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://github.com/mustafaCoskunAgu/GArnoldi-GCN

翻译：谱图卷积网络（GCN）在图机器学习应用中日益受到欢迎，部分原因在于其在网络传播规则设定上的灵活性。这些传播规则通常构建为多项式滤波器，其系数在训练过程中利用标签信息进行学习。与学习的多项式滤波器相比，显式滤波器函数在捕捉网络拓扑结构与标签分布之间的关系方面具有优势。目前已提出了多种结合这两种方法的算法；然而，滤波器函数与多项式近似之间的关系尚未完全厘清。这主要是由于推导滤波器函数的多项式近似时必须求解的线性系统具有病态特性。为应对这一挑战，我们提出了一种基于Arnoldi正交归一化的新颖算法，以及一个统一框架——称为G-Arnoldi-GCN——该框架能够高效且有效地用多项式逼近给定滤波器函数。我们在十个具有不同拓扑特征的数据集上，针对多类别节点分类任务评估了G-Arnoldi-GCN。实验表明，当采用合适的滤波器函数时，G-Arnoldi-GCN始终优于现有最先进方法。总体而言，G-Arnoldi-GCN通过支持多样化滤波器函数的显式设计与应用，为图机器学习开辟了重要的新方向。代码链接：https://github.com/mustafaCoskunAgu/GArnoldi-GCN