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Tree Topologies along a Tropical Line Segment

Abstract

Tropical geometry with the max-plus algebra has been applied to statistical learning models over tree spaces because geometry with the tropical metric over tree spaces has some nice properties such as convexity in terms of the tropical metric. One of the challenges in applications of tropical geometry to tree spaces is the difficulty interpreting outcomes of statistical models with the tropical metric. This paper focuses on combinatorics of tree topologies along a tropical line segment, an intrinsic geodesic with the tropical metric, between two phylogenetic trees over the tree space and we show some properties of a tropical line segment between two trees. Specifically we show that a probability of a tropical line segment of two randomly chosen trees going through the origin (the star tree) is zero if the number of leave is greater than four, and we also show that if two given trees differ only one nearest neighbor interchange (NNI) move, then the tree topology of a tree in the tropical line segment between them is the same tree topology of one of these given two trees with possible zero branch lengths.

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References

  1. Akian, M., Gaubert, S., Niţică, V., Singer, I.: Best approximation in max-plus semimodules. Linear Algebra Appl. 435, 3261–3296 (2011)

    MathSciNet  Article  Google Scholar 

  2. Ardila, F., Klivans, C.J.: The Bergman complex of a matroid and phylogenetic trees. J. Combin. Theory Ser. B 96, 38–49 (2006)

    MathSciNet  Article  Google Scholar 

  3. Billera, L.J., Holmes, S.P., Vogtmann, K.: Geometry of the space of phylogenetic trees. Adv. Appl. Math. 27, 733–767 (2001)

    MathSciNet  Article  Google Scholar 

  4. Buneman, P.: A note on the metric properties of trees. J. Combin. Theory Ser. B 17, 48–50 (1974)

    MathSciNet  Article  Google Scholar 

  5. Cardona, G., Rosselló, F., Valiente, G.: Extended newick: it is time for a standard representation of phylogenetic networks. BMC Bioinforma. 9, 532 (2008)

    Article  Google Scholar 

  6. Cohen, G., Gaubert, S., Quadrat, J.-P.: Duality and separation theorems in idempotent semimodules. Linear Algebra Appl. 379, 395–422 (2004)

    MathSciNet  Article  Google Scholar 

  7. Garba, M.K., Nye, T.M.W., Lueg, J., Huckemann, S.F.: Information geometry for phylogenetic trees. J. Math. Biol. 82, 19 (2021)

    MathSciNet  Article  Google Scholar 

  8. Garba, M.K., Nye, T.M.W., Boys, R.J.: Probabilistic distances between trees. Syst. Biol. 67, 320–327 (2018)

    Article  Google Scholar 

  9. Lin, B., Sturmfels, B., Tang, X., Yoshida, R.: Convexity in tree spaces. SIAM J. Discrete Math. 3, 2015–2038 (2017)

    MathSciNet  Article  Google Scholar 

  10. Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Graduate Studies in Mathematics, vol. 161. American Mathematical Society, Providence (2015)

    Book  Google Scholar 

  11. Maddison, W.P.: Gene trees in species trees. Syst. Biol. 46, 523–536 (1997)

    Article  Google Scholar 

  12. Monod, A., Lin, B., Yoshida, R.: Tropical geometric variation of phylogenetic tree shapes. arXiv:2010.06158 (2020)

  13. Monod, A., Lin, B., Yoshida, R., Kang, Q.: Tropical geometry of phylogenetic tree space:, A statistical perspective. arXiv:1805.12400 (2019)

  14. Nye, T.M.W.: Principal components analysis in the space of phylogenetic trees. Ann. Stat. 39, 2716–2739 (2011)

    MathSciNet  Article  Google Scholar 

  15. Pachter, L., Sturmfels, B.: Tropical geometry of statistical models. Proc. Nat. Acad. Sci. 101, 16132–16137 (2004)

    MathSciNet  Article  Google Scholar 

  16. Page, R., Yoshida, R., Zhang, L.: Tropical principal component analysis on the space of phylogenetic trees. Bioinformatics 36, 4590–4598 (2020)

    Article  Google Scholar 

  17. Speyer, D., Sturmfels, B.: Tropical mathematics. Math. Mag. 82, 163–173 (2009)

    MathSciNet  Article  Google Scholar 

  18. Tang, X., Wang, H., Yoshida, R.: Tropical support vector machines and its applications to phylogenomics. arXiv:1710.02682 (2020)

  19. Yoshida, R.: Tropical balls and its applications to K nearest neighbor over the space of phylogenetic trees. Mathematics 9, 779 (2021)

    Article  Google Scholar 

  20. Yoshida, R.: Tropical data science over the space of phylogenetic trees. In: Arai, K. (ed.) Intelligent Systems and Applications. Lecture Notes in Networks and Systems, vol. 295, pp 340–361. Springer, Cham (2021)

  21. Yoshida, R., Zhang, L., Zhang, X.: Tropical principal component analysis and its application to phylogenetics. arXiv:1710.02682 (2017)

  22. Zwiernik, P., Smith, J.Q.: Tree cumulants and the geometry of binary tree models. Bernoulli 18, 290–321 (2012)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

R.Y. is partially supported by NSF (DMS 1916037). Also the author thank the editor and referees for improving this manuscript.

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Correspondence to Ruriko Yoshida.

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This paper is to dedicated to the 60th birthday of our academic groundfather, Prof. Bernd Sturmfels.

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Yoshida, R., Cox, S. Tree Topologies along a Tropical Line Segment. Vietnam J. Math. 50, 395–419 (2022). https://doi.org/10.1007/s10013-021-00526-3

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  • DOI: https://doi.org/10.1007/s10013-021-00526-3

Keywords

  • Phylogenetic trees
  • Phylogenomics
  • Tree spaces
  • Ultrametrics

Mathematics Subject Classification (2010)

  • 14T05
  • 05C05
  • 58D17