Advertisement

Traversing the Schrödinger Bridge Strait: Robert Fortet’s Marvelous Proof Redux

  • Montacer Essid
  • Michele PavonEmail author
Article
  • 208 Downloads

Abstract

In the early 1930s, Erwin Schrödinger, motivated by his quest for a more classical formulation of quantum mechanics, posed a large deviation problem for a cloud of independent Brownian particles. He showed that the solution to the problem could be obtained through a system of two linear equations with nonlinear coupling at the boundary (Schrödinger system). Existence and uniqueness for such a system, which represents a sort of bottleneck for the problem, was first established by Fortet in 1938/1940 under rather general assumptions by proving convergence of an ingenious but complex approximation method. It is the first proof of what are nowadays called Sinkhorn-type algorithms in the much more challenging continuous case. Schrödinger bridges are also an early example of the maximum entropy approach and have been more recently recognized as a regularization of the important optimal mass transport problem. Unfortunately, Fortet’s contribution is by and large ignored in contemporary literature. This is likely due to the complexity of his approach coupled with an idiosyncratic exposition style and due to missing details and steps in the proofs. Nevertheless, Fortet’s approach maintains its importance to this day as it provides the only existing algorithmic proof, in the continuous setting, under rather mild assumptions. It can be adapted, in principle, to other relevant optimal transport problems. It is the purpose of this paper to remedy this situation by rewriting the bulk of his paper with all the missing passages and in a transparent fashion so as to make it fully available to the scientific community. We consider the problem in \({\mathbb {R}}^d\) rather than in \({\mathbb {R}}\) and use as much as possible his notation to facilitate comparison.

Keywords

Schrödinger system Large deviations Iterative algorithm 

Notes

Acknowledgements

The authors thank Robert V. Kohn for useful suggestions. The second named author would also like to thank the Courant Institute of Mathematical Sciences of the New York University for the hospitality during the time this paper was written. The authors finally wish to thank two anonymous reviewers for very careful reading and providing plenty of general and specific comments/suggestions on how to improve the paper. The second named author was partly supported by the University of Padova Research Project CPDA 140897.

References

  1. 1.
    Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106(4), 620 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Jaynes, E.T.: On the rationale of maximum-entropy methods. Proc. IEEE 70(9), 939–952 (1982)CrossRefGoogle Scholar
  3. 3.
    Burg, J.P.: Maximum entropy spectral analysis. In: 37th Annual International Meeting, Society of Exploration Geophysicists Oklahoma City, Okla, 31 Oct 1967 (1967)Google Scholar
  4. 4.
    Burg, J.P., Luenberger, D.G., Wenger, D.L.: Estimation of structured covariance matrices. Proc. IEEE 70(9), 963–974 (1982)CrossRefGoogle Scholar
  5. 5.
    Dempster, A.P.: Covariance selection. Biometrics 28, 157–175 (1972)CrossRefGoogle Scholar
  6. 6.
    Csiszár, I.: I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3, 146–158 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Csiszár, I.: Sanov property, generalized I-projection and a conditional limit theorem. Ann. Probab. 12, 768–793 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Csiszar, I., et al.: Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann. Stat. 19(4), 2032–2066 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mikami, T.: Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes. Probab. Theory Relat. Fields 129(2), 245–260 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mikami, T., Thieullen, M.: Duality theorem for the stochastic optimal control problem. Stoch. Process. Appl. 116(12), 1815–1835 (2006).  https://doi.org/10.1016/j.spa.2006.04.014 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mikami, T., Thieullen, M.: Optimal transportation problem by stochastic optimal control. SIAM J. Control Optim. 47(3), 1127–1139 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Léonard, C.: A survey of the schrodinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. A 34(4), 1533–1574 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Léonard, C.: From the Schrödinger Problem to the Monge–Kantorovich Problem. arXiv preprint arXiv:1011.2564 (2010)
  14. 14.
    Chen, Y., Georgiou, T.T., Pavon, M.: On the relation between optimal transport and Schrödinger bridges: a stochastic control viewpoint. J. Optim. Theory Appl. 169(2), 671–691 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Peyré, G., Cuturi, M.: Computational Optimal Transport. arXiv preprint arXiv:1803.00567 (2018)
  16. 16.
    Cuturi, M.: Sinkhorn distances: lightspeed computation of optimal transport. In: Advances in Neural Information Processing Systems, pp. 2292–2300 (2013)Google Scholar
  17. 17.
    Benamou, J.D., Carlier, G., Cuturi, M., Nenna, L., Peyré, G.: Iterative Bregman projections for regularized transportation problems. SIAM J. Sci. Comput. 37(2), A1111–A1138 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chen, Y., Georgiou, T.T., Pavon, M.: Optimal transport over a linear dynamical system. IEEE Trans. Autom. Control 62(5), 2137–2152 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chen, Y., Georgiou, T., Pavon, M.: Entropic and displacement interpolation: a computational approach using the Hilbert metric. SIAM J. Appl. Math. 76(6), 2375–2396 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fortet, R.: Résolution d’un système d’équations de M. Schrodinger. Comptes Rendus 206, 721–723 (1938)zbMATHGoogle Scholar
  21. 21.
    Fortet, R.: Résolution d’un système d’équations de M. Schrodinger. J. Math. Pure Appl. IX, 83–105 (1940)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Beurling, A.: An automorphism of product measures. Ann. Math. 72, 189–200 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jamison, B.: The Markov processes of Schrödinger. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 32(4), 323–331 (1975)CrossRefzbMATHGoogle Scholar
  24. 24.
    Zambrini, J.C.: Variational processes and stochastic versions of mechanics. J. Math. Phys. 27, 2307–2330 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Föllmer, H.: Random fields and diffusion processes. In: Hennequin, P.-L. (ed.) École d’Été de Probabilités de Saint-Flour XV-XVII, 1985–87, pp. 101–203. Springer, Berlin (1988)CrossRefzbMATHGoogle Scholar
  26. 26.
    Deming, W.E., Stephan, F.F.: On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Stat. 11(4), 427–444 (1940)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sinkhorn, R.: A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Stat. 35(2), 876–879 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Chizat, L., Peyré, G., Schmitzer, B., Vialard, F.X.: Scaling Algorithms for Unbalanced Transport Problems. arXiv preprint arXiv:1607.05816 (2016)
  29. 29.
    Schrödinger, E.: Uber, : die umkehrung der naturgesetze. Sitzungsberichte der Preuss Akad. Wissen. Berlin. Phys. Math. Klasse 1, 144–153 (1931)Google Scholar
  30. 30.
    Schrödinger, E.: Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique. Ann. Inst. H. Poincaré 2(4), 269–310 (1932)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Sanov, I.N.: On the Probability of Large Deviations of Random Variables, Technical report. Department of Statistics, North Carolina State University (1958)Google Scholar
  32. 32.
    Dudley, R.M.: Real Analysis and Probability. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  33. 33.
    Ellis, R.S.: Entropy, Large Deviations, and Statistical Mechanics. Springer, Berlin (2007)Google Scholar
  34. 34.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Corrected reprint of the second (1998) edition. Stochastic Modelling and Applied Probability, p. 38 (2010)Google Scholar
  35. 35.
    Villani, C.: Topics in Optimal Transportation, vol. 58. American Mathematical Society, Providence (2003)zbMATHGoogle Scholar
  36. 36.
    Wakolbinger, A.: Schrödinger bridges from 1931 to 1991. In: Proceedings of the 4th Latin American Congress in Probability and Mathematical Statistics, Mexico City, pp. 61–79 (1990)Google Scholar
  37. 37.
    Dai Pra, P.: A stochastic control approach to reciprocal diffusion processes. Appl. Math. Optim. 23(1), 313–329 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Dai Pra, P., Pavon, M.: On the Markov processes of Schrödinger, the Feynman–Kac formula and stochastic control. In: Realization and Modelling in System Theory, pp. 497–504. Springer, Berlin (1990)Google Scholar
  39. 39.
    Pavon, M., Wakolbinger, A.: On free energy, stochastic control, and Schrödinger processes. In: Modeling, Estimation and Control of Systems with Uncertainty, pp. 334–348. Springer, Berlin (1991)Google Scholar
  40. 40.
    Mikami, T.: Optimal transportation problem as stochastic mechanics. Sel. Pap. Probab. Stat. 227, 75–94 (2008)Google Scholar
  41. 41.
    Chen, Y., Georgiou, T.T., Pavon, M.: Optimal steering of a linear stochastic system to a final probability distribution, part I. IEEE Trans. Autom. Control 61(5), 1158–1169 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Chen, Y., Georgiou, T.T., Pavon, M.: Optimal steering of a linear stochastic system to a final probability distribution, part II. IEEE Trans. Autom. Control 61(5), 1170–1180 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Chen, Y., Georgiou, T.T., Pavon, M.: Fast cooling for a system of stochastic oscillators. J. Math. Phys. 56(11), 113,302 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84(3), 375–393 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Birkhoff, G.: Extensions of Jentzsch’s theorem. Trans. Am. Math. Soc. 85(1), 219–227 (1957)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Bushell, P.: On the projective contraction ratio for positive linear mappings. J. Lond. Math. Soc. 2(2), 256–258 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Bushell, P.J.: Hilbert’s metric and positive contraction mappings in a Banach space. Arch. Ration. Mech. Anal. 52(4), 330–338 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Birkhoff, G.: Uniformly semi-primitive multiplicative processes. Trans. Am. Math. Soc. 104(1), 37–51 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Lemmens, B., Nussbaum, R.: Birkhoff’s Version of Hilbert’s Metric and Its Applications in Analysis. arXiv preprint arXiv:1304.7921 (2013)
  50. 50.
    Tsitsiklis, J., Bertsekas, D., Athans, M.: Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE Trans. Autom. Control 31(9), 803–812 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Sepulchre, R., Sarlette, A., Rouchon, P.: Consensus in Non-Commutative Spaces. arXiv preprint arXiv:1003.5653 (2010)
  52. 52.
    Bonnabel, S., Astolfi, A., Sepulchre, R.: Contraction and observer design on cones. In: Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference on, pp. 7147–7151. IEEE (2011)Google Scholar
  53. 53.
    Reeb, D., Kastoryano, M.J., Wolf, M.M.: Hilbert’s projective metric in quantum information theory. J. Math. Phys. 52(8), 082,201 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Lemmens, B., Nussbaum, R.: Nonlinear Perron–Frobenius Theory, vol. 189. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  55. 55.
    Georgiou, T.T., Pavon, M.: Positive contraction mappings for classical and quantum Schrödinger systems. J. Math. Phys. 56(3), 033,301 (2015)CrossRefzbMATHGoogle Scholar
  56. 56.
    Franklin, J., Lorenz, J.: On the scaling of multidimensional matrices. Linear Algebra Appl. 114, 717–735 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Schmitzer, B.: Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems. arXiv preprint arXiv:1610.06519 (2016)
  58. 58.
    Galichon, A., Kominers, S.D., Weber, S.: The nonlinear Bernstein–Schrödinger equation in economics. In: International Conference on Networked Geometric Science of Information, pp. 51–59. Springer, Berlin (2015)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Dipartimento di Matematica “Tullio Levi-Civita”Università di PadovaPaduaItaly

Personalised recommendations