Entropy-Regularized Fuzzy Clustering for Non-Euclidean Relational Data and Indefinite Kernel Data
Shibaura Institute of Technology, 3-7-5 Toyosu, Koto, Tokyo 135-8548, Japan
In this paper, an entropy-regularized fuzzy clustering approach for non-Euclidean relational data and indefinite kernel data is developed that has not previously been discussed. It is important because relational data and kernel data are not always Euclidean and positive semi-definite, respectively. It is theoretically determined that an entropy-regularized approach for both non-Euclidean relational data and indefinite kernel data can be applied without using a β-spread transformation, and that two other options make the clustering results crisp for both data types. These results are in contrast to those from the standard approach. Numerical experiments are employed to verify the theoretical results, and the clustering accuracy of three entropy-regularized approaches for non-Euclidean relational data, and three for indefinite kernel data, is compared.
-  J. C. Bezdek, “Pattern Recognition with Fuzzy Objective Function Algorithms,” Plenum, 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.
-  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.
-  M. Filippone, “Dealing with Non-metric Dissimilarities in Fuzzy Central Clustering Algorithms,” Int. J. of Approximate Reasoning, Vol.40, No.2, pp. 363-384, 2009.
-  R. J. Hathaway and J. C. Bezdek, “NERF C-means: Non-Euclidean Relational Fuzzy Clustering,” Pattern Recognition, Vol.27, pp. 429-437, 1994.
-  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.
-  Y. Kanzawa, Y. Endo, and S. Miyamoto, “Indefinite Kernel Fuzzy c-Means Clustering Algorithms,” Lecture Notes in Computer Science, Vol.6408, pp. 116-128, 2010.
-  S. Tamura, S. Higuchi, and K. Tanaka, “Pattern Classification Based on Fuzzy Relations,” IEEE Trans. Syst. Man Cybern., Vol.1, No.1, pp. 61-66, 1971.
-  J. W. Scannell, C. Blakemore, and M. P. Young, “Analysis of Connectivity in the Cat Cerebral Cortex,” J. Neuroscience, Vol.15, No.2, pp. 1463-1483, 1995.
-  G. Ghosh, A. Strehl, and S. Merugu, “A Consensus Framework for Integrating Distributed Clusterings under Limited Knowledge Sharing,” Proc. NSFWorkshop on Next Generation DataMining, pp. 99-108, 2002.
-  J. C. Bezdek, J. Keller, R. Krisnapuram, and N. R. Pal, “Fuzzy Models and Algorithms for Pattern Recognition and Image Processing,” Kluwer Academic Publishing, Boston, 1999.