On Hard and Fuzzy c-Means with Conditionally Positive Definite Kernel

Abstract

In this paper, three types of c-means clustering algorithms are investigated with conditionally positive definite kernel. One is based on hard c-means, and the others are on standard fuzzy $c$-means and entropy regularized one. First, based on that conditionally positive definite kernel describes a squared Euclid distance between data in feature space, these algorithms are indicated from revised optimization problems of the conventional kernel $c$-means. Next, based on the relationship between positive definite kernel and conditionally one, the revised dissimilarity using conditionally positive definite kernel between a datum and a cluster center in feature space is shown. Last, it is shown that conditionally positive definite kernel c-means algorithm and kernel c-means algorithm with positive definite kernel from conditionally one are essentially equal with each other. An explicit mapping for conditionally positive definite kernel is also described geometrically.