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.
-  J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenun, New York, 1981.
-  S. Miyamoto and K. Umayahara, “Methods in Hard and Fuzzy Clustering,” in: Z.-Q. Liu and S. Miyamoto (Eds.), Soft computing and human-centered machines, Springer-Verlag Tokyo, 2000.
-  S.Miyamoto and Y. Nakayama, “Algorithms of Hard c-Means Clustering Using Kernel Functions in Support Vector Machines,” J. Advanced Computational Intelligence and Intelligent Informatics, Vol.7, No.1, pp. 19-24, 2003.
-  S. Miyamoto and D. Suizu, “Fuzzy c-Means Clustering Using Kernel Functions in Support Vector Machines,” J. Advanced Computational Intelligence and Intelligent Informatics, Vol.7, No.1, pp. 25-30, 2003.
-  V. N. Vapnik, “Statistical Learning Theory,” Wiley, New York, 1998.
-  S.Miyamoto, Y. Kawasaki, and K. Sawazaki, “An Explicit Mapping for Kernel Data Analysis and Application to Text Analysis,” Proc. IFSA-EUSFLAT 2009, pp. 618-623, 2009.
-  S. Miyamoto and K. Sawazaki, “An Explicit Mapping for Kernel Data Analysis and Application to c-Means Clustering,” Proc. NOLTA 2009, pp. 556-559, 2009.
-  B. Schoölkopf, “The Kernel Trick for Distances,” In Advances in Neural Information Processing System, Vol.13, MIT Press, 2001.
-  R. J. Hathaway, J. W. Davenport, and J. C. Bezdek, “Relational Duals of the c-means Clustering Algorithms,” Pattern Recognition, Vol.22, No.2, pp. 205-212, 1989.