Discerning Linkage-Based Algorithms among Hierarchical Clustering Methods

Selecting a clustering algorithm is a perplexing task. Yet since different algorithms may yield dramatically different outputs on the same data, the choice of algorithm is crucial. When selecting a clustering algorithm, users tend to focus on cost-related considerations (software purchasing costs, running times, etc). Differences concerning the output of the algorithms are not usually considered. Recently, a formal approach for selecting a clustering algorithm has been proposed [2]. The approach involves distilling abstract properties of the input-output behavior of different clustering paradigms and classifying algorithms based on these properties. In this paper, we extend the approach in [2] into the hierarchical setting. The class of linkagebased algorithms is perhaps the most popular class of hierarchical algorithms. We identify two properties of hierarchical algorithms, and prove that linkage-based algorithms are the only ones that satisfy both of these properties. Our characterization clearly delineates the difference between linkage-based algorithms and other hierarchical algorithms. We formulate an intuitive notion of locality of a hierarchical algorithm that distinguishes between linkagebased and “global” hierarchical algorithms like bisecting k-means, and prove that popular divisive hierarchical algorithms produce clusterings that cannot be produced by any linkage-based algorithm.