Dimension-free Convergence Rate in Sliced Wasserstein Distance for Empirical Measures of Markov Processes

Feng-Yu Wang

REVIEW 2 major objections 5 minor 38 references

Dimension-free Convergence Rate in Sliced Wasserstein Distance for Empirical Measures of Markov Processes

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper claims that the sliced Wasserstein distance W_{p,λ}, averaging one-dimensional projections over a full-support measure on the dual sphere, is topologically stronger than finite-dimensional-distribution convergence on Banach spaces

desk verdict Sliced-Wasserstein rates on Banach spaces hit a real snag: the central topological theorem rests on a false uniform estimate, though the quantitative dimension-free bounds may survive. read the letter →

arxiv 2607.20150 v1 pith:WURQZTD7 submitted 2026-07-22 math.PR

Feng-Yu Wang This is my paper · ORCID

classification math.PR MSC 60A1060J60

keywords slicedWassersteindistancedimension-freeconvergencerateempiricalmeasureMarkovprocessBanachspacefinite-dimensionaldistributionsergodic

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 proposes the sliced Wasserstein distance W_{p,λ} on probability measures over a Banach space, averaging one-dimensional Wasserstein distances over a full-support probability measure λ on the dual unit sphere. Its central theoretical claim is that convergence in W_{p,λ} is equivalent to convergence in finite-dimensional distributions together with a uniform integrability condition, making W_{p,λ} a topologically stronger notion than f.d.d. convergence in infinite dimensions. The pay-off is a collection of dimension-free bounds — often of sharp order t^{-1} or n^{-1} — for how quickly empirical measures of exponentially (or weakly) ergodic Markov processes converge to their invariant measure, with applications to infinite-dimensional SPDEs. The author's stated motivation is that this gives an efficient way to simulate infinite-dimensional distributions without the curse of dimensionality that afflicts classical Wasserstein distances.

What carries the argument

The sliced distance W_{p,λ}(μ,ν):=(∫_S W_p(μ^θ,ν^θ)^p λ(dθ))^{1/p}, with μ^θ the push-forward of μ under the bounded linear functional θ. It turns a difficult infinite-dimensional transport problem into a family of one-dimensional transport problems that can be attacked with empirical-process and spectral-gap methods. The proof of the topological characterization relies on Fourier transforms and Kantorovich duality to relate W_{p,λ} to finite-dimensional distributions; the rate theorems use ergodicity (exponential in Assumption (A1), subexponential in (4.1)) together with Poincaré and heat-kernel bounds for the projected one-dimensional laws.

What would settle it

Evaluate the sup in inequality (2.6) for B=ℓ^2 and ξ a nonzero coordinate functional: for θ≠θ_ξ, sup_{x∈B} |e^{i∥ξ∥θ(x)} - e^{iξ(x)}| = 2, not ≤ε/4, because ∥ξ∥(θ−θ_ξ) is an unbounded linear functional. Exhibiting a sequence θ_k→θ_ξ with λ(D_ε)>0 but W_p distributions converging would make the claimed implication concrete to test.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.1(1): for λ with full support on the dual unit sphere, W_{p,λ}(μ_n,μ)→0 holds if and only if μ_n converges to μ in finite-dimensional distributions and the sliced p-th moments satisfy the uniform integrability condition (2.1). The paper further proves that in finite dimensions W_{p,λ} is topologically equivalent to the usual Wasserstein distance, that W_{p,λ} is complete under a moment bound on a separable Hilbert space, and that it gives dimension-free convergence rates for empirical measures of ergodic Markov processes, including sharp t^{-1}/n^{-1} rates for W_{2,λ} under explicit spectral assumptions.

Load-bearing premise

The theorem presupposes that a tiny change in the projection direction changes the exponential e^{iξ(x)} by at most ε/4 uniformly over all x∈B; for unbounded linear functionals on an infinite-dimensional Banach space this uniform bound is unavailable once the directions differ at all.

Editorial extensions

If this is right

If W_{p,λ} defines a metric equivalent to W_p on R^d and dimension-free on infinite-dimensional spaces, empirical-measure convergence rates no longer degrade with the ambient dimension.
For exponentially ergodic Markov processes with invariant measure having enough moments, the bound (3.11) gives E_μ[W_{p,λ}(μ_t,μ)^{2p}] ≤ C/t, and the analogous discrete-time bound with C/n.
Under the stronger assumptions (A2), the W_{2,λ} rate is sharp: t E_μ[W_{2,λ}(μ_t,μ)^2] stays bounded away from zero and above, giving the t^{-1} order as the exact rate.
For Markov processes with subexponential ergodicity, the rates remain t^{-1} and n^{-1} provided the decay profile ξ(t) is integrable, covering heavy-tailed potentials.
The SPDE application shows the framework handles infinite-dimensional state spaces, so the empirical measures of solutions to partially dissipative semilinear SPDEs converge to the invariant measure at the same dimension-free rate.

Reading between the lines

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

The rate theorems in Sections 3–5 rely on one-dimensional projection bounds and would remain meaningful even if the topological equivalence theorem requires modification; the equivalence is used for motivation and for identifying the limit, not for the rate estimates themselves.
The non-completeness example in Remark 1.1 suggests that for actual simulation or inference, one would likely work inside a completion of the Banach space or restrict attention to measures supported on a smaller subspace, so the practical use of W_{p,λ} may not require a Polish space.
A numerical comparison of W_{p,λ} versus W_p for the empirical measure of a dissipative SPDE projected onto high-dimensional subspaces would test whether the claimed dimension-free rates translate into observable accuracy gains in practice.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper's genuine contribution is a dimension-free convergence-rate result for empirical measures of ergodic Markov processes under a sliced Wasserstein distance on a Banach space. The estimates in Sections 3–4, and the SPDE application in Section 5, are interesting and probably right as statements about the pseudometric W_{p,λ}. That part deserves a serious look. But the paper as written makes a load-bearing claim that I think is false or at least unsupported: Theorem 2.1(1) asserts that W_{p,λ} convergence is equivalent to convergence in finite-dimensional distributions plus uniform integrability. The proof fails at display (2.6), where the author claims that for θ close to θ_ξ in the dual norm, the exponentials e^{i‖ξ‖θ(x)} and e^{iξ(x)} are uniformly close for all x in a bounded set. For θ≠θ_ξ, the difference is a nonzero linear functional, so it is unbounded on any ball; the difference of exponentials can be arbitrarily close to 2. The uniform ε/4 bound is simply wrong. That kills the contradiction argument.

The reader's concrete counterexample with μ_n=N(0,n e_n e_n^T) does not actually work as written: for each fixed θ, only finitely many n give a non-δ_0 projection, so μ_n converges in f.d.d. to δ_0. But the failure of (2.6) is real, and I suspect the theorem is false in the full-support generality. With λ supported on a countable dense set, W_{p,λ} only averages countably many directions; one can choose λ with weights decaying fast and construct μ_n where a carefully chosen θ* outside the atomic set sees N(0,1) infinitely often, while the λ-weighted average goes to zero. The topological claims and the metric/Polish-space assertions in Theorem 2.1(2)–(3) rest on this wobbly foundation.

The quantitative rate theorems are less affected: they are inequalities for a fixed λ, and the integration over θ is honest. If Theorem 2.1(1) is removed or corrected to a weaker statement, those results can still stand as convergence in the pseudometric sense. The assumption that λ has full support also needs a caveat: such λ need not exist when the dual ball is not separable.

Bottom line: the paper has a useful core and the author is clearly capable, but the central topological theorem needs a major rework. I would send it to peer review, with a clear expectation of revision.

Referee Report

2 major / 5 minor

Summary. The paper introduces a sliced Wasserstein distance W_{p,λ} on the space of probability measures with finite p-th moment on a Banach space B, using a full-support probability λ on the unit sphere of the dual space B*. Theorem 2.1 claims that W_{p,λ}-convergence is equivalent to convergence in finite-dimensional distributions plus a uniform p-th integrability condition on the projected tail masses, and that in Hilbert spaces W_{p,λ}-Cauchy sequences with bounded p-th moments converge. The main body uses this framework to derive dimension-free convergence rates for empirical measures of Markov processes: exponential ergodicity is treated in Theorems 3.2 and 3.4, a subexponential ergodicity extension in Theorem 4.1, and an application to partially dissipative SPDEs in Theorem 5.1. The paper also claims sharpness for some of these rates.

Significance. If the topological characterization in Theorem 2.1 were correct, the paper would provide a genuine infinite-dimensional extension of sliced Wasserstein distances and useful dimension-free bounds for Markov-chain simulation. The quantitative estimates are nontrivial and appear to be independent of dimension; the SPDE application is concrete. However, the entire advertised framework rests on Theorem 2.1(1), which is false. The paper's central claim that W_{p,λ} convergence is stronger than finite-dimensional-distribution convergence is contradicted by a counterexample, and the proof relies on an invalid uniform estimate. The rate theorems may still be interesting as estimates for the pseudometric W_{p,λ}, but the paper's stated interpretation and topological conclusions cannot be accepted as they stand.

major comments (2)

[Theorem 2.1(1), Eq. (2.6)] The proof of Theorem 2.1(1)(a) is invalid. For θ ∈ D_ε with θ ≠ θ_ξ, the linear functional ξ − ∥ξ∥_* θ is nonzero and hence unbounded on the infinite-dimensional Banach space B. Therefore the left-hand side of (2.6) is not uniformly bounded by ε/4 in x ∈ B; for any such θ one can choose x making the exponential difference arbitrarily close to 2. Consequently the lower bound in (2.7) does not follow. The theorem is not merely unproved but false. Let B = ℓ^2, let {v_m} be a dense sequence of finite-support unit vectors enumerated so that v_{m,n} ≠ 0 implies m ≥ 2n, and set λ = Σ_m 2^{-m} δ_{v_m}. This λ has full support on the unit sphere, and ∫ θ_n^2 λ(dθ) ≤ 2^{-2n+1}. Take μ_n = δ_{n e_n} and μ = δ_0. Then W_{2,λ}(μ_n,μ)^2 = n^2 ∫ θ_n^2 λ(dθ) → 0. But for ξ = (n^{-3/4}) ∈ ℓ^2, the characteristic function of μ_n at ξ is exp(i n ξ_n) = exp(i n^{1/4}), which does not converge to 1. Hence μ_
[Theorem 2.1(2)(3)] The failure of (2.6) propagates to the rest of Theorem 2.1. The Fourier-Cauchy argument in the proof of Theorem 2.1(2) explicitly invokes 'the argument leading to (2.7)', and the proof of Theorem 2.1(3) uses Theorem 2.1(1) to obtain weak convergence from W_{p,λ} convergence. Both steps are therefore unsupported. The counterexample above also shows that W_{p,λ} does not control convergence of cylindrical Fourier transforms uniformly over B*, so the Bochner-Minlos route cannot work as written. While the finite-dimensional part Theorem 2.1(3) may be true by other arguments, it is not established by the supplied proof. More importantly, the advertised property that W_{p,λ} is topologically stronger than finite-dimensional-distribution convergence is false: the distance can tend to zero while the measures are point masses moving macroscopically far apart in B. The motivation that this distanc

minor comments (5)

[Introduction] Typos: 'Lispchitz' should be 'Lipschitz'; 'machining learning' should be 'machine learning'.
[Proof of Theorem 2.1] The notation S^d and Sd appears in several places where S is clearly intended, e.g., in equations (2.8) and the surrounding text. Since S already denotes the unit sphere, the exponent d is confusing.
[Remark 1.1] Remark 1.1 appears after Theorem 2.1 and should be renumbered as Remark 2.1 or similar.
[Theorem 3.2] Near the end of the proof, 'Then (5.9) holds' should refer to the displayed equation of the theorem, presumably (3.5).
[References] Reference [34] is missing a comma between the author's name and the title; the reference list should be checked for formatting consistency.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the dimension-free rates follow from mixing/spectral assumptions; only minor non-load-bearing self-citations are present.

full rationale

The paper's main estimates (Theorems 3.2, 3.4, 4.1) are upper bounds on E[W_{p,λ}(μ_t,μ)^{2p}]. Their constants come from the explicit ergodicity assumptions (A1)/(4.1) and from the spectral/heat-kernel quantities α(θ), γ(θ,t) in (A2); they are not fitted to the t^{-1} or n^{-1} rates, and the sharpness claim in Remark 3.1 is anchored by an external CLT lower bound [38]. Theorem 2.1 is a topological characterization; even if the reader's objection to estimate (2.6) is correct, that is a proof error, not a reduction of the conclusion to its own inputs. The paper cites several works by the same author ([27], [32], [33], [36]), but these are used for parameter-free supporting lemmas (heat-kernel bounds, a coupling construction) or contextual asymptotics whose assumptions do not include the target rates. Thus there is no fitted-input-called-prediction, no ansatz-smuggling, and no self-citation chain forcing the central claims. The score of 2 reflects the presence of these minor self-citations, not an actual circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities or parameters. Its mathematical results rest on mixing/spectral assumptions about the Markov process and on the existence of a full-support λ, plus standard tools from optimal transport. The invalid inequality (2.6) is a proof error, not an assumed axiom.

assumptions (4)

domain assumption (A1): Exponential convergence of the Markov semigroup in L^2(μ): ||P_t − μ||_{L^2(μ)} ≤ c0 e^{-κ0 t}.
Used in Theorem 3.2 and elsewhere to bound the variance of integrated indicator functions. This is a strong mixing assumption taken as input, not derived.
ad hoc to paper Existence of a probability measure λ on the unit sphere S of B* with full support.
Such a measure exists only if S is separable; the paper states results for general Banach space without this hypothesis. This is needed for W_{p,λ} to be a separating metric.
domain assumption (A2): For each θ, the projected one-dimensional diffusion has a Poincaré inequality, heat-kernel moment bound, and heat-kernel trace bound (3.15)-(3.17).
Assumed in Theorem 3.4 to obtain sharp O(t^{-1}) and O(n^{-1}) rates for W_{2,λ}. Verified only in Gaussian and uniform-ball examples.
standard math The finite-dimensional result [3, Proposition 7.4] bounding 1D empirical Wasserstein distance by integrated sup-differences of CDFs.
External known result used in the derivation of Theorem 3.2; not re-proved.

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Cite this review

Pith. "Pith review of Dimension-free Convergence Rate in Sliced Wasserstein Distance for Empirical Measures of Markov Processes." pith.science (2026). https://pith.science/paper/WURQZTD7

@misc{pith2026260720150,
  author       = {Pith},
  title        = {Pith review of: Dimension-free Convergence Rate in Sliced Wasserstein Distance for Empirical Measures of Markov Processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WURQZTD7}},
  note         = {Machine review of arXiv:2607.20150}
}

abstract

To derive dimension-free convergence rates of empirical measures for Markov processes on a Banach space, we adopt the sliced Wasserstein distance (SW distance) induced by a probability measure with full support on the unit ball of the dual space. This distance is topologically stronger than the convergence in finite-dimensional distributions, and is topologically equivalent to the Wasserstein distance when the Banach space is finite-dimensional. Under this distance, we derive dimension-free convergence rates for the empirical measures of ergodic Markov processes on $\BB$, which can be sharp as illustrated by concrete examples. The study provides an efficient way to simulate infinite-dimensional distributions using sample trajectories of Markov processes, so that the $``$curse of dimensionality" appearing to the classical Wasserstein distance is avoided. The main results apply to a broad class of infinite-dimensional models, and are illustrated by partially dissipative SPDEs in the end of the paper.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Ambrosio, F

L. Ambrosio, F. Stra, D. Trevisan,A PDE approach to a 2-dimensional matching prob- lem,Probab. Theory Related Fields 173(2019), 433-477

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