We establish a connection between tangles, a concept from structural graph theory that plays a central role in Robertson and Seymour's graph minor project, and hierarchical clustering. Tangles cannot only be defined for graphs, but in fact for arbitrary connectivity functions, which are functions defined on the subsets of some finite universe. In typical clustering applications these universes consist of points in some metric space. Connectivity functions are usually required to be submodular. It is our first contribution to show that the central duality theorem connecting tangles with hierarchical decompositions (so-called branch decompositions) also holds if submodularity is replaced by a different property that we call maximum-submodular. We then define a connectivity function on finite data sets in an arbitrary metric space and prove that its tangles are in one-to-one correspondence with the clusters obtained by applying the well-known single linkage clustering algorithms to the same data set. Lastly we generalize this correspondence for any hierarchical clustering. We show that the data structure that represents hierarchical clustering results, called dendograms, are equivalent to maximum-submodular connectivity functions and their tangles. The idea of viewing tangles as clusters has first been proposed by Diestel and Whittle in 2016 as an approach to image segmentation. To the best of our knowledge, our result is the first that establishes a precise technical connection between tangles and clusters.
翻译:我们从结构图学理论中,在Robertson和Seymour的图形小小项目和等级组群中起中心作用的构造图解理论概念和结构组群之间建立起联系。 Tangles无法为图表定义,而实际上不能为任意连接功能而定义,这些功能是某些有限宇宙子子组中界定的。在典型的组合应用中,这些宇宙由某些计量空间的点组成。连接功能通常需要为子模块。连接功能通常需要为子模块。这是我们的第一个贡献,以显示如果亚模式被一个我们称之为最高子模块的不同属性所取代,那么与等级分解组(所谓的分支分解)的交点之间的中心两极性理论关系也仍然有效。我们随后在一个任意的计量空间中定义了有限数据集的连接功能,并证明其串联在与通过对同一数据集应用众所周知的单一链接算法获得的集群的一对一对一对一对一的对应关系中。最后,我们将任何等级组群集(所谓的分支分解剖)的对应关系也存在。我们所展示的数据结构结构结构结构结构结构结构中,称为最大次次组合中的最大分解的链接系系系系系系系系系系系函数,作为我们最先在最精确的端端端端端端端端端端端端端端端端端端的链接函数和图像系系系系。我们所建的端端端端端端端端端端端端端端端端端系函数和端端端端端端端的功能的功能函数的功能和端端端端端端端端端端的功能和端端端端端端端端端端端端端端端端。我们方函数和端端端端端端端端的端的端端端端端端端函数,以方函数。在我们的端端端端端端端端端端端端端端端的端端端端端端端的端端端端。在我们方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方