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On the cohesion and separability of average-link for hierarchical agglomerative clustering

Eduardo Sany Laber, Miguel Bastista

TL;DR

A comprehensive study of the performance of average-link in metric spaces, regarding several natural criteria that capture separability and cohesion and are more interpretable than Dasgupta's cost function and its variants.

Abstract

Average-link is widely recognized as one of the most popular and effective methods for building hierarchical agglomerative clustering. The available theoretical analyses show that this method has a much better approximation than other popular heuristics, as single-linkage and complete-linkage, regarding variants of Dasgupta's cost function [STOC 2016]. However, these analyses do not separate average-link from a random hierarchy and they are not appealing for metric spaces since every hierarchical clustering has a 1/2 approximation with regard to the variant of Dasgupta's function that is employed for dissimilarity measures [Moseley and Yang 2020]. In this paper, we present a comprehensive study of the performance of average-link in metric spaces, regarding several natural criteria that capture separability and cohesion and are more interpretable than Dasgupta's cost function and its variants. We also present experimental results with real datasets that, together with our theoretical analyses, suggest that average-link is a better choice than other related methods when both cohesion and separability are important goals.

On the cohesion and separability of average-link for hierarchical agglomerative clustering

TL;DR

A comprehensive study of the performance of average-link in metric spaces, regarding several natural criteria that capture separability and cohesion and are more interpretable than Dasgupta's cost function and its variants.

Abstract

Average-link is widely recognized as one of the most popular and effective methods for building hierarchical agglomerative clustering. The available theoretical analyses show that this method has a much better approximation than other popular heuristics, as single-linkage and complete-linkage, regarding variants of Dasgupta's cost function [STOC 2016]. However, these analyses do not separate average-link from a random hierarchy and they are not appealing for metric spaces since every hierarchical clustering has a 1/2 approximation with regard to the variant of Dasgupta's function that is employed for dissimilarity measures [Moseley and Yang 2020]. In this paper, we present a comprehensive study of the performance of average-link in metric spaces, regarding several natural criteria that capture separability and cohesion and are more interpretable than Dasgupta's cost function and its variants. We also present experimental results with real datasets that, together with our theoretical analyses, suggest that average-link is a better choice than other related methods when both cohesion and separability are important goals.

Paper Structure

This paper contains 25 sections, 12 theorems, 40 equations, 6 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our results
  3. Related work
  4. Preliminaries
  5. Cohesion and separability
  6. The ${\tt cs\hbox{-}ratio_{AV}}$ criterion
  7. The ${\tt cs\hbox{-}ratio_{DM}}$ criterion
  8. Separability criteria
  9. On the cohesion of average-link
  10. Experiments
  11. Proof of proposition \ref{['prop:triangle-ineq']}
  12. Proofs of section \ref{['sec:separability-cohesion']}
  13. Proof of Theorem \ref{['thm:goodnessav']}
  14. Lower bounds on ${\tt cs\hbox{-}ratio_{AV}}$ for other methods
  15. On the approximation of average-link for ${\tt cs\hbox{-}ratio_{AV}}$
  16. ...and 10 more sections

Key Result

Proposition 2.1

Let $A$, $B$ and $C$ be three clusters. Then,

Figures (6)

  • Figure 1: Results for the max-diam for the different datasets. For interpreting the bars, the lower the better
  • Figure 2: Results for the max-avg for the different datasets. For interpreting the bars, the lower the better
  • Figure 3: Results for the ${\tt sep_{min}}$ for the different datasets. For interpreting the bars, the higher the better
  • Figure 4: Results for the ${\tt sep_{av}}$ for the different datasets. For interpreting the bars, the higher the better
  • Figure 5: Results for the ${\tt cs\hbox{-}ratio_{AV}}$ for the different datasets and methods. For interpreting the bars, the lower the better
  • ...and 1 more figures

Theorems & Definitions (25)

  • Proposition 2.1: Triangle Inequality for averages
  • Theorem 3.1
  • Lemma 3.2
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Claim 1
  • Lemma 3.5
  • Proposition 4.1
  • ...and 15 more