Pith. sign in

REVIEW 3 major objections 8 minor 2 cited by

Distributional Limit Theory for Optimal Transport

T0 review · 3 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves a new central limit theorem for the empirical 1-Wasserstein cost on the real line under finite mean and variance.

desk verdict A genuinely useful survey of OT limit theory with one new p=1 CLT, but the central theorem's display has a load-bearing typo that must be fixed. read the letter →

arxiv 2505.19104 v2 pith:Z5S7MMON submitted 2025-05-25 math.ST stat.TH

Eustasio del Barrio , Alberto González-Sanz , Jean-Michel Loubes , David Rodríguez-Vítores This is my paper
classification math.STstat.TH MSC 62G0562R1062G30
keywords optimaltransportWassersteindistancecentrallimittheoremempiricalmeasureBrownianbridgeEfron-Steininequalityquantileprocessdistributionaltheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper is a review of distributional limit theory for optimal transport, and it also establishes a new result that fills a gap in the one-dimensional theory. For the absolute-value cost p=1, the empirical optimal transport cost, centered at its own expectation, converges weakly to a non-Gaussian limit whenever the reference measure Q has finite mean and the sample measure P has finite variance. Previously, such fluctuation central limit theorems were known only for p>1 in this generality. The paper also assembles the known limit theorems across dimensions, discrete and semi-discrete settings, regularized optimal transport, and sliced Wasserstein distances, and it closes with a list of open problems.

What carries the argument

The load-bearing identity is T1(P,Q) = ∫_R |F(x)-G(x)| dx, expressing the 1-Wasserstein cost as the L1 distance between distribution functions, which reduces the cost fluctuation to a functional of the empirical process alpha_n(x)=sqrt(n)(F_n(x)-F(x)). The proof combines weak convergence of this process to B∘F on bounded intervals, an Efron--Stein variance bound Var(T1(Pn,Q)) ≤ σ²(P)/n, and a truncation argument with the 'classical 3ε argument' to pass from bounded intervals to the whole real line.

What would settle it

Take P=Q=Uniform[0,1] and compute n Var(T1(Pn,P)) for large n; if it does not converge to the variance of ∫$_0^{1}$ |B(t)| dt, where B is a Brownian bridge, then the CLT fails.

Watch

Extended reading notes

Core claim

The paper's central new claim is Theorem 2.1 for p=1 in dimension one: if Q has finite mean and P has finite variance, then sqrt(n)(T1(Pn,Q) - E[T1(Pn,Q)]) converges weakly to the random variable gamma(P,Q) = ∫_R (v_{F,G}(x) - E[v_{F,G}(x)]) dx, where v_{F,G}(x) = sgn(F(x)-G(x)) B(F(x)) 1_{F(x)≠G(x)} + |B(F(x))| 1_{F(x)=G(x)} and B is a standard Brownian bridge. The limiting distribution is Gaussian if and only if the set {F=G} has zero Lebesgue measure. The p>1 case of the same theorem was already known from previous work, and the paper states that the p=1 case in this generality is new.

Load-bearing premise

The new p=1 limit rests on an unstated '3ε argument' that passes from convergence on bounded intervals to convergence on the whole real line; if that passage cannot be supplied, the central claim is not established.

Editorial extensions

If this is right

  • For p=1 on the real line, fluctuation central limit theorems hold under minimal moment conditions: finite mean for Q and finite variance for P, with no compact support or light-tail assumption.
  • Non-Gaussian limits are generic for p=1: whenever F=G on a set of positive Lebesgue measure, the limiting distribution is not Gaussian, in sharp contrast to the p>1 case.
  • Under the additional condition J1(P)<∞ and ℓ(F=G)=0, the centering E[T1(Pn,Q)] can be replaced by T1(P,Q), yielding asymptotically valid confidence intervals for the population cost.
  • The Efron--Stein bound on the variance of T1(Pn,Q) holds in every dimension, giving stochastic boundedness of fluctuations and a template for higher-dimensional fluctuation limit theorems.
  • The review provides a map of which optimal transport objects (costs, plans, maps, potentials) admit distributional limits in which settings, and it identifies open problems such as the higher-dimensional p=1 fluctuation question and the conjecture about empirical optimal plans for distributions with non-unique transports.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The truncation technique used for the p=1 proof likely extends to higher dimensions under an integrability condition analogous to ∫ sqrt(F(x)(1-F(x))) dx < ∞, potentially answering the paper's open Problem 1 with a Gaussian limit in some cases.
  • The non-Gaussian limit gamma(P,Q) is an L1 functional of the empirical process; when P=Q it does not degenerate, so it could yield a goodness-of-fit test that remains non-trivial in the null case where Gaussian fluctuation limits vanish.
  • If the companion preprint [101] supporting Theorem 2.2 for p>1 is correct, then the paper provides the first confidence-interval construction for one-dimensional Wasserstein distances under purely moment-based assumptions, without compact support.
  • The unstated '3ε argument' in the supplement could be made explicit by proving convergence of moments of the truncated integrals; a formal proof would remove the only gap in the p=1 claim.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 8 minor

Summary. The paper is a comprehensive review of distributional limit theory for empirical optimal transport, covering one-dimensional CLTs, general dimensions, discrete and semi-discrete settings, entropic and smooth regularized OT, sliced Wasserstein distances, applications, and open problems. The only original contribution claimed by the authors is the p=1 case of Theorem 2.1: in one dimension, under the assumptions that Q has finite mean and P has finite variance, the fluctuation sqrt(n)(T_1(P_n,Q)-E[T_1(P_n,Q)]) converges weakly to a centered random variable gamma(P,Q) defined through equations (9) and (10). The proof is sketched in the supplementary material, using an Efron-Stein variance bound (Lemma 8.1) and convergence of the empirical process in L^1 on compact intervals. The remaining theorems are presented as reviews or adaptations of known results, with several proofs deferred to the supplementary material or to other papers.

Significance. If the p=1 result is correct as intended, it is a genuine and useful extension of the existing univariate CLT literature, which previously covered the case P=Q or required stronger integrability. The review brings together a large number of recent results in a structured way, and the open problems section is a valuable addition. The paper also includes a helpful discussion of proof techniques, especially the Efron-Stein linearization. However, the central new theorem is presented with a typo in its defining display, and one of the supporting theorems relies on an unpublished preprint; these issues must be addressed before the contribution can be considered solid.

major comments (3)
  1. [Section 2.1, Eq. (10)] The display defining v_{F,G} is internally inconsistent and makes Theorem 2.1 false as printed. Both summands carry the indicator I(F(x)!=G(x)), so when the Lebesgue measure of {F=G} is zero, the second term contributes the non-Gaussian quantity integral |B(F(x))| dx, contradicting the theorem's assertion that gamma(P,Q) is Gaussian in that case. The supplementary proof (display following (41)) defines v^{(1)}_{F,G}=|B∘F|·1_{F=G} and v^{(2)}_{F,G}=sgn(F-G)B∘F·1_{F!=G}, so the theorem statement should have the second term multiplied by I(F(x)=G(x)). The fix is a one-character change, but the error is load-bearing because Theorem 2.1 is the paper's only new result.
  2. [Section 2.2, Theorem 2.2 and supplementary material] The main text states that details of the proof of Theorem 2.2 are given in the supplementary material, but the supplement says only that the p>1 case is considered in [101], an arXiv preprint by the same authors. Since Theorem 2.2 is the basis for replacing E[T_p(P_n,Q)] with T_p(P,Q) and hence for the confidence intervals discussed in Section 2.2, the p>1 part is load-bearing for the paper's inferential claims. Either include the proof in the supplement or state explicitly that this part is deferred to a not-yet-published manuscript; citing an unpublished preprint is not sufficient support for a theorem stated as part of this paper.
  3. [Supplement, proof of Theorem 2.1] The proof concludes with 'a classical 3epsilon argument' after establishing convergence of truncated integrals and moment convergence. This step is what delivers the full limit from the truncated ones and is central to the paper's only new theorem; it should be spelled out, showing how (42), the L2 convergence of the truncated integrals as M grows, and the uniform tightness of sqrt(n)(T_1(P_n,Q)-E[T_1(P_n,Q)]) combine to yield (8). Leaving this to a 'classical' argument is too vague for a central proof.
minor comments (8)
  1. [Abstract] The phrase 'underlying the some of the applications' should be 'underlying some of the applications'.
  2. [Introduction] There are typos such as 'smootness' (should be 'smoothness') and the duplicated phrase 'goodness-of-fit problems in goodness-of-fit problems' in the second paragraph.
  3. [Section 2.1] The word 'satisfes' should be 'satisfies' in the remark following Theorem 2.1.
  4. [Section 2.3] The phrase 'version of (2.3)' refers to a nonexistent equation label; it should refer to the display following Theorem 2.3 or to equation (16).
  5. [Section 3.5.1] In the Hessian formula, 'Lagk(z)∩Lagk(z)' appears to be a typo for 'Lag_i(z)∩Lag_j(z)' with i≠j, and the condition 'n̸=j' should presumably be 'i≠j'; as written the expression is unintelligible.
  6. [Section 4.1] The phrase 'Kulback-Leiber divergence' should be 'Kullback-Leibler divergence'.
  7. [Section 4.2.1] The word 'Sovolev' should be 'Sobolev' in the statement of Theorem 4.6.
  8. [Section 2.2] There is a typo 'appproximate' in the displayed confidence interval; it should be 'approximate'.

Circularity Check

1 steps flagged · score 4.0 of 10

Secondary self-citation carries the centering theorem; the central new p=1 CLT is independently proved.

  1. self citation load bearing [Section 2.2, Theorem 2.2; Supplement, Proof of Theorem 2.2]
    "Proof of Theorem 2.2.The casep> 1 is considered in [101]."

    The main text promises that the proof of Theorem 2.2 is given in the supplementary material, but the supplement's proof paragraph handles only the p=1 case and defers the p>1 centering result to [101], an arXiv preprint by the same authors (Rodríguez-Vítores, del Barrio, Loubes). The p>1 half of Theorem 2.2 is what allows the paper to replace E[Tp(Pn,Q)] by Tp(P,Q) and to build asymptotic confidence intervals, so that claim rests on a same-author citation rather than on a proof supplied in this paper. This self-citation is load-bearing for the centering discussion, although it does not support the paper's only original theorem: the p=1 case of Theorem 2.1 is proved in the supplement using independently published results [26,31].

full rationale

The paper is largely a review, and its only original theorem is the p=1 case of Theorem 2.1. The supplement's proof of that case is not circular: it starts from weak convergence of the empirical process to B∘F, citing the published Theorem 2.1 in [26], combines it with the Efron-Stein variance bound in Lemma 8.1, and then uses dominated convergence and a truncation argument; it does not assume the conclusion. The p>1 case of Theorem 2.1 is explicitly attributed to the published [31]. The one genuinely load-bearing self-citation is in Theorem 2.2: the main text says details are in the supplementary material, but the supplement says the p>1 case is considered in [101], an arXiv preprint by the same authors. This carries the centering theorem used for confidence-interval applications, but it is not the paper's new p=1 result, which remains independently derived; hence the score is 4 rather than higher. Separately, the printed formula (10) is internally inconsistent: both indicators read I(F(x)≠G(x)), whereas the supplement's decomposition uses |B∘F(x)|I(F(x)=G(x)) for the absolute-value term; as printed, γ would not be Gaussian when ℓ(F=G)=0. This is a statement-level error and a correctness risk, not a circularity. The supplement also omits details in the final 'classical 3ε argument' used to pass from truncated integrals to the full limit, but that is an omitted technical detail rather than circular reasoning. No fitted parameter is renamed as a prediction, no ansatz is smuggled in through citation, and no uniqueness theorem is imported from the authors' prior work to force a conclusion.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities appear: this is a survey. The axioms listed are the standard background facts and cited theorems on which the survey and the new p=1 proof rely. The most fragile external input is the unpublished self-cited preprint [101] used for Theorem 2.2.

assumptions (6)
  • standard math Optimal transport duality and Monge formulation for costs cp: primal (1), dual (2), and map representation (3) hold for the measures and costs considered.
    Invoked throughout Section 1; cited to Villani [113, Theorem 5.10] and [114].
  • standard math Efron-Stein inequality bounds the variance of Tp(Pn,Q) as O(1/n) under moment assumptions, as in Lemma 8.1.
    Used in Section 1 and in the proof of Theorem 2.1; attributed to [11] and [28,33].
  • domain assumption One-dimensional empirical process sqrt(n)(Fn-F) converges weakly to B composed with F in L1 over finite intervals, and tail terms vanish under finite second moment; condition (11) gives almost sure L1 trajectories.
    The new p=1 fluctuation CLT in the supplement builds on this, following [26]; condition (11) is discussed after Theorem 2.1.
  • standard math Empirical OT potentials fn converge to f0 almost surely and in L2(P), with a variance bound nVar(Tp(Pn,Q) - integral f0 dPn) bounded by c E[(fn(X1)-f0(X1))^2].
    Sketch of Theorem 3.1 in Section 3.1; results quoted from [33] and [28].
  • standard math For entropy-regularized OT, the Schrodinger system (27) has unique solutions, the linearized operator I+A is invertible on the quotient by constants, and the empirical potentials lie in a Donsker class.
    Theorems 4.2 to 4.5 rely on these facts, stated in Section 4.1 and cited to [59,58,90,48,14].
  • domain assumption The linearized Monge-Ampere equation on the flat torus has a bounded inverse under lambda I <= grad^2 phi <= Lambda I, and the kernel density estimator satisfies the estimates needed for the CLT in Theorem 4.8.
    Section 4.2.2 quotes Theorem 4.8 from [86] and sketches the linearization; this is a strong regularity assumption, not proved in the paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Distributional Limit Theory for Optimal Transport." pith.science (2026). https://pith.science/paper/Z5S7MMON

@misc{pith2026250519104,
  author       = {Pith},
  title        = {Pith review of: Distributional Limit Theory for Optimal Transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z5S7MMON}},
  note         = {Machine review of arXiv:2505.19104}
}
read the original abstract

Optimal Transport (OT) is a resource allocation problem with applications in biology, data science, economics and statistics, among others. In some of the applications, practitioners have access to samples which approximate the continuous measure. Hence the quantities of interest derived from OT -- plans, maps and costs -- are only available in their empirical versions. Statistical inference on OT aims at finding confidence intervals of the population plans, maps and costs. In recent years this topic gained an increasing interest in the statistical community. In this paper we provide a comprehensive review of the most influential results on this research field, underlying the some of the applications. Finally, we provide a list of open problems.

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Empirical optimal transport potentials: fast rates and a functional central limit theorem

    math.ST 2026-08 accept novelty 8.0 of 10

    Empirical Brenier potentials converge in L1(μ) at rate n^{-1/2} for d≤3, n^{-1/2} log^{5/2} n for d=4, and n^{-2/d} log^{(d+2)/d} n for d≥5, with sharp polynomial exponents, an FCLT and consistent bootstrap for d≤3.

  2. Sharp Asymptotics for Regularized Optimal Transport

    math.AP 2026-07 conditional novelty 7.0 of 10

    Sharp small-regularization asymptotics (first-order for EOT, matching-order for p-ROT with 1<p<∞) are established under mild moment/regularity assumptions via a unified quantization-based construction.

Reference graph

Works this paper leans on

120 extracted references · 66 canonical work pages · cited by 2 Pith papers

  1. [101]

    An improved central limit theorem for the empirical sliced Wasserstein distance

    D. Rodríguez-Vítores, E. del Barrio, and J.-M. Loubes. An improved cen- tral limit theorem for the empirical sliced wasserstein distance.arXiv preprint arXiv:2503.18831, 2025

    work page Pith review arXiv 2025

  2. [1]

    Bachoc, L

    F. Bachoc, L. Béthune, A. Gonzalez-Sanz, and J.-M. Loubes. Gaussian processes on distributions based on regularized optimal transport. InProceedings of The 26th International Conference on Artificial Intelligence and Statistics, volume 206 of Proceedings of Machine Learning Research, pages 4986–5010. PMLR, 25–27 Apr 2023. 32 Distributional Limit Theory fo...

  3. [2]

    Bachoc, L

    F. Bachoc, L. Béthune, A. González-Sanz, and J.-M. Loubes. Improved learning theoryforkerneldistributionregressionwithtwo-stagesampling. arXiv:2308.14335, 2023

    arXiv 2023

  4. [3]

    Balakrishnan and T

    S. Balakrishnan and T. Manole. Stability bounds for smooth optimal transport maps and their statistical implications.arXiv:2502.12326, 2025

    arXiv 2025

  5. [4]

    Bayraktar, S

    E. Bayraktar, S. Eckstein, and X. Zhang. Stability and sample complexity of divergence regularized optimal transport.Bernoulli, 31(1):213–239, 2025

    2025

  6. [5]

    B. K. Beare and T. Kaji. Necessary and sufficient conditions for convergence in distribution of quantile and p-p processes inl1(0, 1), 2025

    2025

  7. [6]

    Blondel, V

    M. Blondel, V. Seguy, and A. Rolet. Smooth and sparse optimal transport. volume 84 ofProceedings of Machine Learning Research, pages 880–889, 2018

    2018

  8. [7]

    S. G. Bobkov and M. Ledoux. One-dimensional empirical measures, order statistics, and kantorovich transport distances.Memoirs of the American Mathematical Society, 2019

    2019

Show all 120 references
  1. [8]

    Boissard, T

    E. Boissard, T. Le Gouic, and J.-M. Loubes. Distribution’s template estimate with wasserstein metrics.Bernoulli, 21(2), 2015

    2015

  2. [9]

    Bonneel and J

    N. Bonneel and J. Digne. A survey of optimal transport for computer graphics and computer vision.Computer Graphics Forum, 42:439–460, 2023

    2023

  3. [10]

    Bonneel, J

    N. Bonneel, J. Rabin, G. Peyré, and H. Pfister. Sliced and radon wasserstein barycenters of measures.Journal of Mathematical Imaging and Vision, 51:22 – 45, 2014

    2014

  4. [11]

    Boucheron, G

    S. Boucheron, G. Lugosi, and P. Massart.Concentration inequalities. Oxford University Press, Oxford, 2013. A nonasymptotic theory of independence

    2013

  5. [12]

    Cárcamo, A

    J. Cárcamo, A. Cuevas, and L.-A. Rodríguez. Directional differentiability for supremum-type functionals: Statistical applications.Bernoulli, 26:2143 – 2175, 2020

    2020

  6. [13]

    G. Carlier. On the linear convergence of the multimarginal sinkhorn algorithm. SIAM Journal on Optimization, 32(2):786–794, 2022

    2022

  7. [14]

    Carlier and M

    G. Carlier and M. Laborde. A differential approach to the multi-marginal schrödinger system. SIAM Journal on Mathematical Analysis, 52(1):709–717, 2020

    2020

  8. [15]

    Cazelles, V

    E. Cazelles, V. Seguy, J. Bigot, M. Cuturi, and N. Papadakis. Geodesic pca versus log-pca of histograms in the wasserstein space.SIAM Journal on Scientific Computing, 40(2):B429–B456, 2018

    2018

  9. [16]

    Chang, Y

    W. Chang, Y. Shi, H. Tuan, and J. Wang. Unified optimal transport framework for universal domain adaptation. Advances in Neural Information Processing Systems, 35:29512–29524, 2022

    2022

  10. [17]

    Chernozhukov, A

    V. Chernozhukov, A. Galichon, M. Hallin, and M. Henry. Monge–Kantorovich depth, quantiles, ranks and signs.The Annals of Statistics, 45(1):223 – 256, 2017

    2017

  11. [18]

    Chzhen, C

    E. Chzhen, C. Denis, M. Hebiri, L. Oneto, and M. Pontil. Fair regression with 33 Distributional Limit Theory for Optimal Transport wasserstein barycenters. Advances in Neural Information Processing Systems, 33:7321–7331, 2020

    2020

  12. [19]

    Csörgő and L

    M. Csörgő and L. Horvàth.Weighted Approximations in Probability and Statistics. Chichester: Wiley., 1993

    1993

  13. [20]

    M. Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In NeurIPS, volume 26, 2013

    2013

  14. [21]

    De Lara, A

    L. De Lara, A. González-Sanz, N. Asher, L. Risser, and J.-M. Loubes. Transport- based counterfactual models.Journal of Machine Learning Research, 25:1–59, 2024

    2024

  15. [22]

    N. Deb, B. B. Bhattacharya, and B. Sen. Pitman efficiency lower bounds for multivariate distribution-free tests based on optimal transport, 2023

    2023

  16. [23]

    N. Deb, P. Ghosal, and B. Sen. Rates of estimation of optimal transport maps using plug-in estimators via barycentric projections. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan, editors,Advances in Neural Information Processing Systems, volume 34, page...

    2021

  17. [24]

    Deb and B

    N. Deb and B. Sen. Multivariate rank-based distribution-free nonparametric testing using measure transportation.J. Amer. Statist. Assoc., 118(541):192–207, 2023

    2023

  18. [25]

    del Barrio, J

    E. del Barrio, J. Cuesta-Albertos, C. Matrán, and J. Rodríguez-Rodríguez. Tests of goodness of fit based on theL2-Wasserstein distance.Ann. Statist., 27:1230–1239, 1999

    1999

  19. [26]

    del Barrio, E

    E. del Barrio, E. Giné, and C. Matrán. Central limit theorems for the Wasser- stein distance between the empirical and the true distributions.Ann. Probab., 27(2):1009–1071, 1999

    1999

  20. [27]

    del Barrio, E

    E. del Barrio, E. Giné, and F. Utzet. Asymptotics forL2 functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances. Bernoulli, 11:131–189, 2005

    2005

  21. [28]

    del Barrio, A

    E. del Barrio, A. González-Sanz, and J.-M. Loubes. Central limit theorems for general transportation costs.Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 60(2):847 – 873, 2024

    2024

  22. [29]

    del Barrio, A

    E. del Barrio, A. González Sanz, and J.-M. Loubes. Central limit theorems for semi-discrete Wasserstein distances.Bernoulli, 30(1):554 – 580, 2024

    2024

  23. [30]

    Del Barrio, P

    E. Del Barrio, P. Gordaliza, H. Lescornel, and J.-M. Loubes. Central limit theorem and bootstrap procedure for wasserstein’s variations with an application to structural relationships between distributions.Journal of Multivariate Analysis, 169:341–362, 2019

    2019

  24. [31]

    del Barrio, P

    E. del Barrio, P. Gordaliza, and J.-M. Loubes. A central limit theorem for Lp transportation cost on the real line with application to fairness assessment in machine learning.Information and Inference: A Journal of the IMA, 8(4):817–849, 10 2019. 34 Distributional Limit Theory...

    2019

  25. [32]

    Del Barrio, H

    E. Del Barrio, H. Inouzhe, J.-M. Loubes, C. Matrán, and A. Mayo-Íscar. opti- malflow: optimal transport approach to flow cytometry gating and population matching. BMC bioinformatics, 21:1–25, 2020

    2020

  26. [33]

    del Barrio and J

    E. del Barrio and J. Loubes. Central limit theorems for empirical transportation cost in general dimension.Ann. Probab., 2019

    2019

  27. [34]

    del Barrio, J.-M

    E. del Barrio, J.-M. Loubes, and D. Rodríguez-Vítores. Improved Central Limit Theorems for the empirical sliced Wasserstein distance.arXiv preprint, 2025

    2025

  28. [35]

    del Barrio, A

    E. del Barrio, A. G. Sanz, J.-M. Loubes, and J. Niles-Weed. An improved central limit theorem and fast convergence rates for entropic transportation costs.SIAM Journal on Mathematics of Data Science, 5(3):639–669, 2023

    2023

  29. [36]

    Dupuy, J.-M

    J.-F. Dupuy, J.-M. Loubes, and E. Maza. Non parametric estimation of the struc- tural expectation of a stochastic increasing function.Statistics and Computing, 21:121–136, 2011

    2011

  30. [37]

    Feydy, T

    J. Feydy, T. Séjourné, F.-X. Vialard, S.-i. Amari, A. Trouve, and G. Peyré. Interpolating between optimal transport and MMD using Sinkhorn divergences. In K. Chaudhuri and M. Sugiyama, editors,Proceedings of Machine Learning Research, volume 89 ofProceedings of Machine Learnin...

  31. [38]

    A. Figalli. The Monge-Ampère Equation and Its Applications. Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zurich, 2017

    2017

  32. [39]

    Fournier and A

    N. Fournier and A. Guillin. On the rate of convergence in wasserstein distance of the empirical measure.Probability Theory and Related Fields, 162(3–4):707–738, Oct. 2014

    2014

  33. [40]

    Franklin and J

    J. Franklin and J. Lorenz. On the scaling of multidimensional matrices.Linear Algebra and its Applications, 114-115:717–735, 1989. Special Issue Dedicated to Alan J. Hoffman

    1989

  34. [41]

    Freitag, C

    G. Freitag, C. Czado, and A. Munk. A nonparametric test for similarity of marginals—with applications to the assessment of population bioequivalence. Journal of statistical planning and inference, 137(3):697–711, 2007

    2007

  35. [42]

    Freitag and A

    G. Freitag and A. Munk. On Hadamard differentiability ink-sample semipara- metric models—with applications to the assessment of structural relationships.J. Multivariate Anal., 94:123–158, 2005

    2005

  36. [43]

    Freitag and A

    G. Freitag and A. Munk. On hadamard differentiability in k-sample semiparametric models—with applications to the assessment of structural relationships.Journal of multivariate analysis, 94(1):123–158, 2005

    2005

  37. [44]

    Freulon, J

    P. Freulon, J. Bigot, and B. P. Hejblum. Cytopt: Optimal transport with domain adaptation for interpreting flow cytometry data.Ann. Appl. Statist., 17:1086–1104, 2023

    2023

  38. [45]

    Galichon

    A. Galichon. Optimal Transport Methods in Economics. Princeton University Press, 09 2016

    2016

  39. [46]

    Alagrangianschemeàlabrenierfortheincompressible 35 Distributional Limit Theory for Optimal Transport euler equations

    T.GallouëtandQ.Mérigot. Alagrangianschemeàlabrenierfortheincompressible 35 Distributional Limit Theory for Optimal Transport euler equations. Found. Comput. Math., 18:835–865, 2018

    2018

  40. [47]

    Gangbo and R

    W. Gangbo and R. J. McCann. The geometry of optimal transportation.Acta Mathematica, 177(2):113–161, 1996

    1996

  41. [48]

    Genevay, L

    A. Genevay, L. Chizat, F. Bach, M. Cuturi, and G. Peyré. Sample complexity of sinkhorn divergences. In K. Chaudhuri and M. Sugiyama, editors,Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, volume 89 ofProceedings of Machine ...

  42. [49]

    Genevay, G

    A. Genevay, G. Peyré, and M. Cuturi. Learning generative models with sinkhorn divergences. In A. Storkey and F. Perez-Cruz, editors,Proceedings of the Twenty- First International Conference on Artificial Intelligence and Statistics, volume 84 of Proceedings of Machine Learning...

    2018

  43. [50]

    Goldfeld and K

    Z. Goldfeld and K. Greenewald. Gaussian-smoothed optimal transport: Metric structure and statistical efficiency. In International Conference on Artificial Intelligence and Statistics, pages 3327–3337. PMLR, 2020

    2020

  44. [51]

    Goldfeld, K

    Z. Goldfeld, K. Greenewald, J. Niles-Weed, and Y. Polyanskiy. Convergence of smoothed empirical measures with applications to entropy estimation.IEEE Transactions on Information Theory, 66(7):4368–4391, 2020

    2020

  45. [52]

    Goldfeld, K

    Z. Goldfeld, K. Kato, S. Nietert, and G. Rioux. Limit distribution theory for smooth p-wasserstein distances.The Annals of Applied Probability, 34(2), Apr. 2024

    2024

  46. [53]

    Goldfeld, K

    Z. Goldfeld, K. Kato, G. Rioux, and R. Sadhu. Statistical inference with regularized optimal transport. arXiv:2205.04283, 2022

    2022 arXiv

  47. [54]

    Goldfeld, K

    Z. Goldfeld, K. Kato, G. Rioux, and R. Sadhu. Limit theorems for entropic optimal transport maps and sinkhorn divergence.Electronic Journal of Statistics, 18(1), Jan. 2024

    2024

  48. [55]

    González-Sanz, S

    A. González-Sanz, S. Eckstein, and M. Nutz. Sparse regularized optimal transport without curse of dimensionality.Preprint, 2025

    2025

  49. [56]

    González-Delgado, A

    J. González-Delgado, A. González-Sanz, J. Cortés, and P. Neuvial. Two-sample goodness-of-fit tests on the flat torus based on wasserstein distance and their relevance to structural biology.Electronic Journal of Statistics, 17(1), Jan. 2023

    2023

  50. [57]

    González-Sanz, M

    A. González-Sanz, M. Hallin, and B. Sen. Monotone measure-preserving maps in Hilbert spaces: existence, uniqueness, and stability.arXiv:2305.11751, 2023

    2023 arXiv

  51. [58]

    González-Sanz and S

    A. González-Sanz and S. Hundrieser. Weak limits for empirical entropic optimal transport: Beyond smooth costs.arXiv:2305.09745, 2023

    2023 arXiv

  52. [59]

    González-Sanz, J.-M

    A. González-Sanz, J.-M. Loubes, and J. Niles-Weed. Weak limits of entropy regularized optimal transport; potentials, plans and divergences.arXiv:2207.07427, 2024

    2024 arXiv

  53. [60]

    González-Sanz and M

    A. González-Sanz and M. Nutz. Sparsity of quadratically regularized optimal transport: Scalar case.arXiv:2410.03353, 2024

    2024

  54. [61]

    González-Sanz, M

    A. González-Sanz, M. Nutz, and A. R. Valdevenito. Monotonicity in quadratically 36 Distributional Limit Theory for Optimal Transport regularized linear programs.arXiv:2408.07871, 2024

    2024 arXiv

  55. [62]

    González-Sanz and S

    A. González-Sanz and S. Sheng. Linearization of Monge-Ampère equations and data science applications.arXiv:2408.06534, 2024

    2024

  56. [63]

    Gordaliza, E

    P. Gordaliza, E. Del Barrio, G. Fabrice, and J.-M. Loubes. Obtaining fairness using optimal transport theory. InInternational conference on machine learning, pages 2357–2365. PMLR, 2019

    2019

  57. [64]

    T. L. Gouic, J.-M. Loubes, and P. Rigollet. Projection to fairness in statistical learning. arXiv preprint arXiv:2005.11720, 2020

    2005 arXiv

  58. [65]

    Graf and H

    S. Graf and H. Luschgy.Foundations of Quantization for Probability Distributions. Springer-Verlag, Berlin, Heidelberg, 2000

    2000

  59. [66]

    Ontheconvergencerateofpotentialsofbreniermaps

    F.F.Gunsilius. Ontheconvergencerateofpotentialsofbreniermaps. Econometric Theory, 38(2):381–417, Feb. 2021

    2021

  60. [67]

    Hallin, E

    M. Hallin, E. del Barrio, J. Cuesta-Albertos, and C. Matrán. Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach. The Annals of Statistics, 49(2):1139 – 1165, 2021

    2021

  61. [68]

    Hallin, D

    M. Hallin, D. Hlubinka, and v. S. Hudecová. Efficient fully distribution-free center-outward rank tests for multiple-output regression and MANOVA.J. Amer. Statist. Assoc., 118(543):1923–1939, 2023

    1923

  62. [69]

    Hallin, G

    M. Hallin, G. Mordant, and J. Segers. Multivariate goodness-of-fit tests based on wasserstein distance. Electronic Journal of Statistics, 15(1), Jan. 2021

    2021

  63. [70]

    R. Han, C. Rush, and J. Wiesel. Max-sliced wasserstein concentration and uniform ratio bounds of empirical measures on rkhs.arXiv preprint arXiv:2405.13153, 2024

    2024 arXiv

  64. [71]

    Harchaoui, L

    Z. Harchaoui, L. Liu, and S. Pal. Asymptotics of entropy-regularized optimal transport via chaos decomposition.Preprint arXiv:2011.08963, 2020

    2011 arXiv

  65. [72]

    Hundrieser, M

    S. Hundrieser, M. Klatt, and A. Munk. Limit distributions and sensitivity analysis for empirical entropic optimal transport on countable spaces.Ann. Appl. Probab., 34(1B):1403–1468, 2024

    2024

  66. [73]

    Hundrieser, M

    S. Hundrieser, M. Klatt, A. Munk, and T. Staudt. A unifying approach to distributional limits for empirical optimal transport.Bernoulli, 30(4):2846 – 2877, 2024

    2024

  67. [74]

    Hundrieser, G

    S. Hundrieser, G. Mordant, C. A. Weitkamp, and A. Munk. Empirical optimal transport under estimated costs: Distributional limits and statistical applications. Stochastic Processes and their Applications, 178:104462, 2024

    2024

  68. [75]

    Hundrieser, T

    S. Hundrieser, T. Staudt, and A. Munk. Empirical optimal transport between different measures adapts to lower complexity.Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 60(2), May 2024

    2024

  69. [76]

    Hütter and P

    J.-C. Hütter and P. Rigollet. Minimax estimation of smooth optimal transport maps. The Annals of Statistics, 49(2):1166 – 1194, 2021

    2021

  70. [77]

    Kitagawa, Q

    J. Kitagawa, Q. Mérigot, and B. Thibert. Convergence of a newton algorithm for semi-discrete optimal transport.J. Eur. Math. Soc., 21:2603–2651, 2019. 37 Distributional Limit Theory for Optimal Transport

    2019

  71. [78]

    Klatt, A

    M. Klatt, A. Munk, and Y. Zemel. Limit laws for empirical optimal solutions in random linear programs.Annals of Operations Research, 315(1):251–278, Apr. 2022

    2022

  72. [79]

    Klatt, C

    M. Klatt, C. Tameling, and A. Munk. Empirical regularized optimal transport: Statistical theory and applications.SIAM Journal on Mathematics of Data Science, 2(2):419–443, 2020

    2020

  73. [80]

    D. T. L. Ambrosio, F. Stra. A pde approach to a 2-dimensional matching problem. Probab. Theory Related Fields, 173:433–478, 2019

    2019

  74. [81]

    M. Ledoux. Optimal matching of random samples and rates of convergence of empirical measures. Mathematics going forward—collected mathematical brushstrokes, 2023

    2023

  75. [82]

    Ledoux and M

    M. Ledoux and M. Talagrand.Probability in Banach Spaces. Springer, 1991

    1991

  76. [83]

    B. Levy, R. Mohayaee, and S. von Hausegger. A fast semidiscrete optimal transport algorithm for a unique reconstruction of the early universe.Monthly Notices of the Royal Astronomical Society, 506(1):1165–1185, June 2021

    2021

  77. [84]

    G. Loeper. On the regularity of the polar factorization for time dependent maps. Calc. Var. Partial Differential Equations, 22(3):343–374, 2005

    2005

  78. [85]

    C. Léonard. A survey of the schrödinger problem and some of its connec- tions with optimal transport.Discrete and Continuous Dynamical Systems – A, 34(4):1533–1574, 2014

    2014

  79. [86]

    Manole, S

    T. Manole, S. Balakrishnan, J. Niles-Weed, and L. Wasserman. Central limit theorems for smooth optimal transport maps.arXiv:2312.12407, 2024

    2024 arXiv

  80. [87]

    Manole, S

    T. Manole, S. Balakrishnan, J. Niles-Weed, and L. Wasserman. Plugin estimation of smooth optimal transport maps. The Annals of Statistics, 52(3):966–998, 2024

    2024

  81. [88]

    Manole, S

    T. Manole, S. Balakrishnan, and L. Wasserman. Minimax confidence intervals for the Sliced Wasserstein distance.Electronic Journal of Statistics, 16(1):2252 – 2345, 2022

    2022

  82. [89]

    Manole and J

    T. Manole and J. Niles-Weed. Sharp convergence rates for empirical optimal transport with smooth costs.The Annals of Applied Probability, 34(1B):1108 – 1135, 2024

    2024

  83. [90]

    Mena and J

    G. Mena and J. Niles-Weed. Statistical bounds for entropic optimal transport: Sample complexity and the central limit theorem.Advances in Neural Information Processing Systems, 32, 2019

    2019

  84. [91]

    J. Meyron. Initialization procedures for discrete and semi-discrete optimal trans- port. Computer-Aided Design, 115:13–22, 2019

    2019

  85. [92]

    G. Mordant. The entropic optimal (self-)transport problem: Limit distributions for decreasing regularization with application to score function estimation, 2024

    2024

  86. [93]

    Munk and C

    A. Munk and C. Czado. Nonparametric validation of similar distributions and assessment of goodness of fit.J. R. Stat. Soc. Ser. B Stat. Methodol., 60:223–241, 1998. 38 Distributional Limit Theory for Optimal Transport

    1998

  87. [94]

    Muzellec, R

    B. Muzellec, R. Nock, G. Patrini, and F. Nielsen. Tsallis regularized optimal transport and ecological inference. Proceedings of the AAAI Conference on Artificial Intelligence, 31(1), Feb. 2017

    2017

  88. [95]

    Nietert, Z

    S. Nietert, Z. Goldfeld, and K. Kato. Smoothp-wasserstein distance: Structure, empirical approximation, and statistical applications. In M. Meila and T. Zhang, editors, Proceedings of the 38th International Conference on Machine Learning, volume 139 ofProceedings of Machine Le...

    2021

  89. [96]

    Niles-Weed and P

    J. Niles-Weed and P. Rigollet. Estimation of wasserstein distances in the spiked transport model. Bernoulli, 28(4):2663–2688, 2022

    2022

  90. [97]

    M. Nutz. Quadratically regularized optimal transport: Existence and multiplicity of potentials. SIAM J. Math. Anal., to appear, 2024

    2024

  91. [98]

    Peyré and M

    G. Peyré and M. Cuturi. Computational optimal transport: With applications to data science. Foundations and Trends in Machine Learning, 11:355–607, 2019

    2019

  92. [99]

    Pooladian and J

    A.-A. Pooladian and J. Niles-Weed. Plug-in estimation of schrödinger bridges, 2024

    2024

  93. [100]

    Rigollet and A

    P. Rigollet and A. J. Stromme. On the sample complexity of entropic optimal transport. Preprint arXiv:2206.13472, 2022

    2022 arXiv

  94. [102]

    Sadhu, Z

    R. Sadhu, Z. Goldfeld, and K. Kato. Limit distribution theory for the smooth 1-wasserstein distance with applications.Arxiv:2107.13494, 2022

    2022 arXiv

  95. [103]

    Sadhu, Z

    R. Sadhu, Z. Goldfeld, and K. Kato. Stability and statistical inference for semidiscrete optimal transport maps.The Annals of Applied Probability, 34(6), Dec. 2024

    2024

  96. [104]

    Samworth and O

    R. Samworth and O. Johnson. Convergence of the empirical process in mallows distance, with an application to bootstrap performance. preprint.Centre for Mathematical Sciences, Cambridge., 2004

    2004

  97. [105]

    Santambrogio

    F. Santambrogio. Optimal transport for applied mathematicians . Birkhäuser/Springer, 2015

    2015

  98. [106]

    Schrödinger

    E. Schrödinger. Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique. Annales de l’institut Henri Poincaré, 2(4):269–310, 1932

    1932

  99. [107]

    Seguy, B

    V. Seguy, B. B. Damodaran, R. Flamary, N. Courty, A. Rolet, and M. Blondel. Large-scale optimal transport and mapping estimation. InICLR 2018-International Conference on Learning Representations, pages 1–15, 2018

    2018

  100. [108]

    N. Si, K. Murthy, J. Blanchet, and V. A. Nguyen. Testing group fairness via optimal transport projections. InInternational Conference on Machine Learning, pages 9649–9659. PMLR, 2021

    2021

  101. [109]

    Sinkhorn

    R. Sinkhorn. Diagonal equivalence to matrices with prescribed row and column sums. The American Mathematical Monthly, 74(4):402, Apr. 1967. 39 Distributional Limit Theory for Optimal Transport

    1967

  102. [110]

    Sommerfeld and A

    M. Sommerfeld and A. Munk. Inference for empirical wasserstein distances on finite spaces. Journal of the Royal Statistical Society Series B: Statistical Methodology, 80(1):219–238, 2018

    2018

  103. [111]

    Tameling, M

    C. Tameling, M. Sommerfeld, and A. Munk. Empirical optimal transport on conuntable metric spaces: distributional limits and statistical applcations.The Annals of Applied Probability, 29:2744–2781, 2019

    2019

  104. [112]

    Taskesen, J

    B. Taskesen, J. Blanchet, D. Kuhn, and V. A. Nguyen. A statistical test for probabilistic fairness. InProceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, FAccT ’21, page 648–665, New York, NY, USA,

    2021

  105. [113]

    C. Villani. Topics in Optimal Transportation. Graduate studies in mathematics. American Mathematical Society, 2003

    2003

  106. [114]

    C. Villani. Optimal transport. Old and new.Springer, 2009

    2009

  107. [115]

    Xi and J

    J. Xi and J. Niles-Weed. Distributional convergence of the sliced wasserstein process. In Neural Information Processing Systems, 2022

    2022

  108. [116]

    Xu and Z

    X. Xu and Z. Huang. Central limit theorem for the sliced 1-wasserstein distance and the max-sliced 1-wasserstein distance.arXiv preprint arXiv:2205.14624, 2022

    2022 arXiv

  109. [117]

    Zhang, G

    S. Zhang, G. Mordant, T. Matsumoto, and G. Schiebinger. Manifold learning with sparse regularised optimal transport.arXiv:2307.09816, 2023. 8 supplement Proof of Theorem 2.1.The convergence statement(7) is Theorem 2.1 in [31]. The limiting variance is defined as follows. We se...

    2023 arXiv

  110. [121]

    (44) The assumptionJp(P ) is sufficient (and necessary) to ensure that Bn f◦F−1 is an Lp-valued random element and is also clear from it that ∫ [1/n,1−1/n]c ⏐⏐⏐⏐ Bn(t) f(F−1(t)) ⏐⏐⏐⏐ p dt−→ Pr. 0. Hence, to conclude it only remains to show that ∫ [1/n,1−1/n]c |vn(t)|pdt−→ Pr. ...

  111. [1583]

    PMLR, 16–18 Apr 2019

    2019

  112. [2021]

    Association for Computing Machinery

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.