An improved central limit theorem for the empirical sliced Wasserstein distance

David Rodr\'iguez-V\'itores; Eustasio del Barrio; Jean-Michel Loubes

arxiv: 2503.18831 · v3 · pith:KEH2HH72new · submitted 2025-03-24 · 🧮 math.ST · stat.TH

An improved central limit theorem for the empirical sliced Wasserstein distance

David Rodr\'iguez-V\'itores , Eustasio del Barrio , Jean-Michel Loubes This is my paper

Pith reviewed 2026-05-22 22:49 UTC · model grok-4.3

classification 🧮 math.ST stat.TH

keywords sliced Wasserstein distancecentral limit theoremEfron-Stein inequalityoptimal transportstatistical inferenceempirical processesnon-compact measures

0 comments

The pith

The empirical p-sliced Wasserstein distance obeys a central limit theorem centered at the population distance for p greater than 1.

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

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.

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.

What carries the argument

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

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.

Where Pith is reading between the lines

These are 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.

Load-bearing premise

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

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.

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.

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.

Desk Editor's Note private letter to a colleague

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

The main thing to know is that the authors prove a central limit theorem for the empirical p-sliced Wasserstein distance (p>1) centered at the population sliced distance rather than only at its expectation. This removes a bias obstacle and opens the door to valid inference procedures when the measures lack compact support. They reach this by combining the Efron-Stein inequality with a uniform control on the one-dimensional optimal transport potentials across directions, then add Monte Carlo approximation of the slicing integral and consistent variance estimation as practical follow-ons. The work extends existing one-dimensional CLTs to the sliced setting in higher dimensions and states the result explicitly for non-compact measures. The motivation and the added implementation pieces are handled clearly. The soft spot sits in the uniform control on the potentials. Without compactness the argument rests on moment conditions, and it is not obvious from the abstract how strong those conditions must be or whether the bound stays uniform when dimension grows or tails get heavier. If that control slips, the o_p(n^{-1/2}) claim for the centering difference would not hold. The abstract treats the step as non-trivial but feasible, so the details decide how far the result reaches. This is written for statisticians who need asymptotic justification for sliced Wasserstein distances in applications with unbounded supports. A reader working on rigorous inference for optimal transport distances would find the technical development useful. It deserves a serious referee because the claimed advance targets a real gap that has limited practical use of these distances.

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)

[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.
[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)

[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.
[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

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
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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 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.

axioms (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

pith-pipeline@v0.9.0 · 5680 in / 1336 out tokens · 69276 ms · 2026-05-22T22:49:24.583657+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we establish a central limit theorem for the p-sliced Wasserstein distance, for p>1, centered at the expected empirical cost... using the Efron-Stein inequality and a non-trivial control of the optimal transport potentials across directions
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4.1... SJα(P) < ∞ and moment assumptions... bias term vanishes

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.