Graph Neural Networks (GNNs) show impressive performance in many practical scenarios, which can be largely attributed to their stability properties. Empirically, GNNs can scale well on large size graphs, but this is contradicted by the fact that existing stability bounds grow with the number of nodes. Graphs with well-defined limits can be seen as samples from manifolds. Hence, in this paper, we analyze the stability properties of convolutional neural networks on manifolds to understand the stability of GNNs on large graphs. Specifically, we focus on stability to relative perturbations of the Laplace-Beltrami operator. To start, we construct frequency ratio threshold filters which separate the infinite-dimensional spectrum of the Laplace-Beltrami operator. We then prove that manifold neural networks composed of these filters are stable to relative operator perturbations. As a product of this analysis, we observe that manifold neural networks exhibit a trade-off between stability and discriminability. Finally, we illustrate our results empirically in a wireless resource allocation scenario where the transmitter-receiver pairs are assumed to be sampled from a manifold.
翻译:神经网络图(GNNs)在许多实际情景中表现出了令人印象深刻的性能,这在很大程度上可以归因于它们的稳定性。 简而言之,GNNs可以在大尺寸的图形上大幅缩放,但这与现有的稳定性线条随着节点数的增加而增长的事实相矛盾。 具有明确限制的图表可以被看成是来自多个图段的样本。 因此,在本文中,我们分析了在多个图段上革命性神经网络的稳定性特性,以了解大型图段上GNs的稳定性。 具体地说, 我们侧重于与Laplace- Beltrami操作器相对的渗透性相对的稳定性。 首先,我们建立频率率阈值过滤器,将Laplace-Beltrami操作器的无限维谱分离出来。 然后我们证明,由这些过滤器组成的多重神经网络对于相对操作器的扰动性是稳定的。 作为这一分析的产物,我们观察到, 多个神经网络在稳定性和不均匀性之间显示出一种交易。 最后,我们用实验性的方式在无线资源分配假设中展示了我们的结果, 。