Topological Hierarchical Decompositions

Topological data analysis is an emerging field that applies the study of topological invariants to data. Perhaps the simplest of these invariants is the number of connected components or clusters. In this work, we explore a topological framework for cluster analysis and show how it can be used as a basis for explainability in unsupervised data analysis. Our main object of study will be hierarchical data structures referred to as Topological Hierarchical Decompositions (THDs). We give a number of examples of how traditional clustering algorithms can be topologized, and provide preliminary results on the THDs associated with Reeb graphs and the mapper algorithm. In particular, we give a generalized construction of the mapper functor as a pixelization of a cosheaf in order to generalize multiscale mapper.