Volume 24, 2020
|Page(s)||914 - 934|
|Published online||27 November 2020|
Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho,
2 Department of Mathematical Engineering, Musashino University, 1 Chome-1-20 Shinmachi, Nishitokyo, Tokyo 202-8585, Japan.
3 Instituto de Matemática e Estatística, Universidade Federal da Bahia, Ondina, Salvador–BA, 40.170-115, Brazil.
4 Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canada.
*** Corresponding author: email@example.com
Accepted: 22 June 2020
The k nearest neighbour learning rule (under the uniform distance tie breaking) is universally consistent in every metric space X that is sigma-finite dimensional in the sense of Nagata. This was pointed out by Cérou and Guyader (2006) as a consequence of the main result by those authors, combined with a theorem in real analysis sketched by D. Preiss (1971) (and elaborated in detail by Assouad and Quentin de Gromard (2006)). We show that it is possible to give a direct proof along the same lines as the original theorem of Charles J. Stone (1977) about the universal consistency of the k-NN classifier in the finite dimensional Euclidean space. The generalization is non-trivial because of the distance ties being more prevalent in the non-Euclidean setting, and on the way we investigate the relevant geometric properties of the metrics and the limitations of the Stone argument, by constructing various examples.
Mathematics Subject Classification: 62H30 / 54F45
Key words: k-NN classifier / universal consistency / geometric Stone lemma / distance ties / Nagata dimension / sigma-finite dimensional metric spaces
S.K. would like to thank JICA-FRIENDSHIP scholarship (fellowship D-1590283/ J-1593019) for supporting her doctoral study at Kyoto University and Department of Mathematics, Kyoto University for supporting the Brazil research trip under the KTGU project.
© EDP Sciences, SMAI 2020
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