A Central Limit Theorem for Semidiscrete Wasserstein Distances

Barrio, Eustasio del; González-Sanz, Alberto; Loubes, Jean–Michel

doi:10.48550/arxiv.2105.11721

Cited by 2 publications

(2 citation statements)

References 16 publications

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“…This complements distributional limits for the one-sample estimator Tc,n = T c (μ n , ν) by del Barrio et al (2021). We also derive metric entropy bounds for F c if X is given in terms of the image of sufficiently regular functions on sufficiently nice domains, where we exploit the Lipschitz continuity, semi-concavity, or Hölder continuity of the cost function to obtain novel theoretical guarantees for the convergence rate of Tc,n (Theorems 3.3,3.8 and 3.11).…”

Section: Introductionsupporting

confidence: 57%

Empirical Optimal Transport between Different Measures Adapts to Lower Complexity

Hundrieser¹,

Staudt²,

Munk³

2022

Preprint

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The empirical optimal transport (OT) cost between two probability measures from random data is a fundamental quantity in transport based data analysis. In this work, we derive novel guarantees for its convergence rate when the involved measures are different, possibly supported on different spaces. Our central observation is that the statistical performance of the empirical OT cost is determined by the less complex measure, a phenomenon we refer to as lower complexity adaptation of empirical OT. For instance, under Lipschitz ground costs, we find that the empirical OT cost based on n observations converges at least with rate n −1 d to the population quantity if one of the two measures is concentrated on a d-dimensional manifold, while the other can be arbitrary. For semi-concave ground costs, we show that the upper bound for the rate improves to n −2 d . Similarly, our theory establishes the general convergence rate n −1 2 for semi-discrete OT. All of these results are valid in the two-sample case as well, meaning that the convergence rate is still governed by the simpler of the two measures. On a conceptual level, our findings therefore suggest that the curse of dimensionality only affects the estimation of the OT cost when both measures exhibit a high intrinsic dimension. Our proofs are based on the dual formulation of OT as a maximization over a suitable function class F c and the observation that the c-transform of F c under bounded costs has the same uniform metric entropy as F c itself.

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Section: Introductionsupporting

confidence: 57%

Empirical Optimal Transport between Different Measures Adapts to Lower Complexity

Hundrieser¹,

Staudt²,

Munk³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…. 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.…”

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

confidence: 99%

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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A Central Limit Theorem for Semidiscrete Wasserstein Distances

Cited by 2 publications

References 16 publications

Empirical Optimal Transport between Different Measures Adapts to Lower Complexity

Empirical Optimal Transport between Different Measures Adapts to Lower Complexity

Nonparametric Multiple-Output Center-Outward Quantile Regression

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