Foundations of Computational Mathematics

, Volume 19, Issue 5, pp 1113–1143 | Cite as

Second-Order Models for Optimal Transport and Cubic Splines on the Wasserstein Space

  • Jean-David BenamouEmail author
  • Thomas O. Gallouët
  • François-Xavier Vialard


On the space of probability densities, we extend the Wasserstein geodesics to the case of higher-order interpolation such as cubic spline interpolation. After presenting the natural extension of cubic splines to the Wasserstein space, we propose a simpler approach based on the relaxation of the variational problem on the path space. We explore two different numerical approaches, one based on multimarginal optimal transport and entropic regularization and the other based on semi-discrete optimal transport.


Multimarginal optimal transportation Splines Wasserstein geodesics 

Mathematics Subject Classification

49M99 65D99 



  1. 1.
    Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Luca Nenna, and Gabriel Peyré. Iterative Bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing, 37(2):A1111–A1138, 2015.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Jean-David Benamou, Guillaume Carlier, and Luca Nenna. A Numerical Method to solve Optimal Transport Problems with Coulomb Cost. working paper or preprint, May 2015.Google Scholar
  3. 3.
    Jean-David Benamou, Guillaume Carlier, and Luca Nenna. Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm. working paper or preprint, October 2017.Google Scholar
  4. 4.
    Geir Bogfjellmo, Klas Modin, and Olivier Verdier. A Numerical Algorithm for C2-splines on Symmetric Spaces. arXiv e-prints, page arXiv:1703.09589, Mar 2017.
  5. 5.
    Yann Brenier. The least action principle and the related concept of generalized flows for incompressible perfect fluids. Journal of the American Mathematical Society, 2(2):225–255, 1989.MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Camarinha, F. Silva Leite, and P.Crouch. Splines of class \(\cal{C}^k\) on non-euclidean spaces. IMA Journal of Mathematical Control & Information, 12:399–410, 1995.MathSciNetCrossRefGoogle Scholar
  7. 7.
    L. Chizat, B. Schmitzer, G. Peyré, and F.-X. Vialard. An Interpolating Distance between Optimal Transport and Fisher-Rao. Found. Comp. Math., 2016.Google Scholar
  8. 8.
    P. Crouch and F. Silva Leite. The dynamic interpolation problem: On Riemannian manifold, Lie groups and symmetric spaces. Journal of dynamical & Control Systems, 1:177–202, 1995.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems, pages 2292–2300, 2013.Google Scholar
  10. 10.
    Alfred Galichon and Bernard Salanié. Matching with Trade-offs: Revealed Preferences over Competiting Characteristics. working paper or preprint, April 2010.Google Scholar
  11. 11.
    F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu, and F.-X. Vialard. Invariant Higher-Order Variational Problems. Communications in Mathematical Physics, 309:413–458, January 2012.MathSciNetCrossRefGoogle Scholar
  12. 12.
    F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu, and F.-X. Vialard. Invariant Higher-Order Variational Problems II. Journal of NonLinear Science, 22:553–597, August 2012.MathSciNetCrossRefGoogle Scholar
  13. 13.
    B. Heeren, M. Rumpf, and B. Wirth. Variational time discretization of Riemannian splines. ArXiv e-prints, November 2017.Google Scholar
  14. 14.
    François-Xavier Vialard Jean-David Benamou, Thomas Gallouët. Second order models for optimal transport and cubic splines on the wasserstein space. Preprint arXiv:1801.04144, 2018.
  15. 15.
    B. Khesin and R. Wendt. The geometry of infinite-dimensional groups, volume 51. Springer, Berlin, 2008.zbMATHGoogle Scholar
  16. 16.
    Young-Heon Kim and Brendan Pass. A general condition for monge solutions in the multi-marginal optimal transport problem. SIAM Journal on Mathematical Analysis, 46(2):1538–1550, 2014.MathSciNetCrossRefGoogle Scholar
  17. 17.
  18. 18.
    Lévy, Bruno. A numerical algorithm for l2 semi-discrete optimal transport in 3d. ESAIM: M2AN, 49(6):1693–1715, 2015.CrossRefGoogle Scholar
  19. 19.
    J. Lott. Some geometric calculations on Wasserstein space. Communications in Mathematical Physics, 277(2):423–437, 2008.MathSciNetCrossRefGoogle Scholar
  20. 20.
  21. 21.
    Quentin Mérigot. A multiscale approach to optimal transport. Computer Graphics Forum, 30 (5):1583–1592, 2011.CrossRefGoogle Scholar
  22. 22.
    Quentin Mérigot and Jean-Marie Mirebeau. Minimal geodesics along volume preserving maps, through semi-discrete optimal transport. arXiv preprint arXiv:1505.03306, 2015.
  23. 23.
    Quentin Mérigot and Jean-Marie Mirebeau. Minimal geodesics along volume-preserving maps, through semidiscrete optimal transport. SIAM J. Numer. Anal., 54(6):3465–3492, 2016.MathSciNetCrossRefGoogle Scholar
  24. 24.
    L. Noakes, G. Heinzinger, and B. Paden. Cubic splines on curved spaces. IMA Journal of Mathematical Control & Information, 6:465–473, 1989.MathSciNetCrossRefGoogle Scholar
  25. 25.
    F. Otto. The geometry of dissipative evolution equations: The porous medium equation. Communications in Partial Differential Equations, 26(1-2):101–174, 2001.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pass, Brendan. Multi-marginal optimal transport: Theory and applications. ESAIM: M2AN, 49(6):1771–1790, 2015.MathSciNetCrossRefGoogle Scholar
  27. 27.
    F. Santambrogio. Optimal transport for applied mathematicians. Progress in Nonlinear Differential Equations and their applications, 87, 2015.Google Scholar
  28. 28.
    Nikhil Singh, François-Xavier Vialard, and Marc Niethammer. Splines for diffeomorphisms. Medical Image Analysis, 25(1):56–71, 2015.CrossRefGoogle Scholar
  29. 29.
    R. Sinkhorn. Diagonal equivalence to matrices with prescribed row and column sums. Amer. Math. Monthly, 74:402–405, 1967.MathSciNetCrossRefGoogle Scholar
  30. 30.
    R. Tahraoui and F.-X. Vialard. Riemannian cubics on the group of diffeomorphisms and the Fisher-Rao metric. ArXiv e-prints, June 2016.Google Scholar
  31. 31.
    Alain Trouvé and François-Xavier Vialard. A second-order model for time-dependent data interpolation: Splines on shape spaces. In Proceedings of Miccai workshop, STIA, Beijing, 2010.Google Scholar
  32. 32.
    F.-X. Vialard and A. Trouvé. Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View. Quart. Appl. Math., 2012.Google Scholar
  33. 33.
    Cédric Villani. Optimal transport: old and new, volume 338. Springer, Berlin, 2008.zbMATHGoogle Scholar
  34. 34.
    Tryphon T Georgiou Yongxin Chen, Giovanni Conforti. Measure-valued spline curves: An optimal transport viewpoint. Preprint arXiv:1801.03186, 2018.

Copyright information

© SFoCM 2019

Authors and Affiliations

  • Jean-David Benamou
    • 1
    • 2
    Email author
  • Thomas O. Gallouët
    • 1
    • 2
  • François-Xavier Vialard
    • 3
  1. 1.Project Team MokaplanINRIAParisFrance
  2. 2.CeremadeUniversité Paris-Dauphine, PSL Research UniversityParisFrance
  3. 3.LIGM, UPEMUniversity Paris-EstMarne-la-ValléeFrance

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