Iterative bregman projections for regularized transportation problems

SIAM Journal on Scientific Computing

Volumn 37, Issue 2, 2015, Pages A1111-A1138

  Benamou, Jean David ^a   Carlier, Guillaume ^b   Cuturi, Marco ^c   Nenna, Luca ^a   Peyŕ, Gabriel ^b  

^a INRIA   (France)

^b Centre National de la Recherche Scientifique   (France)

^c KYOTO UNIVERSITY   (Japan)

Author keywords

Bregman projection; Convex optimization; Entropy regularization; Kullback Leibler; Optimal transport; Wasserstein barycenter

Indexed keywords

CONSTRAINT THEORY; LINEAR PROGRAMMING;

BARYCENTERS; BREGMAN PROJECTION; CONVEX OPTIMISATION; ENTROPY REGULARIZATION; KULLBACK-LEIBLER; LINEAR PROGRAMS; OPTIMAL TRANSPORT; REGULARISATION; TRANSPORTATION PROBLEM; WASSERSTEIN BARYCENTER;

CONVEX OPTIMIZATION;

EID: 84928966855     PISSN: 10648275     EISSN: 10957197     Source Type: Journal    
DOI: 10.1137/141000439     Document Type: Article

References (62)

1. I. Abraham, R. Abraham, M. Bergounioux, and G. Carlier, Tomographic Reconstruction from a Few Views: A Multi-Marginal Optimal Transport Approach, Preprint Hal-01065981, 2014.
2. M. Agueh and G. Carlier, Barycenters in the Wasserstein space, SIAM J. Math. Anal., 43 (2011), pp. 904-924.
3. L. Ambrosio and A. Figalli, Geodesics in the space of measure-preserving maps and plans, Arch. Ration. Mech. Anal., 194 (2009), pp. 421-462.
4. M. Bacharach, Estimating nonnegative matrices from marginal data, Internat. Econom. Rev., 6 (1965), pp. 294-310.
5. R. Bapat, d1ad2 theorems for multidimensional matrices, Linear Algebra Appl., 48 (1982), pp. 437-442.
6. H. H. Bauschke and P. L. Combettes, A Dykstra-like algorithm for two monotone operators, Pac. J. Optim., 4 (2008), pp. 383-391.
7. H. H. Bauschke and A. S. Lewis, Dykstra's algorithm with Bregman projections: A convergence proof, Optimization, 48 (2000), pp. 409-427.
8. J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution of the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), pp. 375-393.
9. J.-D. Benamou, B. D. Froese, and A. M. Oberman, Numerical solution of the optimal transportation problem using the Monge-Ampère equation, J. Comput. Phys., 260 (2014), pp. 107-126.
10. B. Bhattacharya, An iterative procedure for general probability measures to obtain Iprojections onto intersections of convex sets, Ann. Statist., 34 (2006), pp. 878-902.
11. J. Bigot and T. Klein, Consistent Estimation of a Population Barycenter in the Wasserstein Space, preprint, arXiv:1212.2562, 2012.
12. N. Bonneel, M. van de Panne, S. Paris, and W. Heidrich, Displacement interpolation using Lagrangian mass transport, ACM Trans. Graph., 30 (2011), 158.
13. L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys., 7 (1967), pp. 200-217.
14. Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc., 2 (1990), pp. 225-255.
15. Y. Brenier, The dual least action problem for an ideal, incompressible fluid, Arch. Rational Mech. Anal., 122 (1993), pp. 323-351.
16. Y. Brenier, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Comm. Pure Appl. Math., 52 (1999), pp. 411-452.
17. Y. Brenier, Generalized solutions and hydrostatic approximation of the Euler equations, Phys. D, 237 (2008), pp. 1982-1988.
18. R. Burkard, M. Dell'Amico, and S. Martello, Assignment Problems, SIAM, Philadelphia, 2009.
19. L. A. Caffarelli and R. J. McCann, Free boundaries in optimal transport and Monge-Ampère obstacle problems, Ann. of Math. (2), 171 (2010), pp. 673-730.
20. G. Carlier and I. Ekeland, Matching for teams, Econom. Theory, 42 (2010), pp. 397-418.
21. G. Carlier, A. Oberman, and E. Oudet, Numerical methods for matching for teams and Wasserstein barycenters, ESAIM Math. Model. Numer. Anal., to appear.
22. Y. Censor and S. Reich, The Dykstra algorithm with Bregman projections, Commun. Appl. Anal., 2 (1998), pp. 407-419.
23. P-A. Chiappori, R. J. McCann, and L. P. Nesheim, Hedonic price equilibria, stable matching, and optimal transport: Equivalence, topology, and uniqueness, Econom. Theory, 42 (2010), pp. 317-354.
24. R. Cominetti and J. San Martin, Asymptotic analysis of the exponential penalty trajectory in linear programming, Math. Programming, 67 (1992), pp. 169-187.
25. C. Cotar, G. Friesecke, and C. Klüppelberg, Density functional theory and optimal transportation with Coulomb cost, Comm. Pure Appl. Math., 66 (2013), pp. 548-599.
26. I. Csiszár, I-divergence geometry of probability distributions and minimization problems, Ann. Probability, 3 (1975), pp. 146-158.
27. M. Cuturi, Sinkhorn distances: Lightspeed computation of optimal transport., in Advances in Neural Information Processing Systems (NIPS) 26, MIT Press, Cambridge, MA, 2013, pp. 2292-2300.
28. M. Cuturi and A. Doucet, Fast computation of Wasserstein barycenters, in Proceedings of the 31st International Conference on Machine Learning, 2014.
29. W. E. Deming and F. F. Stephan, On a least squares adjustment of a sampled frequency table when the expected marginal totals are known, Ann. Math. Statistics, 11 (1940), pp. 427-444.
30. R. L. Dykstra, An algorithm for restricted least squares regression, J. Amer. Stat., 78 (1983), pp. 839-842.
31. R. L. Dykstra, An iterative procedure for obtaining I-projections onto the intersection of convex sets, Ann. Probab., 13 (1985), pp. 975-984.
32. S. Erlander and N. F. Stewart, The Gravity Model in Transportation Analysis: Theory and Extensions, VSP, Utrecht, 1990.
33. A. Figalli, The optimal partial transport problem, Arch. Ration. Mech. Anal., 195 (2010), pp. 533-560.
34. J. Franklin and J. Lorentz, On the scaling of multidimensional matrices, Linear Algebra Appl., 114/115 (1989), pp. 717-735.
35. U. Frisch, S. Matarrese, R. Mohayaee, and A. Sobolevski, A reconstruction of the initial conditions of the Universe by optimal mass transportation, Nature, 417 (2002), pp. 260-262.
36. A. Galichon and B. Salanié, Matching with Trade-Offs: Revealed Preferences over Competing Characteristics, Tech. report, preprint SSRN-1487307, 2009.
37. W. Gangbo and A. Šwięch, Optimal maps for the multidimensional Monge-Kantorovich problem, Comm. Pure Appl. Math., 51 (1998), pp. 23-45.
38. G. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980.
39. L. Kantorovich, On the transfer of masses, Dokl. Akad. Nauk, 37 (1942), pp. 227-229 (in Russian).
40. E. Klann, A Mumford-Shah-like method for limited data tomography with an application to electron tomography, SIAM J. Imaging Sci., 4 (2011), pp. 1029-1048.
41. J. Korman and R. J. McCann, Optimal Transportation with Capacity Constraints, preprint, arXiv:1201.6404, 2012.
42. J. Korman and R. J. McCann, Insights into capacity constrained optimal transport, Proc. Natl. Acad. Sci. USA, 110 (2013), pp. 10064-10067.
43. C. Leonard, A survey of the Schrodinger problem and some of its connections with optimal transport, Discrete Contin. Dyn. Syst. A, 34 (2014), pp. 1533-1574.
44. G. Loeper and F. Rapetti, Numerical solution of the Monge-Ampère equation by a Newton's algorithm, C. R. Math. Acad. Sci. Paris, 340 (2005), pp. 319-324.
45. Q. Merigot, A multiscale approach to optimal transport, Comput. Graph. Forum, 30 (2011), pp. 1583-1592.
46. Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Methods in Convex Programming: Theory and Algorithms, SIAM, Philadelphia, 1994.
47. N. Papadakis, G. Peyré, and E. Oudet, Optimal transport with proximal splitting, SIAM J. Imaging Sci., 7 (2014), pp. 212-238.
48. B. Pass, Uniqueness and Monge solutions in the multimarginal optimal transportation problem, SIAM J. Math. Anal., 43 (2011), pp. 2758-2775.
49. B. Pass, On the local structure of optimal measures in the multi-marginal optimal transportation problem, Calc. Var. Partial Differential Equations, 43 (2012), pp. 529-536.
50. B. Pass and J. Kitagawa, The Multi-Marginal Optimal Partial Transport Problem, preprint, arXiv:1401.7255, 2014.
51. T. E. S. Raghavan, On pairs of multidimensional matrices, Linear Algebra Appl., 62 (1984), pp. 263-268.
52. Y. Rubner, C. Tomasi, and L. J. Guibas, The earth mover's distance as a metric for image retrieval, Int. J. Comput. Vis., 40 (2000), pp. 99-121.
53. L. Rüschendorf, Convergence of the iterative proportional fitting procedure, Ann. Statist., 23 (1995), pp. 1160-1174.
54. L. Rüschendorf and W. Thomsen, Closedness of sum spaces and the generalized Schrödinger problem, Theory Probab. Appl., 42 (1998), pp. 483-494.
55. E. Schrödinger, Uber die umkehrung der naturgesetze, Sitzungsberichte Preuss. Akad. Wiss. Berlin. Phys. Math., 144 (1931), pp. 144-153.
56. R. Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist., 35 (1964), pp. 876-879.
57. R. Sinkhorn, Diagonal equivalence to matrices with prescribed row and column sums, Amer. Math. Monthly, 74 (1967), pp. 402-405.
58. R. Sinkhorn and P . Knopp, Concerning nonnegative matrices and doubly stochastic matrices, Pacific J. Math., 21 (1967), pp. 343-348.
59. J. Solomon, R. M. Rustamov, L. Guibas, and A. Butscher, Wasserstein propagation for semi-supervised learning, in Proceedings of the 31st International Conference on Machine Learning, 2014.
60. C. Villani, Topics in Optimal Transportation, Grad. Stud. Math. 58, American Mathematical Society, Providence, RI, 2003.
61. A. G. Wilson, The use of entropy maximizing models, in the theory of trip distribution, mode split and route split, J. Transport Econ. Pol., 3 (1969), pp. 108-126.
62. G.-S. Xia, S. Ferradans, G. Peyré, and J.-F. Aujol, Synthesizing and mixing stationary Gaussian texture models, SIAM J. Imaging Sci., 7 (2014), pp. 476-508.