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

Hartmann, Valentin; Schuhmacher, Dominic

doi:10.1007/s00186-020-00703-z

Cited by 22 publications

(21 citation statements)

References 44 publications

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“…when one of both probabilities is supported on a discrete set. Such a problem has been studied in many contexts, including on resource allocation problem, points versus demand distribution, positions of sites such that the mean allocation cost is minimal ( [Hartmann and Schuhmacher, 2020]), resolution of the incompressible Euler equation using Lagrangian methods ( [Gallouët and Mérigot, 2018]), non-imaging optics; matching between a point cloud and a triangulated surface; seismic imaging ( [Meyron, 2019]), generation of blue noise distributions with applications for instance to low-level hardware implementation in printers ( [de Goes et al, 2012]), in astronomy ( [Lévy et al, 2020]). But in a more statistical point of view it can also be used to implement Goodness-of-fit-tests, in detecting deviations from a density map to have P = Q, by using the fluctuations of W(P n , Q), see [Hartmann and Schuhmacher, 2020] and to the new transport based generalization of the distribution function, proposed by [del Barrio et al, 2020], when the probability is discrete.…”

Section: Introductionmentioning

confidence: 99%

A Central Limit Theorem for Semidiscrete Wasserstein Distances

Barrio¹,

González-Sanz²,

Loubes³

2021

Preprint

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We address the problem of proving a Central Limit Theorem for the empirical optimal transport cost, √ n{Tc(Pn, Q) − Wc(P, Q)}, in the semi discrete case, i.e when the distribution P is finitely supported. We show that the asymptotic distribution is the supremun of a centered Gaussian process which is Gaussian under some additional conditions on the probability Q and on the cost. Such results imply the central limit theorem for the p-Wassertein distance, for p ≥ 1. Finally, the semidiscrete framework provides a control on the second derivative of the dual formulation, which yields the first central limit theorem for the optimal transport potentials.

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Section: Introductionmentioning

confidence: 99%

A Central Limit Theorem for Semidiscrete Wasserstein Distances

Barrio¹,

González-Sanz²,

Loubes³

2021

Preprint

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“…In recent years, semi-discrete optimal transport theory has seen significant expansion in its theoretical foundations (see [5,15,17,30,32,36,38,[42][43][44][45]). It has also been applied to many diverse problems in the sciences, both within fluid dynamics [28,37] and elsewhere such as materials science [6,7,33], economics [27,Chapter 5], crowd dynamics [34] and image interpolation [36].…”

Section: Sg In Geostrophic Coordinates and Semi-discrete Optimal Tran...mentioning

confidence: 99%

Semi-discrete optimal transport methods for the semi-geostrophic equations

et al. 2022

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We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.

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“…Moreover, ν ε is absolutely continuous with respect to the Lebesgue measure. Thus, according to Hartmann and Schuhmacher (2020), there exists w ε = (w ε1 , . .…”

Section: Kmentioning

confidence: 99%

Optimal 1-Wasserstein Distance for WGANs

Stéphanovitch¹,

Tanielian²,

Cadre³

et al. 2022

Preprint

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The mathematical forces at work behind Generative Adversarial Networks raise challenging theoretical issues. Motivated by the important question of characterizing the geometrical properties of the generated distributions, we provide a thorough analysis of Wasserstein GANs (WGANs) in both the finite sample and asymptotic regimes. We study the specific case where the latent space is univariate and derive results valid regardless of the dimension of the output space. We show in particular that for a fixed sample size, the optimal WGANs are closely linked with connected paths minimizing the sum of the squared Euclidean distances between the sample points. We also highlight the fact that WGANs are able to approach (for the 1-Wasserstein distance) the target distribution as the sample size tends to infinity, at a given convergence rate and provided the family of generative Lipschitz functions grows appropriately. We derive in passing new results on optimal transport theory in the semi-discrete setting.

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

Cited by 22 publications

References 44 publications

A Central Limit Theorem for Semidiscrete Wasserstein Distances

A Central Limit Theorem for Semidiscrete Wasserstein Distances

Semi-discrete optimal transport methods for the semi-geostrophic equations

Optimal 1-Wasserstein Distance for WGANs

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