arXiv:2605.01110cs.LGcs.SI2026-05

提出拓扑神经正切核,让图模型能捕捉更高阶关系。

Topological Neural Tangent Kernel

论文配图:Topological Neural Tangent Kernel
图 1 · 摘自论文原文
  • 基于单纯形结构设计新型无限宽核,融合上下Hodge交互。
  • 可区分同构图但填充不同单纯形的复杂结构,提升表达能力。
  • 揭示学习速度与拓扑成分的关联,适合关注可解释性的研究者。

图神经正切核为图神经网络提供了无限宽度的理论基础,但仅能捕捉成对结构。许多关系系统包含更自然地由单纯形复形表示的高阶交互。我们提出拓扑神经正切核(TopoNTK),一种用于边特征上单纯形消息传递的无限宽度核。TopoNTK结合下Hodge交互(通过共享顶点的图式耦合)与上Hodge交互(通过填充单纯形的耦合),使核对图核无法察觉的拓扑结构敏感,从而区分具有相同图结构但填充不同单纯形的复形。除表达性外,Hodge结构赋予核可解释的学习几何:边信号分解为梯度类、调和与局部环流分量,且TopoNTK谱决定各分量的学习速度。这带来拓扑形式的谱偏置:与大特征值模式对齐的成分学习快,而全局调和模式通过残差通道保留,常处于小特征值,学习较慢。我们证明了表达性、Hodge对齐、谱学习与稳定性,并在合成单纯形任务和DBLP高阶链接预测中验证。结果表明,拓扑不仅是额外结构,更能提供坐标,使关系学习更忠实、可解释且高效。

原文摘要 · Abstract (English)

Graph neural tangent kernels give a principled infinite-width theory for graph neural networks, but inherit a basic limitation of graph models: they see only pairwise structure. Many relational systems contain higher-order interactions that are more naturally represented by simplicial complexes. We introduce the Topological Neural Tangent Kernel (TopoNTK), an infinite-width kernel for simplicial message passing on edge features. TopoNTK combines lower Hodge interactions, capturing graph-like coupling through shared vertices, with upper Hodge interactions, capturing coupling through filled simplices. This makes the kernel sensitive to topology invisible to graph kernels, allowing complexes with the same graph but different filled simplices to induce different kernels. Beyond expressivity, the Hodge structure gives the kernel an interpretable learning geometry. Edge signals decompose into gradient-like, harmonic, and local circulation components, and the spectrum of the TopoNTK determines how quickly each component is learned. This yields a topological form of spectral bias: components aligned with large-eigenvalue modes are learned quickly, while global harmonic modes, retained through the residual channel, often lie at smaller eigenvalues and are learned more slowly. We prove expressivity, Hodge-alignment, spectral learning, and stability properties, and validate them on synthetic simplicial tasks and DBLP higher-order link prediction. The results show that topology is not merely extra structure; it can provide coordinates that make relational learning more faithful, interpretable, and effective.

拓扑学习图神经网络核方法高阶关系

Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品;中文卡片由大模型生成,请以原文为准。