An Axiomatic Definition of Hierarchical Clustering
This work provides a foundational framework for hierarchical clustering in statistics and machine learning, addressing a theoretical gap for general densities.
The authors tackled the problem of defining hierarchical clustering for general densities by proposing an axiomatic approach, extending it from piecewise constant to more general densities and showing that under mild conditions it aligns with Hartigan's cluster tree definition.
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.