Advertisement

Optimal Transport to a Variety

  • Türkü Özlüm ÇelikEmail author
  • Asgar Jamneshan
  • Guido Montúfar
  • Bernd Sturmfels
  • Lorenzo Venturello
Conference paper
  • 36 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11989)

Abstract

We study the problem of minimizing the Wasserstein distance between a probability distribution and an algebraic variety. We consider the setting of finite state spaces and describe the solution depending on the choice of the ground metric and the given distribution. The Wasserstein distance between the distribution and the variety is the minimum of a linear functional over a union of transportation polytopes. We obtain a description in terms of the solutions of a finite number of systems of polynomial equations. The case analysis is based on the ground metric. A detailed analysis is given for the two bit independence model.

Keywords

Algebraic statistics Linear programming Optimal transport estimator Polynomial optimization Transportation Polytope Triangulation Wasserstein distance 

Notes

Acknowledgments

GM has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no 757983).

References

  1. 1.
    Allman, E., et al.: Maximum likelihood estimation of the latent class model through model boundary decomposition. J. Algebraic Stat. 34, 51–84 (2019)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arjovsky, M., Chintala, S., Bottou, L.: Wasserstein GAN. arXiv:1701.07875
  3. 3.
    Bassetti, F., Bodini, A., Regazzini, E.: On minimum Kantorovich distance estimators. Stat. Probab. Lett. 76, 1298–1302 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bernton, E., Jacob, P., Gerber, M., Robert, C.: On parameter estimation with the Wasserstein distance. Inf. Infer. J. IMA 8(4), 657–676 (2019)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cuturi, M.: Sinkhorn distances: lightspeed computation of optimal transport. In: Advances in Neural Information Processing Systems, Proceedings NIPS 2013, pp. 2292–2300 (2013)Google Scholar
  6. 6.
    De Loera, J.A., Rambau, J., Santos, F.: Triangulations: Structures for Algorithms and Applications, Algorithms and Computation in Mathematics, vol. 25. Springer-Verlag, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-12971-1CrossRefzbMATHGoogle Scholar
  7. 7.
    Duarte, E., Marigliano, O., Sturmfels, B.: Discrete statistical models with rational maximum likelihood estimator. arXiv:1903.06110
  8. 8.
    Kulas, K., Joswig, M.: Tropical and ordinary convexity combined. Adv. Geom. 10, 333–352 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Lasserre, J.: An Introduction to Polynomial and Semi-Algebraic Optimization, Texts in Applied Mathematics. Cambridge University Press, Cambridge (2015)CrossRefGoogle Scholar
  10. 10.
    Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227. Springer-Verlag, New York (2005).  https://doi.org/10.1007/b138602CrossRefzbMATHGoogle Scholar
  11. 11.
    Pele, O., Werman, M.: Fast and robust earth mover’s distances. In: 2009 IEEE 12th International Conference on Computer Vision, pp. 460–467, September 2009Google Scholar
  12. 12.
    Peyre, G., Cuturi, M.: Computational optimal transport. Found. Trends Mach. Learn. 11, 355–607 (2019)CrossRefGoogle Scholar
  13. 13.
    Rostalski, P., Sturmfels, B.: Dualities in convex algebraic geometry. Rendiconti di Matematica 30, 285–327 (2010)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Seigal, A., Montúfar, G.: Mixtures and products in two graphical models. J. Alg. Stat. 9, 1–20 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence (1996)zbMATHGoogle Scholar
  16. 16.
    Sullivant, S.: Algebraic Statistics. Graduate Studies in Math. American Mathematical Society, Providence (2018)CrossRefGoogle Scholar
  17. 17.
    Villani, C.: Optimal Transport: Old and New. Grundlehren Series, vol. 338. Springer Verlag, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-71050-9CrossRefzbMATHGoogle Scholar
  18. 18.
    Weed, J., Bach, F.: Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance. arXiv:1707.00087

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Türkü Özlüm Çelik
    • 1
    Email author
  • Asgar Jamneshan
    • 3
  • Guido Montúfar
    • 1
    • 3
  • Bernd Sturmfels
    • 1
    • 2
  • Lorenzo Venturello
    • 1
  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.University of California at BerkeleyBerkeleyUSA
  3. 3.University of California at Los AngelesLos AngelesUSA

Personalised recommendations