层次聚类可以同时满足丰富性、一致性和尺度不变性
Hierarchical Clustering Can Jointly Satisfy Richness, Consistency, and Scale Invariance
- École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
- Université Gustave Eiffel（古斯塔夫·埃菲尔大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明层次聚类可同时满足尺度不变性、丰富性和一致性公理，构造了多种可容许方法，并揭示了其偏序结构中的多样性与共同主干。
AI中文摘要：
尽管聚类无处不在，但对于什么是簇，缺乏普遍接受的定义。Kleinberg 的不可能性定理通过证明没有一种平面聚类方法能同时满足三个自然公理（尺度不变性、丰富性和一致性）来形式化这一困难。在本文中，我们探讨当输出是层次结构而非单一划分时，这种不可能性是否依然存在。我们证明，与平面聚类设置相反，这些公理的层次对应版本是可以同时满足的。事实上，存在不可数多个满足这些公理的层次聚类方法，我们称之为可容许方法。我们明确构造了几种可容许方法，包括基于良好分离簇的方法以及单连接的非二叉版本。对于某些可容许方法对，一种方法产生的层次结构总是细化另一种方法产生的层次结构。这种细化关系在可容许方法类上定义了一个偏序。这个偏序集没有最大元素，并包含不可数多个两两不相容的极大元素，揭示了可容许方法之间的显著多样性。然而，这种多样性受到约束：每种可容许方法都包含一个由充分良好分离的簇构成的层次结构，并且每个有限的可容许方法集合都共享这样一个非平凡的共同主干。
英文摘要：
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.
