On hard and fuzzy c-means clustering with conditionally positive definite kernel

In this paper, we investigate three types of c-means clustering algorithms with a conditionally positive definite kernel. One is based on hard c-means, and the others are based on standard and entropy-regularized fuzzy c-means. First, based on a conditionally positive definite kernel describing a squared Euclidean distance between data in the feature space, these algorithms are derived from revised optimization problems of the conventional kernel c-means. Next, based on the relationship between the positive definite kernel and conditionally positive definite kernel, the revised dissimilarity between a datum and a cluster center in the feature space is shown. Finally, it is shown that a conditionally positive definite kernel c-means algorithm and a kernel c-means algorithm with a positive definite kernel derived from the conditionally positive definite kernel are essentially identical to each other. An explicit mapping for a conditionally positive definite kernel is also described geometrically.