pith. sign in

arxiv: 2606.10767 · v1 · pith:U5UTVMFUnew · submitted 2026-06-09 · 📊 stat.ME

Two-Sample Homogeneity Test via Entropic Optimal Transport

Yiming Ma , Hang Liu , Weiwei Zhuang This is my paper

Pith reviewed 2026-06-27 12:26 UTC · model grok-4.3

classification 📊 stat.ME
keywords two-sample testentropic optimal transporthomogeneity testingfunctional central limit theoremmultiplier bootstrapmap discrepancyconsistencylocal power
0
0 comments X

The pith

The squared L2 distance between empirical entropic optimal transport maps from a uniform reference provides a consistent test for equality of two distributions.

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

This paper shows that the squared L2 distance between empirical entropic optimal transport maps, each obtained by pushing forward a fixed uniform distribution on the unit ball, can serve as a test statistic for whether two unknown distributions are identical. For a fixed regularization parameter the population maps differ if and only if the distributions differ, and under the null the centered empirical map process converges to a Gaussian process whose quadratic form gives the null limit. A multiplier bootstrap approximates the distribution of that quadratic form, the test is consistent against fixed alternatives, and its local power is characterized for contiguous alternatives. The approach also supplies map-based diagnostics that reveal the nature of any detected difference.

Core claim

The authors establish that the population map discrepancy is identifiable, derive the functional CLT for the empirical map difference under the null yielding a Gaussian quadratic-form limit, prove consistency against fixed alternatives, characterize local asymptotic power under contiguous alternatives, and validate a weighted multiplier bootstrap for the non-pivotal null distribution.

What carries the argument

The entropic optimal transport map from the uniform law on the unit ball to each target distribution; the squared L2 norm of the difference between two such maps forms the test statistic.

Load-bearing premise

The entropic regularization parameter is fixed in advance and the reference measure is the uniform distribution on the unit ball, with the data distributions admitting well-behaved EOT maps.

What would settle it

Empirical rejection rates under the null that deviate substantially from the nominal level, or failure to reject with high probability when the two distributions are known to differ by a fixed amount.

Figures

Figures reproduced from arXiv: 2606.10767 by Hang Liu, Weiwei Zhuang, Yiming Ma.

Figure 1
Figure 1. Figure 1: Illustration of the proposed common-reference map comparison. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Empirical power under fixed alternatives. The nominal level is [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Empirical rejection rates under location alternatives [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Visualization of the estimated EOT-map discrepancy in dimension [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Spatial distributions of Citi Bike starting locations. Left: member [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Empirical EOT-map discrepancy on the source unit disk. The color [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗
read the original abstract

This paper proposes a two-sample homogeneity test based on entropic optimal transport (EOT) maps from a common reference distribution -- the uniform law on the unit ball. The test statistic is the squared $L^2$-distance between the two empirical EOT maps. For fixed entropic regularization parameter, we prove that the population map discrepancy is identifiable, derive a functional central limit theorem for the empirical map difference under the null, and establish the Gaussian quadratic-form null limit. We also prove consistency against fixed alternatives and characterize local asymptotic power under contiguous alternatives. A weighted multiplier bootstrap is proposed to calibrate the non-pivotal null distribution, and its validity is established. Extensive simulations demonstrate that the proposed EOT-map test has reliable finite-sample size control and exhibits competitive power compared with other existing methods. The method is particularly powerful for location alternatives and, beyond a single scalar discrepancy, it provides additional diagnostic information on how the two distributions differ. Finally, a real data application concludes the paper.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes a two-sample homogeneity test using the squared L² distance between empirical entropic optimal transport (EOT) maps from a fixed common reference (uniform on the unit ball). For fixed regularization parameter, it claims to prove identifiability of the population map discrepancy, a functional CLT for the empirical map difference under the null yielding a Gaussian quadratic-form limit, consistency against fixed alternatives, local asymptotic power under contiguous alternatives, and validity of a weighted multiplier bootstrap for the non-pivotal limit. Simulations and a real-data example are included.

Significance. If the regularity conditions hold, the method supplies both a calibrated test statistic and diagnostic information on distributional differences beyond a scalar p-value, with competitive power for location shifts. The combination of identifiability, FCLT, bootstrap validity, and local-power characterization is a substantive contribution to nonparametric two-sample testing.

major comments (2)
  1. [FCLT theorem statement and proof] The functional CLT (and hence the Gaussian quadratic-form null limit and bootstrap validity) rests on tightness of the centered empirical EOT-map process in the relevant function space. The manuscript must explicitly state the moment/support conditions on P and Q (relative to the fixed uniform reference on the unit ball) that guarantee this tightness; without them the central limit theorem does not follow from standard empirical-process arguments.
  2. [Identifiability and consistency sections] The identifiability claim and consistency result are stated for the population map discrepancy under the fixed reference measure. The paper should clarify whether the unit-ball support of the reference is essential or whether the results extend to other compactly supported references without altering the test statistic's form.
minor comments (1)
  1. [Notation and bootstrap section] Notation for the EOT map functional and the precise definition of the weighted multiplier bootstrap weights should be introduced earlier and used consistently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [FCLT theorem statement and proof] The functional CLT (and hence the Gaussian quadratic-form null limit and bootstrap validity) rests on tightness of the centered empirical EOT-map process in the relevant function space. The manuscript must explicitly state the moment/support conditions on P and Q (relative to the fixed uniform reference on the unit ball) that guarantee this tightness; without them the central limit theorem does not follow from standard empirical-process arguments.

    Authors: We agree that the conditions guaranteeing tightness of the empirical EOT-map process must be stated explicitly for the FCLT to be fully rigorous. The current manuscript assumes P and Q admit densities with respect to Lebesgue measure and possess finite moments of sufficiently high order (to control the Lipschitz constants of the EOT potentials), but these are not collected in a single assumption. In the revision we will add an explicit Assumption (new Assumption 2.2) stating that P and Q are supported on a fixed compact convex set containing the unit ball in its interior and have finite (2+δ)-moments for some δ>0; we will then verify in the proof of Theorem 3.1 that these conditions imply the required entropy-integrability condition for tightness in the Hölder space used for the functional CLT. revision: yes

  2. Referee: [Identifiability and consistency sections] The identifiability claim and consistency result are stated for the population map discrepancy under the fixed reference measure. The paper should clarify whether the unit-ball support of the reference is essential or whether the results extend to other compactly supported references without altering the test statistic's form.

    Authors: The unit ball is selected for computational convenience and to ensure an explicit characterization of the reference measure, but it is not essential to the theoretical results. Identifiability of the map discrepancy follows from the strict convexity of the entropic transport cost and the uniqueness of the Brenier potential for any reference measure that is absolutely continuous with respect to Lebesgue measure on a compact convex set with nonempty interior; the same argument applies verbatim to consistency. We will insert a short remark after Definition 2.1 clarifying that all results in Sections 3 and 4 continue to hold, with no change to the form of the test statistic, when the reference is replaced by any other fixed compactly supported probability measure satisfying the same absolute-continuity condition. revision: yes

Circularity Check

0 steps flagged

No circularity: standard empirical-process arguments applied to EOT map functional

full rationale

The paper states that for fixed entropic regularization it proves identifiability of the population map discrepancy, derives a functional CLT for the empirical map difference under the null, obtains a Gaussian quadratic-form limit, proves consistency, and characterizes local power, all under the maintained assumption that the data-generating distributions admit well-defined EOT maps from the fixed uniform reference on the unit ball. These are presented as consequences of standard tightness and convergence conditions in the relevant function space rather than as quantities fitted or defined in terms of the target test statistic. No self-definitional loop, fitted-input prediction, or load-bearing self-citation chain appears in the derivation chain; the central claims rest on external empirical-process theory applied to the EOT functional.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The development relies on standard assumptions from optimal transport and empirical process theory; the regularization parameter is treated as fixed rather than estimated.

free parameters (1)
  • entropic regularization parameter
    Held fixed throughout the asymptotic analysis; its specific value is not derived from data but chosen by the user.
axioms (2)
  • domain assumption The EOT map functional satisfies the conditions for a functional central limit theorem under the null (tightness, finite-dimensional convergence).
    Invoked to obtain the Gaussian quadratic-form null limit.
  • domain assumption The population map discrepancy is identifiable when the two distributions differ.
    Stated as proved for fixed regularization.

pith-pipeline@v0.9.1-grok · 5697 in / 1433 out tokens · 25992 ms · 2026-06-27T12:26:27.470339+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

71 extracted references · 28 canonical work pages

  1. [1]

    The Econometrics Journal , volume =

    R-estimators in GARCH models: asymptotics and applications , author =. The Econometrics Journal , volume =. 2022 , doi =

    2022

  2. [2]

    2025 , eprint=

    Multidimensional Stochastic Dominance Test Based on Center-outward Quantiles , author=. 2025 , eprint=

    2025

  3. [3]

    2024 , eprint=

    Quantiles and Quantile Regression on Riemannian Manifolds: a measure-transportation-based approach , author=. 2024 , eprint=

    2024

  4. [4]

    Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics , series =

    Sample Complexity of Sinkhorn Divergences , author =. Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics , series =. 2019 , url =

    2019

  5. [5]

    Advances in Neural Information Processing Systems , volume =

    Statistical Bounds for Entropic Optimal Transport: Sample Complexity and the Central Limit Theorem , author =. Advances in Neural Information Processing Systems , volume =. 2019 , url =

    2019

  6. [6]

    Proceedings of the Thirty-Ninth International Conference on Machine Learning , series =

    Debiaser Beware: Pitfalls of Centering Regularized Transport Maps , author =. Proceedings of the Thirty-Ninth International Conference on Machine Learning , series =. 2022 , url =

    2022

  7. [7]

    Advances in Neural Information Processing Systems , volume =

    Sinkhorn Distances: Lightspeed Computation of Optimal Transport , author =. Advances in Neural Information Processing Systems , volume =. 2013 , url =

    2013

  8. [8]

    Journal of Machine Learning Research , volume =

    A Kernel Two-Sample Test , author =. Journal of Machine Learning Research , volume =. 2012 , url =

    2012

  9. [10]

    2003 , doi =

    Topics in Optimal Transportation , author =. 2003 , doi =

    2003

  10. [11]

    2009 , doi =

    Optimal Transport: Old and New , author =. 2009 , doi =

    2009

  11. [12]

    2008 , doi =

    Introduction to Empirical Processes and Semiparametric Inference , author =. 2008 , doi =

    2008

  12. [13]

    Journal of the American Statistical Association , volume =

    Multivariate Two-Sample Tests Based on Nearest Neighbors , author =. Journal of the American Statistical Association , volume =. 1986 , doi =

    1986

  13. [14]

    The Annals of Statistics , volume =

    A multivariate two-sample test based on the number of nearest neighbor type coincidences , author =. The Annals of Statistics , volume =. 1988 , doi =

    1988

  14. [15]

    1998 , doi =

    Asymptotic Statistics , author =. 1998 , doi =

    1998

  15. [16]

    1996 , doi =

    Weak Convergence and Empirical Processes: With Applications to Statistics , author =. 1996 , doi =

    1996

  16. [17]

    The Annals of Statistics , volume =

    Multivariate Generalizations of the Wald-Wolfowitz and Smirnov Two-Sample Tests , author =. The Annals of Statistics , volume =. 1979 , doi =

    1979

  17. [18]

    Foundations and Trends in Machine Learning , volume =

    Computational Optimal Transport: With Applications to Data Science , author =. Foundations and Trends in Machine Learning , volume =. 2019 , publisher =

    2019

  18. [19]

    2015 , doi =

    Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling , author =. 2015 , doi =

    2015

  19. [20]

    Proceedings of the American Mathematical Society , volume=

    Anderson inequality is strict for Gaussian and stable measures , author=. Proceedings of the American Mathematical Society , volume=. 1995 , doi=

    1995

  20. [21]

    Annales de l'Institut Henri Poincar

    Central Limit Theorems for General Transportation Costs , author =. Annales de l'Institut Henri Poincar. 2024 , doi =

    2024

  21. [22]

    SIAM Journal on Mathematics of Data Science , volume =

    Empirical Regularized Optimal Transport: Statistical Theory and Applications , author =. SIAM Journal on Mathematics of Data Science , volume =. 2020 , doi =

    2020

  22. [23]

    Journal of the American Statistical Association , volume=

    Multivariate rank-based distribution-free nonparametric testing using measure transportation , author=. Journal of the American Statistical Association , volume=. 2023 , publisher=

    2023

  23. [24]

    A Survey of the

    L. A Survey of the. Discrete and Continuous Dynamical Systems - A , volume =. 2014 , doi =

    2014

  24. [25]

    Distribution and quantile functions, ranks and signs in dimension

    Hallin, Marc and del Barrio, Eustasio and Cuesta-Albertos, Juan and Matr. Distribution and quantile functions, ranks and signs in dimension. The Annals of Statistics , volume=. 2021 , doi=

    2021

  25. [26]

    Journal of Multivariate Analysis , volume=

    On a new multivariate two-sample test , author=. Journal of Multivariate Analysis , volume=. 2004 , doi=

    2004

  26. [27]

    Journal of the Royal Statistical Society: Series B , volume=

    An exact distribution-free test comparing two multivariate distributions based on adjacency , author=. Journal of the Royal Statistical Society: Series B , volume=. 2005 , doi=

    2005

  27. [28]

    Journal of the American Statistical Association , volume=

    A new graph-based two-sample test for multivariate and object data , author=. Journal of the American Statistical Association , volume=. 2017 , doi=

    2017

  28. [29]

    2020 , doi =

    An Invitation to Statistics in Wasserstein Space , author =. 2020 , doi =

    2020

  29. [30]

    The Annals of Probability , volume =

    Exchangeably Weighted Bootstraps of the General Empirical Process , author =. The Annals of Probability , volume =. 1993 , doi =

    1993

  30. [31]

    The Annals of Statistics , volume=

    On the sample complexity of entropic optimal transport , author=. The Annals of Statistics , volume=. 2025 , publisher=

    2025

  31. [32]

    Stability of Schr

    Nutz, Marcel and Wiesel, Johannes , journal =. Stability of Schr. 2023 , doi =

    2023

  32. [33]

    Journal of Statistical Planning and Inference , volume =

    Energy Statistics: A Class of Statistics Based on Distances , author =. Journal of Statistical Planning and Inference , volume =. 2013 , doi =

    2013

  33. [34]

    Entropy , volume =

    On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests , author =. Entropy , volume =. 2017 , doi =

    2017

  34. [35]

    Electronic Journal of Statistics , volume =

    Multivariate Goodness-of-Fit Tests Based on Wasserstein Distance , author =. Electronic Journal of Statistics , volume =. 2021 , doi =

    2021

  35. [36]

    Electronic Journal of Statistics , volume =

    Limit Theorems for Entropic Optimal Transport Maps and Sinkhorn Divergence , author =. Electronic Journal of Statistics , volume =. 2024 , doi =

    2024

  36. [37]

    On a new multivariate two-sample test

    Ludwig Baringhaus and Carsten Franz. On a new multivariate two-sample test. Journal of Multivariate Analysis, 88 0 (1): 0 190--206, 2004. doi:10.1016/S0047-259X(03)00079-4

    work page doi:10.1016/s0047-259x(03)00079-4 2004

  37. [38]

    doi: 10.1080/01621459.2016.1211016

    Hao Chen and Jerome H. Friedman. A new graph-based two-sample test for multivariate and object data. Journal of the American Statistical Association, 112 0 (517): 0 397--409, 2017. doi:10.1080/01621459.2016.1147356

    work page doi:10.1080/01621459.2016.1147356 2017

  38. [39]

    Sinkhorn distances: Lightspeed computation of optimal transport

    Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems, volume 26, pages 2292--2300, 2013. URL https://papers.nips.cc/paper_files/paper/2013/hash/af21d0c97db2e27e13572cbf59eb343d-Abstract.html

    2013

  39. [40]

    Journal of the American Statistical Association , volume =

    Nabarun Deb and Bodhisattva Sen. Multivariate rank-based distribution-free nonparametric testing using measure transportation. Journal of the American Statistical Association, 118 0 (541): 0 192--207, 2023. doi:10.1080/01621459.2021.1923508

    work page doi:10.1080/01621459.2021.1923508 2023

  40. [41]

    Central limit theorems for general transportation costs

    Eustasio del Barrio, Alberto Gonz \'a lez-Sanz, and Jean-Michel Loubes. Central limit theorems for general transportation costs. Annales de l'Institut Henri Poincar \'e , Probabilit \'e s et Statistiques , 60 0 (2): 0 847--873, 2024. doi:10.1214/22-AIHP1356

    work page doi:10.1214/22-aihp1356 2024

  41. [42]

    Bootstrap methods: Another look at the jackknife,

    Jerome H. Friedman and Lawrence C. Rafsky. Multivariate generalizations of the wald-wolfowitz and smirnov two-sample tests. The Annals of Statistics, 7 0 (4): 0 697--717, 1979. doi:10.1214/aos/1176344722

    work page doi:10.1214/aos/1176344722 1979

  42. [43]

    Sample complexity of sinkhorn divergences

    Aude Genevay, L \'e na \"i c Chizat, Francis Bach, Marco Cuturi, and Gabriel Peyr \'e . Sample complexity of sinkhorn divergences. In Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, volume 89 of Proceedings of Machine Learning Research, pages 1574--1583, 2019. URL http://proceedings.mlr.press/v89/geneva...

    2019

  43. [44]

    Limit theorems for entropic optimal transport maps and sinkhorn divergence

    Ziv Goldfeld, Kengo Kato, Gabriel Rioux, and Ritwik Sadhu. Limit theorems for entropic optimal transport maps and sinkhorn divergence. Electronic Journal of Statistics, 18 0 (1): 0 980--1041, 2024. doi:10.1214/24-EJS2217

    work page doi:10.1214/24-ejs2217 2024

  44. [45]

    Borgwardt, Malte J

    Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Sch \"o lkopf, and Alexander Smola. A kernel two-sample test. Journal of Machine Learning Research, 13 0 (25): 0 723--773, 2012. URL http://jmlr.org/papers/v13/gretton12a.html

    2012

  45. [46]

    Quantiles and quantile regression on riemannian manifolds: a measure-transportation-based approach, 2024

    Marc Hallin and Hang Liu. Quantiles and quantile regression on riemannian manifolds: a measure-transportation-based approach, 2024. URL https://arxiv.org/abs/2410.15711

    arXiv 2024

  46. [47]

    Distribution and quantile functions, ranks and signs in dimension d : A measure transportation approach

    Marc Hallin, Eustasio del Barrio, Juan Cuesta-Albertos, and Carlos Matr \'a n. Distribution and quantile functions, ranks and signs in dimension d : A measure transportation approach. The Annals of Statistics, 49 0 (2): 0 1139--1165, 2021 a . doi:10.1214/20-AOS1996

    work page doi:10.1214/20-aos1996 2021

  47. [48]

    Multivariate goodness-of-fit tests based on wasserstein distance

    Marc Hallin, Gilles Mordant, and Johan Segers. Multivariate goodness-of-fit tests based on wasserstein distance. Electronic Journal of Statistics, 15 0 (1): 0 1328--1371, 2021 b . doi:10.1214/21-EJS1816

    work page doi:10.1214/21-ejs1816 2021

  48. [49]

    A multivariate two-sample test based on the number of nearest neighbor type coincidences

    Norbert Henze. A multivariate two-sample test based on the number of nearest neighbor type coincidences. The Annals of Statistics, 16 0 (2): 0 772--783, 1988. doi:10.1214/aos/1176350835

    work page doi:10.1214/aos/1176350835 1988

  49. [50]

    Empirical regularized optimal transport: Statistical theory and applications

    Marcel Klatt, Carla Tameling, and Axel Munk. Empirical regularized optimal transport: Statistical theory and applications. SIAM Journal on Mathematics of Data Science, 2 0 (2): 0 419--443, 2020. doi:10.1137/19M1278788

    work page doi:10.1137/19m1278788 2020

  50. [51]

    Michael R. Kosorok. Introduction to Empirical Processes and Semiparametric Inference. Springer, New York, 2008. doi:10.1007/978-0-387-74978-5

    work page doi:10.1007/978-0-387-74978-5 2008

  51. [52]

    A survey of the S chr \"o dinger problem and some of its connections with optimal transport

    Christian L \'e onard. A survey of the S chr \"o dinger problem and some of its connections with optimal transport. Discrete and Continuous Dynamical Systems - A, 34 0 (4): 0 1533--1574, 2014. doi:10.3934/dcds.2014.34.1533

    work page doi:10.3934/dcds.2014.34.1533 2014

  52. [53]

    Anderson inequality is strict for gaussian and stable measures

    Maciej Lewandowski, Micha Ryznar, and Tomasz \.Z ak. Anderson inequality is strict for gaussian and stable measures. Proceedings of the American Mathematical Society, 123 0 (12): 0 3875--3880, 1995. doi:10.1090/S0002-9939-1995-1264821-6

    work page doi:10.1090/s0002-9939-1995-1264821-6 1995

  53. [54]

    R-estimators in garch models: asymptotics and applications

    Hang Liu and Kanchan Mukherjee. R-estimators in garch models: asymptotics and applications. The Econometrics Journal, 25 0 (1): 0 98--113, 2022. doi:10.1093/ectj/utab026

    work page doi:10.1093/ectj/utab026 2022

  54. [55]

    Multidimensional stochastic dominance test based on center-outward quantiles, 2025

    Yiming Ma, Hang Liu, and Weiwei Zhuang. Multidimensional stochastic dominance test based on center-outward quantiles, 2025. URL https://arxiv.org/abs/2512.19966

    arXiv 2025

  55. [56]

    Statistical bounds for entropic optimal transport: Sample complexity and the central limit theorem

    Gonzalo Mena and Jonathan Niles-Weed. Statistical bounds for entropic optimal transport: Sample complexity and the central limit theorem. In Advances in Neural Information Processing Systems, volume 32, pages 4541--4551, 2019. URL https://proceedings.neurips.cc/paper_files/paper/2019/file/5acdc9ca5d99ae66afdfe1eea0e3b26b-Paper.pdf

    2019

  56. [57]

    Stability of schr \"o dinger potentials and convergence of sinkhorn's algorithm

    Marcel Nutz and Johannes Wiesel. Stability of schr \"o dinger potentials and convergence of sinkhorn's algorithm. The Annals of Probability, 51 0 (2): 0 699--722, 2023. doi:10.1214/22-AOP1611

    work page doi:10.1214/22-aop1611 2023

  57. [58]

    Panaretos and Yoav Zemel

    Victor M. Panaretos and Yoav Zemel. An Invitation to Statistics in Wasserstein Space. SpringerBriefs in Probability and Mathematical Statistics. Springer, Cham, 2020. doi:10.1007/978-3-030-38438-8

    work page doi:10.1007/978-3-030-38438-8 2020

  58. [59]

    and Cuturi, M

    Gabriel Peyr \'e and Marco Cuturi. Computational optimal transport: With applications to data science. Foundations and Trends in Machine Learning, 11 0 (5--6): 0 355--607, 2019. doi:10.1561/2200000073

    work page doi:10.1561/2200000073 2019

  59. [60]

    Entropic estimation of optimal transport maps

    Aram-Alexandre Pooladian and Jonathan Niles-Weed. Entropic estimation of optimal transport maps. arXiv preprint arXiv:2109.12004, 2021. URL https://arxiv.org/abs/2109.12004

    arXiv 2021

  60. [61]

    Debiaser beware: Pitfalls of centering regularized transport maps

    Aram-Alexandre Pooladian, Marco Cuturi, and Jonathan Niles-Weed. Debiaser beware: Pitfalls of centering regularized transport maps. In Proceedings of the Thirty-Ninth International Conference on Machine Learning, volume 162 of Proceedings of Machine Learning Research, pages 17830--17847, 2022. URL https://proceedings.mlr.press/v162/pooladian22a.html

    2022

  61. [62]

    Jens Pr stgaard and Jon A. Wellner. Exchangeably weighted bootstraps of the general empirical process. The Annals of Probability, 21 0 (4): 0 2053--2086, 1993. doi:10.1214/aop/1176989011

    work page doi:10.1214/aop/1176989011 2053

  62. [63]

    On Wasserstein two-sample testing and related families of nonparametric tests,

    Aaditya Ramdas, Nicol \'a s Garc \'i a Trillos, and Marco Cuturi. On wasserstein two-sample testing and related families of nonparametric tests. Entropy, 19 0 (2): 0 47, 2017. doi:10.3390/e19020047

    work page doi:10.3390/e19020047 2017

  63. [64]

    On the sample complexity of entropic optimal transport

    Philippe Rigollet and Austin J Stromme. On the sample complexity of entropic optimal transport. The Annals of Statistics, 53 0 (1): 0 61--90, 2025. doi:10.1214/24-AOS2455

    work page doi:10.1214/24-aos2455 2025

  64. [65]

    Journal of the Royal Statistical Society: Series B (Statistical Methodology) , author =

    Paul R. Rosenbaum. An exact distribution-free test comparing two multivariate distributions based on adjacency. Journal of the Royal Statistical Society: Series B, 67 0 (4): 0 515--530, 2005. doi:10.1111/j.1467-9868.2005.00513.x

    work page doi:10.1111/j.1467-9868.2005.00513.x 2005

  65. [66]

    Birkhäuser Cham, 1 edition, 2015

    Filippo Santambrogio. Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, volume 87 of Progress in Nonlinear Differential Equations and Their Applications. Birkh \"a user, Cham, 2015. doi:10.1007/978-3-319-20828-2

    work page doi:10.1007/978-3-319-20828-2 2015

  66. [67]

    Schilling

    Mark F. Schilling. Multivariate two-sample tests based on nearest neighbors. Journal of the American Statistical Association, 81 0 (395): 0 799--806, 1986. doi:10.1080/01621459.1986.10478337

    work page doi:10.1080/01621459.1986.10478337 1986

  67. [68]

    Sz \'e kely and Maria L

    G \'a bor J. Sz \'e kely and Maria L. Rizzo. Energy statistics: A class of statistics based on distances. Journal of Statistical Planning and Inference, 143 0 (8): 0 1249--1272, 2013. doi:10.1016/j.jspi.2013.03.018

    work page doi:10.1016/j.jspi.2013.03.018 2013

  68. [69]

    Cambridge university press, Cambridge

    Aad W. van der Vaart. Asymptotic Statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 1998. doi:10.1017/CBO9780511802256

    work page doi:10.1017/cbo9780511802256 1998

  69. [70]

    and Wellner, Jon A

    Aad W. van der Vaart and Jon A. Wellner. Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York, 1996. doi:10.1007/978-1-4757-2545-2

    work page doi:10.1007/978-1-4757-2545-2 1996

  70. [71]

    Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics

    C \'e dric Villani. Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003. doi:10.1090/gsm/058

    work page doi:10.1090/gsm/058 2003

  71. [72]

    Optimal transport, volume 338 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    C \'e dric Villani. Optimal Transport: Old and New, volume 338 of Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 2009. doi:10.1007/978-3-540-71050-9

    work page doi:10.1007/978-3-540-71050-9 2009