Consistency constraints for overlapping data clustering

TL;DR

This paper explores functorial constraints in overlapping clustering, demonstrating that such clustering schemes are inherently limited to refining single-linkage clusters and being refined by maximal-linkage clusters within metric spaces.

Contribution

It introduces a functorial framework for overlapping clustering, revealing fundamental constraints and relationships among clustering methods.

Findings

Clustering functors are constrained to refine single-linkage clusters.

Clustering functors are refined by maximal-linkage clusters.

The framework applies to data modeled in metric spaces with non-expansive maps.

Abstract

We examine overlapping clustering schemes with functorial constraints, in the spirit of Carlsson--Memoli. This avoids issues arising from the chaining required by partition-based methods. Our principal result shows that any clustering functor is naturally constrained to refine single-linkage clusters and be refined by maximal-linkage clusters. We work in the context of metric spaces with non-expansive maps, which is appropriate for modeling data processing which does not increase information content.