arXiv:2606.20932cs.LG2026-06

提出可学习的分层池化方法,让谱图神经网络能构建层次化表示。

Hierarchical Pooling for Sheaf Neural Networks

  • 基于局部谱粗化构造分层结构,保留图内局部模式。
  • 通过低频特征投影实现细粒度信号到粗粒度的映射。
  • 支持批处理与多层架构,适用于复杂图结构建模。

Sheaf Neural Networks (SNNs) 通过将节点信号推广为茎值信号并利用限制映射衡量边上的相容性,扩展了图神经网络(GNNs)。与标准图扩散不同,谱扩散通过限制映射促进一致性,能建模更广泛的邻接关系。然而现有 SNN 架构通常在固定图分辨率下运行,缺乏可解释的分层池化机制。本文提出分层谱池化(HiSP),基于局部谱粗化构建分层表示。给定图划分,HiSP 将每个粗粒度茎值通过投影至簇内谱沙拉拉普拉斯的低频本征模式得到。这些局部模式定义了上延算子,使细粒度沙拉拉普拉斯能量可通过伽辽金算子在粗粒度空间表示。我们进一步分离粗化带来的截断误差(因丢弃局部模式)和实现误差(因投影算子表示为粗粒度沙拉)。最后,我们实现了兼容 SNN 的 HiSP 池化层,并提供 PyG 版本,支持批量处理、提升后的沙拉拉普拉斯及层级架构。

原文摘要 · Abstract (English)

Sheaf Neural Networks (SNNs) generalize Graph Neural Networks (GNNs) by replacing scalar node signals with stalk-valued signals and by using restriction maps to measure compatibility across edges. Unlike standard graph diffusion, which encourages neighboring node features to become similar, sheaf diffusion promotes consistency through the restriction maps and can therefore model more general relationships between neighboring nodes. However, existing sheaf neural architectures mainly operate at a fixed graph resolution and do not provide a principled pooling mechanism for building hierarchical representations. In this paper, we introduce Hierarchical Sheaf Pool (HiSP), a sheaf-aware pooling framework based on local spectral coarsening. Given a partition of the graph, HiSP constructs each coarse stalk by projecting fine stalk-valued features onto the low-frequency eigenmodes of the cluster-internal sheaf Laplacian. These local modes define a cochain-level prolongation map, which allows the fine sheaf energy to be represented on the coarse space through a Galerkin operator. We further analyze the approximation induced by coarsening by separating truncation loss, due to discarded local modes, from realization loss, due to representing the projected operator as a coarse sheaf. Finally, we implement HiSP as a GNN pooling layer compatible with SNNs and provide a PyG implementation supporting batching, lifted sheaf Laplacians, and hierarchical architectures.

图神经网络分层建模谱方法

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