[论文解读] Graph Signal Processing -- Part III: Machine Learning on Graphs, from Graph Topology to Applications

Ljubiša Stanković, Danilo P. Mandic+5 more|arXiv (Cornell University)|Jan 2, 2020

Advanced Graph Neural Networks被引用 7

一句话总结

本文提出了一种统一框架，用于从未知先验的观测数据中学习图拓扑结构，整合了相关性、精度矩阵以及稀疏性和平滑性等结构约束。该框架引入了图LASSO和拉普拉斯学习方法用于拓扑推断，并将其应用于物理系统和机器学习（GNNs/GCNs），在金融领域（投资组合切割）和地铁网络分析中展示了优于标准方法的鲁棒性和性能。

ABSTRACT

Many modern data analytics applications on graphs operate on domains where graph topology is not known a priori, and hence its determination becomes part of the problem definition, rather than serving as prior knowledge which aids the problem solution. Part III of this monograph starts by addressing ways to learn graph topology, from the case where the physics of the problem already suggest a possible topology, through to most general cases where the graph topology is learned from the data. A particular emphasis is on graph topology definition based on the correlation and precision matrices of the observed data, combined with additional prior knowledge and structural conditions, such as the smoothness or sparsity of graph connections. For learning sparse graphs (with small number of edges), the least absolute shrinkage and selection operator, known as LASSO is employed, along with its graph specific variant, graphical LASSO. For completeness, both variants of LASSO are derived in an intuitive way, and explained. An in-depth elaboration of the graph topology learning paradigm is provided through several examples on physically well defined graphs, such as electric circuits, linear heat transfer, social and computer networks, and spring-mass systems. As many graph neural networks (GNN) and convolutional graph networks (GCN) are emerging, we have also reviewed the main trends in GNNs and GCNs, from the perspective of graph signal filtering. Tensor representation of lattice-structured graphs is next considered, and it is shown that tensors (multidimensional data arrays) are a special class of graph signals, whereby the graph vertices reside on a high-dimensional regular lattice structure. This part of monograph concludes with two emerging applications in financial data processing and underground transportation networks modeling.

研究动机与目标

• 解决在现代数据分析中图拓扑未知时学习图结构的挑战，尤其当缺乏物理约束时。
• 开发一种系统化方法，利用信号相似性、协方差/精度矩阵以及稀疏性和平滑性等结构先验来推断图结构。
• 通过将图神经网络（GNNs）和图卷积网络（GCNs）建模为扩散过程，实现图拓扑学习与机器学习模型的集成。
• 在金融投资组合优化和地下交通网络建模中展示实际应用，证明其在鲁棒性和性能上的改进。
• 建立用于网格结构图的张量表示，支持多线性代数方法对多维数据和内在关系进行建模。

提出的方法

• 利用观测数据的相关性和精度矩阵作为图拓扑学习的主要度量，结合先验知识和结构条件。
• 采用LASSO及其图专用变体——图LASSO，以在学习到的图连接中强制实现稀疏性，并提供了直观的推导过程。
• 对图拉普拉斯矩阵施加平滑性和能量约束，以确保拓扑结构在物理上合理且稳定。
• 将GNNs和GCNs建模为图上的扩散过程，通过图信号扩散的视角重新解释其架构改进。
• 使用张量表示高维网格结构图，支持对多维图信号进行张量分解和多线性分析。
• 提出基于图划分的组合切割方法，用于建模资产依赖关系，并在不求逆奇异协方差矩阵的情况下实现稳健、低方差的投资组合配置。

实验结果

研究问题

• RQ1 当图拓扑未知且缺乏物理约束时，如何从未知数据中学习图拓扑？
• RQ2 相关性和精度矩阵在图结构学习中起什么作用？如何强制实现稀疏性和平滑性？
• RQ3 如何通过图信号扩散的视角重新诠释图神经网络（GNNs）和图卷积网络（GCNs）的设计？
• RQ4 张量表示在多维图结构数据建模中能发挥哪些作用？
• RQ5 图拓扑学习如何改善金融投资组合构建和地铁网络脆弱性分析？

主要发现

• 图LASSO和基于LASSO的拓扑学习方法显著提升了从高维数据中识别稀疏且有意义的图连接的能力。
• 所提出的图资产配置方案在样本外方差上始终低于传统等权重和最小方差投资组合，并获得了更高的夏普比率。
• 在伦敦地铁网络中，基于图论的中心性度量成功识别出若关闭将对整体服务水平造成最严重影响的站点。
• 基于图划分的组合切割方法减少了虚假依赖关系，即使在数据有限或协方差矩阵奇异的情况下，也能实现稳健的资产配置。
• 将GCNs和GNNs解释为扩散过程，为未来架构设计提供了原则性基础，超越了启发式改进。
• 基于张量的网格结构图建模方法，相比传统不平衡邻接矩阵，在发现多维数据内在关系方面表现更优。