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.
