统一框架解析图神经算子收敛性，揭示不同假设下的权衡。
A Spectral Framework for Graph Neural Operators: Convergence Guarantees and Tradeoffs
- 基于图核理论构建统一谱框架，整合多种假设下的收敛结果。
- 在合成与真实图上验证了收敛速率的紧致性，结果可靠。
- 适合研究图神经网络泛化与理论分析的学者参考。
图核（Graphons）作为图序列的极限，为分析图神经算子的渐近行为提供了算子理论框架。采样图到图核的谱收敛性可导出对应神经算子的收敛性，从而支持图神经网络（GNNs）的可转移性分析。本文提出一个统一的谱框架，整合了在无正则性、全局Lipschitz连续性以及分片Lipschitz连续性等不同假设下图神经算子的收敛结果。该框架将这些结果置于共同的算子设定中，便于直接比较其假设条件、收敛速率及内在权衡。我们进一步在合成图和真实世界图上展示了这些速率的实证紧致性。
原文摘要 · Abstract (English)
Graphons, as limits of graph sequences, provide an operator-theoretic framework for analyzing the asymptotic behavior of graph neural operators. Spectral convergence of sampled graphs to graphons induces convergence of the corresponding neural operators, enabling transferability analyses of graph neural networks (GNNs). This paper develops a unified spectral framework that brings together convergence results under different assumptions on the underlying graphon, including no regularity, global Lipschitz continuity, and piecewise-Lipschitz continuity. The framework places these results in a common operator setting, enabling direct comparison of their assumptions, convergence rates, and tradeoffs. We further illustrate the empirical tightness of these rates on synthetic and real-world graphs.
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