# An Axiomatic Definition of Hierarchical Clustering

arXiv: https://arxiv.org/abs/2407.03574 · HTML: https://mlsanity.com/paper/2407.03574/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 38 references that are in the map.
- First posted on arXiv 4 Jul 2024.

## Builds on (most cited first, 9 of 9)

- [A cost function for similarity-based hierarchical clustering](https://mlsanity.com/paper/1510.05043/) (2015)
- [Consistent procedures for cluster tree estimation and pruning](https://mlsanity.com/paper/1406.1546/) (2014)
- [The modal age of Statistics](https://mlsanity.com/paper/1807.02789/) (2018)
- [Mixture model modal clustering](https://mlsanity.com/paper/1609.04721/) (2016)
- [Possibility results for graph clustering: A novel consistency axiom](https://mlsanity.com/paper/1806.06142/) (2018)
- [Cluster Trees on Manifolds](https://mlsanity.com/paper/1307.6515/) (2013)
- [Beyond Hartigan Consistency: Merge Distortion Metric for Hierarchical Clustering](https://mlsanity.com/paper/1506.06422/) (2015)
- [Statistical Inference for Cluster Trees](https://mlsanity.com/paper/1605.06416/) (2016)
- [Moving Up the Cluster Tree with the Gradient Flow](https://mlsanity.com/paper/2109.08362/) (2021)

## Built on by (most cited first, 3 of 3)

- [Hierarchical Linkage Clustering Beyond Binary Trees and Ultrametrics](https://mlsanity.com/paper/2511.18056/) (2025)
- [Neighborhood Stability as a Measure of Nearest Neighbor Searchability](https://mlsanity.com/paper/2602.16673/) (2026)
- [Classification Fields: Arbitrarily Fine Recursive Hierarchical Clustering From Few Examples](https://mlsanity.com/paper/2605.07119/) (2026)

## Abstract (arXiv)

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.