On the Existence of Kernel Function for Kernel-Trick of k-Means in the Light of Gower Theorem

Article type: Research Article

Authors: Kłopotek, Mieczysław A.^{; *}

Affiliations: Institute of Computer Science of the Polish Academy of Sciences, ul. Jana Kazimierza 5, 01-248 Warsaw, Poland. klopotek@ipipan.waw.pl

Correspondence: [*] Address for correspondence: Institute of Computer Science of the Polish Academy of Sciences, ul. Jana Kazimierza 5, 01-248 Warsaw, Poland.

Abstract: This paper, constituting an extension to the conference paper [1], corrects the proof of the Theorem 2 from the Gower’s paper [2, page 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.

Keywords: kernel trick, k-means, Gower Theorem, Euclidean embedding, kernel function

DOI: 10.3233/FI-2019-1822

Journal: Fundamenta Informaticae, vol. 168, no. 1, pp. 25-43, 2019