arXiv:2305.04281 · v6
Analysing Multiscale Clusterings with Persistent Homology
摘要
In data clustering, it is often desirable to find not just a single partition into clusters but a sequence of partitions that describes the data at different scales (or levels of coarseness). A natural problem then is to analyse and compare, as a whole, the (not necessarily hierarchical) sequences of partitions that underpin such multiscale descriptions. Using tools from topological data analysis, we introduce here the Multiscale Clustering Filtration (MCF), a filtration of abstract simplicial complexes that encodes arbitrary cluster assignments in a sequence of partitions across scales of increasing coarseness. We prove the stability of the MCF with respect to small changes in the sequence of partitions, and establish that its persistence diagram (PD) provides an interpretable representation of the multiscale clustering: at dimension 0, it captures violations of the refinement order, and at higher dimensions, it tracks the emergence and resolution of higher-order inconsistencies between cluster assignments. We also introduce an equivalent construction of the MCF via a nerve complex filtration, and show that this nerve-based MCF has computational advantages in certain settings, aiding scalability. Further, we show that in the hierarchical case, the MCF reduces to a Vietoris-Rips filtration of an ultrametric space, and we establish that pairwise measures, such as conditional entropy or zigzag persistence, are fully determined by a graph representation of the Sankey diagram and thus fail to describe the higher-order inconsistencies across scales that MCF captures. Using synthetic and real-world data from stochastic block models, single-cell transcriptomics and temporal social interactions, we illustrate how the MCF PD serves to characterise and compare multiscale clusterings beyond pairwise or hierarchical baseline methods.
作者备注：Extensively rewritten: new theoretical results, additional numerical experiments on single-cell data and temporal social interactions, and comparison to baseline methods