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Projections of Tropical Fermat-Weber Points

Author

Listed:
  • Weiyi Ding

    (School of Mathematical Sciences, Beihang University, Beijing 100191, China
    These authors contributed equally to this work.)

  • Xiaoxian Tang

    (School of Mathematical Sciences, Beihang University, Beijing 100191, China
    These authors contributed equally to this work.)

Abstract

This paper is motivated by the difference between the classical principal component analysis (PCA) in a Euclidean space and the tropical PCA in a tropical projective torus as follows. In Euclidean space, the projection of the mean point of a given data set on the principle component is the mean point of the projection of the data set. However, in tropical projective torus, it is not guaranteed that the projection of a Fermat-Weber point of a given data set on a tropical polytope is a Fermat-Weber point of the projection of the data set. This is caused by the difference between the Euclidean metric and the tropical metric. In this paper, we focus on the projection on the tropical triangle (the three-point tropical convex hull), and we develop one algorithm and its improved version, such that for a given data set in the tropical projective torus, these algorithms output a tropical triangle, on which the projection of a Fermat-Weber point of the data set is a Fermat-Weber point of the projection of the data set. We implement these algorithms in R language and test how they work with random data sets. We also use R language for numerical computation. The experimental results show that these algorithms are stable and efficient, with a high success rate.

Suggested Citation

  • Weiyi Ding & Xiaoxian Tang, 2021. "Projections of Tropical Fermat-Weber Points," Mathematics, MDPI, vol. 9(23), pages 1-23, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:23:p:3102-:d:692866
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    References listed on IDEAS

    as
    1. Tom M W Nye & Xiaoxian Tang & Grady Weyenberg & Ruriko Yoshida, 2017. "Principal component analysis and the locus of the Fréchet mean in the space of phylogenetic trees," Biometrika, Biometrika Trust, vol. 104(4), pages 901-922.
    2. Ruriko Yoshida & Kenji Fukumizu & Chrysafis Vogiatzis, 2019. "Multilocus phylogenetic analysis with gene tree clustering," Annals of Operations Research, Springer, vol. 276(1), pages 293-313, May.
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