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JACIII Vol.16 No.7 pp. 825-830
doi: 10.20965/jaciii.2012.p0825
(2012)

Paper:

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

Received:
December 16, 2011
Accepted:
September 25, 2012
Published:
November 20, 2012
Keywords:
clustering, fuzzy c-means, conditionally positive definite kernel
Abstract

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.

Cite this article as:
Y. Kanzawa, Y. Endo, and S. Miyamoto, “Hard and Fuzzy c-Means Clustering with Conditionally Positive Definite Kernel,” J. Adv. Comput. Intell. Intell. Inform., Vol.16, No.7, pp. 825-830, 2012.
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References
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Last updated on Jan. 21, 2019