# On the Necessity of Learnable Sheaf Laplacians

arXiv: https://arxiv.org/abs/2603.05395 · HTML: https://mlsanity.com/paper/2603.05395/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 15 references that are in the map.
- First posted on arXiv 5 Mar 2026.

## Builds on (most cited first, 10 of 10)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [A Note on Over-Smoothing for Graph Neural Networks](https://mlsanity.com/paper/2006.13318/) (2020)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Sheaf Neural Networks with Connection Laplacians](https://mlsanity.com/paper/2206.08702/) (2022)
- [Residual Connections and Normalization Can Provably Prevent Oversmoothing in GNNs](https://mlsanity.com/paper/2406.02997/) (2024)
- [Can Classic GNNs Be Strong Baselines for Graph-level Tasks? Simple Architectures Meet Excellence](https://mlsanity.com/paper/2502.09263/) (2025)
- [Understanding Heterophily for Graph Neural Networks](https://mlsanity.com/paper/2401.09125/) (2024)
- [Cooperative Sheaf Neural Networks](https://mlsanity.com/paper/2507.00647/) (2025)
- [Joint Diffusion Processes as an Inductive Bias in Sheaf Neural Networks](https://mlsanity.com/paper/2407.20597/) (2024)

## Built on by (most cited first, 5 of 5)

- [Do Sheaf Neural Networks Use Holonomy? A Measure--Intervene--Control Study](https://mlsanity.com/paper/2607.19514/) (2026)
- [Learned, Relied Upon, or Necessary? Separating Checkpoint Dependence from Task-Level Value in Sheaf GNNs](https://mlsanity.com/paper/2607.25387/) (2026)
- [Demystifying Oversmoothing in Sheaf Neural Networks: An Index-Theoretic Criterion](https://mlsanity.com/paper/2608.16180/) (2026)
- [Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs](https://mlsanity.com/paper/2608.28853/) (2026)
- [Let the Heads Talk: Beyond Diagonal Graph Attention](https://mlsanity.com/paper/2610.01494/) (2026)

## Abstract (arXiv)

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.