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Solving Semi-Discrete Optimal Transport Problems: star shapedeness and Newton's method (2310.07489v1)

Published 11 Oct 2023 in math.NA, cs.NA, and math.OC

Abstract: In this work, we propose a novel implementation of Newton's method for solving semi-discrete optimal transport (OT) problems for cost functions which are a positive combination of $p$-norms, $1<p<\infty$. It is well understood that the solution of a semi-discrete OT problem is equivalent to finding a partition of a bounded region in Laguerre cells, and we prove that the Laguerre cells are star-shaped with respect to the target points. By exploiting the geometry of the Laguerre cells, we obtain an efficient and reliable implementation of Newton's method to find the sought network structure. We provide implementation details and extensive results in support of our technique in 2-d problems, as well as comparison with other approaches used in the literature.

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References (38)
  1. Luigi Ambrosio. Lecture Notes on Optimal Transport Problems, pages 1–52. Springer Berlin Heidelberg, Berlin, Heidelberg, 2003.
  2. F. Aurenhammer. Power diagrams: Properties, algorithms and applications. SIAM Journal on Computing, 16(1):78–96, 1987.
  3. A computational fluid mechanics solution to the monge-kantorovich mass transfer problem. Numerische Mathematik, 84(3):375–393, 2000.
  4. Numerical solution of the optimal transportation problem using the monge–ampère equation. Journal of Computational Physics, 260:107–126, 2014.
  5. Béla Bollobás. Linear Analysis, Cambridge Mathematical Textbooks, page 21. Cambridge University Press,, 1999.
  6. Boundaries and hulls of Euclidean graphs: From theory to practice, page 34. CRC Press, 2018.
  7. Centroidal power diagrams, Lloyd’s algorithm, and applications to optimal location problems. SIAM Journal on Numerical Analysis, 53(6):2545–2569, 2015.
  8. Semi-discrete unbalanced optimal transport and quantization. arXiv: Optimization and Control, 2018-08.
  9. A characterization for the solution of the Monge-Kantorovich mass transference problem. Statistics and Probability Letters, 16(2):147–152, 1993.
  10. Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems, volume 26, pages 2292–2300. Curran Associates, Inc., 2013.
  11. Differentiation and regularity of semi-discrete optimal transport with respect to the parameters of the discrete measure. Numerische Mathematik, 141:429–453, 2019.
  12. Techniques for continuous optimal transport problem. Computers & Mathematics with Applications, 146:176–191, 2023.
  13. Luca Dieci and J.D. Walsh III. The boundary method for semi-discrete optimal transport partitions and Wasserstein distance computation. Journal of Computational and Applied Mathematics, 353:318–344, 2019.
  14. Adam Dobrin. A review of properties and variations of Voronoi diagrams. Whitman College, 2005.
  15. Centroidal Voronoi tessellations: Applications and algorithms. SIAM Review, 41(4):637–676, 1999.
  16. Steven Fortune. A sweepline algorithm for Voronoi diagrams. In Proceedings of the Second Annual Symposium on Computational Geometry, SCG ’86, pages 313–322, New York, 1986. ACM.
  17. Brittany D Froese. A numerical method for the elliptic monge–ampère equation with transport boundary conditions. SIAM Journal on Scientific Computing, 34(3):A1432–A1459, 2012.
  18. The geometry of optimal transportation. Acta Mathematica, 177(2):113–161, 1996.
  19. Efficient proximity search for 3-d cuboids. In Vipin Kumar, Marina L. Gavrilova, Chih Jeng Kenneth Tan, and Pierre L’Ecuyer, editors, Computational Science and Its Applications — ICCSA 2003, pages 817–826, Berlin, Heidelberg, 2003. Springer Berlin Heidelberg.
  20. Optimally solving a transportation problem using Voronoi diagrams. Computational Geometry, 46(8):1009–1016, 2013.
  21. Algebraic Graph Theory, volume 207, page 279. Springer Science & Business Media, 2001.
  22. Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case. Mathematical Methods of Operations Research, pages 1–31, 2020.
  23. Leonid V. Kantorovich. On the translocation of masses. C.R. (Doklady) Acad. Sci. URSS (N.S.), 37:199–201, 1942.
  24. Leonid V. Kantorovich. On a problem of Monge. Uspekhi Mat. Nauk, 3:225–226, 1948.
  25. Herb B Keller. Numerical methods in bifurcation problems, volume Tata Institute of Fundamental Research. Springer-Verlag, Bombay, 1987.
  26. Convergence of a Newton algorithm for semi-discrete optimal transport. Journal of the European Mathematical Society, 21(9):2603–2651, 2019.
  27. Bruno Lévy. A numerical algorithm for L2subscript𝐿2L_{2}italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT semi-discrete optimal transport in 3D. ESAIM: Mathematical Modelling and Numerical Analysis, 49(6):1693–1715, 2015.
  28. Quentin Mérigot. A comparison of two dual methods for discrete optimal transport. In Frank Nielsen and Frédéric Barbaresco, editors, GSI 2013 — Geometric Science of Information, Aug 2013, Paris, France, volume 8085 of Lecture Notes in Computer Science, pages 389–396. Springer, 1781.
  29. Jocelyn Meyron. Initialization procedures for discrete and semi-discrete optimal transport. Computer-Aided Design, 115:13–22, 2019.
  30. Gaspard Monge. Mémoire sur la théorie des déblais et des remblais. In Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, pages 666–704. Académie des sciences (France)., 1781. In French.
  31. Computational optimal transport: With applications to data science. Foundations and Trends® in Machine Learning, 11(5-6):355–607, 2019.
  32. Aldo Pratelli. On the equality between Monge’s infimum and Kantorovich’s minimum in optimal mass transportation. Annales de l’Institut Henri Poincare (B) Probability and Statistics, 43(1):1–13, 2007.
  33. Numerical and analytical results for the transportation problem of Monge-Kantorovich. Metrika, 51(3):245–258, 2000.
  34. Filippo Santambrogio. Optimal transport for applied mathematicians, volume 55, page 94. Springer, 2015.
  35. Micha Sharir. Intersection and closest-pair problems for a set of planar discs. SIAM Journal on Computing, 14(2):448–468, 1985.
  36. Richard Sinkhorn. Diagonal equivalence to matrices with prescribed row and column sums. The American Mathematical Monthly, 74(4):402–405, 1967.
  37. Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains. ACM Transactions on Graphics, 34, 07 2015.
  38. Cédric Villani. Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, R.I., 2003.

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