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Empirical optimal transport potentials converge at root-n speed in low dimensions, and at sharp dimension-dependent rates above, with a function-space central limit theorem in dimensions up to three.

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2026-08-04 01:16 UTC pith:YOXL526D

load-bearing objection Potentials converge faster than maps; the stability inequality is the real contribution, but the endpoint BMO regularity theorem is the part to check.

arxiv 2608.00649 v1 pith:YOXL526D submitted 2026-08-01 math.ST stat.TH

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

Alberto Gonz\'alez-Sanz , Gilles Mordant , Shunan Sheng This is my paper
classification math.ST stat.TH MSC 62G3562G30
keywords optimal transportempirical Brenier potentialfunctional central limit theoremsemidiscrete transportconvex potential stabilityshadow pricesentropic regularizationconvex approximation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies how well one can estimate the quadratic optimal transport potential when one side of the transport is fixed and the other is known only through n i.i.d. samples. Its central claim is that potentials converge faster than transport maps: in dimension 3 or less the exact semidiscrete Brenier potential is within O(n^{-1/2}) in L1(μ) modulo constants, in dimension 4 the rate is n^{-1/2}(log n)^{5/2}, and in dimension at least 5 it is n^{-2/d}(log n)^{(d+2)/d}, with sharp polynomial exponents. The mechanism is a stability inequality that isolates the leading empirical-process error in a weak dual norm and relegates the second-order Wasserstein term to a remainder. In low dimensions the paper also proves a nondegenerate function-space central limit theorem, bootstrap consistency, and joint root-n inference for location-dependent shadow prices; for entropic regularization it matches upper and lower bounds of order ε log(1/ε) for the sum of the dual potentials.

Core claim

The paper establishes that the empirical Brenier potential, defined by pushing the fixed absolutely continuous reference measure μ to the empirical measure of the sample, is a much faster estimator of the population potential φ than the transport map is of ∇φ. Its main result bounds the L1(μ) error modulo constants by the dual-C^{1,LogLip} norm of the empirical discrepancy plus a logarithmic Wasserstein remainder, then combines that bound with empirical-process entropy estimates to obtain n^{-1/2} rates for d≤3, n^{-1/2}(log n)^{5/2} for d=4, and n^{-2/d}(log n)^{(d+2)/d} for d≥5. In dimensions d≤3, √n(bφ_n−φ) converges in L^{p′}(Ω)/⟨1⟩ to a centered Gaussian random element whose variance is

What carries the argument

The key object is a stability inequality (Theorem 2.1) comparing any convex potential φ̃ with the strongly convex reference φ: the L1(μ) distance modulo constants is controlled by |ν̃−ν|′_{1,LogLip} + W2(ν̃,ν) ω_{LogLip}(W2(ν̃,ν)), where ν=(∇φ)#μ and ν̃=(∇φ̃)#μ. The proof linearizes the push-forward identity around ∇φ and inverts the linearized Monge–Ampère operator with Neumann boundary conditions, using an endpoint elliptic regularity estimate: bounded centered data yield second derivatives in BMO, and the embedding W^{2,BMO}(Ω)↪C^{1,LogLip}(Ω) converts the dual norm into a usable L1 bound. This separation is what makes the potential rate faster than the map rate, because the leading term

Load-bearing premise

The load-bearing premise is a regularity estimate for the linearized transport equation: whenever the right-hand side is bounded, the solution's second derivatives remain within a controlled, uniform amount of oscillation; if this estimate gives way, the fast convergence rates collapse.

What would settle it

Solve, numerically or analytically, the linearized Neumann problem div(μ[∇²φ]^{-1}∇u)=f on a smooth bounded domain with C^{1,α} coefficient and bounded centered f, and check whether the oscillation of ∇²u stays proportional to ∥f∥∞ uniformly. A sequence of coefficients for which this ratio diverges would disprove the endpoint estimate and undermine the n^{-1/2} rates; alternatively, a Monte Carlo simulation in d=4 comparing empirical L1 error with n^{-1/2}(log n)^{5/2} would test the predicted dimension threshold.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • In dimensions d≤3, the empirical potential is a root-n consistent estimator with a nondegenerate Gaussian limit, allowing confidence intervals and tests for any fixed finite collection of normalization-invariant contrasts.
  • The L1 potential estimate improves on the indirect Poincaré–Wasserstein route, whose high-dimensional rate is essentially n^{-1/d}; the paper's high-dimensional rate n^{-2/d} is the correct polynomial scale.
  • Because the limiting process is function-valued, the same theorem can be projected onto new linear functionals without repeating the first-order analysis.
  • For entropic optimal transport, the sum of the two dual potentials (modulo additive normalization) converges at the sharp rate ε log(1/ε), with matching upper and lower bounds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the stability inequality likely applies to deterministic approximation errors—such as truncated Laguerre cells or convex neural fits—not only to sampling errors, so the same fast rates may describe approximation bias.
  • A testable extension is whether the d=4 logarithmic factor n^{-1/2}(log n)^{5/2} is removable; the paper proves sharp polynomial exponents but leaves logarithmic gaps, so a finer empirical-process analysis might close them.
  • The regional shadow-premium interpretation invites a local distribution-shift audit: one could estimate first-order effects of arbitrary fixed reweightings of the reference measure, provided the weight function lies in the required integrability class.
  • The endpoint W^{2,BMO} regularity result may carry over to other strongly convex costs or Riemannian settings, giving analogous fast rates for potential estimation in those geometries.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper studies estimation of the quadratic optimal transport potential when a fixed absolutely continuous reference μ is transported to an unknown ν replaced by its empirical measure. The main ingredient is a stability inequality (Theorem 2.1) bounding the L1(μ) distance between two convex potentials, modulo constants, by a weak dual norm of the pushforward discrepancy plus a second-order Wasserstein remainder. The proof linearizes the push-forward relation and uses an endpoint BMO regularity theorem for the linearized Neumann problem (Theorem B.2) and the embedding W^{2,BMO}→C^{1,LogLip}. This yields empirical potential rates n^{-1/2} for d≤3, n^{-1/2}(log n)^{5/2} for d=4, and n^{-2/d}(log n)^{(d+2)/d} for d≥5, with sharp polynomial exponents. For d≤3 the paper proves a nondegenerate functional central limit theorem in L^{p'}(Ω)/⟨1⟩, bootstrap consistency, and joint root-n inference for weighted contrasts. A companion result gives matching upper and lower bounds of order ε log(1/ε) for the normalization-invariant sum of entropic dual potentials. The theoretical results are illustrated by numerical experiments and by an application to regional shadow premia in maximal-correlation risk.

Significance. If correct, these results are a substantial advance: they show that empirical OT potentials converge faster than the corresponding transport maps and identify the exact polynomial scale in low and high dimensions. The sharpness arguments are constructive, the function-space CLT and bootstrap are nondegenerate, and the entropic-potential bounds are two-sided. Strengths include the self-contained proof of the stability inequality, the entropy calculations that give the dimension threshold, and the explicit polyhedral lower bound. The endpoint BMO regularity theorem is the pivot of the paper; I found no concrete flaw in the written proof, but this is the least local step and deserves the authors' explicit attention in revision.

minor comments (5)
  1. [§2.4 / Corollary 2.11] The corollary says 'Assume the setting of Theorem 2.10 (ii)' but Theorem 2.10 has no part (ii); it should refer to Theorem 2.4(ii) or simply to Theorem 2.10.
  2. [§2.2 / Eq. (5)] Redundant phrase 'a strategy also that is also useful' should be corrected.
  3. [Proposition 3.1] The envelope-theorem step is sketched in one sentence. The proposition is secondary, but since it motivates the data application, a few additional lines explaining why the normalized dual optimizers are stable as t→0 and how uniqueness modulo constants suffices would be helpful.
  4. [Appendices B–C / Theorem B.2] The endpoint BMO regularity is the pivot of the paper and the proof is long. I found no concrete gap, but I recommend adding a short 'road map' at the start of Appendix B stating exactly where Lemma C.1, Lemma C.2, and Lemma D.1 are used and why the comparison estimates are uniform up to the boundary. This would make the least local step easier to audit.
  5. [Appendix B / Remark B.3] Minor: 'When g∈L∞_0' should likely be 'When f∈L∞_0'.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained, with the endpoint BMO regularity as the main (non-circular) verification risk.

full rationale

The derivation is self-contained. Theorem 2.1's stability bound is proved from a Taylor expansion of the push-forward relation (Eq. (5), Proposition 4.1), the linearized Neumann well-posedness/W^{2,BMO} estimate (Theorem B.2, proved in Appendices B-C), and the embedding W^{2,BMO}->C^{1,LogLip} (Lemma D.1). The statistical rates in Theorem 2.4(i) are obtained by inserting the empirical-process estimate of Lemma 2.2 (derived from the entropy bound of Lemma E.1 for C^{1,LogLip}) and the cited W_2^2 map-rate alpha(n,d) (Manole et al. 2024; Weed and Bach), which is external to this paper. The CLT uses the negative-Sobolev empirical CLT (Lemma 2.3) proved by bracketing entropy and the continuous mapping theorem. Bootstrap consistency (Theorem 2.10) uses the standard Donsker bootstrap and the same linearization, with remainders vanishing under p > max{d, 2d/(4-d)}. The EOT lower bound invokes Pal (2024) and Malamut and Sylvestre (2025), external; the upper bound follows from the same stability inequality. No parameter is fitted to the target quantities; no claimed prediction is identical to an input by construction. The self-citations (Gonzalez-Sanz et al. 2026; del Barrio et al. 2026) are contextual or backed by external results and are not load-bearing. The endpoint BMO regularity is lengthy and not machine-checked, but that is a correctness/verification risk, not a circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

The central claim rests on standard elliptic regularity, empirical-process theory, and explicit smoothness assumptions. No free parameters are fitted to data in the theoretical results, and no new entities are introduced. The BMO endpoint estimate is proved inside the paper but is technically demanding and not machine-checked, which is the main residual risk.

axioms (7)
  • standard math Brenier's theorem gives a unique (up to constants) convex potential whose gradient pushes μ to ν, and the semidiscrete analogue for atomic targets.
    Used throughout the paper; the estimator bφ_n is defined by this theorem.
  • standard math Uniform oblique elliptic regularity: for centered f∈L^p(Ω), the Neumann problem div(A∇u)=f with conormal boundary condition has a unique solution in W^{2,p}(Ω)/⟨1⟩ with Schauder/Calderón–Zygmund estimates.
    Invoked in Theorem B.5 and throughout the proof of Theorem 2.4 and Lemma 2.3.
  • standard math Sobolev/Orlicz-Sobolev embeddings: W^{2,BMO}(Ω) embeds into C^{1,LogLip}(Ω), and W^{2,p}(Ω) embeds into C^{1,α}(Ω) for p>d.
    Necessary to convert the elliptic estimates into the C^{1,LogLip} dual-norm bound in Theorem 2.1.
  • standard math Empirical-process Donsker theory for classes with finite bracketing entropy (van der Vaart–Wellner).
    Used in Lemma 2.3 and the bootstrap consistency proof, Theorem 2.10.
  • domain assumption Known entropic-cost asymptotics: K_ε − OT/ε = (H(ν|L^d)−H(μ|L^d))/2 + o(1) from Pal (2024), and the √ε bound from Malamut and Sylvestre (2025, Remark 3.10).
    The matching ε log(1/ε) bounds in Theorem 2.12 depend on these external asymptotic results for entropic regularization.
  • domain assumption Smoothness and compactness assumptions on the setting: Ω bounded with C^{2,α} boundary, μ has C^{1,α} density bounded below, φ is κ-strongly convex and C^{3,α}, and the relevant gradients lie in a fixed ball B_R.
    These hypotheses are explicit in Theorems 2.1 and 2.4 but are load-bearing: without them the linearized Monge–Ampère equation and the BMO estimates do not hold.
  • domain assumption For the functional CLT, d≤3 and p>max(d,2d/(4-d)); for the bootstrap, the same condition.
    These conditions are necessary for the entropy integral to converge and for the Wasserstein remainder to be o_P(n^{-1/2}).

pith-pipeline@v1.3.0-daily-deepseek · 43916 in / 13675 out tokens · 153124 ms · 2026-08-04T01:16:54.212317+00:00 · methodology

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read the original abstract

Optimal transport potentials are fundamental objects in statistics, economics, and machine learning: their gradients generate optimal transport maps, while the potentials themselves act as location-dependent dual prices and sensitivity variables. We study the estimation of the quadratic optimal transport potential when a fixed absolutely continuous reference distribution $\mu$ is transported to an unknown distribution $\nu$, accessed to via its empirical measure. Our main ingredient is a stability inequality that controls the $L^1(\mu)$ distance, modulo additive constants, between a strongly convex potential $\varphi$ and a convex potential $\widetilde\varphi$ by a weak dual norm of $(\nabla\widetilde\varphi)_\#\mu-(\nabla\varphi)_\#\mu,$ together with a second-order Wasserstein remainder of logarithmic type. This separation between the leading empirical-process term and the Wasserstein remainder yields faster convergence for potentials than for the corresponding transport maps. Under smoothness and uniform convexity assumptions, the exact semidiscrete Brenier potential converges in $L^1(\mu)$ at rate $n^{-1/2}$ for $d\leq3$, at rate $n^{-1/2}(\log n)^{5/2}$ for $d=4$, and at rate $n^{-2/d}(\log n)^{(d+2)/d}$ for $d\geq5$. The polynomial exponents are sharp. In dimensions $d\leq3$, we further establish a nondegenerate function-space central limit theorem and prove consistency of the nonparametric bootstrap. These results yield joint root-$n$ inference for every fixed finite collection of normalization-invariant weighted contrasts of the potential, including regional shadow premia in reference-based risk problems. Finally, we prove matching upper and lower bounds of order $\varepsilon\log(1/\varepsilon)$ for the normalization-invariant sum of the entropic dual potentials.

Figures

Figures reproduced from arXiv: 2608.00649 by Alberto Gonz\'alez-Sanz, Gilles Mordant, Shunan Sheng.

Figure 1
Figure 1. Figure 1: Log-log plot of E[infa∈R ∥φbn − φ − a∥L1(µ) ] for dimensions d ∈ {2, 3, 4, 6, 8, 10}. The black dots report averages over 100 independent Monte Carlo trials for each n ∈ {100, 300, 1000, 3000, 10000}. The semi-discrete Brenier potentials are computed by constructing the associated Laguerre cells, follow￾ing the procedure described in (Peyré and Cuturi, 2019, Section 5.2). The L 1 (µ) integral is approximat… view at source ↗
Figure 2
Figure 2. Figure 2: Regional shadow premia for the Loss–ALAE claims. (a) Log-transformed [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

discussion (0)

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Works this paper leans on

107 extracted references · 4 canonical work pages

  1. [1]

    arXiv preprint arXiv:2607.19080 , year=

    The Influence Function of Transport-based Quantiles , author=. arXiv preprint arXiv:2607.19080 , year=

    Pith/arXiv arXiv

  2. [2]

    Journal of Multivariate Analysis , volume=

    Central limit theorem and bootstrap procedure for Wasserstein’s variations with an application to structural relationships between distributions , author=. Journal of Multivariate Analysis , volume=. 2019 , publisher=

    2019

  3. [3]

    arXiv preprint arXiv:2411.04908 , year=

    Gluing methods for quantitative stability of optimal transport maps , author=. arXiv preprint arXiv:2411.04908 , year=

    Pith/arXiv arXiv

  4. [4]

    2026 , journal=

    The Influence Function of Transport-based Quantiles , author=. 2026 , journal=

    2026

  5. [5]

    Pal, Soumik , TITLE =. Ann. Appl. Probab. , FJOURNAL =. 2024 , NUMBER =. doi:10.1214/23-aap1983 , URL =

    work page doi:10.1214/23-aap1983 2024

  6. [6]

    arXiv:2505.04721 , year =

    Gonz\'alez-Sanz, Alberto and Eckstein, Stephan and Nutz, Marcel , title =. arXiv:2505.04721 , year =

    Pith/arXiv arXiv

  7. [7]

    2025 , journal=

    Sample Complexity of Quadratically Regularized Optimal Transport , author=. 2025 , journal=

    2025

  8. [8]

    Electron

    Limit theorems for entropic optimal transport maps and Sinkhorn divergence , volume =. Electron. J. Statist. , author =. 2024 , pages=

    2024

  9. [9]

    and Beckmann, Martin , title =

    Koopmans, Tjalling C. and Beckmann, Martin , title =. Econometrica , year =

  10. [10]

    and Shubik, Martin , title =

    Shapley, Lloyd S. and Shubik, Martin , title =. International Journal of Game Theory , year =

  11. [11]

    , title =

    Becker, Gary S. , title =. Journal of Political Economy , year =

  12. [12]

    and Ostroy, Joseph M

    Gretsky, Neil E. and Ostroy, Joseph M. and Zame, William R. , title =. Economic Theory , year =

  13. [13]

    Journal of Political Economy , year =

    Choo, Eugene and Siow, Aloysius , title =. Journal of Political Economy , year =

  14. [14]

    Journal of Political Economy , year =

    Dupuy, Arnaud and Galichon, Alfred , title =. Journal of Political Economy , year =

  15. [15]

    The Econometrics of Matching Models , journal =

    Chiappori, Pierre-Andr. The Econometrics of Matching Models , journal =. 2016 , volume =

    2016

  16. [16]

    Journal of Political Economy , year =

    Galichon, Alfred and Kominers, Scott Duke and Weber, Simon , title =. Journal of Political Economy , year =

  17. [17]

    Like Attract Like?

    Ciscato, Edoardo and Galichon, Alfred and Gouss. Like Attract Like?. Journal of Political Economy , year =

  18. [18]

    Cupid's Invisible Hand: Social Surplus and Identification in Matching Models , journal =

    Galichon, Alfred and Salani. Cupid's Invisible Hand: Social Surplus and Identification in Matching Models , journal =. 2022 , volume =

    2022

  19. [19]

    The Review of Economic Studies , year =

    Lindenlaub, Ilse , title =. The Review of Economic Studies , year =

  20. [20]

    , title =

    Boerma, Job and Tsyvinski, Aleh and Zimin, Alexander P. , title =. Journal of Political Economy , year =

  21. [21]

    , title =

    Boerma, Job and Tsyvinski, Aleh and Zimin, Alexander P. , title =

  22. [22]

    and Nesheim, Lars , title =

    Ekeland, Ivar and Heckman, James J. and Nesheim, Lars , title =. Journal of Political Economy , year =

  23. [23]

    Hedonic Price Equilibria, Stable Matching, and Optimal Transport: Equivalence, Topology, and Uniqueness , journal =

    Chiappori, Pierre-Andr. Hedonic Price Equilibria, Stable Matching, and Optimal Transport: Equivalence, Topology, and Uniqueness , journal =. 2010 , volume =

    2010

  24. [24]

    Economic Theory , year =

    Carlier, Guillaume and Ekeland, Ivar , title =. Economic Theory , year =

  25. [25]

    and Matzkin, Rosa L

    Heckman, James J. and Matzkin, Rosa L. and Nesheim, Lars , title =. Econometrica , year =

  26. [26]

    Journal of Political Economy , year =

    Chernozhukov, Victor and Galichon, Alfred and Henry, Marc and Pass, Brendan , title =. Journal of Political Economy , year =

  27. [27]

    The Annals of Statistics , year =

    Carlier, Guillaume and Chernozhukov, Victor and Galichon, Alfred , title =. The Annals of Statistics , year =

  28. [28]

    Distribution and Quantile Functions, Ranks and Signs in Dimension d : A Measure Transportation Approach , journal =

    Hallin, Marc and del Barrio, Eustasio and Cuesta-Albertos, Juan and Matr. Distribution and Quantile Functions, Ranks and Signs in Dimension d : A Measure Transportation Approach , journal =. 2021 , volume =

    2021

  29. [29]

    Mathematical Finance , year =

    Ekeland, Ivar and Galichon, Alfred and Henry, Marc , title =. Mathematical Finance , year =

  30. [30]

    Journal of Economic Theory , year =

    Galichon, Alfred and Henry, Marc , title =. Journal of Economic Theory , year =

  31. [31]

    Electronic Journal of Statistics , year =

    Bercu, Bernard and Bigot, J. Electronic Journal of Statistics , year =

  32. [32]

    Ironing, Sweeping, and Multidimensional Screening , journal =

    Rochet, Jean-Charles and Chon. Ironing, Sweeping, and Multidimensional Screening , journal =. 1998 , volume =

    1998

  33. [33]

    Econometrica , year =

    Daskalakis, Constantinos and Deckelbaum, Alan and Tzamos, Christos , title =. Econometrica , year =

  34. [34]

    Quantitative Economics , year =

    Chiong, Khai Xiang and Galichon, Alfred and Shum, Matt , title =. Quantitative Economics , year =

  35. [35]

    Galichon, Alfred , title =

  36. [36]

    The Econometrics Journal , year =

    Galichon, Alfred , title =. The Econometrics Journal , year =

  37. [37]

    , title =

    Makkuva, Ashok Vardhan and Taghvaei, Amirhossein and Oh, Sewoong and Lee, Jason D. , title =. Proceedings of the 37th International Conference on Machine Learning , series =. 2020 , url =

    2020

  38. [38]

    Regularity as Regularization: Smooth and Strongly Convex

    Paty, Fran. Regularity as Regularization: Smooth and Strongly Convex. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics , series =. 2020 , url =

    2020

  39. [39]

    Communications on Pure and Applied Mathematics , year =

    Brenier, Yann , title =. Communications on Pure and Applied Mathematics , year =

  40. [40]

    SIAM Journal on Applied Mathematics , year =

    Benamou, Jean-David and Brenier, Yann , title =. SIAM Journal on Applied Mathematics , year =

  41. [41]

    and Egan, Charlie P

    Bourne, David P. and Egan, Charlie P. and Pelloni, Beatrice and Wilkinson, Mark , title =. Calculus of Variations and Partial Differential Equations , year =

  42. [42]

    A Fast Semidiscrete Optimal Transport Algorithm for a Unique Reconstruction of the Early Universe , journal =

    L. A Fast Semidiscrete Optimal Transport Algorithm for a Unique Reconstruction of the Early Universe , journal =. 2021 , volume =

    2021

  43. [43]

    Optimal Transport Reconstruction of Biased Tracers in Redshift Space , journal =

    Nikakhtar, Farnik and Padmanabhan, Nikhil and L. Optimal Transport Reconstruction of Biased Tracers in Redshift Space , journal =. 2023 , volume =

    2023

  44. [44]

    The Annals of Probability , year =

    del Barrio, Eustasio and Loubes, Jean-Michel , title =. The Annals of Probability , year =

  45. [45]

    Advances in Neural Information Processing Systems , volume =

    Mena, Gonzalo and Niles-Weed, Jonathan , title =. Advances in Neural Information Processing Systems , volume =

  46. [46]

    Minimax Estimation of Smooth Optimal Transport Maps , journal =

    H. Minimax Estimation of Smooth Optimal Transport Maps , journal =. 2021 , volume =

    2021

  47. [47]

    , title =

    Kosorok, Michael R. , title =. 2008 , doi =

    2008

  48. [48]

    Manole, Tudor and Balakrishnan, Sivaraman and Niles-Weed, Jonathan and Wasserman, Larry , title =

  49. [49]

    Center-Outward Quantiles and the Measurement of Multivariate Risk , journal =

    Beirlant, Jan and Buitendag, Sven and del Barrio, Eustasio and Hallin, Marc and Kamper, Fran. Center-Outward Quantiles and the Measurement of Multivariate Risk , journal =. 2020 , volume =

    2020

  50. [50]

    Journal of Business & Economic Statistics , year =

    Hallin, Marc and Mordant, Gilles , title =. Journal of Business & Economic Statistics , year =

  51. [51]

    , title =

    Fan, Yanqin and Henry, Marc and Pass, Brendan and Rivero, Jorge A. , title =. The Review of Economics and Statistics , year =

  52. [52]

    Snowflake universality of Wasserstein spaces , author=. Ann. Sci

  53. [53]

    Beirlant and S

    J. Beirlant and S. Buitendag and E. Center-outward quantiles and the measurement of multivariate risk , journal =. 2020 , issn =. doi:https://doi.org/10.1016/j.insmatheco.2020.08.005 , url =

    work page doi:10.1016/j.insmatheco.2020.08.005 2020

  54. [54]

    Borgwardt and Malte J

    Arthur Gretton and Karsten M. Borgwardt and Malte J. Rasch and Bernhard Sch. A Kernel Two-Sample Test , journal =. 2012 , volume =

    2012

  55. [55]

    Journal of Machine Learning Research , year =

    Krishnakumar Balasubramanian and Tong Li and Ming Yuan , title =. Journal of Machine Learning Research , year =

  56. [56]

    Electronic Journal of Statistics , number =

    Marc Hallin and Gilles Mordant and Johan Segers , title =. Electronic Journal of Statistics , number =. 2021 , doi =

    2021

  57. [57]

    Bernoulli , number =

    Estate Khmaladze , title =. Bernoulli , number =. 2016 , doi =

    2016

  58. [58]

    Gigli, Nicola , journal=. On H. 2011 , publisher=

    2011

  59. [59]

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

    Inference for Empirical Wasserstein Distances on Finite Spaces , volume =. Journal of the Royal Statistical Society Series B: Statistical Methodology , author =. 2018 , pages =

    2018

  60. [60]

    Fefferman, Charles , TITLE =. Rev. Mat. Iberoam. , FJOURNAL =. 2009 , NUMBER =. doi:10.4171/RMI/568 , URL =

    work page doi:10.4171/rmi/568 2009

  61. [61]

    2019 , publisher=

    Computational optimal transport: With applications to data science , author=. 2019 , publisher=

    2019

  62. [62]

    and Zemel, Yoav , title =

    Panaretos, Victor M. and Zemel, Yoav , title =. Annual Review of Statistics and Its Application , year =

  63. [63]

    arXiv preprint arXiv:2312.12407 , year=

    Central limit theorems for smooth optimal transport maps , author=. arXiv preprint arXiv:2312.12407 , year=

    Pith/arXiv arXiv

  64. [64]

    Bernoulli , FJOURNAL =

    Weed, Jonathan and Bach, Francis , TITLE =. Bernoulli , FJOURNAL =. 2019 , NUMBER =. doi:10.3150/18-BEJ1065 , URL =

    doi:10.3150/18-bej1065 2019

  65. [65]

    Dudley, R. M. , TITLE =. Ann. Math. Statist. , FJOURNAL =. 1968 , PAGES =. doi:10.1214/aoms/1177697802 , URL =

    arXiv 1968

  66. [66]

    arXiv preprint arXiv:2504.05412 , year=

    Stability of optimal transport maps on Riemannian manifolds , author=. arXiv preprint arXiv:2504.05412 , year=

    Pith/arXiv arXiv

  67. [67]

    Journal of Machine Learning Research , volume=

    Distance-based classification with Lipschitz functions , author=. Journal of Machine Learning Research , volume=

  68. [68]

    2011 , publisher=

    Functional analysis, Sobolev spaces and partial differential equations , author=. 2011 , publisher=

    2011

  69. [69]

    and Nirenberg, L

    John, F. and Nirenberg, L. , TITLE =. Comm. Pure Appl. Math. , FJOURNAL =. 1961 , PAGES =. doi:10.1002/cpa.3160140317 , URL =

    doi:10.1002/cpa.3160140317 1961

  70. [70]

    , title =

    Jones, Peter W. , title =. Indiana University Mathematics Journal , year =

  71. [71]

    Annali di Matematica Pura ed Applicata , year =

    Peetre, Jaak , title =. Annali di Matematica Pura ed Applicata , year =

  72. [72]

    and da Silva, Jo

    Bessa, Junior da S. and da Silva, Jo. Sharp Hessian Estimates for Fully Nonlinear Elliptic Equations under Relaxed Convexity Assumptions, Oblique Boundary Conditions and Applications , journal =. 2023 , volume =

    2023

  73. [73]

    2011 , PAGES =

    Han, Qing and Lin, Fanghua , TITLE =. 2011 , PAGES =

    2011

  74. [74]

    Duke Math

    Delalande, Alex and M\'erigot, Quentin , TITLE =. Duke Math. J. , FJOURNAL =. 2023 , NUMBER =. doi:10.1215/00127094-2022-0106 , URL =

    doi:10.1215/00127094-2022-0106 2023

  75. [75]

    2019 , publisher=

    High-dimensional statistics: A non-asymptotic viewpoint , author=. 2019 , publisher=

    2019

  76. [76]

    arXiv preprint arXiv:2505.19104 , year=

    Distributional limit theory for optimal transport , author=. arXiv preprint arXiv:2505.19104 , year=

    Pith/arXiv arXiv

  77. [77]

    R. M. Dudley , title =. The Annals of Mathematical Statistics , number =. 1969 , doi =

    1969

  78. [78]

    Probability theory and related fields , volume=

    On the rate of convergence in Wasserstein distance of the empirical measure , author=. Probability theory and related fields , volume=. 2015 , publisher=

    2015

  79. [79]

    The Annals of Statistics , number =

    Victor Chernozhukov and Alfred Galichon and Marc Hallin and Marc Henry , title =. The Annals of Statistics , number =. 2017 , doi =

    2017

  80. [80]

    arXiv preprint arXiv:1508.06465 , year=

    A statistical analysis of a deformation model with Wasserstein barycenters: estimation procedure and goodness of fit test , author=. arXiv preprint arXiv:1508.06465 , year=

    Pith/arXiv arXiv

Showing first 80 references.