发表机构
Chern Institute of Mathematics; Nankai University; School of Physical and Mathematical Sciences; Nanyang Technological University; National University of Singapore（陈省身数学研究所; 南开大学; 物理与数学科学学院; 南洋理工大学; 新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对层神经网络的过度平滑问题，提出指标理论比较判据，引入GyroSheaf扩展至非线性场景，经十个模型实验验证了该判据的有效性。
AI 中文摘要
为了应对图卷积网络中的过度平滑问题，层神经网络（SNNs）被提出作为一种泛化模型，其通过为图配备层结构并将图拉普拉斯算子替换为层拉普拉斯算子$\boldsymbol{\textit{L}}$来实现。现有分析通过调和空间（$\boldsymbol{\text{ker}}\boldsymbol{\textit{L}}$）将层扩散与过度平滑联系起来，将其绝对维度作为抗过度平滑能力的指标。然而，仅绝对维度并非可靠的衡量标准：某些层配置会增大$\text{dim}\boldsymbol{\text{ker}}\boldsymbol{\textit{L}}$，但其调和截面完全保持恒定，并未增强判别能力。我们转而引入首个相对几何方法，对抗过度平滑能力进行精确刻画。在茎传输和全局层结构的自然条件下，我们建立了一种指标理论比较判据，表明某一层的调和空间真正包含另一层的调和空间，而非单纯的膨胀。我们用一个具体实例说明这一点，并进一步引入GyroSheaf，一种具有弯曲 gyrovector-space 茎的层，通过局部切空间线性化将该判据扩展至非线性设置。在十个模型上开展的实验证实了该理论判据：违反该判据的层模型尽管存在指标跳跃仍会崩溃，而符合该判据的模型则能保持深度稳定的表示。
英文摘要
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.
