Understanding the organisation of high-dimensional data is of primary interest for many branches of applied sciences. It is typically achieved by applying dimensionality reduction techniques which, while preserving local features, often miss the global structure of the dataset. Clustering techniques are another class of methods operating in the ambient space, grouping together similar points. However, unlike dimensionality reduction techniques, they do not provide information about the organisation of the data. Leveraging ideas from Topological Data Analysis, in this paper we provide an additional layer on the output of any clustering algorithm. Such a data structure, ClusterGraph, provides information about the global layout of clusters, obtained from the chosen clustering algorithm. Appropriate measures are provided to assess the quality and usefulness of the obtained representation. Subsequently, the ClusterGraph, possibly with an appropriate structure-preserving simplification, can be visualised and used in synergy with state-of-the-art exploratory data analysis techniques.
Dłotko et al. (Sat,) studied this question.