Graph neural networks (GNNs) are commonly divided into message-passing neural networks (MPNNs) and spectral GNNs, reflecting two largely separate research traditions in machine learning and signal processing. While MPNNs have a precise definition, there is no widely accepted criterion for what makes a mapping a spectral GNN. Most existing work restricts spectral GNNs to layered architectures based on linear spectral filters. Under this restriction, we show that spectral and spatial GNNs have largely equivalent expressive power. To promote progress in the field, we propose a precise definition of spectral GNNs based on eigenbasis symmetries, in contrast to the definition of MPNNs via neighborhood permutation symmetries. We further argue that the two perspectives offer complementary strengths. MPNNs provide a natural language for discrete structure and expressivity analysis through tools from logic and graph isomorphism, while the spectral perspective offers principled tools for understanding smoothing, bottlenecks, stability, and community structure. Overall, we argue that progress in graph learning will be accelerated by clarifying the similarities and differences between these perspectives and by moving toward a unified theoretical framework.

翻译：图神经网络（GNN）通常分为消息传递神经网络（MPNN）和频谱图神经网络，这反映了机器学习和信号处理中两个相对独立的研究传统。尽管MPNN有精确的定义，但对于何种映射构成频谱GNN，目前尚无广泛接受的标准。大多数现有工作将频谱GNN限定为基于线性频谱滤波器的分层架构。在此限制下，我们证明了频谱GNN与空间GNN的表达能力大致等价。为促进该领域的发展，我们提出了一种基于特征基对称性的频谱GNN精确定义，以区别于通过邻域置换对称性定义的MPNN。我们进一步论证两种视角具有互补优势：MPNN通过逻辑和图同构工具为离散结构和表达能力分析提供自然语言，而频谱视角则为理解平滑性、瓶颈、稳定性和社区结构提供原则性工具。总体而言，我们认为通过厘清这两种视角的异同并迈向统一的理论框架，将加速图学习的进步。