Hard and fuzzy c-means clustering with conditionally positive definite kernel

Yuchi Kanzawa, Yasunori Yasunori Endo, Sadaaki Miyamoto

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)825-830
Number of pages6
JournalJournal of Advanced Computational Intelligence and Intelligent Informatics
Issue number7
Publication statusPublished - 2012 Nov


  • Clustering
  • Conditionally positive definite kernel
  • Fuzzy c-means

ASJC Scopus subject areas

  • Human-Computer Interaction
  • Computer Vision and Pattern Recognition
  • Artificial Intelligence


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