Spectral Graph Convolutional Networks (GCNs) have gained popularity in graph machine learning applications due, in part, to their flexibility in specification of network propagation rules. These propagation rules are often constructed as polynomial filters whose coefficients are learned using label information during training. In contrast to learned polynomial filters, explicit filter functions are useful in capturing relationships between network topology and distribution of labels across the network. A number of algorithms incorporating either approach have been proposed; however the relationship between filter functions and polynomial approximations is not fully resolved. This is largely due to the ill-conditioned nature of the linear systems that must be solved to derive polynomial approximations of filter functions. To address this challenge, we propose a novel Arnoldi orthonormalization-based algorithm, along with a unifying approach, called G-Arnoldi-GCN that can efficiently and effectively approximate a given filter function with a polynomial. We evaluate G-Arnoldi-GCN in the context of multi-class node classification across ten datasets with diverse topological characteristics. Our experiments show that G-Arnoldi-GCN consistently outperforms state-of-the-art methods when suitable filter functions are employed. Overall, G-Arnoldi-GCN opens important new directions in graph machine learning by enabling the explicit design and application of diverse filter functions. Code link: https://anonymous.4open.science/r/GArnoldi-GCN-F7E2/README.md

本研究针对谱图卷积网络中滤波器系数学习的问题，提出了一种新颖的Arnoldi正交化算法及其统一方法G-Arnoldi-GCN，以有效地用多项式逼近给定的滤波器函数。实验表明，G-Arnoldi-GCN在多类节点分类任务中表现优于现有的最先进方法，推动了图机器学习领域的发展，尤其是在滤波器函数的显式设计与应用方面。

广义谱图卷积网络中的系数学习