An Axiomatic Definition of Hierarchical Clustering

TL;DR

This paper introduces an axiomatic framework for hierarchical clustering that generalizes existing definitions, ensuring consistency with classical cluster tree concepts under mild density conditions.

Contribution

It provides a formal axiomatic foundation for population hierarchical clustering applicable to a broad class of densities, extending Hartigan's cluster tree definition.

Findings

Axiomatic definition aligns with Hartigan's cluster tree under mild conditions

Extension to more general densities beyond piecewise constant cases

Framework ensures consistency with classical clustering concepts

Abstract

In this paper, we take an axiomatic approach to defining a population hierarchical clustering for piecewise constant densities, and in a similar manner to Lebesgue integration, extend this definition to more general densities. When the density satisfies some mild conditions, e.g., when it has connected support, is continuous, and vanishes only at infinity, or when the connected components of the density satisfy these conditions, our axiomatic definition results in Hartigan's definition of cluster tree.