Advertisement

On the Existence of Kernel Function for Kernel-Trick of k-Means

  • Mieczysław A. Kłopotek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10352)

Abstract

This paper corrects the proof of the Theorem 2 from the Gower’s paper [1, p. 5]. The correction is needed in order to establish the existence of the kernel function used commonly in the kernel trick e.g. for k-means clustering algorithm, on the grounds of distance matrix. The correction encompasses the missing if-part proof and dropping unnecessary conditions.

References

  1. 1.
    Gower, J.C.: Euclidean distance geometry. Math. Sci. 7, 1–14 (1982)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Dhillon, I.S., Guan, Y., Kulis, B.: Kernel k-means: spectral clustering and normalized cuts. In: Proceedings of the Tenth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2004, pp. 551–556. ACM, New York (2004)Google Scholar
  3. 3.
    Balaji, R., Bapat, R.B.: On Euclidean distance matrices. Linear Algebra Appl. 424(1), 108–117 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Schoenberg, I.J.: Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espace distanciés vectoriellement applicable sur l’ espace de Hilbert”. Ann. Math. 36(3), 724–732 (1935)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gower, J.C., Legendre, P.: Metric and Euclidean properties of dissimilarity coefficients. J. Classif. 3(1), 5–48 (1986). (Here Gower: 1982 is cited in theorem 4, but with a different form of condditions for D and s)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Computer Science of the Polish Academy of SciencesWarszawaPoland

Personalised recommendations