从傅里叶域分析谱GNN，揭示深度与阶数对泛化的影响。
Generalization Bounds for Spectral GNNs via Fourier Domain Analysis
- 在图傅里叶域中，每层为频域元素的独立更新，显式分离谱与参数。
- 推导出依赖数据、深度和多项式阶数的泛化界，线性情况下更紧。
- 实图实验显示数据项与泛化差距相关，指导避免频率放大。
谱图神经网络学习图滤波器，但其深度和多项式阶数增加时的行为尚不明确。本文在图傅里叶域中分析这些模型，使每一层变为频域元素的逐点更新，将固定谱与可训练参数分离，使深度和阶数显式化。在此框架下，我们证明高斯复杂度在图傅里叶变换下不变，从而导出依赖数据、深度和阶数的泛化界及稳定性估计。在线性情形下，我们的边界更紧；在真实图上，数据依赖项与不同多项式基下的泛化差距呈正相关，凸显了避免层间频率放大的实际选择。
原文摘要 · Abstract (English)
Spectral graph neural networks learn graph filters, but their behavior with increasing depth and polynomial order is not well understood. We analyze these models in the graph Fourier domain, where each layer becomes an element-wise frequency update, separating the fixed spectrum from trainable parameters and making depth and order explicit. In this setting, we show that Gaussian complexity is invariant under the Graph Fourier Transform, which allows us to derive data-dependent, depth, and order-aware generalization bounds together with stability estimates. In the linear case, our bounds are tighter, and on real graphs, the data-dependent term correlates with the generalization gap across polynomial bases, highlighting practical choices that avoid frequency amplification across layers.
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