Advertisement

Diffeomorphic Random Sampling Using Optimal Information Transport

  • Martin Bauer
  • Sarang Joshi
  • Klas ModinEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10589)

Abstract

In this article we explore an algorithm for diffeomorphic random sampling of nonuniform probability distributions on Riemannian manifolds. The algorithm is based on optimal information transport (OIT)—an analogue of optimal mass transport (OMT). Our framework uses the deep geometric connections between the Fisher-Rao metric on the space of probability densities and the right-invariant information metric on the group of diffeomorphisms. The resulting sampling algorithm is a promising alternative to OMT, in particular as our formulation is semi-explicit, free of the nonlinear Monge–Ampere equation. Compared to Markov Chain Monte Carlo methods, we expect our algorithm to stand up well when a large number of samples from a low dimensional nonuniform distribution is needed.

Keywords

Density matching Information geometry Fisher–Rao metric Optimal transport Image registration Diffeomorphism groups Random sampling 

MSC2010

58E50 49Q10 58E10 

References

  1. 1.
    Amari, S., Nagaoka, H.: Methods of information geometry. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  2. 2.
    Bauer, M., Bruveris, M., Michor, P.W.: Uniqueness of the Fisher-Rao metric on the space of smooth densities. Bull. Lond. Math. Soc. 48(3), 499–506 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bauer, M., Joshi, S., Modin, K.: Diffeomorphic density matching by optimal information transport. SIAM J. Imaging Sci. 8(3), 1718–1751 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Friedrich, T.: Die Fisher-information und symplektische strukturen. Math. Nachr. 153(1), 273–296 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. (N.S.) 7(1), 65–222 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97–109 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Khesin, B., Lenells, J., Misiołek, G., Preston, S.C.: Geometry of diffeomorphism groups, complete integrability and geometric statistics. Geom. Funct. Anal. 23(1), 334–366 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Khesin, B., Wendt, R.: The Geometry of Infinite-dimensional Groups. A Series of Modern Surveys in Mathematics, vol. 51. Springer, Berlin (2009)zbMATHGoogle Scholar
  9. 9.
    Marzouk, Y., Moselhy, T., Parno, M., Spantini, A.: Sampling via measure transport: An introduction. In: Ghanem, R., Higdon, D., Owhadi, H. (eds.) Handbook of Uncertainty Quantification, pp. 1–14. Springer International Publishing, Cham (2016). doi: 10.1007/978-3-319-11259-6_23-1 Google Scholar
  10. 10.
    Miller, M.I., Trouvé, A., Younes, L.: On the metrics and euler-lagrange equations of computational anatomy. Annu. Rev. Biomed. Eng. 4, 375–405 (2002)CrossRefGoogle Scholar
  11. 11.
    Modin, K.: Generalized Hunter-Saxton equations, optimal information transport, and factorization of diffeomorphisms. J. Geom. Anal. 25(2), 1306–1334 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Moselhy, T.A.E., Marzouk, Y.M.: Bayesian inference with optimal maps. J. Comput. Phys. 231(23), 7815–7850 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Moser, J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Comm. Partial Diff. Eqn. 26(1–2), 101–174 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Reich, S.: A nonparametric ensemble transform method for Bayesian inference. SIAM J. Sci. Comput. 35(4), A2013–A2024 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Villani, C.: Optimal Transport: Old and New, Grundlehren der Mathematischen Wissenschaften, vol. 338. Springer, Berlin (2009)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA
  2. 2.Department of Bioengineering, Scientific Computing and Imaging InstituteUniversity of UtahSalt Lake CityUSA
  3. 3.Mathematical SciencesChalmers and University of GothenburgGothenburgSweden

Personalised recommendations