,43,44,45,46,47} , evaluated on test graphs generated from 3 independent test seeds{100,101,102}

monitored on validation accuracy; the best checkpoint is restored at the end of training. Results are reported as mean ± std over 6 random train seeds {42,43,44,45,46,47} , evaluated on test graphs generated from 3 independent test seeds{100,101,102}. 11 Deep Neural Sheaf Diffusion Model complexity.Table 5 reports parameter counts at each model’s selected...

work page 2023

[14] [14]

Training.We use the Adam optimiser

Best hyperparameters are selected per dataset–model combination based on validation accuracy. Training.We use the Adam optimiser. The learning rate is reduced on plateau (factor 0.5, patience 20 epochs). Early stopping is applied with a patience of 100 epochs monitored on validation accuracy; the best checkpoint is restored at the end of training. All res...

work page 2021