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

• URL: http://arxiv.org/abs/2608.16180v1
• Date: Mon, 17 Aug 2026 06:52:55 GMT
• Title: Demystifying Oversmoothing in Sheaf Neural Networks: An Index-Theoretic Criterion
• Abstract summary: 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 $mathcalL$.<n>Existing analyses connect sheaf diffusion to oversmoothing via the harmonic space ($kermathcalL$), taking its absolute dimension as an indicator of anti-oversmoothing capacity.<n>We introduce the first relative, geometric approach, yielding a precise characterisation of anti-oversmoothing capacity.
• Score: 7.425166211928442
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Across the Loss Landscape with Progressive Growth [51.366966420881646]
  We show how grow-and-optimize strategies to train toward flatter regions by viewing growth as progressive relaxation.<n>We empirically validate these predictions in controlled toy basins and in a realistic ResNet/CI-100 setting.<n>Although progressive subspace growth reliably produces flatter solutions, curvature reductions do not universally translate into improved test performance, highlighting subtleties in the flatness-generalization connection.
  arXiv  Detail & Related papers  (2026-08-25T13:50:48Z)
• CSympNet-ID: conformal-symplectic map learning for linearly damped Hamiltonian systems [6.683674534366321]
  For linearly damped Hamiltonian systems, the exact flow is generally not symplectic but conformally symplectic.<n>We propose Conformal Symplectic Networks with damping identification (CSympNet-ID)<n>CSympNet-ID learns the one-step flow map directly from snapshot pairs.
  arXiv  Detail & Related papers  (2026-07-03T13:56:41Z)
• Oversmoothing as Representation Degeneracy in Neural Sheaf Diffusion [0.0]
  Neural Sheaf Diffusion (NSD) generalizes diffusion-based Graph Neural Networks by replacing scalar graph Laplacians with sheaf Laplacians.<n>We develop a quiver-theoretic interpretation of NSD by identifying cellular sheaves on graphs with representations of the associated incidence.<n>We show that direct-sum decompositions of the underlying incidence-quiver representation induce decompositions of the harmonic space reached in the diffusion limit.
  arXiv  Detail & Related papers  (2026-05-11T19:41:56Z)
• A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning [52.452902154360565]
  We introduce Mean Curvature Boundary Points (MCBP), a novel geometric framework grounded in Geometric Machine Learning.<n>MCBP explicitly models the intrinsic curvature of the data manifold and computes pointwise mean curvature without requiring principled manifold parametrization.<n>Experiments on synthetic and real-world datasets demonstrate that MCBP consistently improves clustering performance.
  arXiv  Detail & Related papers  (2026-05-05T20:19:09Z)
• Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities [0.0]
  We investigate the differential equations governing the evolution of a temporal network within Random Dot Product Graphs.<n>We identify three fundamental obstructions: gauge freedom from rotational ambiguity in latent positions, realizability constraints from the manifold structure of the probability matrix, and trajectory recovery artifacts from spectral embedding.<n>We prove an identifiability principle: symmetric dynamics cannot absorb skew-symmetric gauge contamination, so dynamics structure can resolve obstructions.
  arXiv  Detail & Related papers  (2026-03-05T21:56:07Z)
• The Inductive Bias of Convolutional Neural Networks: Locality and Weight Sharing Reshape Implicit Regularization [57.37943479039033]
  We study how architectural inductive bias reshapes the implicit regularization induced by the edge-of-stability phenomenon in gradient descent.<n>We show that locality and weight sharing fundamentally change this picture.
  arXiv  Detail & Related papers  (2026-03-05T04:50:51Z)
• Symmetry-protected topology and deconfined solitons in a multi-link $\mathbb{Z}_2$ gauge theory [45.88028371034407]
  We study a $mathbbZ$ lattice gauge theory defined on a multi-graph with links that can be visualized as great circles of a spherical shell.<n>We show that this leads to state-dependent tunneling amplitudes underlying a phenomenon analogous to the Peierls instability.<n>By performining a detailed analysis based on matrix product states, we prove that charge deconfinement emerges as a consequence of charge-fractionalization.
  arXiv  Detail & Related papers  (2026-03-02T22:59:25Z)
• Manifold limit for the training of shallow graph convolutional neural networks [1.2744523252873352]
  We study the consistency of the training of shallow graph convolutional neural networks (GCNNs) on proximity graphs of sampled point clouds.<n>We prove $$-convergence of regularized empirical risk minimization functionals and corresponding convergence of their global minimizers.
  arXiv  Detail & Related papers  (2026-01-09T18:59:20Z)
• Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization [48.25304391127552]
  Estimating the tangent spaces of a data manifold is a fundamental problem in data analysis.<n>We propose a method, Laplacian Eigenvector Gradient Orthogonalization (LEGO), that utilizes the global structure of the data to guide local tangent space estimation.
  arXiv  Detail & Related papers  (2025-10-02T17:59:45Z)
• Learning Latent Graph Geometry via Fixed-Point Schrödinger-Type Activation: A Theoretical Study [1.1745324895296467]
  We develop a unified theoretical framework for neural architectures with internal representations evolving as stationary states of dissipative Schr"odinger-type dynamics on learned latent graphs.<n>We prove existence, uniqueness, and smooth dependence of equilibria, and show that the dynamics are equivalent under the Bloch map to norm-preserving Landau--Lifshitz flows.<n>The resulting model class provides a compact, geometrically interpretable, and analytically tractable foundation for learning latent graph geometry via fixed-point Schr"odinger-type activations.
  arXiv  Detail & Related papers  (2025-07-27T00:35:15Z)
• High-probability Convergence Bounds for Nonlinear Stochastic Gradient Descent Under Heavy-tailed Noise [59.25598762373543]
  We show that wetailed high-prob convergence guarantees of learning on streaming data in the presence of heavy-tailed noise. We demonstrate analytically and that $ta$ can be used to the preferred choice of setting for a given problem.
  arXiv  Detail & Related papers  (2023-10-28T18:53:41Z)
• Boundary theories of critical matchgate tensor networks [59.433172590351234]
  Key aspects of the AdS/CFT correspondence can be captured in terms of tensor network models on hyperbolic lattices. For tensors fulfilling the matchgate constraint, these have previously been shown to produce disordered boundary states. We show that these Hamiltonians exhibit multi-scale quasiperiodic symmetries captured by an analytical toy model.
  arXiv  Detail & Related papers  (2021-10-06T18:00:03Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.