Families of Triangular Norm-Based Kernel Functions and Their Application to Kernel <i>k</i>-Means

Kazushi Okamoto

doi:10.20965/jaciii.2017.p0534

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JACIII Vol.21 No.3 pp. 534-542

doi: 10.20965/jaciii.2017.p0534

(2017)

Paper:

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Families of Triangular Norm-Based Kernel Functions and Their Application to Kernel k-Means

Kazushi Okamoto

Department of Informatics, Graduate School of Informatics and Engineering, The University of Electro-Communications
1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan

Received:

August 4, 2016

Accepted:

February 27, 2017

Online released:

May 19, 2017

Published:

May 20, 2017

Keywords:

adjusted rand index, k-means, kernel method, positive-definite kernel, t-norm

Abstract

This study proposes the concept of families of triangular norm (t-norm)-based kernel functions, and discusses their positive-definite property and the conditions for applicable t-norms. A clustering experiment with kernel k-means is performed in order to analyze the characteristics of the proposed concept, as well as the effects of the t-norm and parameter selections. It is evaluated that the clusters obtained in terms of the adjusted rand index and the experimental results suggested the following : (1) the adjusted rand index values obtained by the proposed method were almost the same or higher than those produced using the linear kernel for all of the data sets; (2) the proposed method slightly improved the adjusted rand index values for some data sets compared with the radial basis function (RBF) kernel; (3) the proposed method tended to map data to a higher dimensional feature space than the linear kernel but the dimension was lower than that using the RBF kernel.

Cite this article as:

K. Okamoto, “Families of Triangular Norm-Based Kernel Functions and Their Application to Kernel k-Means,” J. Adv. Comput. Intell. Intell. Inform., Vol.21 No.3, pp. 534-542, 2017.

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References

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This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] Y. Rubner, J. Puzicha, C. Tomasi, and J. M. Buhmann, “Empirical evaluation of dissimilarity measures for color and texture,” Computer Vision and Image Understanding, Vol.84, No.1, pp. 25-43, 2001.

[2] [2] A. Barla, F. Odone, and A. Verri, “Histogram intersection kernel for image classification,” Proc. of 2003 Int. Conf. on Image Processing, Vol.2, No.III-513-16, 2003.

[3] [3] A. Vedaldi and A. Zisserman, “Efficient additive kernels via explicit feature maps,” IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol.34, No.3, pp. 480-492, 2012.

[4] [4] M. Mizumoto, “Pictorial representations of fuzzy connectives, part I: cases of t-norms, t-conorms and averaging operators,” Fuzzy Sets and Systems, Vol.31, No.2, pp. 217-242, 1989.

[5] [5] E. P. Klement, R. Mesiar, and E. Pap, “Triangular norms. position paper I: basic analytical and algebraic properties,” Fuzzy Sets and Systems, Vol.143, No.1, pp. 5-26, 2004.

[6] [6] C. Alsina and M. S. Tomas, “On positive semidefinite strict t-norms,” General Inequalities, Vol.6, pp. 215-225, 1992.

[7] [7] L. Fu and E. Medico, “FLAME, a novel fuzzy clustering method for the analysis of DNA microarray data,” BMC Bioinformatics, Vol.8, No.1, pp. 1-15, 2007.

[8] [8] A. K. Jain and M. H. C. Law, “Data clustering: a user’s dilemma,” Lecture Notes in Computer Science, Vol.3776, pp. 1-10, 2005.

[9] [9] H. Chang and D. Y. Yeung, “Robust path-based spectral clustering,” Pattern Recognition, Vol.41, No.1, pp. 191-203, 2008.

[10] [10] L. Hubert and P. Arabie, “Comparing partitions,” J. of Classification, Vol.2, No.1, pp. 193-218, 1985.