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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
For an overview of kernel k-means algorithm see e.g. [2].
- 2.
Gower does not consider flipping.
- 3.
This means that there exists a matrix X such that rows are coordinates of objects in an Euclidean space with distances as in D.
References
Gower, J.C.: Euclidean distance geometry. Math. Sci. 7, 1–14 (1982)
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)
Balaji, R., Bapat, R.B.: On Euclidean distance matrices. Linear Algebra Appl. 424(1), 108–117 (2007)
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)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Kłopotek, M.A. (2017). On the Existence of Kernel Function for Kernel-Trick of k-Means. In: Kryszkiewicz, M., Appice, A., Ślęzak, D., Rybinski, H., Skowron, A., Raś, Z. (eds) Foundations of Intelligent Systems. ISMIS 2017. Lecture Notes in Computer Science(), vol 10352. Springer, Cham. https://doi.org/10.1007/978-3-319-60438-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-60438-1_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-60437-4
Online ISBN: 978-3-319-60438-1
eBook Packages: Computer ScienceComputer Science (R0)