图形信号处理--第一部分：图形、图形谱和谱聚类
Graph Signal Processing -- Part I: Graphs, Graph Spectra, and Spectral Clustering
摘要：
图上的数据分析领域有望实现范式转变，因为我们接近数据类的信息处理，这些数据通常是在不规则但结构化的域（社交网络，各种ad-hoc传感器网络）上获得的。然而，尽管其历史悠久，目前的方法主要集中在优化图本身，而不是直接推断学习策略，如检测，估计，统计和概率推理，聚类和分离从图上获取的信号和数据。为了填补这一空白，我们首先从数据分析的角度重新审视图拓扑，并通过图拓扑的线性代数形式（顶点，连接，方向性）建立图网络的分类。这是图的谱分析的基础，图的拉普拉斯算子和邻接矩阵的特征值和特征向量被用来表达与图的拓扑结构和高阶图属性（如切割、行走、路径和邻域）相关的物理意义。接下来，为了说明对图形信号执行的估计策略，通过图形的数学描述符的特征分析并且以通用的方式引入图形的谱分析。最后，一个框架的顶点聚类和图分割的基础上建立图的谱表示（特征分析），说明了各种数据关联任务的图形的力量。支持的例子展示了图数据分析在建模结构和功能/语义推理方面的前景。同时，第一部分是第二部分和第三部分的基础，第二部分和第三部分涉及处理图上的数据和从数据中学习图拓扑的理论，方法和应用。
英文摘要：
The area of Data Analytics on graphs promises a paradigm shift as we approach information processing of classes of data, which are typically acquired on irregular but structured domains (social networks, various ad-hoc sensor networks). Yet, despite its long history, current approaches mostly focus on the optimization of graphs themselves, rather than on directly inferring learning strategies, such as detection, estimation, statistical and probabilistic inference, clustering and separation from signals and data acquired on graphs. To fill this void, we first revisit graph topologies from a Data Analytics point of view, and establish a taxonomy of graph networks through a linear algebraic formalism of graph topology (vertices, connections, directivity). This serves as a basis for spectral analysis of graphs, whereby the eigenvalues and eigenvectors of graph Laplacian and adjacency matrices are shown to convey physical meaning related to both graph topology and higher-order graph properties, such as cuts, walks, paths, and neighborhoods. Next, to illustrate estimation strategies performed on graph signals, spectral analysis of graphs is introduced through eigenanalysis of mathematical descriptors of graphs and in a generic way. Finally, a framework for vertex clustering and graph segmentation is established based on graph spectral representation (eigenanalysis) which illustrates the power of graphs in various data association tasks. The supporting examples demonstrate the promise of Graph Data Analytics in modeling structural and functional/semantic inferences. At the same time, Part I serves as a basis for Part II and Part III which deal with theory, methods and applications of processing Data on Graphs and Graph Topology Learning from data.
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。
