Ultrametric Cluster Hierarchies: I Want 'em All!

TL;DR

This paper introduces a method to efficiently find optimal center-based clusterings within any hierarchical structure, enabling quick access to multiple meaningful hierarchies for data analysis.

Contribution

It proves that optimal center-based clusterings can be computed quickly for any hierarchy and that these solutions form new hierarchies, expanding hierarchical clustering capabilities.

Findings

Efficient algorithms for optimal center-based clustering in arbitrary hierarchies.

Ability to generate multiple meaningful hierarchies from a given cluster tree.

Validated techniques across various datasets and clustering schemes.

Abstract

Hierarchical clustering is a powerful tool for exploratory data analysis, organizing data into a tree of clusterings from which a partition can be chosen. This paper generalizes these ideas by proving that, for any reasonable hierarchy, one can optimally solve any center-based clustering objective over it (such as $k$-means). Moreover, these solutions can be found exceedingly quickly and are themselves necessarily hierarchical. Thus, given a cluster tree, we show that one can quickly access a plethora of new, equally meaningful hierarchies. Just as in standard hierarchical clustering, one can then choose any desired partition from these new hierarchies. We conclude by verifying the utility of our proposed techniques across datasets, hierarchies, and partitioning schemes.