图信号处理--部分III：图上的机器学习，从图拓扑到应用
Graph Signal Processing -- Part III: Machine Learning on Graphs, from Graph Topology to Applications
摘要：
图上的许多现代数据分析应用程序在图拓扑结构先验未知的域上操作，因此其确定成为问题定义的一部分，而不是作为帮助问题解决方案的先验知识。本专题的第三部分首先讨论学习图拓扑的方法，从问题的物理学已经表明可能的拓扑的情况，到从数据中学习图拓扑的最一般情况。一个特别强调的是基于观察到的数据的相关性和精度矩阵的图形拓扑定义，结合额外的先验知识和结构条件，如图连接的平滑性或稀疏性。为了学习稀疏图（具有少量边），采用最小绝对收缩和选择算子（称为LASSO）及其特定于图的变体（图形LASSO）。为了完整起见，LASSO的两个变体都是以直观的方式导出的，并进行了解释。通过几个物理上定义良好的图形，如电路，线性传热，社会和计算机网络，弹簧质量系统的例子，提供了一个深入的阐述的图形拓扑学习范式。随着许多图神经网络（GNN）和卷积图网络（GCN）的出现，我们还从图信号滤波的角度回顾了GNN和GCN的主要趋势。张量表示的格子结构的图形被认为是下一个，它表明，张量（多维数据数组）是一类特殊的图形信号，从而图形顶点驻留在一个高维的规则的晶格结构。这部分专题论文的结论是两个新兴的应用在金融数据处理和地下交通网络建模。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
