Fuzzy approaches to hard c-means clustering

Abstract

A popular clustering model is hard c-means (HCM). For many data sets the HCM objective function has local extrema, so HCM optimization often yields suboptimal clusterings. The effect of local extrema can be reduced by fuzzification, leading to the well-known fuzzy c-means (FCM) model with the fuzziness parameter m > 1. In this paper we use FCM to optimize the HCM model, even though we actually optimize a different objective function. This work is motivated by a popular approach to avoid local extrema in HCM which approximates the minimum operator in HCM by the harmonic means, leading to c-harmonic means (CHM), which was recently shown to be equivalent to FCM for m = 2. Generalizing the harmonic means in CHM to generalized means yields a clustering model that we call c-generalized means (CGM), which is equivalent to FCM for arbitrary m > 1. Numerical experiments with the BIRCH and Lena data sets show that FCM/CGM (with optimal m) often yields significantly better HCM clusterings than HCM itself or CHM.