Spectral graph neural networks are proposed to harness spectral information inherent in graph-structured data through the application of polynomial-defined graph filters, recently achieving notable success in graph-based web applications. Existing studies reveal that various polynomial choices greatly impact spectral GNN performance, underscoring the importance of polynomial selection. However, this selection process remains a critical and unresolved challenge. Although prior work suggests a connection between the approximation capabilities of polynomials and the efficacy of spectral GNNs, there is a lack of theoretical insights into this relationship, rendering polynomial selection a largely heuristic process. To address the issue, this paper examines polynomial selection from an error-sum of function slices perspective. Inspired by the conventional signal decomposition, we represent graph filters as a sum of disjoint function slices. Building on this, we then bridge the polynomial capability and spectral GNN efficacy by proving that the construction error of graph convolution layer is bounded by the sum of polynomial approximation errors on function slices. This result leads us to develop an advanced filter based on trigonometric polynomials, a widely adopted option for approximating narrow signal slices. The proposed filter remains provable parameter efficiency, with a novel Taylor-based parameter decomposition that achieves streamlined, effective implementation. With this foundation, we propose TFGNN, a scalable spectral GNN operating in a decoupled paradigm. We validate the efficacy of TFGNN via benchmark node classification tasks, along with an example graph anomaly detection application to show its practical utility.

翻译：谱图神经网络通过应用多项式定义的图滤波器来利用图结构数据中固有的谱信息，近年来在图基网络应用中取得了显著成功。现有研究表明，不同的多项式选择会极大影响谱图神经网络的性能，这凸显了多项式选择的重要性。然而，该选择过程仍是一个关键且尚未解决的挑战。尽管先前工作表明多项式的逼近能力与谱图神经网络效能之间存在关联，但缺乏对这一关系的理论见解，使得多项式选择在很大程度上仍是一个启发式过程。为解决这一问题，本文从函数切片误差和的角度审视多项式选择。受传统信号分解的启发，我们将图滤波器表示为不相交函数切片之和。在此基础上，通过证明图卷积层的构建误差受限于多项式在各函数切片上逼近误差之和，从而建立了多项式能力与谱图神经网络效能之间的理论桥梁。这一结果促使我们开发一种基于三角多项式的高级滤波器，三角多项式是逼近窄信号切片时广泛采用的选项。所提出的滤波器在理论上保持参数高效性，并采用一种新颖的基于泰勒展开的参数分解方法，实现了精简而有效的实现。基于此，我们提出了TFGNN，一种在解耦范式下运行的可扩展谱图神经网络。我们通过基准节点分类任务验证了TFGNN的有效性，并辅以一个图异常检测应用实例以展示其实际效用。