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Conference Papers Year : 2018

Hierarchical Clustering: Objective Functions and Algorithms

Abstract

Hierarchical clustering is a recursive partitioning of a dataset into clusters at an increasingly finer granularity. Motivated by the fact that most work on hierarchical clustering was based on providing algorithms, rather than optimizing a specific objective, Dasgupta framed similarity-based hierarchical clustering as a combinatorial optimization problem, where a `good' hierarchical clustering is one that minimizes some cost function. He showed that this cost function has certain desirable properties. We take an axiomatic approach to defining `good' objective functions for both similarity and dissimilarity-based hierarchical clustering. We characterize a set of "admissible" objective functions (that includes Dasgupta's one) that have the property that when the input admits a `natural' hierarchical clustering, it has an optimal value. Equipped with a suitable objective function, we analyze the performance of practical algorithms, as well as develop better algorithms. For similarity-based hierarchical clustering, Dasgupta showed that the divisive sparsest-cut approach achieves an O(log3/2n)-approximation. We give a refined analysis of the algorithm and show that it in fact achieves an O(logn−−−−√)-approx. (Charikar and Chatziafratis independently proved that it is a O(logn−−−−√)-approx.). This improves upon the LP-based O(logn)-approx. of Roy and Pokutta. For dissimilarity-based hierarchical clustering, we show that the classic average-linkage algorithm gives a factor 2 approx., and provide a simple and better algorithm that gives a factor 3/2 approx.. Finally, we consider `beyond-worst-case' scenario through a generalisation of the stochastic block model for hierarchical clustering. We show that Dasgupta's cost function has desirable properties for these inputs and we provide a simple 1 + o(1)-approximation in this setting.

Dates and versions

hal-02169539 , version 1 (01-07-2019)

Identifiers

Cite

Vincent Cohen-Addad, Varun Kanade, Frederik Mallmann-Trenn, Claire Mathieu. Hierarchical Clustering: Objective Functions and Algorithms. SODA 2018 - Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, Jan 2018, New Orleans, LA, United States. pp.378-397, ⟨10.1137/1.9781611975031.26⟩. ⟨hal-02169539⟩
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