Hierarchical Clustering better than Average-Linkage

TL;DR

This paper demonstrates that average-linkage hierarchical clustering has significant worst-case limitations for certain objectives and introduces new semidefinite programming algorithms that outperform it.

Contribution

The paper provides tight worst-case bounds for average-linkage and proposes new SDP-based algorithms with better approximation guarantees.

Findings

Average-linkage cannot surpass 1/3 and 2/3 approximation ratios for specific HC objectives.

Counterexamples show the limitations of average-linkage in worst-case scenarios.

New SDP algorithms achieve better approximation ratios than average-linkage.

Abstract

Hierarchical Clustering (HC) is a widely studied problem in exploratory data analysis, usually tackled by simple agglomerative procedures like average-linkage, single-linkage or complete-linkage. In this paper we focus on two objectives, introduced recently to give insight into the performance of average-linkage clustering: a similarity based HC objective proposed by [Moseley and Wang, 2017] and a dissimilarity based HC objective proposed by [Cohen-Addad et al., 2018]. In both cases, we present tight counterexamples showing that average-linkage cannot obtain better than 1/3 and 2/3 approximations respectively (in the worst-case), settling an open question raised in [Moseley and Wang, 2017]. This matches the approximation ratio of a random solution, raising a natural question: can we beat average-linkage for these objectives? We answer this in the affirmative, giving two new algorithms based on semidefinite programming with provably better guarantees.