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

2026-09-10

AI Summary

本文解决了平面聚类方法无法同时满足三个自然公理的问题，通过构建层次聚类方法来共同满足丰富性、一致性和尺度不变性。

Abstract

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 methods satisfying these axioms, which we call admissible. We explicitly construct several admissible methods, including methods based on well-separated clusters and a non-binary version of single linkage. For certain pairs of admissible methods, the hierarchy produced by one always refines that produced by the other. This refinement relation defines a partial order on the class of admissible methods. This partially ordered set has no greatest element and contains uncountably many pairwise incompatible maximal elements, revealing substantial diversity among admissible methods. Nevertheless, this diversity is constrained: every admissible method contains a hierarchy of sufficiently well-separated clusters, and every finite collection of admissible methods shares such a nontrivial common backbone.

Problem

• Hierarchical Clustering
• Richness
• Consistency
• Scale Invariance

Innovation

• hierarchical clustering
• scale invariance
• richness
• consistency