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 pro...

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Vydáno v:Computer graphics forum Ročník 30; číslo 5; s. 1583 - 1592
Hlavní autor: Merigot, Quentin
Médium: Journal Article
Jazyk:angličtina
Vydáno: Oxford, UK Blackwell Publishing Ltd 01.08.2011
Wiley
Témata:
Algorithms
and systems
Computational Geometry
Computer Science
Density
I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms
I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms, languages, and systems
languages
Least squares method
Mathematical analysis
Mathematical models
Numerical Analysis
Optimization
Planes
Studies
Transport
Transportation problem (Operations research)
optimal transport
convex programming
Power diagram
wasserstein distance
ISSN:0167-7055, 1467-8659
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Shrnutí: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.
Bibliografie:istex:ABA8CCD6A85D97390A94FDCE35E56624C358F707
ark:/67375/WNG-H13Q3HG2-M
ArticleID:CGF2032
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ISSN:0167-7055
1467-8659
DOI:10.1111/j.1467-8659.2011.02032.x