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Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case

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  • Valentin Hartmann

    (University of Goettingen
    EPFL)

  • Dominic Schuhmacher

    (University of Goettingen)

Abstract

We consider the problem of finding an optimal transport plan between an absolutely continuous measure and a finitely supported measure of the same total mass when the transport cost is the unsquared Euclidean distance. We may think of this problem as closest distance allocation of some resource continuously distributed over Euclidean space to a finite number of processing sites with capacity constraints. This article gives a detailed discussion of the problem, including a comparison with the much better studied case of squared Euclidean cost. We present an algorithm for computing the optimal transport plan, which is similar to the approach for the squared Euclidean cost by Aurenhammer et al. (Algorithmica 20(1):61–76, 1998) and Mérigot (Comput Graph Forum 30(5):1583–1592, 2011). We show the necessary results to make the approach work for the Euclidean cost, evaluate its performance on a set of test cases, and give a number of applications. The later include goodness-of-fit partitions, a novel visual tool for assessing whether a finite sample is consistent with a posited probability density.

Suggested Citation

  • Valentin Hartmann & Dominic Schuhmacher, 2020. "Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 133-163, August.
    • Handle: RePEc:spr:mathme:v:92:y:2020:i:1:d:10.1007_s00186-020-00703-z
    DOI: 10.1007/s00186-020-00703-z
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    References listed on IDEAS

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    1. Hundrieser, Shayan & Mordant, Gilles & Weitkamp, Christoph A. & Munk, Axel, 2024. "Empirical optimal transport under estimated costs: Distributional limits and statistical applications," Stochastic Processes and their Applications, Elsevier, vol. 178(C).

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