A note on the regularity of optimal-transport-based center-outward distribution and quantile functions

Barrio, Eustasio del; González-Sanz, Alberto; Hallin, Marc

doi:10.1016/j.jmva.2020.104671

Cited by 23 publications

(10 citation statements)

References 41 publications

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“…But note that such probability measure does not satisfy the regularity condition of continuous density over a convex set, in consequence some additional work should be done which is left as a future work. In the same way as the regularity of the transport can be derived in the continuous case by a careful treatment of the Monge-Ampére equation, see [del Barrio et al, 2020], we conjecture that Theorem 4.2 could hold also in that discrete framework.…”

Section: A Central Limit Theorem For the Potentialsmentioning

confidence: 55%

“…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%

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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: A Central Limit Theorem For the Potentialsmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 99%

A Central Limit Theorem for Semidiscrete Wasserstein Distances

Barrio¹,

González-Sanz²,

Loubes³

2021

Preprint

Self Cite

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“…This choice introduces a complication in the regularity theory of the transport maps, as the density of P S is unbounded at the center zero. Chernozhukov et al [3] require a homeomorphism condition (C) for the validity of Theorem A.2, which is studied in Figalli [10] and del Barrio et al [1]. Alternatively, it would be possible to avoid this difficulty by replacing the spherical P S by, e.g., the uniform distribution on the unit ball: the latter is also absolutely continuous, with the same bounded convex support, but without a singularity in its density.…”

Section: Remarkmentioning

confidence: 99%

Functional, randomized and smoothed multivariate quantile regions

Faugeras

Rüschendorf

2021

Journal of Multivariate Analysis

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The mass transportation approach to multivariate quantiles in Chernozhukov et al. [3] was modified in Faugeras and Rüschendorf [8] by a two steps procedure. In the first step, a mass transportation problem from a spherical reference measure to the copula is solved and combined in the second step with a marginal quantile transformation in the sample space. Also, generalized quantiles given by suitable Markov morphisms are introduced there.In the present paper, this approach is further extended by a functional approach in terms of membership functions, and by the introduction of randomized quantile regions. In addition, in the case of continuous marginals, a smoothed version of the empirical quantile regions is obtained by smoothing the empirical copula. All three extended approaches give empirical quantile ares of exact level and improved stability. The resulting depth areas give a valid representation of the central quantile areas of a multivariate distribution and provide a valuable tool for their analysis.

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“…More general cases are studied in del Barrio et al (2020), but require more cautious and less intuitive definitions which, for the sake of simplicity, we do not consider here. Denote by U d the spherical uniform distribution over S d , that is, the product of a uniform measure over the hypersphere S d−1 and a uniform over the unit interval of distances to the origin.…”

Section: Center-outward Ranks and Signsmentioning

confidence: 99%

“…The requirement that f has support R d can be relaxed to a requirement of a convex support (see delBarrio et al (2020)) at the expense, however, of a less direct definition of center-outward distribution and quantile functions. For the sake of simplicity, we are sticking to the assumption made here.…”

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confidence: 99%

Rank-Based Testing for Semiparametric VAR Models: a measure transportation approach

Hallin¹,

Vecchia²,

Liu³

2020

Preprint

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We develop a class of tests for semiparametric vector autoregressive (VAR) models with unspecified innovation densities, based on the recent measure-transportation-based concepts of multivariate center-outward ranks and signs. We show that these concepts, combined with Le Cam's asymptotic theory of statistical experiments, yield novel testing procedures, which (a) are valid under a broad class of innovation densities (possibly non-elliptical, skewed, and/or with infinite moments), (b) are optimal (locally asymptotically maximin or most stringent) at selected ones, and (c) are robust against additive outliers. In order to do so, we establish a Hájek asymptotic representation result, of independent interest, for a general class of center-outward rank-based serial statistics. As an illustration, we consider the problems of testing the absence of serial correlation in multiple-output and possibly non-linear regression (an extension of the classical Durbin-Watson problem) and the sequential identification of the order p of a vector autoregressive (VAR(p)) model. A Monte Carlo comparative study of our tests and their routinely-applied Gaussian competitors demonstrates the benefits (in terms of size, power, and robustness) of our methodology; these benefits are particularly significant in the presence of asymmetric and leptokurtic innovation densities. A real data application concludes the paper.

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A note on the regularity of optimal-transport-based center-outward distribution and quantile functions

Cited by 23 publications

References 41 publications

A Central Limit Theorem for Semidiscrete Wasserstein Distances

A Central Limit Theorem for Semidiscrete Wasserstein Distances

Functional, randomized and smoothed multivariate quantile regions

Rank-Based Testing for Semiparametric VAR Models: a measure transportation approach

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