  In many applications in data clustering, it is desirable to find not just a
single partition into clusters but a sequence of partitions describing the data
at different scales, or levels of coarseness. A natural problem then is to
analyse and compare the (not necessarily hierarchical) sequences of partitions
that underpin such multiscale descriptions of data. Here, we introduce a
filtration of abstract simplicial complexes, denoted the Multiscale Clustering
Filtration (MCF), which encodes arbitrary patterns of cluster assignments
across scales, and we prove that the MCF produces stable persistence diagrams.
We then show that the zero-dimensional persistent homology of the MCF measures
the degree of hierarchy in the sequence of partitions, and that the
higher-dimensional persistent homology tracks the emergence and resolution of
conflicts between cluster assignments across the sequence of partitions. To
broaden the theoretical foundations of the MCF, we also provide an equivalent
construction via a nerve complex filtration, and we show that in the
hierarchical case, the MCF reduces to a Vietoris-Rips filtration of an
ultrametric space. We briefly illustrate how the MCF can serve to characterise
multiscale clustering structures in numerical experiments on synthetic data.
