MGNN：受距离几何问题启发的图神经网络
MGNN: Graph Neural Networks Inspired by Distance Geometry Problem
AI总结：
提出受距离几何问题启发的空间图神经网络MGNN，通过优化弹簧网络和MDS能量函数实现普适性，可处理同质和异质图，并在合成与真实数据集上验证有效性。
摘要：
图神经网络（GNNs）已成为机器学习领域的一个突出研究课题。现有的GNN模型通常分为两类：基于多项式图滤波器设计的谱域GNN，以及以消息传递机制为模型基础的空间GNN。对于谱域GNN的表达能力和普适性，一种自然的方法是改进基函数的设计以获得更好的逼近能力。至于空间GNN，诸如图同构网络（GIN）等模型基于图同构测试分析其表达能力。近年来，已有尝试将空间GNN与曲率、细胞层等几何概念以及振荡器等物理现象建立联系。然而，尽管近期取得了进展，从几何和物理角度对空间GNN的普适性进行全面分析仍然缺乏。在本文中，我们提出了MetricGNN（MGNN），这是一种受GNN分类阶段中分类器全等不敏感特性启发的空间GNN模型。我们证明，如果一个GNN模型能够生成与任意给定嵌入矩阵全等的嵌入矩阵，那么它在空间域中是普适的。这一性质与距离几何问题（DGP）密切相关。由于DGP是一个NP难组合优化问题，我们提出优化一个源自弹簧网络和多维缩放（MDS）问题的能量函数。这种方法还使我们的模型能够处理同质图和异质图。最后，我们提出采用迭代方法来优化我们的能量函数。我们通过在合成数据集和真实世界数据集上进行的实验广泛评估了我们模型的有效性。我们的代码可在 https://github.com/GuanyuCui/MGNN 获取。
英文摘要：
Graph Neural Networks (GNNs) have emerged as a prominent research topic in the field of machine learning. Existing GNN models are commonly categorized into two types: spectral GNNs, which are designed based on polynomial graph filters, and spatial GNNs, which utilize a message-passing scheme as the foundation of the model. For the expressive power and universality of spectral GNNs, a natural approach is to improve the design of basis functions for better approximation ability. As for spatial GNNs, models like Graph Isomorphism Networks (GIN) analyze their expressive power based on Graph Isomorphism Tests. Recently, there have been attempts to establish connections between spatial GNNs and geometric concepts like curvature and cellular sheaves, as well as physical phenomena like oscillators. However, despite the recent progress, there is still a lack of comprehensive analysis regarding the universality of spatial GNNs from the perspectives of geometry and physics. In this paper, we propose MetricGNN (MGNN), a spatial GNN model inspired by the congruent-insensitivity property of classifiers in the classification phase of GNNs. We demonstrate that a GNN model is universal in the spatial domain if it can generate embedding matrices that are congruent to any given embedding matrix. This property is closely related to the Distance Geometry Problem (DGP). Since DGP is an NP-Hard combinatorial optimization problem, we propose optimizing an energy function derived from spring networks and the Multi-Dimensional Scaling (MDS) problem. This approach also allows our model to handle both homophilic and heterophilic graphs. Finally, we propose employing the iteration method to optimize our energy function. We extensively evaluate the effectiveness of our model through experiments conducted on both synthetic and real-world datasets. Our code is available at: https://github.com/GuanyuCui/MGNN.
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