Clustering I: Basic Clustering Models and Algorithms

  • Ke-Lin DuEmail author
  • M. N. S. Swamy


Clustering is an unsupervised classification technique that identifies some inherent structure present in a set of objects based on a similarity measure. Clustering methods can be derived from statistical models or competitive learning and correspondingly they can be classified into generative (or model-based) and discriminative (or similarity-based) approaches. A clustering problem can also be modeled as a COP. Clustering neural networks are statistical models, where a probability density function (pdf) for data is estimated by learning its parameters. In this chapter, our emphasis is placed on a number of competitive learning-based neural networks and clustering algorithms. We describe the SOM, learning vector quantization (LVQ), and ART models, as well as C-means, subtractive, and fuzzy clustering algorithms.


  1. 1.
    Alex, N., Hasenfuss, A., & Hammer, B. (2009). Patch clustering for massive data sets. Neurocomputing, 72, 1455–1469.CrossRefGoogle Scholar
  2. 2.
    Anderson, I. A., Bezdek, J. C., & Dave, R. (1982). Polygonal shape description of plane boundaries. In L. Troncale (Ed.), Systems science and science (Vol. 1, pp. 295–301). Louisville, KY: SGSR.Google Scholar
  3. 3.
    Angelov, P. P., & Filev, D. P. (2004). An approach to online identification of Takagi–Sugeno fuzzy models. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 34(1), 484–498.CrossRefGoogle Scholar
  4. 4.
    Aoki, T., & Aoyagi, T. (2007). Self-organizing maps with asymmetric neighborhood function. Neural Computation, 19, 2515–2535.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Atiya, A. F. (2005). Estimating the posterior probabilities using the \(K\)-nearest neighbor rule. Neural Computation, 17, 731–740.zbMATHCrossRefGoogle Scholar
  6. 6.
    Baraldi, A., & Blonda, P. (1999). A survey of fuzzy clustering algorithms for pattern recognition—Part II. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 29(6), 786–801.CrossRefGoogle Scholar
  7. 7.
    Bax, E. (2012). Validation of \(k\)-nearest neighbor classifiers. IEEE Transactions on Information Theory, 58(5), 3225–3234.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Beliakov, G., & Li, G. (2012). Improving the speed and stability of the \(k\)-nearest neighbors method. Pattern Recognition Letters, 33, 1296–1301.CrossRefGoogle Scholar
  9. 9.
    Beni, G., & Liu, X. (1994). A least biased fuzzy clustering method. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(9), 954–960.CrossRefGoogle Scholar
  10. 10.
    Berglund, E., & Sitte, J. (2006). The parameterless self-organizing map algorithm. IEEE Transactions on Neural Networks, 17(2), 305–316.CrossRefGoogle Scholar
  11. 11.
    Bermejo, S. (2006). The regularized LVQ1 algorithm. Neurocomputing, 70, 475–488.CrossRefGoogle Scholar
  12. 12.
    Bezdek, J. (1981). Pattern recognition with fuzzy objective function algorithms. New York: Plenum Press.Google Scholar
  13. 13.
    Biehl, M., Ghosh, A., & Hammer, B. (2007). Dynamics and generalization ability of LVQ algorithms. Journal of Machine Learning Research, 8, 323–360.MathSciNetzbMATHGoogle Scholar
  14. 14.
    Brodal, A. (1981). Neurological anatomy in relation to clinical medicine (3rd ed.). New York: Oxford University Press.Google Scholar
  15. 15.
    Burke, L. I. (1991). Clustering characterization of adaptive resonance. Neural Networks, 4(4), 485–491.CrossRefGoogle Scholar
  16. 16.
    Carpenter, G., Grossberg, S., & Rosen, D. B. (1991). ART 2-A: An adaptive resonance algorithm for rapid category learning and recognition. Neural Networks, 4, 493–504.CrossRefGoogle Scholar
  17. 17.
    Carpenter, G. A., & Grossberg, S. (1987). A massively parallel architecture for a self-organizing neural pattern recognition machine. Computer Vision, Graphics, and Image Processing, 37, 54–115.zbMATHCrossRefGoogle Scholar
  18. 18.
    Carpenter, G. A., & Grossberg, S. (1987). ART 2: Self-organization of stable category recognition codes for analog input patterns. Applied Optics, 26, 4919–4930.CrossRefGoogle Scholar
  19. 19.
    Carpenter, G. A., Grossberg, S., & Reynolds, J. H. (1991). ARTMAP: Supervised real-time learning and classification of nonstationary data by a self-organizing neural network. Neural Networks, 4(5), 565–588.CrossRefGoogle Scholar
  20. 20.
    Carpenter, G. A., Grossberg, S., Markuzon, N., Reynolds, J. H., & Rosen, D. B. (1992). Fuzzy ARTMAP: A neural network architecture for incremental supervised learning of analog multidimensional maps. IEEE Transactions on Neural Networks, 3, 698–713.CrossRefGoogle Scholar
  21. 21.
    Cheng, Y. (1997). Convergence and ordering of Kohonen’s batch map. Neural Computation, 9, 1667–1676.CrossRefGoogle Scholar
  22. 22.
    Chinrunrueng, C., & Sequin, C. H. (1995). Optimal adaptive \(k\)-means algorithm with dynamic adjustment of learning rate. IEEE Transactions on Neural Networks, 6(1), 157–169.CrossRefGoogle Scholar
  23. 23.
    Chiu, S. (1994). Fuzzy model identification based on cluster estimation. Journal of Intelligent and Fuzzy Systems, 2(3), 267–278.Google Scholar
  24. 24.
    Chiu, S. L. (1994). A cluster estimation method with extension to fuzzy model identification. In Proceedings of the IEEE International Conference on Fuzzy Systems (Vol. 2, pp. 1240–1245). Orlando, FL.Google Scholar
  25. 25.
    Cho, J., Paiva, A. R. C., Kim, S.-P., Sanchez, J. C., & Principe, J. C. (2007). Self-organizing maps with dynamic learning for signal reconstruction. Neural Networks, 20, 274–284.zbMATHCrossRefGoogle Scholar
  26. 26.
    Choy, C. S. T., & Siu, W. C. (1998). Fast sequential implementation of “neural-gas” network for vector quantization. IEEE Transactions on Communications, 46(3), 301–304.CrossRefGoogle Scholar
  27. 27.
    Cottrell, M., Hammer, B., Hasenfuss, A., & Villmann, T. (2006). Batch and median neural gas. Neural Networks, 19, 762–771.zbMATHCrossRefGoogle Scholar
  28. 28.
    Cover, T. M., & Hart, P. E. (1967). Nearest neighbor pattern classification. IEEE Transactions on Information Theory, 13, 21–27.zbMATHCrossRefGoogle Scholar
  29. 29.
    Dave, R. N., & Krishnapuram, R. (1997). Robust clustering methods: A unified view. IEEE Transactions on Fuzzy Systems, 5(2), 270–293.CrossRefGoogle Scholar
  30. 30.
    de Carvalho, F. A. T., Tenorio, C. P., & Cavalcanti, N. L, Jr. (2006). Partitional fuzzy clustering methods based on adaptive quadratic distances. Fuzzy Sets and Systems, 157, 2833–2857.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Demartines, P., & Herault, J. (1997). Curvilinear component analysis: A self-organizing neural network for nonlinear mapping of data sets. IEEE Transactions on Neural Networks, 8(1), 148–154.CrossRefGoogle Scholar
  32. 32.
    Desarbo, W. S., Carroll, J. D., Clark, L. A., & Green, P. E. (1984). Synthesized clustering: A method for amalgamating clustering bases with differential weighting variables. Psychometrika, 49, 57–78.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Ding, C., & He, X. (2004). Cluster structure of \(k\)-means clustering via principal component analysis. In Proceedings of the 8th Pacific-Asia Conference on Advances in Knowledge Discovery and Data Mining (PAKDD), LNCS (Vol. 3056, pp. 414–418). Sydney, Australia; Berlin: Springer.Google Scholar
  34. 34.
    Du, K.-L., & Swamy, M. N. S. (2006). Neural networks in a softcomputing framework. London: Springer.Google Scholar
  35. 35.
    Du, K.-L. (2010). Clustering: A neural network approach. Neural Networks, 23(1), 89–107.zbMATHCrossRefGoogle Scholar
  36. 36.
    Dunn, J. C. (1974). Some recent investigations of a new fuzzy partitioning algorithm and its applicatiopn to pattern classification problems. Journal of Cybernetics, 4, 1–15.CrossRefGoogle Scholar
  37. 37.
    Estevez, P. A., & Figueroa, C. J. (2006). Online data visualization using the neural gas network. Neural Networks, 19, 923–934.zbMATHCrossRefGoogle Scholar
  38. 38.
    Fayed, H. A., & Atiya, A. F. (2009). A novel template reduction approach for the \(K\)-nearest neighbor method. IEEE Transactions on Neural Networks, 20(5), 890–896.CrossRefGoogle Scholar
  39. 39.
    Fix, E., & Hodges, J. L., Jr. (1951). Discriminatory analysis—Nonparametric discrimination: Consistency properties. Project No. 2-49-004, Report No. 4. Randolph Field, TX: USAF School of Aviation (Reprinted in International Statistical Review, 57(3), 238–247 (1989)).Google Scholar
  40. 40.
    Flanagan, J. A. (1996). Self-organization in Kohonen’s SOM. Neural Networks, 9(7), 1185–1197.CrossRefGoogle Scholar
  41. 41.
    Fort, J. C. (1988). Solving a combinatorial problem via self-organizing process: An application of Kohonen-type neural networks to the travelling salesman problem. Biology in Cybernetics, 59, 33–40.zbMATHCrossRefGoogle Scholar
  42. 42.
    Fritzke, B. (1997). The LBG-U method for vector quantization—An improvement over LBG inspired from neural networks. Neural Processing Letters, 5(1), 35–45.CrossRefGoogle Scholar
  43. 43.
    Fukushima, K. (2010). Neocognitron trained with winner-kill-loser rule. Neural Networks, 23, 926–938.CrossRefGoogle Scholar
  44. 44.
    Gath, I., & Geva, A. B. (1989). Unsupervised optimal fuzzy clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(7), 773–781.zbMATHCrossRefGoogle Scholar
  45. 45.
    Gersho, A. (1979). Asymptotically optimal block quantization. IEEE Transactions on Information Theory, 25(4), 373–380.MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Ghosh, A., Biehl, M., & Hammer, B. (2006). Performance analysis of LVQ algorithms: A statistical physics approach. Neural Networks, 19, 817–829.zbMATHCrossRefGoogle Scholar
  47. 47.
    Gottlieb, L.-A., Kontorovich, A., & Krauthgamer, R. (2014). Efficient classification for metric data. IEEE Transactions on Information Theory, 60(9), 5750–5759.MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Grossberg, S. (1976). Adaptive pattern classification and universal recording: I. Parallel development and coding of neural feature detectors; II. Feedback, expectation, olfaction, and illusions. Biological Cybernetics, 23, 121–34, 187–202.MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Guo, Y., & Sengur, A. (2015). NCM: Neutrosophic \(c\)-means clustering algorithm. Pattern Recognition, 48(8), 2710–2724.CrossRefGoogle Scholar
  50. 50.
    Gustafson, D. E., & Kessel, W. (1979). Fuzzy clustering with a fuzzy covariance matrix. In Proceedings of the IEEE Conference on Decision and Control (pp. 761–766). San Diego, CA.Google Scholar
  51. 51.
    Hall, L. O., & Goldgof, D. B. (2011). Convergence of the single-pass and online fuzzy \(C\)-means algorithms. IEEE Transactions on Fuzzy Systems, 19(4), 792–794.CrossRefGoogle Scholar
  52. 52.
    Hammer, B., Strickert, M., & Villmann, T. (2005). Supervised neural gas with general similarity measure. Neural Processing Letters, 21(1), 21–44.CrossRefGoogle Scholar
  53. 53.
    Hara, T., & Hirose, A. (2004). Plastic mine detecting radar system using complex-valued self-organizing map that deals with multiple-frequency interferometric images. Neural Networks, 17, 1201–1210.CrossRefGoogle Scholar
  54. 54.
    Hart, P. E. (1968). The condensed nearest neighbor rule. IEEE Transactions on Information Theory, 14(3), 515–516.CrossRefGoogle Scholar
  55. 55.
    Hathaway, R. J., & Bezdek, J. C. (1993). Switching regression models and fuzzy clustering. IEEE Transactions on Fuzzy Systems, 1, 195–204.CrossRefGoogle Scholar
  56. 56.
    Hathaway, R. J., & Bezdek, J. C. (2000). Generalized fuzzy \(c\)-means clustering strategies using \(L_p\) norm distances. IEEE Transactions on Fuzzy Systems, 8(5), 576–582.CrossRefGoogle Scholar
  57. 57.
    He, J., Tan, A. H., & Tan, C. L. (2004). Modified ART 2A growing network capable of generating a fixed number of nodes. IEEE Transactions on Neural Networks, 15(3), 728–737.CrossRefGoogle Scholar
  58. 58.
    Huang, J. Z., Ng, M. K., Rong, H., & Li, Z. (2005). Automated variable weighting in k-means type clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(5), 657–668.CrossRefGoogle Scholar
  59. 59.
    Huang, P., & Zhang, D. (2010). Locality sensitive \(C\)-means clustering algorithms. Neurocomputing, 73, 2935–2943.CrossRefGoogle Scholar
  60. 60.
    Hueter, G. J. (1988). Solution of the traveling salesman problem with an adaptive ring. In Proceedings of International Conference on Neural Networks (pp. 85–92). San Diego, CA.Google Scholar
  61. 61.
    Hwang, C., & Rhee, F. (2007). Uncertain fuzzy clustering: Interval type-2 fuzzy approach to \(C\)-means. IEEE Transactions on Fuzzy Systems, 15(1), 107–120.CrossRefGoogle Scholar
  62. 62.
    Kanungo, T., Mount, D. M., Netanyahu, N. S., Piatko, C. D., Silverman, R., & Wu, A. Y. (2002). An efficient \(k\)-means clustering algorithm: Analysis and implementation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(7), 881–892.zbMATHCrossRefGoogle Scholar
  63. 63.
    Karayiannis, N. B., & Randolph-Gips, M. M. (2003). Soft learning vector quantization and clustering algorithms based on non-Euclidean norms: Multinorm algorithms. IEEE Transactions on Neural Networks, 14(1), 89–102.CrossRefGoogle Scholar
  64. 64.
    Kaymak, U., & Setnes, M. (2002). Fuzzy clustering with volume prototypes and adaptive cluster merging. IEEE Transactions on Fuzzy Systems, 10(6), 705–712.CrossRefGoogle Scholar
  65. 65.
    Kersten, P. R. (1999). Fuzzy order statistics and their application to fuzzy clustering. IEEE Transactions on Fuzzy Systems, 7(6), 708–712.CrossRefGoogle Scholar
  66. 66.
    Kohonen, T. (1982). Self-organized formation of topologically correct feature maps. Biological Cybernetics, 43, 59–69.MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Kohonen, T. (1989). Self-organization and associative memory. Berlin: Springer.zbMATHCrossRefGoogle Scholar
  68. 68.
    Kohonen, T. (1990). The self-organizing map. Proceedings of the IEEE, 78, 1464–1480.CrossRefGoogle Scholar
  69. 69.
    Kohonen, T. (1990). Improved versions of learning vector quantization. In Proceedings of the International Joint Conference on Neural Networks (IJCNN) (Vol. 1, pp. 545–550). San Diego, CA.Google Scholar
  70. 70.
    Kohonen, T. (1990). Derivation of a class of training algorithms. IEEE Transactions on Neural Networks, 1, 229–232.CrossRefGoogle Scholar
  71. 71.
    Kohonen, T. (1996). Emergence of invariant-feature detectors in the adaptive-subspace self-organizing map. Biological Cybernetics, 75, 281–291.zbMATHCrossRefGoogle Scholar
  72. 72.
    Kohonen, T. (2001). Self-organizing maps (3rd ed.). Berlin: Springer.zbMATHCrossRefGoogle Scholar
  73. 73.
    Kohonen, T., Kangas, J., Laaksonen, J., & Torkkola, K. (1992). LVQPAK: A program package for the correct application of learning vector quantization algorithms. In Proceedings of the International Joint Conference on Neural Networks (IJCNN) (Vol. 1, pp. 725–730). Baltimore, MD.Google Scholar
  74. 74.
    Kohonen, T. (2006). Self-organizing neural projections. Neural Networks, 19, 723–733.zbMATHCrossRefGoogle Scholar
  75. 75.
    Kolen, J., & Hutcheson, T. (2002). Reducing the time complexity of the fuzzy \(C\)-means algorithm. IEEE Transactions on Fuzzy Systems, 10(2), 263–267.CrossRefGoogle Scholar
  76. 76.
    Krishnapuram, R., & Kim, J. (2000). Clustering algorithms based on volume criteria. IEEE Transactions on Fuzzy Systems, 8(2), 228–236.CrossRefGoogle Scholar
  77. 77.
    Kusumoto, H., & Takefuji, Y. (2006). \(O(\log _2 M)\) self-organizing map algorithm without learning of neighborhood vectors. IEEE Transactions on Neural Networks, 17(6), 1656–1661.CrossRefGoogle Scholar
  78. 78.
    Laaksonen, J., & Oja, E. (1996). Classification with learning k-nearest neighbors. In Proceedings of the International Conference on Neural Networks (pp. 1480–1483). Washington, DC.Google Scholar
  79. 79.
    Lai, J. Z. C., Huang, T.-J., & Liaw, Y.-C. (2009). A fast \(k\)-means clustering algorithm using cluster center displacement. Pattern Recognition, 42, 2551–2556.zbMATHCrossRefGoogle Scholar
  80. 80.
    Leski, J. (2003). Towards a robust fuzzy clustering. Fuzzy Sets and Systems, 137, 215–233.MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Li, M. J., Ng, M. K., Cheung, Y.-M., & Huang, J. Z. (2008). Agglomerative fuzzy \(K\)-means clustering algorithm with selection of number of clusters. IEEE Transactions on Knowledge and Data Engineering, 20(11), 1519–1534.CrossRefGoogle Scholar
  82. 82.
    Likas, A., Vlassis, N., & Verbeek, J. J. (2003). The global \(k\)-means clustering algorithm. Pattern Recognition, 36(2), 451–461.CrossRefGoogle Scholar
  83. 83.
    Linda, O., & Manic, M. (2012). General type-2 fuzzy \(C\)-means algorithm for uncertain fuzzy clustering. IEEE Transactions on Fuzzy Systems, 20(5), 883–897.CrossRefGoogle Scholar
  84. 84.
    Linde, Y., Buzo, A., & Gray, R. M. (1980). An algorithm for vector quantizer design. IEEE Transactions on Communications, 28, 84–95.CrossRefGoogle Scholar
  85. 85.
    Lippman, R. P. (1987). An introduction to computing with neural nets. IEEE ASSP Magazine, 4(2), 4–22.MathSciNetCrossRefGoogle Scholar
  86. 86.
    Lo, Z. P., & Bavarian, B. (1991). On the rate of convergence in topology preserving neural networks. Biological Cybernetics, 65, 55–63.MathSciNetzbMATHCrossRefGoogle Scholar
  87. 87.
    Luttrell, S. P. (1990). Derivation of a class of training algorithms. IEEE Transactions on Neural Networks, 1, 229–232.CrossRefGoogle Scholar
  88. 88.
    Luttrell, S. P. (1994). A Bayesian analysis of self-organizing maps. Neural Computation, 6, 767–794.zbMATHCrossRefGoogle Scholar
  89. 89.
    MacQueen, J. B. (1967). Some methods for classification and analysis of multivariate observations. In Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability (pp. 281–297). Berkeley, CA: University of California Press.Google Scholar
  90. 90.
    Martinetz, T. M. (1993). Competitive Hebbian learning rule forms perfectly topology preserving maps. In Proceedings of the International Conference on Artificial Neural Networks (ICANN) (pp. 427–434). Amsterdam, The Netherlands.Google Scholar
  91. 91.
    Martinetz, T. M., Berkovich, S. G., & Schulten, K. J. (1993). Neural-gas network for vector quantization and its application to time-series predictions. IEEE Transactions on Neural Networks, 4(4), 558–569.CrossRefGoogle Scholar
  92. 92.
    Martinetz, T. M., & Schulten, K. J. (1994). Topology representing networks. Neural Networks, 7, 507–522.CrossRefGoogle Scholar
  93. 93.
    Moore, B. (1988). ART and pattern clustering. In D. Touretzky, G. Hinton & T. Sejnowski (Eds.), Proceedings of the 1988 Connectionist Model Summer School (pp. 174–183). San Mateo, CA: Morgan Kaufmann.Google Scholar
  94. 94.
    Mulder, S. A., & Wunsch, D. C, I. I. (2003). Million city traveling salesman problem solution by divide and conquer clustering with adaptive resonance neural networks. Neural Networks, 16, 827–832.CrossRefGoogle Scholar
  95. 95.
    Obermayer, K., Ritter, H., & Schulten, K. (1991). Development and spatial structure of cortical feature maps: A model study. In R. P. Lippmann, J. E. Moody, & D. S. Touretzky (Eds.), Advances in neural information processing systems (Vol. 3, pp. 11–17). San Mateo, CA: Morgan Kaufmann.Google Scholar
  96. 96.
    Odorico, R. (1997). Learning vector quantization with training count (LVQTC). Neural Networks, 10(6), 1083–1088.CrossRefGoogle Scholar
  97. 97.
    Pal, N. R., Bezdek, J. C., & Tsao, E. C. K. (1993). Generalized clustering networks and Kohonen’s self-organizing scheme. IEEE Transactions on Neural Networks, 4(2), 549–557.CrossRefGoogle Scholar
  98. 98.
    Pal, N. R., & Chakraborty, D. (2000). Mountain and subtractive clustering method: Improvements and generalizations. International Journal of Intelligent Systems, 15, 329–341.zbMATHCrossRefGoogle Scholar
  99. 99.
    Patane, G., & Russo, M. (2001). The enhanced LBG algorithm. Neural Networks, 14(9), 1219–1237.CrossRefGoogle Scholar
  100. 100.
    Pedrycz, W., & Waletzky, J. (1997). Fuzzy clustering with partial supervision. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 27(5), 787–795.CrossRefGoogle Scholar
  101. 101.
    Richardt, J., Karl, F., & Muller, C. (1998). Connections between fuzzy theory, simulated annealing, and convex duality. Fuzzy Sets and Systems, 96, 307–334.MathSciNetCrossRefGoogle Scholar
  102. 102.
    Rodriguez, A., & Laio, A. (2014). Clustering by fast search and find of density peaks. Science, 344(6191), 1492–1496.CrossRefGoogle Scholar
  103. 103.
    Rose, K. (1998). Deterministic annealing for clustering, compression, classification, regression, and related optimization problems. Proceedings of the IEEE, 86(11), 2210–2239.CrossRefGoogle Scholar
  104. 104.
    Rose, K., Gurewitz, E., & Fox, G. C. (1990). A deterministic annealing approach to clustering. Pattern Recognition Letters, 11(9), 589–594.zbMATHCrossRefGoogle Scholar
  105. 105.
    Sato, A., & Yamada, K. (1995). Generalized learning vector quantization. In G. Tesauro, D. Touretzky, & T. Leen (Eds.), Advances in neural information processing systems (Vol. 7, pp. 423–429). Cambridge, MA: MIT Press.Google Scholar
  106. 106.
    Serrano-Gotarredona, T., & Linares-Barranco, B. (1996). A modified ART 1 algorithm more suitable for VLSI implementations. Neural Networks, 9(6), 1025–1043.CrossRefGoogle Scholar
  107. 107.
    Seo, S., & Obermayer, K. (2003). Soft learning vector quantization. Neural Computation, 15, 1589–1604.zbMATHCrossRefGoogle Scholar
  108. 108.
    Shalev-Shwartz, S., & Ben-David, S. (2014). Understanding machine learning: From theory to algorithms. Cambridge, UK: Cambridge University Press.Google Scholar
  109. 109.
    Su, Z., & Denoeux, T. (2019). BPEC: Belief-peaks evidential clustering. IEEE Transactions on Fuzzy Systems, 27(1), 111–123.CrossRefGoogle Scholar
  110. 110.
    Tsekouras, G., Sarimveis, H., Kavakli, E., & Bafas, G. (2004). A hierarchical fuzzy-clustering approach to fuzzy modeling. Fuzzy Sets and Systems, 150(2), 245–266.MathSciNetzbMATHCrossRefGoogle Scholar
  111. 111.
    Verzi, S. J., Heileman, G. L., Georgiopoulos, M., & Anagnostopoulos, G. C. (2003). Universal approximation with fuzzy ART and fuzzy ARTMAP. In Proceedings of the International Joint Conference on Neural Networks (IJCNN) (Vol. 3, pp. 1987–1892). Portland, OR.Google Scholar
  112. 112.
    von der Malsburg, C. (1973). Self-organizing of orientation sensitive cells in the striata cortex. Kybernetik, 14, 85–100.CrossRefGoogle Scholar
  113. 113.
    Wilson, D. L. (1972). Asymptotic properties of nearest neighbor rules using edited data. IEEE Transactions on Systems, Man, and Cybernetics, 2(3), 408–420.MathSciNetzbMATHCrossRefGoogle Scholar
  114. 114.
    Witoelar, A., Biehl, M., Ghosh, A., & Hammer, B. (2008). Learning dynamics and robustness of vector quantization and neural gas. Neurocomputing, 71, 1210–1219.CrossRefGoogle Scholar
  115. 115.
    Wu, K.-L., Yang, M.-S., & Hsieh, J.-N. (2010). Mountain \(c\)-regressions method. Pattern Recognition, 43, 86–98.zbMATHCrossRefGoogle Scholar
  116. 116.
    Yager, R. R., & Filev, D. (1994). Approximate clustering via the mountain method. IEEE Transactions on Systems, Man, and Cybernetics, 24(8), 1279–1284.CrossRefGoogle Scholar
  117. 117.
    Yair, E., Zeger, K., & Gersho, A. (1992). Competitive learning and soft competition for vector quantizer design. IEEE Transactions on Signal Processing, 40(2), 294–309.CrossRefGoogle Scholar
  118. 118.
    Yang, M. S. (1993). On a class of fuzzy classification maximum likelihood procedures. Fuzzy Sets and Systems, 57, 365–375.MathSciNetzbMATHCrossRefGoogle Scholar
  119. 119.
    Yang, M.-S., Wu, K.-L., Hsieh, J.-N., & Yu, J. (2008). Alpha-cut implemented fuzzy clustering algorithms and switching regressions. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 38(3), 588–603.CrossRefGoogle Scholar
  120. 120.
    Yin, H., & Allinson, N. M. (1995). On the distribution and convergence of the feature space in self-organizing maps. Neural Computation, 7(6), 1178–1187.Google Scholar
  121. 121.
    Yu, J., Cheng, Q., & Huang, H. (2004). Analysis of the weighting exponent in the FCM. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 34(1), 634–639.CrossRefGoogle Scholar
  122. 122.
    Yu, J., & Yang, M. S. (2005). Optimality test for generalized FCM and its application to parameter selection. IEEE Transactions on Fuzzy Systems, 13(1), 164–176.CrossRefGoogle Scholar
  123. 123.
    Zhao, W.-L., Deng, C.-H., & Ngo, C.-W. (2018). \(k\)-means: A revisit. Neurocomputing, 291, 195–206.CrossRefGoogle Scholar
  124. 124.
    Zheng, H., Lefebvre, G., & Laurent, C. (2008). Fast-learning adaptive-subspace self-organizing map: An application to saliency-based invariant image feature construction. IEEE Transactions on Neural Networks, 19(5), 746–757.Google Scholar

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© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringConcordia UniversityMontrealCanada
  2. 2.Xonlink Inc.HangzhouChina

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