An Axiomatic Definition of Hierarchical Clustering

Clustering, informally understood as the grouping of data, is a central task in statistics and computer science with broad applications. Modern clustering algorithms originated in the work of numerical taxonomists, who developed methods to identify hierarchical structures in the classification of plant and animal species. Since then clustering has been used in disciplines such as medicine, astronomy, anthropology, economics, etc., with aims such as exploratory analysis, data summarization, the identification of salient structures in data, and information organization. The notion of a "good" or "accurate" clustering varies between fields and applications. For example, to some computer scientists, the correct clustering of a dataset is often defined as the solution to an optimization problem (think K-means) and a good algorithm either solves or approximates a solution to this problem, ideally with some guarantees [14, 31].