Persistent Homology of the Multiscale Clustering Filtration

In many applications in data clustering, it is desirable to find not just a single partition but a sequence of partitions that describes the data at different scales, or levels of coarseness, leading naturally to Sankey diagrams as descriptors of the data. The problem of multiscale clustering then becomes how to to select robust intrinsic scales, and how to analyse and compare the (not necessarily hierarchical) sequences of partitions. Here, we define a novel filtration, the Multiscale Clustering Filtration (MCF), which encodes arbitrary patterns of cluster assignments across scales. We prove that the MCF is a proper filtration, give an equivalent construction via nerves, and show that in the hierarchical case the MCF reduces to the Vietoris-Rips filtration of an ultrametric space. We also show that the zero-dimensional persistent homology of the MCF provides a measure of the level of hierarchy in the sequence of partitions, whereas the higher-dimensional persistent homology tracks the emergence and resolution of conflicts between cluster assignments across scales. We briefly illustrate numerically how the structure of the persistence diagram can serve to characterise multiscale data clusterings.