Tangles and Hierarchical Clustering

SIAM Journal on Discrete Mathematics, vol. 38, pp. 75–92

Abstract

Abstract. 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. Last, 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 was first proposed by Diestel and Whittle in 2016 as an approach to image segmentation.

Authors 1

1. Eva Fluck corresponding Aachen Chair for Computer Science 7

   RWTH Aachen University

   Affiliation as printed

   Chair for Computer Science 7, Logic and Theory of Discrete Systems, RWTH Aachen University, 52074 Aachen, Germany