The boundary method for semi-discrete optimal transport partitions and Wasserstein distance computation

Abstract We introduce a new technique, which we call the boundary method, for solving semi-discrete optimal transport problems with a wide range of cost functions. The boundary method reduces the effective dimension of the problem, thus improving complexity. For cost functions equal to a p -norm with p ∈ ( 1 , ∞ ) , we provide mathematical justification, convergence analysis, and algorithmic development. Our testing supports the boundary method with these p -norms, as well as other, more general cost functions.

[1]  Joseph Donald Walsh,et al.  The boundary method and general auction for optimal mass transportation and Wasserstein distance computation , 2017 .

[2]  Michael Muskulus,et al.  Fluctuations and determinism of respiratory impedance in asthma and chronic obstructive pulmonary disease. , 2010, Journal of applied physiology.

[3]  Marco Cuturi,et al.  Sinkhorn Distances: Lightspeed Computation of Optimal Transport , 2013, NIPS.

[4]  Dimitri P. Bertsekas,et al.  A generic auction algorithm for the minimum cost network flow problem , 1993, Comput. Optim. Appl..

[5]  W. Gangbo,et al.  The geometry of optimal transportation , 1996 .

[6]  J. A. Cuesta-Albertos,et al.  A characterization for the solution of the Monge--Kantorovich mass transference problem , 1993 .

[7]  Bernhard Schmitzer,et al.  A Sparse Multiscale Algorithm for Dense Optimal Transport , 2015, Journal of Mathematical Imaging and Vision.

[8]  S. Haker,et al.  Optimal mass transport and image registration , 2001, Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision.

[9]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

[10]  Quentin Mérigot,et al.  A Newton algorithm for semi-discrete optimal transport , 2016, ArXiv.

[11]  K. Parthasarathy,et al.  Probability measures on metric spaces , 1967 .

[12]  L. Kantorovich.  On the Translocation of Masses , 2006 .

[13]  Vladimir Oliker,et al.  On the numerical solution of the equation $$\frac{{\partial ^2 z}}{{\partial x^2 }}\frac{{\partial ^2 z}}{{\partial y^2 }} - \left( {\frac{{\partial ^2 z}}{{\partial x\partial y}}} \right)^2 = f$$ and its discretizations, I , 1989 .

[14]  Mathieu Desbrun,et al.  Blue noise through optimal transport , 2012, ACM Trans. Graph..

[15]  Xavier Dupuis.  Semi-discrete principal-agent problem , 2016 .

[16]  Péter Kovács,et al.  Minimum-cost flow algorithms: an experimental evaluation , 2015, Optim. Methods Softw..

[17]  Adam M. Oberman,et al.  Convergent Difference Schemes for Degenerate Elliptic and Parabolic Equations: Hamilton-Jacobi Equations and Free Boundary Problems , 2006, SIAM J. Numer. Anal..

[18]  D. Kinderlehrer,et al.  THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION , 1996 .

[19]  Stanley Osher,et al.  Fast Algorithms for Earth Mover's Distance Based on Optimal Transport and L1 Type Regularization I , 2016 .

[20]  L. D. Prussner,et al.  On the numerical solution of the equation ∂2z/∂x2 ∂2z/∂y2−(∂2z/∂x∂y)2=f and its discretizations. I , 1988 .

[21]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[22]  L. Caffarelli,et al.  On the Numerical Solution of the Problem of Reflector Design with Given Far-Field Scattering Data , 1999 .

[23]  L. Rüschendorf,et al.  Numerical and analytical results for the transportation problem of Monge-Kantorovich , 2000 .

[24]  Adam M. Oberman,et al.  Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge-Ampère Equation in Dimensions Two and Higher , 2010, SIAM J. Numer. Anal..

[25]  V. Oliker,et al.  Optical Design of Single Reflector Systems and the Monge–Kantorovich Mass Transfer Problem , 2003 .

[26]  Dimitri P. Bertsekas,et al.  Network optimization : continuous and discrete models , 1998 .

[27]  J. Barrett,et al.  A MIXED FORMULATION OF THE MONGE-KANTOROVICH EQUATIONS , 2007 .

[28]  G. Carlier.  A general existence result for the principal-agent problem with adverse selection , 2001 .

[29]  S. Rachev,et al.  Mass transportation problems , 1998 .

[30]  A. Pratelli,et al.  On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation , 2007 .

[31]  B. Lévy.  A Numerical Algorithm for L2 Semi-Discrete Optimal Transport in 3D , 2014, 1409.1279.

[32]  Ludger Rüschendorf,et al.  Monge – Kantorovich transportation problem and optimal couplings , 2006 .

[33]  Gabriel Peyré,et al.  Iterative Bregman Projections for Regularized Transportation Problems , 2014, SIAM J. Sci. Comput..

[34]  James B. Orlin,et al.  Max flows in O(nm) time, or better , 2013, STOC '13.

[35]  N. Trudinger,et al.  Regularity of Potential Functions of the Optimal Transportation Problem , 2005 .

[36]  J. Mirebeau.  Discretization of the 3D Monge-Ampere operator, between Wide Stencils and Power Diagrams , 2015, 1503.00947.

[37]  Franz Aurenhammer,et al.  Minkowski-Type Theorems and Least-Squares Clustering , 1998, Algorithmica.

[38]  Quentin Mérigot,et al.  A Multiscale Approach to Optimal Transport , 2011, Comput. Graph. Forum.

[39]  Arthur Cayley,et al.  The Collected Mathematical Papers: On Monge's “Mémoire sur la théorie des déblais et des remblais” , 2009 .

[40]  L. Kantorovich.  On a Problem of Monge , 2006 .

[41]  Franz Aurenhammer,et al.  Minkowski-type theorems and least-squares partitioning , 1992, SCG '92.

[42]  C. Villani.  Topics in Optimal Transportation , 2003 .

[43]  P. Chiappori,et al.  Hedonic price equilibria, stable matching, and optimal transport: equivalence, topology, and uniqueness , 2007 .

[44]  P. Chiappori,et al.  Transition to nestedness in multi- to one-dimensional optimal transport , 2018, European journal of applied mathematics.

[45]  L. Kantorovitch,et al.  On the Translocation of Masses , 1958 .

[46]  Wilfrid Gangbo,et al.  Solution of a Model Boltzmann Equation via Steepest Descent in the 2-Wasserstein Metric , 2004 .

[47]  Adam M. Oberman,et al.  Numerical solution of the Optimal Transportation problem using the Monge-Ampère equation , 2012, J. Comput. Phys..

[48]  Pierre Seppecher,et al.  Mathématiques/Mathematics Shape optimization solutions via Monge-Kantorovich equation , 1997 .

[49]  Quentin Mérigot.  A Comparison of Two Dual Methods for Discrete Optimal Transport , 2013, GSI.

[50]  Cristian E. Gutiérrez,et al.  An iterative method for generated Jacobian equations , 2017 .

[51]  Jun Kitagawa.  An iterative scheme for solving the optimal transportation problem , 2012 .

[52]  Luca Dieci,et al.  General auction method for real-valued optimal transport , 2017, ArXiv.