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.
Highlights
Optimal transport and Wasserstein metrics are nowadays among the major tools for analyzing complex data
It is sometimes considered problematic that optimal transport plans for p = 1 are in general not unique
It will be desirable to know in what way we may approximate the continuous and discrete Monge–Kantorovich problems by the semi-discrete problem we investigate here
Summary
Optimal transport and Wasserstein metrics are nowadays among the major tools for analyzing complex data. In the semi-discrete setting we can represent a solution to Monge’s problem as a partition of Rd , where each cell is the pre-image of a support point of ν under the optimal transport map. 2.3 of Crippa et al (2009) and independently in Geiß et al (2013), which both treat more general cost functions, that an optimal transport partition always exists, is essentially unique and takes the form of a weighted Voronoi tessellation, or more precisely an Apollonius diagram. We extend this result somewhat within the case p = 1 in Theorems 1 and 2 below. This will be further substantiated in the application Sect. 6
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