Minkowski Metric Based Soft Subspace Clustering with Different Minkowski Exponent and Feature Weight Exponent

  • Xiaobin ZhiEmail author
  • Longtao BiEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1075)


Soft subspace clustering (SSC) methods can simultaneously performance clustering and find the subspace where each cluster lie in. A Minkowski metric based SSC (MSSC) algorithm recently is proposed to improve the adaptability of SSC to data. The empirical results have shown its favorable performance in comparison with several other popular clustering algorithms. However, this algorithm has the following two main defects: (1) The role that the Minkowski exponent \(\beta \) in MSSC plays is not clear. And the Minkowski exponent \(\beta \) is set as the same as the feature weight exponent \(\alpha \) in MSSC that may lead to MSSC missing better clustering performance. (2) the steepest descent method based MSSC (SD-MSSC) is computationally expensive for large data. In this paper, a general formulation for MSSC is presented, in which the Minkowski exponent \(\beta \) can be set not equal to the feature weight exponent \(\alpha \). A novel algorithm for computing the clustering centroids in MSSC is presented using the fixed-point iteration (FPI) method. The FPI based MSSC (FPI-MSSC) algorithm is more efficient than the SD-MSSC algorithm, and it becomes a noise-robust SSC procedure when the \(\beta <2\). Extensive experiments on real-world data sets are presented to show the effectiveness of the proposed algorithm.


Soft subspace clustering Minkowski metric Fixed-point iteration 



This work is supported by the National Science Foundation of China (Grant nos. 61671377, 61102095, 61571361 and 11401045), and the Science Plan Foundation of the Education Bureau of Shaanxi Province (No. 18JK0719), and New Star Team of Xi’an University of Posts and Telecommunications (Grant no. xyt2016-01).


  1. 1.
    Jain, A.K., Dubes, R.C.: Algorithms for Clustering Data. Prentice Hall, Englewood Cliffs (1988)zbMATHGoogle Scholar
  2. 2.
    Mac Queen, J.: Some methods for classification and analysis of multivariate observations. In: Fifth Berkeley Symposium on Mathematical Statistics and Probability, pp. 281–297 (1967)Google Scholar
  3. 3.
    Chan, E.Y., Ching, W., Ng, M.K., Huang, J.Z.: An optimization algorithm for clustering using weighted dissimilarity measures. Pattern Recogn. 37, 943–952 (2004)CrossRefGoogle Scholar
  4. 4.
    Frigui, H., Nasraoui, O.: Unsupervised learning of prototypes and attribute weights. Pattern Recogn. 37, 567–581 (2004)CrossRefGoogle Scholar
  5. 5.
    Deng, Z., Choi, K., Chung, F., Wang, S.: Enhanced soft subspace clustering integrating within-cluster and between-cluster information. Pattern Recogn. 43, 767–781 (2010)CrossRefGoogle Scholar
  6. 6.
    Amorim, R.C., Mirkin, B.: Minkowski metric, feature weighting and anomalous cluster initializing in K-Means clustering. Pattern Recogn. 45, 1061–1075 (2012)CrossRefGoogle Scholar
  7. 7.
    Bezdek, J.C.: Pattern Recognition with Fuzzy Objective Function Algorithms, pp. 95–107. Plenum Press, New York (1981)CrossRefGoogle Scholar
  8. 8.
    Chiang, M.M., Mirkin, B.: Intelligent choice of the number of clusters in k-means clustering: an experimental study with different cluster spreads. J. Classif. 27(1), 1C38 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Maronna, R.A., Martin, D.R., Yohai, V.J.: Robust Statistics: Theory and Methods. Wiley, New York (2006)CrossRefGoogle Scholar
  10. 10.
    Asuncion, A., Newman, D.J.: UCI machine learning repository. University of California, School of Information and Computer Science, Irvine (2007).

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.School of ScienceXi’an University of Posts and TelecommunicationsXi’anChina
  2. 2.School of Communication and Information EngineeringXi’an University of Posts and TelecommunicationsXi’anChina

Personalised recommendations