Hard and Fuzzy c-Means Clustering with Conditionally Positive Definite Kernel
Yuchi Kanzawa*, Yasunori Endo**, and Sadaaki Miyamoto**
*Shibaura Institute of Technology, 3-7-5 Toyosu, Koto-ku, Tokyo 135-8548, Japan
**University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan
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.
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