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 (cpd) kernel. One is based on hard c -means and two are based on standard and entropy-regularized fuzzy c -means. First, based on a cpd kernel describing a squared Euclidean distance between data in 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 (pd) kernel and cpd kernel, the revised dissimilarity between a datum and a cluster center in the feature space is shown. Finally, it is shown that a cpd kernel c -means algorithm and a kernel c -means algorithm with a pd kernel derived from the cpd kernel are essentially identical to each other. Explicit mapping for a cpd kernel is also described geometrically.