A Multiscale Approach to Optimal Transport

In this paper, we propose an improvement of an algorithm of Aurenhammer, Hoffmann and Aronov to find a least square matching between a probability density and finite set of sites with mass constraints, in the Euclidean plane. Our algorithm exploits the multiscale nature of this optimal transport problem. We iteratively simplify the target using Lloyd's algorithm, and use the solution of the simplified problem as a rough initial solution to the more complex one. This approach allows for fast estimation of distances between measures related to optimal transport (known as Earth‐mover or Wasserstein distances). We also discuss the implementation of these algorithms, and compare the original one to its multiscale counterpart.

[1]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[2]  D. Bertsekas.  The auction algorithm: A distributed relaxation method for the assignment problem , 1988 .

[3]  Y. Brenier.  Polar Factorization and Monotone Rearrangement of Vector-Valued Functions , 1991 .

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

[5]  David J. Thuente,et al.  Line search algorithms with guaranteed sufficient decrease , 1994, TOMS.

[6]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

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

[8]  Yann Brenier,et al.  A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , 2000, Numerische Mathematik.

[9]  Steven Haker,et al.  Minimizing Flows for the Monge-Kantorovich Problem , 2003, SIAM J. Math. Anal..

[10]  V. Oliker.  Mathematical Aspects of Design of Beam Shaping Surfaces in Geometrical Optics , 2003 .

[11]  Leonidas J. Guibas,et al.  The Earth Mover's Distance as a Metric for Image Retrieval , 2000, International Journal of Computer Vision.

[12]  Édouard Oudet,et al.  Minimizing within Convex Bodies Using a Convex Hull Method , 2005, SIAM J. Optim..

[13]  G. Loeper,et al.  Numerical Analysis/Partial Differential Equations Numerical solution of the Monge-Ampère equation by a Newton's algorithm , 2005 .

[14]  Pierre Alliez,et al.  Computational geometry algorithms library , 2008, SIGGRAPH '08.

[15]  C. Villani.  Optimal Transport: Old and New , 2008 .

[16]  O. Deussen,et al.  Capacity-constrained point distributions: a variant of Lloyd's method , 2009, SIGGRAPH 2009.

[17]  O. Deussen,et al.  Capacity-constrained point distributions: a variant of Lloyd's method , 2009, ACM Trans. Graph..

[18]  Pierre Alliez,et al.  Filtering Relocations on a Delaunay Triangulation , 2009, Comput. Graph. Forum.

[19]  Pierre Alliez,et al.  Signing the Unsigned: Robust Surface Reconstruction from Raw Pointsets , 2010, Comput. Graph. Forum.

[20]  Julie Delon,et al.  Fast Transport Optimization for Monge Costs on the Circle , 2009, SIAM J. Appl. Math..

[21]  Pierre Alliez,et al.  An Optimal Transport Approach to Robust Reconstruction and Simplification of 2d Shapes , 2022 .

[22]  I. Daubechies,et al.  Conformal Wasserstein distances: Comparing surfaces in polynomial time , 2011, 1103.4408.

[23]  Mathieu Desbrun,et al.  HOT: Hodge-optimized triangulations , 2011, SIGGRAPH 2011.

[24]  Frédéric Chazal,et al.  Geometric Inference for Probability Measures , 2011, Found. Comput. Math..