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Chamfer-Linkage for Hierarchical Agglomerative Clustering

Kishen N Gowda, Willem Fletcher, MohammadHossein Bateni, Laxman Dhulipala, D Ellis Hershkowitz, Rajesh Jayaram, Jakub Łącki

TL;DR

Chamfer-linkage addresses HAC by introducing an asymmetric cluster distance $Ch(A,B)=sum_{a in A} min_{b in B} d(a,b)$, which better captures concept representation and promotes balanced dendrograms. The authors show an exact $O(n^2)$-time HAC algorithm within the HAC-NN framework and provide a space–time trade-off that reduces memory to $O(n^2/t)$ at the cost of $O(n^2 t)$ time, with $t\in[1,n]$. Empirically, Chamfer-linkage beats classical linkages (e.g., average, Ward) on 19 diverse datasets in ARI and NMI while producing well-balanced hierarchies, and the authors release optimized C++ implementations and Python bindings. The work demonstrates that Chamfer-linkage is a practical drop-in replacement for traditional linkages and motivates scalable parallel and approximate variants for very large datasets.

Abstract

Hierarchical Agglomerative Clustering (HAC) is a widely-used clustering method based on repeatedly merging the closest pair of clusters, where inter-cluster distances are determined by a linkage function. Unlike many clustering methods, HAC does not optimize a single explicit global objective; clustering quality is therefore primarily evaluated empirically, and the choice of linkage function plays a crucial role in practice. However, popular classical linkages, such as single-linkage, average-linkage and Ward's method show high variability across real-world datasets and do not consistently produce high-quality clusterings in practice. In this paper, we propose \emph{Chamfer-linkage}, a novel linkage function that measures the distance between clusters using the Chamfer distance, a popular notion of distance between point-clouds in machine learning and computer vision. We argue that Chamfer-linkage satisfies desirable concept representation properties that other popular measures struggle to satisfy. Theoretically, we show that Chamfer-linkage HAC can be implemented in $O(n^2)$ time, matching the efficiency of classical linkage functions. Experimentally, we find that Chamfer-linkage consistently yields higher-quality clusterings than classical linkages such as average-linkage and Ward's method across a diverse collection of datasets. Our results establish Chamfer-linkage as a practical drop-in replacement for classical linkage functions, broadening the toolkit for hierarchical clustering in both theory and practice.

Chamfer-Linkage for Hierarchical Agglomerative Clustering

TL;DR

Chamfer-linkage addresses HAC by introducing an asymmetric cluster distance , which better captures concept representation and promotes balanced dendrograms. The authors show an exact -time HAC algorithm within the HAC-NN framework and provide a space–time trade-off that reduces memory to at the cost of time, with . Empirically, Chamfer-linkage beats classical linkages (e.g., average, Ward) on 19 diverse datasets in ARI and NMI while producing well-balanced hierarchies, and the authors release optimized C++ implementations and Python bindings. The work demonstrates that Chamfer-linkage is a practical drop-in replacement for traditional linkages and motivates scalable parallel and approximate variants for very large datasets.

Abstract

Hierarchical Agglomerative Clustering (HAC) is a widely-used clustering method based on repeatedly merging the closest pair of clusters, where inter-cluster distances are determined by a linkage function. Unlike many clustering methods, HAC does not optimize a single explicit global objective; clustering quality is therefore primarily evaluated empirically, and the choice of linkage function plays a crucial role in practice. However, popular classical linkages, such as single-linkage, average-linkage and Ward's method show high variability across real-world datasets and do not consistently produce high-quality clusterings in practice. In this paper, we propose \emph{Chamfer-linkage}, a novel linkage function that measures the distance between clusters using the Chamfer distance, a popular notion of distance between point-clouds in machine learning and computer vision. We argue that Chamfer-linkage satisfies desirable concept representation properties that other popular measures struggle to satisfy. Theoretically, we show that Chamfer-linkage HAC can be implemented in time, matching the efficiency of classical linkage functions. Experimentally, we find that Chamfer-linkage consistently yields higher-quality clusterings than classical linkages such as average-linkage and Ward's method across a diverse collection of datasets. Our results establish Chamfer-linkage as a practical drop-in replacement for classical linkage functions, broadening the toolkit for hierarchical clustering in both theory and practice.
Paper Structure (14 sections, 3 theorems, 7 equations, 3 figures, 7 tables, 2 algorithms)

This paper contains 14 sections, 3 theorems, 7 equations, 3 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quadratic Time Algorithm
  4. Space--Time Trade-off Algorithm
  5. Empirical Evaluation
  6. Quality Evaluation
  7. Scaling to large datasets
  8. Dendrogram Height
  9. Related Work on HAC
  10. Conclusion
  11. Empirical Evaluation continued
  12. Datasets
  13. Running Times
  14. Quality Evaluation

Key Result

Theorem 1

Algorithm alg:HAC_NN, when instantiated with the Chamfer-linkage-specific Merge subroutine in Algorithm alg:Chamfer_HAC, correctly computes the Chamfer-linkage dendrogram in $O(n^2)$ time and space.

Figures (3)

  • Figure 1: An example of HAC.
  • Figure 2: Non-reducibility: Consider six points equally spaced along a circle. They are partitioned into three clusters of points (red, green, and purple clusters in the figure). When the red and green clusters merge, they create a blue cluster. But $\mathtt{Ch}\xspace(\text{purple}, \text{blue}) < \min\{ \mathtt{Ch}\xspace(\text{purple}, \text{red}), \mathtt{Ch}\xspace(\text{purple}, \text{green}) \}$ since both vertices of the purple cluster have a neighbor (along the circle) in the blue cluster.
  • Figure 3: Demonstration of Observation \ref{['obs:min-monotone']}

Theorems & Definitions (4)

  • Definition 1: Chamfer Distance
  • Theorem 1
  • Theorem 2
  • Theorem 3