Multiparameter Hierarchical Clustering Methods

We propose an extension of hierarchical clustering methods, called multiparameter hierarchical clustering methods which are designed to exhibit sensitivity to density while retaining desirable theoretical properties. The input of the method we propose is a triple pX, d, fq, where pX, dq is a finite metric space and f : X Ñ R is a function defined on the data X, which could be a density estimate or could represent some other type of information. The output of our method is more general than dendrograms in that we track two parameters: the usual scale parameter and a parameter related to the function f . Our construction is motivated by the methods of persistent topology [6], the Reeb graph and Cluster Trees [16]. We present both a characterization, and a stability theorem.