Contour approximation of data: A duality theory

Abstract

Given a dataset D partitioned in clusters, the joint distance function (JDF) J(x) at any point x is the harmonic mean of the distances of x from the cluster centers. The JDF is a continuous function capturing the data points in its lower level sets (a property called contour approximation), and is a useful concept in probabilistic clustering and data analysis. The JDF of the whole dataset, J(D) := {J(x) : x ∈ D}, is a measure of the classifiability of D, and can be used to determine the "right" number of clusters for D. A duality theory for the JDF of a dataset is given, in analogy with Kuhn's geometric duality theory for the Fermat-Weber location problem.