Ultrametric Cluster Hierarchies: I Want ‘em All!

Andrew Draganov, Pascal Weber, Rasmus Skibdahl Melanchton Jørgensen, Anna Beer, Claudia Plant, Ira Assent

Keywords: Ultrametric, Center-Based Clustering, Hierarchical Clustering, DBSCAN, HDBSCAN, k-means, k-median

TL;DR: We prove clean optimality results for clustering in ultrametrics, identify ways to take advantage of this theory and thoroughly evaluate the resulting techniques.

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.

Primary Area: General machine learning (supervised, unsupervised, online, active, etc.)

Submission Number: 7243