[论文解读] Transferability of Spectral Graph Convolutional Neural Networks

Ron Levie, Wei Huang+3 more|arXiv (Cornell University)|Jul 30, 2019

Advanced Graph Neural Networks参考文献 45被引用 49

一句话总结

该论文证明了光谱图卷积神经网络在离散化同一潜在空间的图之间具有可迁移性，使用一般的可迁移性不等式和基于 DSP 的框架。

ABSTRACT

This paper focuses on spectral graph convolutional neural networks (ConvNets), where filters are defined as elementwise multiplication in the frequency domain of a graph. In machine learning settings where the dataset consists of signals defined on many different graphs, the trained ConvNet should generalize to signals on graphs unseen in the training set. It is thus important to transfer ConvNets between graphs. Transferability, which is a certain type of generalization capability, can be loosely defined as follows: if two graphs describe the same phenomenon, then a single filter or ConvNet should have similar repercussions on both graphs. This paper aims at debunking the common misconception that spectral filters are not transferable. We show that if two graphs discretize the same "continuous" space, then a spectral filter or ConvNet has approximately the same repercussion on both graphs. Our analysis is more permissive than the standard analysis. Transferability is typically described as the robustness of the filter to small graph perturbations and re-indexing of the vertices. Our analysis accounts also for large graph perturbations. We prove transferability between graphs that can have completely different dimensions and topologies, only requiring that both graphs discretize the same underlying space in some generic sense.

研究动机与目标

• 在数据分布在多图上的设置中，激发并形式化可迁移性。
• 当它们离散化相同的潜在空间时，展示光谱图滤波器能够在不同大小和拓扑结构的图之间泛化。
• 提供理论保证和实际洞见，表明光谱方法在不需要额外迁移组件的情况下仍具可迁移性。

提出的方法

• 采用光谱滤波器的函数微积分视角，以避免特征分解并确保置换等变性。
• 引入可迁移性不等式，将滤波器可迁移性界限于拉普拉斯可迁移性再加上采样-一致性误差。
• 将图建模为潜在连续/拓扑空间的离散化，并通过采样与插值算子进行比较。
• 构建一个受 DSP 启发的框架，通过采样和插值将连续空间信号与图信号联系起来。
• 证明对离散化拓扑空间的图的可迁移性，并在求积（quadrature）假设下扩展到随机采样的图。

实验结果

研究问题

• RQ1 在应用于离散化相同潜在空间的图时，光谱图滤波器在何种条件下表现出可迁移性？
• RQ2 当图在大小和拓扑结构上不同时，如何量化和界定可迁移性？
• RQ3 光谱卷积网络是否可以在不经过对多张图的训练下在图之间迁移（原则可迁移性）？

主要发现

• 通用的可迁移性不等式将滤波器的可迁移性界限为拉普拉斯可迁移性加上采样一致性误差。
• 光谱卷积网络在离散化同一拓扑空间的图之间具有可迁移性，证明基于 DSP 框架。
• 在类似求积的采样假设下，来自拓扑空间的图的可迁移性被确立。
• 当图从潜在空间随机采样时，在所提框架下以高概率成立可迁移性。
• 实验示例表明光谱方法的原则可迁移性甚至在零-shot 设置下也成立。