Data-Driven Optimal Transport

Communications on Pure and Applied Mathematics

Volumn 69, Issue 4, 2016, Pages 613-648

  Trigila, Giulio ^a   Tabak, Esteban G ^b  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

^b NEW YORK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84958665136     PISSN: 00103640     EISSN: 10970312     Source Type: Journal    
DOI: 10.1002/cpa.21588     Document Type: Article

References (28)

1. Benamou, J.-D.; Brenier, Y. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84 (2000), no. 3, 375-393. doi: 10.1007/s002110050002
2. Brenier, Y. Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44 (1991), no. 4, 375-417. doi: 10.1002/cpa.3160440402
3. Caffarelli, L. The Monge-Ampère equation and optimal transportation, an elementary review. Optimal transportation and applications (Martina Franca, 2001), 1-10. Lecture Notes in Mathematics, 1813. Springer, Berlin, 2003. doi: 10.1007/978-3-540-44857-0_1
4. Chartrand, R.; Wohlberg, B.; Vixie, K. R.; Bollt, E. M. A gradient descent solution to the Monge-Kantorovich problem. Appl. Math. Sci. (Ruse) 3 (2009), no. 21-24, 1071-1080.
5. Cotar, C.; Friesecke, G.; Klüppelberg, C. Density functional theory and optimal transportation with Coulomb cost. Comm. Pure Appl. Math. 66 (2013), no. 4, 548-599. doi: 10.1002/cpa.21437
6. Cuturi, M. Sinkhorn distances: lightspeed computation of optimal transportation. Advances in Neural Information Processing Systems (2013) 26, 2292-2300.
7. Dowson, D. C.; Landau, B. V. The Fréchet distance between multivariate normal distributions. J. Multivariate Anal. 12 (1982), no. 3, 450-455. doi: 10.1016/0047-259X(82)90077-X
8. Evans, L. C. Partial differential equations and Monge-Kantorovich mass transfer. Current developments in mathematics, 1997 (Cambridge, MA), 65-126. International Press, Boston, 1999.
9. Froese, B. D. A numerical method for the elliptic Monge-Ampère equation with transport boundary conditions. SIAM J. Sci. Comput. 34 (2012), no. 3, A1432-A1459. doi: 10.1137/110822372
10. Froese, B. D.; Oberman, A. M. Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation. J. Comput. Phys. 230 (2011), no. 3, 818-834. doi: 10.1016/j.jcp.2010.10.020
11. Gangbo, W. An elementary proof of the polar factorization of vector-valued functions. Arch. Rational Mech. Anal. 128 (1994), no. 4, 381-399. doi: 10.1007/BF00387715
12. Gangbo, W.; McCann, R. J. The geometry of optimal transportation. Acta Math. 177 (1996), no. 2, 113-161. doi: 10.1007/BF02392620
13. Greengard, L.; Lee, J.-Y. Accelerating the nonuniform fast Fourier transform. SIAM Rev. 46 (2004), no. 3, 443-454. doi: 10.1137/S003614450343200X
14. 2 optimal mass transfer problem. SIAM J. Sci. Comput. 32 (2010), no. 1, 197-211. doi: 10.1137/080730238
15. Haker, S.; Zhu, L.; Tannenbaum, A.; Angenent, S. Optimal mass transport for registration and warping. International Journal of Computer Vision 60 (2004), no. 3, 225-240. doi: 10.1023/B:VISI.0000036836.66311.97
16. Iollo, A.; Lombardi, D. A Lagrangian scheme for the solution of the optimal mass transfer problem. J. Comput. Phys. 230 (2011), no. 9, 3430-3442. doi: 10.1016/j.jcp.2011.01.037
17. Jones, M. C.; Marron, J. S.; Sheather, S. J. A brief survey of bandwidth selection for density estimation. J. Amer. Statist. Assoc. 91 (1996), no. 433, 401-407. doi: 10.2307/2291420
18. Kantorovitch, L. On the translocation of masses. C. R. (Doklady) Acad. Sci. URSS (N.S.) 37 (1942), 199-201.
19. Laurence, P.; Pignol, R. J.; Tabak, E. G. Constrained density estimation. Quantitative energy finance, 259-284. Springer, New York, 2014. doi: 10.1007/978-1-4614-7248-3_10
20. McCann, R. J. A convexity principle for interacting gases. Adv. Math. 128 (1997), no. 1, 153-179. doi: 10.1006/aima.1997.1634
21. McCann, R. J.; Oberman, A. M. Exact semi-geostrophic flows in an elliptical ocean basin. Nonlinearity 17 (2004), no. 5, 1891-1922. doi: 10.1088/0951-7715/17/5/017
22. Monge, G. Mémoire sur la théorie des déblais et des remblais. De l'Imprimerie Royale, 1781.
23. Recca, N. A new methodology for importance sampling. Master's thesis, Courant Institute, New York University, 2011.
24. Sulman, M. M.; Williams, J. F.; Russell, R. D. An efficient approach for the numerical solution of the Monge-Ampère equation. Appl. Numer. Math. 61 (2011), no. 3, 298-307. doi: 10.1016/j.apnum.2010.10.006
25. Tabak, E. G.; Turner, C. V. A family of nonparametric density estimation algorithms. Comm. Pure Appl. Math. 66 (2013), no. 2, 145-164. doi: 10.1002/cpa.21423
26. Trudinger, N. S.; Wang, X.-J. The Monge-Ampère equation and its geometric applications. Handbook of geometric analysis, no. 1, 467-524. Advanced Lectures in Mathematics (ALM), 7. International Press, Somerville, Mass., 2008.
27. Villani, C. Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, R.I., 2003.
28. Wasserman, L. All of nonparametric statistics. Springer Texts in Statistics. Springer, New York, 2006.