Initialization Procedures for Discrete and Semi-Discrete Optimal Transport

Meyron, Jocelyn

doi:10.1016/j.cad.2019.05.037

Cited by 7 publications

(4 citation statements)

References 27 publications

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“…when one of both probabilities is supported on a discrete set. Such a problem is inspired by a large variety of applications, including resource allocation problem, points versus demand distribution, positions of sites such that the mean allocation cost is minimal ( [18]), resolution of the incompressible Euler equation using Lagrangian methods ( [11]), non-imaging optics; matching between a point cloud and a triangulated surface; seismic imaging( [25]), generation of blue noise distributions with applications for instance to low-level hardware implementation in printers( [5]), in astronomy ( [22]). From a statistical point of view, Goodness-of-fit-tests based on semi-discrete optimal transport enable to detect deviations from a density map to have P = Q, by using the fluctuations of W(P n , Q), see [18] and to provide a new generalization of distribution functions and quantile, proposed for instance by [17], when the probability is discrete.…”

Section: Introductionmentioning

confidence: 99%

Central Limit Theorems for Semidiscrete Wasserstein Distances

Barrio¹,

González-Sanz²,

Loubes³

2022

Preprint

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We prove a Central Limit Theorem for the empirical optimal transport cost, nm n+m {Tc(Pn, Qm)− Tc(P, Q)}, in the semi discrete case, i.e when the distribution P is supported in N points, but without assumptions on Q. 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. This means that, for fixed N , the curse of dimensionality is avoided. To better understand the influence of such N , we provide bounds of E|W1(P, Qm) − W1(P, Q)| depending on m and N . 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. The results are supported by simulations that help to visualize the given limits and bounds. We analyse also the cases where classical bootstrap works.

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Section: Introductionmentioning

confidence: 99%

Central Limit Theorems for Semidiscrete Wasserstein Distances

Barrio¹,

González-Sanz²,

Loubes³

2022

Preprint

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“…From a numerical point of view, however, this would lead to the computation of a semi-discrete optimal transportation plan, which has complexity O(n d/2 ), hence is unfeasible even for moderate d. While the computational complexity of our procedure does not depend on the dimension, its statistical performance does (see Fournier and Guillin (2015)) and, in that sense, we do not escape the curse of dimensionality-up to the case where P is finitely supported, see del Barrio, González-Sanz and Loubes (2021). Despite the fact that the literature on the computation of such maps is growing quite fastly (see Lévy, Mohayaee and von Hausegger (2020); Gallouët and Mérigot (2018); Meyron (2019); de Goes et al ( 2012)), the exisiting methods are restricted to dimension two, sometimes three. A further issue is that the solution of the semi-discrete problem does not produce quantile contours but creates a Voronoi tessellation of S d , each piece of which is mapped to a single sample point.…”

Section: Relation To the Recent Literature On Numerical Optimal Trans...mentioning

confidence: 92%

Nonparametric Multiple-Output Center-Outward Quantile Regression

Barrio¹,

Sanz²,

Hallin³

2022

Preprint

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Based on the novel concept of multivariate center-outward quantiles introduced recently in Chernozhukov et al. (2017) and Hallin et al. ( 2021), we are considering the problem of nonparametric multiple-output quantile regression. Our approach defines nested conditional center-outward quantile regression contours and regions with given conditional probability content irrespective of the underlying distribution; their graphs constitute nested center-outward quantile regression tubes. Empirical counterparts of these concepts are constructed, yielding interpretable empirical regions and contours which are shown to consistently reconstruct their population versions in the Pompeiu-Hausdorff topology. Our method is entirely non-parametric and performs well in simulations including heteroskedasticity and nonlinear trends; its power as a data-analytic tool is illustrated on some real datasets.

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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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Initialization Procedures for Discrete and Semi-Discrete Optimal Transport

Cited by 7 publications

References 27 publications

Central Limit Theorems for Semidiscrete Wasserstein Distances

Central Limit Theorems for Semidiscrete Wasserstein Distances

Nonparametric Multiple-Output Center-Outward Quantile Regression

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

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