Skip to main content
SpringerLink
Log in

An Interpolating Distance Between Optimal Transport and Fisher–Rao Metrics

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

Abstract

This paper defines a new transport metric over the space of nonnegative measures. This metric interpolates between the quadratic Wasserstein and the Fisher–Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher–Rao metric and is averaged with the transportation term. This gives rise to a convex variational problem defining the new metric. Our first contribution is a proof of the existence of geodesics (i.e., solutions to this variational problem). We then show that (generalized) optimal transport and Hellinger metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Dirac measures are made of translating mixtures of Dirac masses. Lastly, we propose a numerical scheme making use of first-order proximal splitting methods and we show an application of this new distance to image interpolation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. https://github.com/lchizat/optimal-transport/.

References

  1. L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media, 2008.

  2. N. Ay, J. Jost, H. V. Lê, and L. Schwachhöfer. Information geometry and sufficient statistics. Probability Theory and Related Fields, 162(1):327–364, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Bauer, M. Bruveris, and P. W. Michor. Uniqueness of the Fisher–Rao metric on the space of smooth densities. Bull. Lond. Math. Soc., 48(3):499–506, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. F. Beg, M. I. Miller, A. Trouvé, and L. Younes. Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International journal of computer vision, 61(2):139–157, 2005.

    Article  Google Scholar 

  5. J.-D. Benamou. Numerical resolution of an “unbalanced” mass transport problem. ESAIM: Mathematical Modelling and Numerical Analysis, 37(05):851–868, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  6. J.-D. Benamou and Y. Brenier. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik, 84(3):375–393, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  7. J.-D. Benamou and Y. Brenier. Mixed L2-Wasserstein optimal mapping between prescribed density functions. Journal of Optimization Theory and Applications, 111(2):255–271, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Bouchitté and G. Buttazzo. New lower semicontinuity results for nonconvex functionals defined on measures. Nonlinear Analysis: Theory, Methods & Applications, 15(7):679–692, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Caffarelli and R. J. McCann. Free boundaries in optimal transport and Monge-Ampere obstacle problems. Annals of mathematics, 171(2):673–730, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Cardaliaguet, G. Carlier, and B. Nazaret. Geodesics for a class of distances in the space of probability measures. Calculus of Variations and Partial Differential Equations, 48(3-4):395–420, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Combettes and J.-C. Pesquet. Proximal splitting methods in signal processing. In Fixed-point algorithms for inverse problems in science and engineering, pages 185–212. Springer, 2011.

  12. P. L. Combettes and J.-C. Pesquet. A Douglas–Rachford splitting approach to nonsmooth convex variational signal recovery. Selected Topics in Signal Processing, IEEE Journal of, 1(4):564–574, 2007.

    Article  Google Scholar 

  13. J. Dolbeault, B. Nazaret, and G. Savaré. A new class of transport distances between measures. Calculus of Variations and Partial Differential Equations, 34(2):193–231, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Figalli. The optimal partial transport problem. Archive for rational mechanics and analysis, 195(2):533–560, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Figalli and N. Gigli. A new transportation distance between non-negative measures, with applications to gradients flows with dirichlet boundary conditions. Journal de mathématiques pures et appliquées, 94(2):107–130, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Guittet. Extended Kantorovich norms: a tool for optimization. Technical report, Tech. Rep. 4402, INRIA, 2002.

  17. S. Haker, L. Zhu, A. Tannenbaum, and S. Angenent. Optimal mass transport for registration and warping. International Journal of computer vision, 60(3):225–240, 2004.

    Article  Google Scholar 

  18. L. G. Hanin. Kantorovich-Rubinstein norm and its application in the theory of Lipschitz spaces. Proceedings of the American Mathematical Society, 115(2):345–352, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  19. C. Jimenez. Dynamic formulation of optimal transport problems. Journal of Convex Analysis, 15(3):593, 2008.

    MathSciNet  MATH  Google Scholar 

  20. L. Kantorovich. On the transfer of masses (in russian). Doklady Akademii Nauk, 37(2):227–229, 1942.

    Google Scholar 

  21. S. Kondratyev, L. Monsaingeon, and D. Vorotnikov. A new optimal transport distance on the space of finite Radon measures. Technical report, Pre-print, 2015.

    MATH  Google Scholar 

  22. D. Lombardi and E. Maitre. Eulerian models and algorithms for unbalanced optimal transport. ESAIM: M2AN, 49(6):1717 – 1744, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Maas, M. Rumpf, C. Schönlieb, and S. Simon. A generalized model for optimal transport of images including dissipation and density modulation. ESAIM: M2AN, 49(6):1745–1769, 2015.

    Article  MathSciNet  MATH  Google Scholar 

  24. N. Papadakis, G. Peyré, and E. Oudet. Optimal transport with proximal splitting. SIAM Journal on Imaging Sciences, 7(1):212–238, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  25. B. Piccoli and F. Rossi. On properties of the Generalized Wasserstein distance. arXiv:1304.7014, 2013.

  26. B. Piccoli and F. Rossi. Generalized Wasserstein distance and its application to transport equations with source. Archive for Rational Mechanics and Analysis, 211(1):335–358, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  27. C. Rao. Information and accuracy attainable in the estimation of statistical parameters. Bulletin of the Calcutta Mathematical Society, 37(3):81–91, 1945.

    MathSciNet  MATH  Google Scholar 

  28. R. Rockafellar. Duality and stability in extremum problems involving convex functions. Pacific Journal of Mathematics, 21(1):167–187, 1967.

    Article  MathSciNet  MATH  Google Scholar 

  29. R. Rockafellar. Integrals which are convex functionals. ii. Pacific Journal of Mathematics, 39(2):439–469, 1971.

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Trouvé and L. Younes. Metamorphoses through lie group action. Foundations of Computational Mathematics, 5(2):173–198, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  31. C. Villani. Topics in optimal transportation. Number 58. American Mathematical Soc., 2003.

Download references

Acknowledgments

The work of Bernhard Schmitzer has been supported by the Fondation Sciences Mathématiques de Paris. The work of Gabriel Peyré has been supported by the European Research Council (ERC project SIGMA-Vision). We would like to thank Yann Brenier and Jean-David Benamou for stimulating discussions.

Author information

Authors and Affiliations

  1. Project team Mokaplan, CEREMADE, CNRS, INRIA, Université Paris-Dauphine, Paris, France

    Lénaïc Chizat, Gabriel Peyré, Bernhard Schmitzer & François-Xavier Vialard

Authors
  1. Lénaïc Chizat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gabriel Peyré
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bernhard Schmitzer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. François-Xavier Vialard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to François-Xavier Vialard.

Additional information

Communicated by Hans Munthe-Kaas Suh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chizat, L., Peyré, G., Schmitzer, B. et al. An Interpolating Distance Between Optimal Transport and Fisher–Rao Metrics. Found Comput Math 18, 1–44 (2018). https://doi.org/10.1007/s10208-016-9331-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-016-9331-y

Keywords

Mathematics Subject Classification

Search

Navigation