Kernel-Based Clustering: A Comparative Study

We present a comprehensive comparative analysis of classical Fuzzy C-Means (FCM) clustering and kernel ?based Fuzzy C-Means clustering. While Fuzzy C-Means is a popular soft-clustering method, its effectiveness is largely limited to hyper spherical clusters that are linearly separable. If the clusters are not linearly separable, the FCM is not particularly effective. By applying the kernel trick, the kernelised fuzzy c-means algorithm attempts to address this problem. The Kernel FCM operates by first non-linearly mapping the data to appropriate and sufficiently high-dimensional feature spaces, where according to the Mercer’s theorem the data are likely to be linearly separable, and then applying the classical FCM algorithm. We first present the Hard C-Means clustering algorithm and a generalization of it called the Fuzzy C-Means. Then we explore the mathematical basis behind the kernel trick, both in general and especially in the setting of clustering. Following this we evaluate the performance gains provided by kernelised FCM and its classical counterpart, which is the main objective of the present work. It is shown that kernelised FCM does provide significant improvements for several popular Machine Learning data sets. However, it is observed that the performance of kerneized FCM depends greatly on the selection of the kernel parameters. We do a short comparative study of Kernelized FCM using different kernels.