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An improved central limit theorem for the empirical sliced Wasserstein distance

T0 review · 2 major / 2 minor · reviewed 2026-05-22 · grok-4.3

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Pith's one-line read The empirical p-sliced Wasserstein distance obeys a central limit theorem centered at the population distance for p greater than 1.

desk verdict This paper gives a CLT for empirical p-sliced Wasserstein that supports centering at the population distance for non-compact measures, but the uniform potential control is the step that needs the closest check. read the letter →

arxiv 2503.18831 v3 pith:KEH2HH72 submitted 2025-03-24 math.ST stat.TH

David Rodríguez-Vítores , Eustasio del Barrio , Jean-Michel Loubes This is my paper · ORCID
classification math.STstat.TH
keywords slicedWassersteindistancecentrallimittheoremEfron-Steininequalityoptimaltransportstatisticalinferenceempiricalprocessesnon-compactmeasures
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

The paper proves a central limit theorem for the empirical p-sliced Wasserstein distance when p exceeds 1. The limiting normal law can be centered at the true population value of the distance rather than at the expected value of the empirical estimator. This centering property, which fails for the ordinary Wasserstein distance, opens the door to asymptotically valid confidence intervals and tests. The argument combines the Efron-Stein inequality with a uniform bound on the optimal transport potentials taken over all projection directions. The result covers measures that need not have compact support and supplies consistent estimators for the asymptotic variance together with a Monte Carlo scheme for the slicing integral.

What carries the argument

The Efron-Stein inequality applied after a uniform control on the optimal transport potentials across all slicing directions.

What would settle it

A sequence of non-compact measures for which the empirical p-sliced Wasserstein distance, properly normalized, fails to converge in distribution to a centered normal random variable.

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Extended reading notes

Core claim

We establish a central limit theorem for the p-sliced Wasserstein distance, for p>1, centered at the expected empirical cost. Unlike for the general Wasserstein distance, the centering can be replaced by the population cost, enabling valid statistical inference. This generalizes and refines existing one-dimensional results, providing the first asymptotically valid inference framework for the sliced Wasserstein distance between possibly non-compact measures.

Load-bearing premise

The optimal transport potentials admit a uniform bound that does not deteriorate when the projection direction varies.

Editorial extensions

If this is right

  • Asymptotically valid confidence intervals for the sliced Wasserstein distance can be formed without bootstrap.
  • Consistent estimators of the limiting variance are available from the same samples.
  • Monte Carlo approximation of the integral over directions preserves the central limit theorem.
  • The framework applies directly to measures with unbounded support.

Reading between the lines

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

  • Hypothesis tests that treat the sliced distance as a test statistic become feasible at the usual asymptotic level.
  • The same proof pattern may extend to other sliced integral probability metrics whose one-dimensional projections satisfy a one-dimensional CLT.
  • In applications the result reduces computational cost by replacing resampling methods with direct normal approximation.
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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

2 major / 2 minor

Summary. The manuscript establishes a central limit theorem for the empirical p-sliced Wasserstein distance (p>1) centered at the population sliced Wasserstein distance rather than only at its expectation. The argument combines the Efron-Stein inequality with a uniform control on one-dimensional optimal transport potentials (or their derivatives) over random directions on the sphere. This enables asymptotically valid inference and generalizes existing one-dimensional results to possibly non-compact measures; the paper also treats Monte Carlo approximation of the slicing integral and consistent variance estimation.

Significance. If the uniform control on transport potentials holds under the paper's moment assumptions, the result supplies the first asymptotically valid inference framework for sliced Wasserstein distances between non-compact measures. This is a meaningful advance for high-dimensional applications where full Wasserstein distances are intractable.

major comments (2)
  1. [Proof of Theorem 2.1 / Section 3] The non-trivial uniform control on the optimal transport potentials across directions (invoked to obtain E[SW_n] - SW = o_p(n^{-1/2})) is the load-bearing step that permits population centering. The manuscript should state the precise moment conditions on the measures that guarantee this control is uniform in the slicing measure and does not deteriorate with dimension or tail heaviness; without explicit bounds or a counter-example check, it is unclear whether the o_p(n^{-1/2}) claim holds for the full range of non-compact measures advertised.
  2. [Section 2.2 and Theorem 2.1] The Efron-Stein application yields a CLT centered at the expectation; the passage from expectation to population sliced distance therefore rests entirely on the potential-control argument. If that argument requires stronger integrability than the CLT itself, the claimed improvement over the general Wasserstein case is narrower than stated.
minor comments (2)
  1. [Section 4] Notation for the slicing measure and the Monte Carlo approximation error should be introduced earlier and kept consistent between the theoretical statements and the numerical section.
  2. [Section 5] The variance estimator is asserted to be consistent, but the rate or the conditions under which the plug-in estimator converges are not displayed; a short remark or reference to an auxiliary lemma would clarify this.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough review and constructive feedback on our work. We address each of the major comments in detail below, outlining how we plan to revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Proof of Theorem 2.1 / Section 3] The non-trivial uniform control on the optimal transport potentials across directions (invoked to obtain E[SW_n] - SW = o_p(n^{-1/2})) is the load-bearing step that permits population centering. The manuscript should state the precise moment conditions on the measures that guarantee this control is uniform in the slicing measure and does not deteriorate with dimension or tail heaviness; without explicit bounds or a counter-example check, it is unclear whether the o_p(n^{-1/2}) claim holds for the full range of non-compact measures advertised.

    Authors: We thank the referee for this observation. The manuscript's moment assumptions (as stated prior to Theorem 2.1) are sufficient for the uniform control, but we acknowledge that the bound was not made fully explicit. In the revision, we will add a lemma deriving the uniform bound on the potentials, showing it holds uniformly over directions and does not deteriorate with dimension under these assumptions. We will also include a brief discussion confirming the o_p(n^{-1/2}) rate for the non-compact measures considered. revision: yes

  2. Referee: [Section 2.2 and Theorem 2.1] The Efron-Stein application yields a CLT centered at the expectation; the passage from expectation to population sliced distance therefore rests entirely on the potential-control argument. If that argument requires stronger integrability than the CLT itself, the claimed improvement over the general Wasserstein case is narrower than stated.

    Authors: The potential-control argument uses precisely the same moment conditions as the Efron-Stein step and the CLT. We will add a remark in Section 2.2 clarifying that no additional integrability is needed, thereby maintaining the claimed scope of the improvement over the general Wasserstein setting. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation uses external inequalities plus new non-referential control argument

full rationale

The abstract and context describe the CLT as obtained from the Efron-Stein inequality together with a new uniform control on OT potentials across directions. No equations or steps are shown that reduce a claimed prediction to a fitted parameter, a self-citation chain, or a definitional tautology. The population-centering improvement is presented as following from the new control, which is not described as imported from prior self-work or as an ansatz. This is the normal non-circular case.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The claim rests on standard results from probability (Efron-Stein inequality) and optimal transport theory plus one domain-specific control on transport potentials; no free parameters or new entities are introduced in the abstract.

assumptions (2)
  • standard math Efron-Stein inequality applies to the sliced Wasserstein functional
    Explicitly invoked to obtain variance control for the CLT
  • domain assumption Optimal transport potentials admit uniform control across projection directions
    Described as the non-trivial technical ingredient enabling the result for p>1

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Cite this review

Pith. "Pith review of An improved central limit theorem for the empirical sliced Wasserstein distance." pith.science (2026). https://pith.science/paper/KEH2HH72

@misc{pith2026250318831,
  author       = {Pith},
  title        = {Pith review of: An improved central limit theorem for the empirical sliced Wasserstein distance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KEH2HH72}},
  note         = {Machine review of arXiv:2503.18831}
}
read the original abstract

Wasserstein distances are widely used in modern data analysis but pose significant computational and statistical challenges in high dimensions. The sliced Wasserstein distance alleviates these challenges by leveraging one-dimensional projections. Building on the Efron-Stein inequality-a technique proven effective in related problems-and a non-trivial control of the optimal transport potentials across directions, we establish a central limit theorem for the p-sliced Wasserstein distance, for p>1, centered at the expected empirical cost. Unlike for the general Wasserstein distance, the centering can be replaced by the population cost, enabling valid statistical inference. This generalizes and refines existing one-dimensional results, providing the first asymptotically valid inference framework for the sliced Wasserstein distance between possibly non-compact measures. Finally, we address other practical aspects crucial for inference, including Monte Carlo approximation of the slicing integral and consistent variance estimation.

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Forward citations

Cited by 1 Pith paper

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  1. Distributional Limit Theory for Optimal Transport

    math.ST 2025-05 conditional novelty 5.0 of 10

    A survey of central limit theorems for empirical optimal transport, with a new one-dimensional L1 cost fluctuation CLT and a list of open problems.

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