Well-Separated Clusters and Optimal Fuzzy Partitions

Abstract Two separation indices are considered for partitions P = {X1, …, Xk} of a finite data set X in a general inner product space. Both indices increase as the pairwise distances between the subsets Xi become large compared to the diameters of Xi Maximally separated partitions p' are defined and it is shown that as the indices of p' increase without bound, the characteristic functions of Xi' in P' are approximated more and more closely by the membership functions in fuzzy partitions which minimize certain fuzzy extensions of the k-means squared error criterion function.