# Versatile linkage: a family of space-conserving strategies for agglomerative hierarchical clustering

## TL;DR

This paper introduces versatile linkage, a new family of space-conserving agglomerative hierarchical clustering strategies based on generalized means, encompassing various existing methods and novel linkages.

## Contribution

The paper proposes versatile linkage, a comprehensive family of clustering strategies that unify and extend existing methods using generalized means, with space-conservation properties.

## Key findings

- Versatile linkage includes single, complete, average, geometric, and harmonic linkages.
- It is space-conserving, unlike the β-flexible clustering system.
- Different strategies are compared using cophenetic correlation, mean absolute error, and new measures of tree balance and space distortion.

## Abstract

Agglomerative hierarchical clustering can be implemented with several strategies that differ in the way elements of a collection are grouped together to build a hierarchy of clusters. Here we introduce versatile linkage, a new infinite system of agglomerative hierarchical clustering strategies based on generalized means, which go from single linkage to complete linkage, passing through arithmetic average linkage and other clustering methods yet unexplored such as geometric linkage and harmonic linkage. We compare the different clustering strategies in terms of cophenetic correlation, mean absolute error, and also tree balance and space distortion, two new measures proposed to describe hierarchical trees. Unlike the $\beta$-flexible clustering system, we show that the versatile linkage family is space-conserving.