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Hierarchical Clustering Can Jointly Satisfy Richness, Consistency, and Scale Invariance

arXiv · AI, language, vision and robotics · article · Sep 10, 2026 · UTC

Despite its ubiquity, clustering lacks a universally accepted definition of what is a cluster. Kleinberg's Impossibility Theorem formalizes this difficulty by showing that no flat clustering method can simultaneously satisfy three natural axioms: scale invariance, richness, and consistency. In this paper, we ask whether this impossibility persists when the output is a hierarchy rather than a single partition. We show that, in contrast to the flat clustering setting, the hierarchical analog of these axioms are jointly satisfiable. In fact, there exist uncountably many hierarchical clustering me

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