Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case
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Semi-discrete optimal transport: a solution procedure for the unsquared Euclidean distance case Valentin Hartmann1,2
· Dominic Schuhmacher1
Received: 22 May 2019 / Revised: 20 December 2019 © The Author(s) 2020
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. Keywords Monge–Kantorovich problem · Spatial resource allocation · Wasserstein metric · Weighted Voronoi tessellation Mathematics Subject Classification Primary 65D18; Secondary 51N20 · 62-09
VH was partially supported by Deutsche Forschungsgemeinschaft RTG 2088. We thank Marcel Klatt for helpful discussions and three anonymous referees for comments that led to an improvement of the paper.
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Dominic Schuhmacher [email protected] Valentin Hartmann [email protected]
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Institute for Mathematical Stochastics, University of Goettingen, Goldschmidtstr. 7, 37077 Goettingen, Germany
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Present Address: IC IINFCOM DLAB, EPFL, Station 14, 1015 Lausanne, Switzerland
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V. Hartmann, D. Schuhmacher
1 Introduction Optimal transport and Wasserstein metrics are nowadays among the major tools for analyzing complex data. Theoretical advances in the last decades characterize existence, uniqueness, representation and smoothness properties of optimal transport plans in a variety of different settings. Recent algorithmic advances (Peyré and Cuturi 2018) make it possible to compute exact transport plans and Wasserstein distances between discrete measures on regular grids of tens of thousands of support points, see e.g. Schmitzer (2016, Sect. 6), and to approximate such distances (to some extent) on larger and/or irregular structures, see Altschuler et al. (2017) and references therein. The development of new methodology for data analysis based on optimal transport is a booming research topic in statistics and machine learning, see e.g. Sommerfeld and Munk (2018), Schmitz et al. (2018), Arjovsky et al. (2017), Genevay et al. (2018), and Flamary et al. (2018). Applications are abundant throughout al
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