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Dynamical optimal transport on discrete surfaces

Authors:

Sebastian Claici,

Justin SolomonAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 37, Issue 6

Article No.: 250, Pages 1 - 16

https://doi.org/10.1145/3272127.3275064

Published: 04 December 2018 Publication History

Abstract

We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finite-dimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows.

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References

[1]

Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. 2008. Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Springer Science & Business Media.

[2]

Franz Aurenhammer, Friedrich Hoffmann, and Boris Aronov. 1998. Minkowski-type theorems and least-squares clustering. Algorithmica 20, 1 (1998), 61--76.

[3]

Omri Azencot, Orestis Vantzos, and Mirela Ben-Chen. 2016. Advection-based function matching on surfaces. In Computer Graphics Forum, Vol. 35. Wiley Online Library, 55--64.

[4]

Jean-David Benamou and Yann Brenier. 2000. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84, 3 (2000), 375--393.

[5]

Jean-David Benamou and Guillaume Carlier. 2015. Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations. Journal of Optimization Theory and Applications 167, 1 (2015), 1--26.

Digital Library

[6]

Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Luca Nenna, and Gabriel Peyré. 2015. Iterative Bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing 37, 2 (2015), A1111--A1138.

Digital Library

[7]

Jean-David Benamou, Guillaume Carlier, and Maxime Laborde. 2016. An augmented Lagrangian approach to Wasserstein gradient flows and applications. ESAIM: Proceedings and Surveys 54 (2016), 1--17.

[8]

Jean-David Benamou, Guillaume Carlier, and Filippo Santambrogio. 2017. Variational mean field games. In Active Particles, Volume 1. Springer, 141--171.

[9]

Nicolas Bonneel, Michiel Van De Panne, Sylvain Paris, and Wolfgang Heidrich. 2011. Displacement interpolation using Lagrangian mass transport. In ACM Transactions on Graphics (TOG), Vol. 30. ACM, 158.

Digital Library

[10]

Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein, et al. 2011. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine Learning 3, 1 (2011), 1--122.

Digital Library

[11]

Stephen Boyd and Lieven Vandenberghe. 2004. Convex Optimization. Cambridge University Press.

Digital Library

[12]

Yann Brenier. 1991. Polar factorization and monotone rearrangement of vector-valued functions. Communications on Pure and Applied Mathematics 44, 4 (1991), 375--417.

[13]

Susanne Brenner and Ridgway Scott. 2007. The Mathematical Theory of Finite Element Methods. Vol. 15. Springer Science & Business Media.

[14]

Shui-Nee Chow, Luca Dieci, Wuchen Li, and Haomin Zhou. 2016. Entropy dissipation semi-discretization schemes for Fokker-Planck equations. arXiv:1608.02628 (2016).

[15]

Sebastian Claici, Edward Chien, and Justin Solomon. 2018. Stochastic Wasserstein barycenters. In Proceedings of the 35th International Conference on Machine Learning, ICML 2018 (to appear).

[16]

Marco Cuturi. 2013. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems, NIPS 2013. 2292--2300.

Digital Library

[17]

Marco Cuturi and Arnaud Doucet. 2014. Fast computation of Wasserstein barycenters. In International Conference on Machine Learning. 685--693.

Digital Library

[18]

Fernando De Goes, Katherine Breeden, Victor Ostromoukhov, and Mathieu Desbrun. 2012. Blue noise through optimal transport. ACM Transactions on Graphics (TOG) 31, 6 (2012), 171.

Digital Library

[19]

Fernando de Goes, David Cohen-Steiner, Pierre Alliez, and Mathieu Desbrun. 2011. An optimal transport approach to robust reconstruction and simplification of 2d shapes. In Computer Graphics Forum, Vol. 30. Wiley Online Library, 1593--1602.

[20]

Fernando de Goes, Mathieu Desbrun, and Yiying Tong. 2015a. Vector field processing on triangle meshes. In SIGGRAPH Asia 2015 Courses. ACM, 17.

Digital Library

[21]

Fernando de Goes, Pooran Memari, Patrick Mullen, and Mathieu Desbrun. 2014. Weighted Triangulations for Geometry Processing. ACM Trans. Graph. 33, 3 (June 2014), 28:1--28:13.

Digital Library

[22]

Fernando de Goes, Corentin Wallez, Jin Huang, Dmitry Pavlov, and Mathieu Desbrun. 2015b. Power particles: an incompressible fluid solver based on power diagrams. ACM Trans. Graph. 34, 4 (2015), 50--1.

Digital Library

[23]

Julie Digne, David Cohen-Steiner, Pierre Alliez, Fernando De Goes, and Mathieu Desbrun. 2014. Feature-preserving surface reconstruction and simplification from defect-laden point sets. Journal of Mathematical Imaging and Vision 48, 2 (2014), 369--382.

Digital Library

[24]

Jack Edmonds and Richard M Karp. 1972. Theoretical improvements in algorithmic efficiency for network flow problems. Journal of the ACM (JACM) 19, 2 (1972), 248--264.

Digital Library

[25]

Matthias Erbar, Martin Rumpf, Bernhard Schmitzer, and Stefan Simon. 2017. Computation of Optimal Transport on Discrete Metric Measure Spaces. arXiv:1707.06859 (2017).

[26]

Thomas O Gallouët and Quentin Mérigot. 2017. A Lagrangian scheme à la Brenier for the incompressible Euler equations. Foundations of Computational Mathematics (2017), 1--31.

[27]

Wilfrid Gangbo and Robert J McCann. 1996. The geometry of optimal transportation. Acta Mathematica 177, 2 (1996), 113--161.

[28]

Nicola Gigli and Jan Maas. 2013. Gromov-Hausdorff convergence of discrete transportation metrics. SIAM Journal on Mathematical Analysis 45, 2 (2013), 879--899.

Digital Library

[29]

Peter Gladbach, Eva Kopfer, and Jan Maas. 2018. Scaling limits of discrete optimal transport. arXiv:1809.01092 (2018).

[30]

Michael Grant and Stephen Boyd. 2008. Graph implementations for nonsmooth convex programs. In Recent Advances in Learning and Control, V. Blondel, S. Boyd, and H. Kimura (Eds.). Springer-Verlag Limited, 95--110.

[31]

Michael Grant and Stephen Boyd. 2014. CVX: Matlab Software for Disciplined Convex Programming, version 2.1. http://cvxr.com/cvx.

[32]

Kevin Guittet. 2003. On the time-continuous mass transport problem and its approximation by augmented Lagrangian techniques. SIAM J. Numer. Anal. 41, 1 (2003), 382--399.

Digital Library

[33]

Behrend Heeren, Martin Rumpf, Max Wardetzky, and Benedikt Wirth. 2012. Time-Discrete Geodesics in the Space of Shells. In Computer Graphics Forum, Vol. 31. Wiley Online Library, 1755--1764.

Digital Library

[34]

Romain Hug, Nicolas Papadakis, and Emmanuel Maitre. 2015. On the convergence of augmented Lagrangian method for optimal transport between nonnegative densities. (2015).

[35]

Richard Jordan, David Kinderlehrer, and Felix Otto. 1998. The variational formulation of the Fokker-Planck equation. SIAM Journal on Mathematical Analysis 29, 1 (1998), 1--17.

Digital Library

[36]

Jürgen Jost. 2008. Riemannian geometry and geometric analysis. Vol. 42005. Springer.

[37]

Leonid V Kantorovich. 1942. On the translocation of masses. In Dokl. Akad. Nauk. USSR (NS), Vol. 37. 199--201.

[38]

Jun Kitagawa, Quentin Mérigot, and Boris Thibert. 2016. Convergence of a Newton algorithm for semi-discrete optimal transport. arXiv preprint arXiv:1603.05579 (2016).

[39]

Morton Klein. 1967. A primal method for minimal cost flows with applications to the assignment and transportation problems. Management Science 14, 3 (1967), 205--220.

Digital Library

[40]

Hugo Lavenant. 2017. Harmonic mappings valued in the Wasserstein space. arXiv:1712.07528 (2017).

[41]

Bruno Lévy. 2015. A numerical algorithm for L<sub>2</sub> semi-discrete optimal transport in 3D. ESAIM: Mathematical Modelling and Numerical Analysis 49, 6 (2015), 1693--1715.

[42]

Wuchen Li, Ernest K Ryu, Stanley Osher, Wotao Yin, and Wilfrid Gangbo. 2018. A parallel method for earth mover's distance. Journal of Scientific Computing 75, 1 (2018), 182--197.

Digital Library

[43]

Zhuoran Lu. 2017. Properties of Soft Maps on Riemannian Manifolds. Ph.D. Dissertation. New York University.

[44]

Jan Maas. 2011. Gradient flows of the entropy for finite Markov chains. Journal of Functional Analysis 261, 8 (2011), 2250--2292.

[45]

Bertrand Maury, Aude Roudneff-Chupin, and Filippo Santambrogio. 2010. A macroscopic crowd motion model of gradient flow type. Mathematical Models and Methods in Applied Sciences 20, 10 (2010), 1787--1821.

[46]

Robert J McCann. 1997. A convexity principle for interacting gases. Advances in Mathematics 128, 1 (1997), 153--179.

[47]

Quentin Mérigot. 2011. A multiscale approach to optimal transport. In Computer Graphics Forum, Vol. 30. Wiley Online Library, 1583--1592.

[48]

Quentin Mérigot, Jocelyn Meyron, and Boris Thibert. 2018. An algorithm for optimal transport between a simplex soup and a point cloud. SIAM Journal on Imaging Sciences 11, 2 (2018), 1363--1389.

Digital Library

[49]

Quentin Mérigot and Jean-Marie Mirebeau. 2016. Minimal geodesics along volume-preserving maps, through semidiscrete optimal transport. SIAM J. Numer. Anal. 54, 6 (2016), 3465--3492.

Digital Library

[50]

MOSEK ApS. 2017. The MOSEK optimization toolbox for MATLAB manual.

[51]

James B Orlin. 1997. A polynomial time primal network simplex algorithm for minimum cost flows. Mathematical Programming 78, 2 (1997), 109--129.

Digital Library

[52]

Felix Otto. 2001. The geometry of dissipative evolution equations: the porous medium equation. (2001).

[53]

Daniele Panozzo, Ilya Baran, Olga Diamanti, and Olga Sorkine-Hornung. 2013. Weighted averages on surfaces. ACM Transactions on Graphics (TOG) 32, 4 (2013), 60.

Digital Library

[54]

Nicolas Papadakis, Gabriel Peyré, and Edouard Oudet. 2014. Optimal transport with proximal splitting. SIAM Journal on Imaging Sciences 7, 1 (2014), 212--238.

Digital Library

[55]

Ulrich Pinkall and Konrad Polthier. 1993. Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2, 1 (1993), 15--36.

[56]

Konrad Polthier and Markus Schmies. 2006. Straightest geodesics on polyhedral surfaces. ACM.

[57]

Filippo Santambrogio. 2015. Optimal transport for applied mathematicians. Birkäuser, NY (2015), 99--102.

[58]

Filippo Santambrogio. 2018. Crowd motion and evolution PDEs under density constraints.

[59]

Justin Solomon, Fernando De Goes, Gabriel Peyré, Marco Cuturi, Adrian Butscher, Andy Nguyen, Tao Du, and Leonidas Guibas. 2015. Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains. ACM Transactions on Graphics (TOG) 34, 4 (2015), 66.

Digital Library

[60]

Justin Solomon, Leonidas Guibas, and Adrian Butscher. 2013. Dirichlet energy for analysis and synthesis of soft maps. In Computer Graphics Forum, Vol. 32. Wiley Online Library, 197--206.

Digital Library

[61]

Justin Solomon, Andy Nguyen, Adrian Butscher, Mirela Ben-Chen, and Leonidas Guibas. 2012. Soft maps between surfaces. In Computer Graphics Forum, Vol. 31. Wiley Online Library, 1617--1626.

Digital Library

[62]

Justin Solomon, Raif Rustamov, Leonidas Guibas, and Adrian Butscher. 2014. Earth mover's distances on discrete surfaces. ACM Transactions on Graphics (TOG) 33, 4 (2014), 67.

Digital Library

[63]

Nicolas Garcia Trillos. 2017. Gromov-Hausdorff limit of Wasserstein spaces on point clouds. arXiv:1702.03464 (2017).

[64]

Juan Luis Vázquez. 2007. The Porous Medium Equation: Mathematical Theory. Oxford University Press.

[65]

Cédric Villani. 2003. Topics in Optimal Transportation. Number 58. American Mathematical Soc.

[66]

Cédric Villani. 2008. Optimal Transport: Old and New. Vol. 338. Springer Science & Business Media.

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Index Terms

Dynamical optimal transport on discrete surfaces
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Shape analysis
2. Mathematics of computing
  1. Mathematical analysis

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Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 37, Issue 6

December 2018

1401 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3272127

Editor:
Takeo Igarashi
The University of Tokyo, Japan

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Copyright © 2018 ACM.

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Publication History

Published: 04 December 2018

Published in TOG Volume 37, Issue 6

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Shi YZheng LQuan PXiao YNiu L(2025)A universal network strategy for lightspeed computation of entropy-regularized optimal transportNeural Networks10.1016/j.neunet.2024.107038184(107038)Online publication date: Apr-2025
https://doi.org/10.1016/j.neunet.2024.107038
Dong GGuo HJiang CShi Z(2025)Gradient enhanced ADMM Algorithm for dynamic optimal transport on surfacesJournal of Computational Physics10.1016/j.jcp.2025.113805(113805)Online publication date: Feb-2025
https://doi.org/10.1016/j.jcp.2025.113805
Todeschi GMétivier LMirebeau J(2025)Unbalanced L1 optimal transport for vector valued measures and application to Full Waveform InversionJournal of Computational Physics10.1016/j.jcp.2024.113657523(113657)Online publication date: Feb-2025
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Facca ETodeschi GNatale ABenzi M(2024)Efficient Preconditioners for Solving Dynamical Optimal Transport via Interior Point MethodsSIAM Journal on Scientific Computing10.1137/23M157043046:3(A1397-A1422)Online publication date: 2-May-2024
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