Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Alessio Borgi, Fabrizio Silvestri, Pietro Liò

TL;DR

This work introduces PolyNSD, a spectral polynomial diffusion framework operating on cellular sheaves to address oversmoothing and heterophily in graph-structured data. By applying a learnable degree-K polynomial in the sheaf Laplacian, computed via a stable recurrence, PolyNSD provides an explicit K-hop receptive field per layer and a tunable spectral response without eigen-decompositions, while maintaining transport-aware inductive biases. The method achieves state-of-the-art results across both homophilic and heterophilic benchmarks, using diagonal restriction maps and modest stalk dimensions to reduce parameters and runtime. Theoretical and empirical analyses establish stability, locality, and energy-nonincreasing properties, making PolyNSD a depth-efficient, transport-aware diffusion approach for neural sheaf models.

Abstract

Sheaf Neural Networks equip graph structures with a cellular sheaf: a geometric structure which assigns local vector spaces (stalks) and a linear learnable restriction/transport maps to nodes and edges, yielding an edge-aware inductive bias that handles heterophily and limits oversmoothing. However, common Neural Sheaf Diffusion implementations rely on SVD-based sheaf normalization and dense per-edge restriction maps, which scale with stalk dimension, require frequent Laplacian rebuilds, and yield brittle gradients. To address these limitations, we introduce Polynomial Neural Sheaf Diffusion (PolyNSD), a new sheaf diffusion approach whose propagation operator is a degree-K polynomial in a normalised sheaf Laplacian, evaluated via a stable three-term recurrence on a spectrally rescaled operator. This provides an explicit K-hop receptive field in a single layer (independently of the stalk dimension), with a trainable spectral response obtained as a convex mixture of K+1 orthogonal polynomial basis responses. PolyNSD enforces stability via convex mixtures, spectral rescaling, and residual/gated paths, reaching new state-of-the-art results on both homophilic and heterophilic benchmarks, inverting the Neural Sheaf Diffusion trend by obtaining these results with just diagonal restriction maps, decoupling performance from large stalk dimension, while reducing runtime and memory requirements.

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

TL;DR

This work introduces PolyNSD, a spectral polynomial diffusion framework operating on cellular sheaves to address oversmoothing and heterophily in graph-structured data. By applying a learnable degree-K polynomial in the sheaf Laplacian, computed via a stable recurrence, PolyNSD provides an explicit K-hop receptive field per layer and a tunable spectral response without eigen-decompositions, while maintaining transport-aware inductive biases. The method achieves state-of-the-art results across both homophilic and heterophilic benchmarks, using diagonal restriction maps and modest stalk dimensions to reduce parameters and runtime. Theoretical and empirical analyses establish stability, locality, and energy-nonincreasing properties, making PolyNSD a depth-efficient, transport-aware diffusion approach for neural sheaf models.

Abstract

Sheaf Neural Networks equip graph structures with a cellular sheaf: a geometric structure which assigns local vector spaces (stalks) and a linear learnable restriction/transport maps to nodes and edges, yielding an edge-aware inductive bias that handles heterophily and limits oversmoothing. However, common Neural Sheaf Diffusion implementations rely on SVD-based sheaf normalization and dense per-edge restriction maps, which scale with stalk dimension, require frequent Laplacian rebuilds, and yield brittle gradients. To address these limitations, we introduce Polynomial Neural Sheaf Diffusion (PolyNSD), a new sheaf diffusion approach whose propagation operator is a degree-K polynomial in a normalised sheaf Laplacian, evaluated via a stable three-term recurrence on a spectrally rescaled operator. This provides an explicit K-hop receptive field in a single layer (independently of the stalk dimension), with a trainable spectral response obtained as a convex mixture of K+1 orthogonal polynomial basis responses. PolyNSD enforces stability via convex mixtures, spectral rescaling, and residual/gated paths, reaching new state-of-the-art results on both homophilic and heterophilic benchmarks, inverting the Neural Sheaf Diffusion trend by obtaining these results with just diagonal restriction maps, decoupling performance from large stalk dimension, while reducing runtime and memory requirements.

Paper Structure

This paper contains 66 sections, 5 theorems, 50 equations, 7 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Related Work
  4. Preliminaries
  5. Cellular Sheaves on Graphs
  6. Neural Sheaf Diffusion (NSD)
  7. Polynomial Neural Sheaf Diffusion (PolyNSD)
  8. Polynomial Filters on Sheaf Laplacians
  9. K-hop Locality.
  10. Commutation and Dirichlet Energy.
  11. Choice of Basis and Chebyshev Parameterisation.
  12. PolyNSD Layer and Architecture
  13. High-Pass Skip and Gated Residual.
  14. PolyNSD and NSD Comparison.
  15. Experimental Evaluation
  16. ...and 51 more sections

Key Result

proposition 1

Let $L$ be a sparse block Laplacian on a graph $G=(V,E)$ such that the off-diagonal block $(v,u)$ is nonzero only when $(v,u)\in E$, and diagonal blocks are arbitrary (PSD). Let $p(L)$ be as in eq:polysd-poly-filter. Then:

Figures (7)

  • Figure 1: PolyNSD Layer (degree $K$). The layer first rescales the sheaf Laplacian $L$ to $\widetilde{L}$, evaluates a degree-$K$ Chebyshev polynomial via the three-term recurrence, and adds a learnable high-pass correction and gated residual.
  • Figure 2: End-to-end PolyNSD Architecture. (1) Node features are lifted to stalks. (2) A sheaf learner predicts restriction maps (diagonal, bundle, or general). (3) The vertex sheaf Laplacian $L_{\mathcal{F}}$ (or $\Delta_{\mathcal{F}}$) is assembled and its spectral scale $\lambda_{\max}$ is obtained analytically or via power iteration. (4) A Chebyshev-PolyNSD block performs polynomial spectral diffusion with high-pass correction and gated residual. (5) A linear head produces task outputs. A more detailed description is given in \ref{['app:polynsd-architecture']}.
  • Figure 3: Test accuracy vs. stalk dimension $d\in\{2,3,4,5\}$. We sweep $d$ on four real-world datasets (Cora, PubMed, Texas, Film) for Diagonal, Bundle, and General PolyNSD. All other hyper-parameters are kept fixed to the default configuration used in the main real-world experiments. Error bars show mean$\pm$std over the $10$ fixed data splits.
  • Figure 4: Chebyshev order $K$ sweep. Test accuracy vs. $K$ for the nine real-world benchmarks. Each panel corresponds to one dataset and overlays the six configurations given by the three PolyNSD variants crossed with analytic vs. iterative estimates of $\lambda_{\max}$. Error bars denote mean$\pm$std over the $10$ fixed splits.
  • Figure 5: Synthetic heterophily sweeps. Each row corresponds to a different number of classes; columns distinguish the RiSNN and Diff regimes. We sweep $het\in\{0,0.25,0.5,0.75,1.0\}$; error bars show mean$\pm$std over multiple random graph realisations.
  • ...and 2 more figures

Theorems & Definitions (10)

  • definition 1
  • definition 2
  • proposition 1: K-hop Locality
  • proposition 2: Commutation and non-increasing Dirichlet Energy
  • proposition 3: Spectral Enclosure for the Normalised Sheaf Laplacian
  • proof
  • proposition 4: Gershgorin-Type Bound for Unnormalised Sheaf Laplacians
  • proof
  • proposition 5: Chebyshev-PolyNSD with $K{=}1$ Induces a First-Order Polynomial in $L$
  • proof