Due to their flexibility to represent almost any kind of relational data, graph-based models have enjoyed a tremendous success over the past decades. While graphs are inherently only combinatorial objects, however, many prominent analysis tools are based on the algebraic representation of graphs via matrices such as the graph Laplacian, or on associated graph embeddings. Such embeddings associate to each node a set of coordinates in a vector space, a representation which can then be employed for learning tasks such as the classification or alignment of the nodes of the graph. As the geometric picture provided by embedding methods enables the use of a multitude of methods developed for vector space data, embeddings have thus gained interest both from a theoretical as well as a practical perspective. Inspired by trace-optimization problems, often encountered in the analysis of graph-based data, here we present a method to derive ellipsoidal embeddings of the nodes of a graph, in which each node is assigned a set of coordinates on the surface of a hyperellipsoid. Our method may be seen as an alternative to popular spectral embedding techniques, to which it shares certain similarities we discuss. To illustrate the utility of the embedding we conduct a case study in which analyse synthetic and real world networks with modular structure, and compare the results obtained with known methods in the literature.

翻译：由于其表示几乎任何类型关系数据的灵活性，基于图的模型在过去几十年中取得了巨大成功。然而，尽管图本质上只是组合对象，许多重要的分析工具却基于通过矩阵（如图拉普拉斯矩阵）对图进行代数表示，或基于相关的图嵌入。此类嵌入为每个节点关联一组向量空间中的坐标，这种表示随后可用于学习任务，例如图中节点的分类或对齐。由于嵌入方法提供的几何图景能够利用为向量空间数据开发的众多方法，因此嵌入无论在理论还是实践角度都引起了广泛兴趣。受图数据分析中常见的迹优化问题启发，本文提出了一种推导图节点椭球嵌入的方法，其中每个节点被分配一组位于超椭球表面上的坐标。我们的方法可被视为流行谱嵌入技术的一种替代方案，二者共享某些我们讨论的相似性。为说明该嵌入的实用性，我们开展了一项案例研究，分析了具有模块结构的合成网络和真实世界网络，并将结果与文献中已知方法进行了比较。