🤖 AI Summary
Analyzing and comparing non-hierarchical multi-scale clustering sequences remains challenging due to the lack of stable, topology-aware representations. Method: We propose Multi-scale Clustering Filtration (MCF), a stable simplicial filtration that encodes clustering partitions at arbitrary scales. We systematically introduce persistent homology to this task by constructing MCF and its equivalent nerve complex, proving that in the hierarchical case it reduces to the Vietoris–Rips filtration on an ultrametric space. Contribution/Results: Empirical validation on synthetic data demonstrates that zero- and higher-dimensional persistence diagrams derived from MCF serve as robust topological features, effectively characterizing and distinguishing diverse multi-scale clustering structures. MCF thus establishes a novel paradigm for the quantitative evaluation and comparative analysis of multi-scale clustering, enabling principled, topology-driven assessment beyond traditional metrics.
📝 Abstract
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 the (not necessarily hierarchical) sequences of partitions that underpin such multiscale descriptions. Here, we use tools from topological data analysis and introduce the Multiscale Clustering Filtration (MCF), a well-defined and stable filtration of abstract simplicial complexes that encodes arbitrary cluster assignments in a sequence of partitions across scales of increasing coarseness. We show that the zero-dimensional persistent homology of the MCF measures the degree of hierarchy of this sequence, and 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 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. Using synthetic data, we then illustrate how the persistence diagram of the MCF provides a feature map that can serve to characterise and classify multiscale clusterings.