I Want 'Em All (At Once) -- Ultrametric Cluster Hierarchies

2025-02-19

AI Summary

Existing centroid-based objectives (e.g., k-means) lack efficient and exact optimization over arbitrary valid hierarchical clusterings, hindering flexible, interpretable hierarchical modeling. Method: We propose a tree-structured optimization framework grounded in ultrametrics and dynamic programming. Crucially, we prove that—given any initial clustering tree—the optimal hierarchical solution for centroid-based objectives (e.g., k-means) at all granularities can be computed exactly in *O*(*nk*) time. The resulting hierarchy inherently satisfies hierarchical consistency and enables rapid enumeration of semantically equivalent hierarchical structures from the same base tree. Results: Extensive experiments across multiple datasets demonstrate substantial improvements in flexibility, computational efficiency, and generalization performance of hierarchical clustering. Our framework establishes a new paradigm unifying exact objective optimization with interpretable, semantics-aware hierarchical representation learning.

Technology Category

• Machine Learning: Clustering
• Search and Optimization: Heuristic Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

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.

Problem

• Generalizes hierarchical clustering techniques
• Optimizes center-based clustering objectives
• Verifies utility across diverse datasets

Innovation

• Generalizes hierarchical clustering
• Optimizes center-based clustering objectives
• Quickly accesses meaningful hierarchies