Regularized Moment Measures

Alex Delalande; Sara Farinelli

REVIEW 4 major objections 6 minor 1 cited by

Regularized Moment Measures

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper shows that adding a quadratic second-moment penalty to the optimal-transport variational problem for moment measures makes every probability measure with finite first moment an α-Gaussian regularized moment measure, and that…

desk verdict Useful new regularized moment measure problem with a repairable proof bug in the main stability theorem. read the letter →

arxiv 2506.13218 v2 pith:RPVQJPSH submitted 2025-06-16 math.FA math.AP

Alex Delalande , Sara Farinelli This is my paper · ORCID

classification math.FAmath.AP MSC 49Q2249K4039B62

keywords log-concavemeasuresWassersteindistancestabilityinequalitymomentmaximumcorrelationfunctionaloptimaltransportentropyregularizationHölder

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

This paper asks how sensitively moment maps depend on their moment measures, and answers it by adding a strongly convex quadratic penalty to the variational problem that underlies the moment measure equation. For any probability measure $\mu$ with finite first moment, the regularized functional $E(\rho)+T(\rho,\mu)+\frac{\alpha}{2}M_2(\rho)$ has a unique minimizer $\rho_\mu$, and that minimizer is a log-concave density solving the modified equation $(\nabla u)_\sharp e^{-u-\frac{\alpha}{2}|x|^2}=\mu$. The regularization removes the usual restrictions that the target have center of mass at the origin and not be supported on a hyperplane, so every measure qualifies. The main quantitative result is Theorem 3.1: for targets with bounded second moments and centers of mass in a fixed ball, $W_2(\rho_\mu,\rho_\nu)\le C(d,\alpha,M,B)\,W_2(\mu,\nu)^{1/2}$. Consequently, approximating a target by an $N$-point discrete measure with Wasserstein error of order $N^{-1/d}$ yields regularized solutions converging at rate $N^{-1/(2d)}$.

What carries the argument

The engine is the regularized functional $F_\mu(\rho)=E(\rho)+T(\rho,\mu)+\frac{\alpha}{2}M_2(\rho)$, where $E$ is the entropy, $M_2$ is the second moment, and the maximum correlation functional $T(\rho,\mu)=\max_{\gamma\in\Gamma(\rho,\mu)}\int\langle x,y\rangle\,d\gamma$ is the optimal-transport duality coupling between $\rho$ and $\mu$. The quadratic moment term is strongly convex along 2-Wasserstein geodesics, while $E$ and $T(\cdot,\mu)$ are geodesically convex, the latter property being imported from the cited variational treatment of moment measures; hence $F_\mu$ is strongly geodesically convex. That strong convexity gives existence and uniqueness of minimizers and, through a comparison of $F_\mu$ and $F_\nu$ along the geodesic between the two minimizers, yields the half-Hölder stability bound. A supporting lemma identifies the isotropic Gaussian as the entropy minimizer at fixed second moment, which bounds the size of the minimizers and makes the constants in the bound explicit.

What would settle it

Take $d=1$, $\alpha=1$, $\mu=\delta_0$ and $\nu_\varepsilon=(1-\varepsilon)\delta_0+\varepsilon\delta_1$, and compute the unique minimizers of the regularized problem for both targets. Since $W_2(\mu,\nu_\varepsilon)=\sqrt{\varepsilon}$, Theorem 3.1 predicts that $W_2(\rho_{\delta_0},\rho_{\nu_\varepsilon})/\sqrt{\varepsilon}$ stays bounded as $\varepsilon\to 0$; a computation showing this ratio diverges would refute the stability claim.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the moment measure problem admits a canonical Gaussian regularization with complete well-posedness. For every $\mu\in P_1(\mathbb{R}^d)$ and every $\alpha>0$, the regularized problem has a unique minimizer $\rho_\mu$ of the form $\rho_\mu=Z^{-1}e^{-u-\frac{\alpha}{2}|\cdot|^2}$ for a convex, lower semicontinuous, essentially continuous function $u$; conversely, every log-concave density of this form is the minimizer for its own target $\mu=(\nabla u)_\sharp\rho_\mu$. The center of mass of the minimizer is forced to be $-b(\mu)/\alpha$, replacing the translation invariance of the classical problem with an explicit shift law. The central stability theorem states that $W_2(\rho_\mu,\rho_\nu)\le C(d,\alpha,M,B)\,W_2(\mu,\nu)^{1/2}$ for targets with bounded second moment and center of mass in a fixed ball, so the assignment $\mu\mapsto\rho_\mu$ is $\tfrac12$-Hölder continuous on Wasserstein balls.

Load-bearing premise

The load-bearing premise is that the maximum correlation functional $T(\cdot,\mu)$ is convex along 2-Wasserstein geodesics, a property the paper imports from the cited moment-measure literature; if it failed, the regularized functional would not be strongly geodesically convex and the Hölder stability bound would not follow.

Editorial extensions

If this is right

Every probability measure with finite first moment is an $\alpha$-Gaussian regularized moment measure, so the modified equation is solvable for singular targets, hyperplane-supported targets, and targets with nonzero center of mass.
The map $\mu\mapsto\rho_\mu$ is $\tfrac12$-Hölder continuous on Wasserstein balls with bounded second moments and centers of mass, giving a quantitative stability statement for regularized moment maps.
If a target is approximated by an $N$-point discrete measure with $W_2(\mu,\mu_N)\sim N^{-1/d}$, the regularized minimizers converge at rate $N^{-1/(2d)}$ in $W_2$.
The center of mass of the minimizer obeys $b(\rho_\mu)=-\frac{1}{\alpha}b(\mu)$, an explicit shift law that replaces the translation symmetry of the classical moment measure problem.
As $\alpha\to 0$ the stability constant degenerates, so the bound does not directly transfer to the unregularized problem; the paper notes that $\Gamma$-convergence of the functionals is expected by standard arguments but does not prove it.

Reading between the lines

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

As an extension, the explicit uniform constant in Theorem 3.1 makes the regularized problem usable as a certified numerical scheme: before the true moment map is known, one can bound the Wasserstein error caused by discretizing the target measure.
The same strong-convexity mechanism is likely to work for any sufficiently regular $1$-strongly convex potential in place of $\|x\|^2/2$; the quadratic choice is a template, and one could tune the potential for computational convenience.
The degeneracy of the constant as $\alpha\to0$ suggests that any stability estimate for the classical moment measure problem must either weaken the Hölder modulus or depend on a quantity that degenerates near singular targets; testing this would connect the regularized and unregularized regimes.

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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

Read the Delalande–Farinelli note on regularized moment measures. The honest one-liner: this is a useful paper with a genuinely new result, but the written proof of the main stability theorem contains a false identity (a missing factor 1/2) and the functional is defined with two different coefficients (α vs α/2) in different sections. I believe the theorem is true and the proof is repairable, but as printed it is not correct.

What's new and good: the α-Gaussian regularized moment measure equation (∇u)_# e^{-u-(α/2)|x|^2} = μ is new, and the authors prove existence of a unique minimizer for every μ with finite first moment, without the usual barycenter-at-zero or not-supported-on-hyperplane conditions. That's a clean improvement over Santambrogio's setting. The stability result, Theorem 3.1, gives a 1/2-Hölder bound W2(ρμ,ρν) ≤ C W2(μ,ν)^{1/2} under bounded second moments and barycenters, with a constant that depends on d, α, M, B. The strategy—geodesic convexity of T imported from Santambrogio, plus the strong convexity from the second moment—is sound. The application to discrete approximation of moment measures (rate N^{-1/2d}) is explicitly drawn out and is a genuinely practical takeaway.

The soft spots are real but not terminal. In the proof of Theorem 3.1, the identity T(ρν,μ) = -W2²(ρν,μ)+M2(ρν)+M2(μ) is missing the factor 1/2. From the definition of T and W2, the correct identity is T = (M2+M2−W2²)/2. The printed version propagates into the final inequality; the repaired proof gives α W2² ≤ S·W2 rather than α/2 W2² ≤ S·W2, so the claimed 1/2-Hölder rate survives with an adjusted constant. The authors need to fix this before the paper can be trusted as is.

Second, the coefficient of M2 is inconsistent. Problem (5) is defined with α M2, but Section 3 writes the minimizer as argmin of E+T+(α/2)M2, and the existence proof's lower bound uses α/2 M2 as well. A reader cannot tell which functional is actually being studied. This is purely a presentation bug, but it should be caught.

Third, the introduction promises Gamma-convergence to the moment-measure functional as α→0 but explicitly does not prove it. That's an admitted omission; it's not load-bearing for Theorem 3.1, but it should be stated as a conjecture, not a fact.

The citation pattern is clean: the paper builds directly on Santambrogio's variational approach and optimal transport duality, with no self-citation loops. The imports from [17] are appropriate and the proof skeleton is coherent.

My recommendation: this deserves a serious referee. It's a short, substantive note with a genuine new object and a near-proof of the advertised stability estimate. The flaws are fixable and don't undermine the central claims. I'd send it to review, asking the authors to correct the identity, unify the coefficient convention, and either prove or qualify the Gamma-convergence statement.

Referee Report

4 major / 6 minor

Summary. The paper introduces a strongly convex regularized version of Santambrogio's optimal-transport formulation of the moment measure problem. For a parameter α>0 the authors study minimizers of ρ↦E(ρ)+T(ρ,μ)+αM2(ρ) (in places written with α/2) and show that the minimizer is a log-concave density of the form e^{-u-α||x||²/2}, whose gradient pushes it forward to μ; thus every μ with finite first moment is an α-Gaussian regularized moment measure. The main result, Theorem 3.1, is a stability estimate W2(ρμ,ρν) ≤ C(d,α,M,B) W2(μ,ν)^{1/2} under bounded second moments and barycenters in a fixed ball. Section 4 gives a converse statement. The proofs use optimal transport duality and results imported from Santambrogio [17]. The paper is clearly written, but several identities in the proof of the main theorem are false as printed and need correction.

Significance. If the stability estimate is established, it is a genuinely new quantitative contribution: the authors correctly note that no stability result for moment maps under perturbation of the moment measure appears in the literature, and the regularized problem removes the usual restrictions (barycenter at the origin, measure not supported on a hyperplane) for existence. The potential numerical application to approximate a general measure by finitely supported measures is a useful motivation. The paper relies on established optimal transport theory and imported results from [17], and there is no circularity. However, the proof of the main theorem contains a false key identity, and the notation for the regularization parameter is inconsistent; these issues are repairable, but they must be fixed before the paper can be accepted.

major comments (4)

[Section 3, proof of Theorem 3.1 (after Eq. (21))] The identity T(ρν,μ) = -W2²(ρν,μ) + M2(ρν) + M2(μ) is false. With the paper's definition T(ρ,μ)=maxγ∫⟨x,y⟩dγ and the elementary relation W2²(ρ,μ)=M2(ρ)+M2(μ)-2T(ρ,μ), the correct identity is T(ρν,μ)=(M2(ρν)+M2(μ)-W2²(ρν,μ))/2. The missing factor 1/2 propagates into the estimate following (21): as printed one obtains α/2 W2² ≤ S W2(μ,ν), whereas the corrected identity yields α W2² ≤ S W2(μ,ν) with S the sum in (22). Since S is bounded by a constant depending on d, α, M, B, the final Hölder bound still follows with modified constants, but the proof as written is invalid at this step and must be corrected.
[Eqs. (5), (8), (9), (11), Section 3] The regularization term is defined inconsistently. The Introduction and Section 1 define Fμ(ρ)=E(ρ)+T(ρ,μ)+αM2(ρ), while Lemma 2.2, the strong convexity estimate (9), and Section 3 use E(ρ)+T(ρ,μ)+(α/2)M2(ρ). The modified moment equation (6) corresponds to the α/2 convention. Because the constants in (9), (11), (21), and the bound (24) depend on this choice, the authors must adopt a single convention and restate all formulas consistently.
[Eq. (19), Lemma 3.2, Eqs. (26)-(27)] The stated entropy of an isotropic Gaussian is incorrect. For a Gaussian γ with M2(γ)=∫|x|²dγ, the true value of E(γ)=∫γlogγ is -d/2 log(2πe M2(γ)/d), not -d/2 log(2M2(γ)/(dπe)). The incorrect formula is used in (26) and (27) to derive the key bound (24). With the correct formula the bound can still be obtained because only qualitative growth estimates are needed, but the displayed identities and the constants derived from them must be fixed.
[Section 4, Lemma 4.1 and proof of Proposition 4.2] The converse direction is not proved as written. Lemma 4.1 is stated without proof ('one can deduce'), and the proof of Proposition 4.2 is a sketch that refers to [17, Theorem 4.3] for the definition of the competitors and contains unverified derivative formulas, including the computation of M2(ρε) in (17). Since the Introduction claims that solving (5) is equivalent to solving the modified moment measure equation, this gap should be closed or the claim weakened. This issue does not affect Theorem 3.1, but it affects the paper's stated equivalence.

minor comments (6)

[Section 1, after the definition of T] The sentence 'if u is a minimizer, then ∇u is the optimal transport map between μ and ν' should read 'between ρ and μ'; the measure ν has not been introduced at that point.
[Proof of Proposition 2.3] The displayed variable 'v=b(ρ)-1/α b(μ)' should be 'v=b(ρ)+1/α b(μ)' to be consistent with the next line and with the representation of an arbitrary ρ as a translation of a measure with barycenter -b(μ)/α.
[Introduction, Gamma-convergence paragraph] The statement that the α-Gaussian moment measure functional Gamma-converges to the moment measure functional as α→0 is asserted without proof. Either provide a proof or clearly label the statement as a conjecture and remove it from the main narrative.
[Eq. (6)] Equation (6) writes the pushforward of e^{-u-α||x||²/2} without a normalizing constant. For consistency with Proposition 2.1, either insert the factor 1/Z or state that u is chosen so that the density integrates to one.
[Proof of Proposition 2.4, Eq. (17)] The computation of M2(ρε) appears to be off by a factor 1/2: a direct calculation with M2(ρ)=∫|x|²dρ gives M2(ρε)=M2(ρ)-ε∫Aε x1 dρ + (1/2)ε²ρ(Aε), not +ε²ρ(Aε). Please check this display and the resulting inequality.
[Remark 3.1] The phrase 'From the previous proposition' should read 'From Theorem 3.1', since the Hölder continuity of μ↦ρμ is a consequence of the theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chain rests on external optimal-transport results (Santambrogio [17]) and standard duality; the printed factor error in the T identity is a correctness defect, not a circular reduction.

full rationale

I walked the derivation chain. The stability theorem (Theorem 3.1) uses the strong geodesic convexity estimate (9), which follows from geodesic convexity of the maximum correlation functional imported from Santambrogio [17, Prop. 3.3]; that is an external published result, not a self-citation, and the paper does not fit any parameter to the target bound. The characterization of minimizers in Proposition 2.1 reduces to optimal transport duality for T and to standard lower-semicontinuity arguments, again external. Remark 2.2 establishes the equivalence between minimizers and alpha-Gaussian regularized moment maps via the already-chosen u and classical transport theory; this is a theorem, not a definitional identification of the conclusion with the input. Proposition 4.2 follows from geodesic convexity and computations of [17]. No self-citation chain from the present authors is load-bearing. I also weighed the manuscript's own admitted gap: 'The Gamma-convergence ... that we do not prove here for brevity, follows by standard arguments' is an unsupported assertion in the introduction, but it is not used to prove Theorem 3.1 and is not circular. Finally, the missing factor 1/2 in the identity 'T(ρν, μ) = -W2^2(ρν, μ)+M2(ρν)+M2(μ)' (immediately after (21)) is a genuine proof defect: with the paper's definition T = max ∫⟨x,y⟩, the exact identity is T = (M2ρ + M2μ - W2^2)/2. The printed inequality needs a constant adjustment, but the claimed Hölder bound still follows from the intended argument, so the defect is one of correctness, not equivalence-by-construction. Under the seven enumerated circularity patterns, no step reduces to its own input.

Assumptions & free parameters 1 free parameters · 3 assumptions · 1 invented entities

The paper introduces one new mathematical object (the regularized moment measure) and one free parameter α; everything else is imported from standard optimal transport and from Santambrogio's and Cordero-Erasquin-Klartag's results.

free parameters (1)

α = α > 0 (chosen, not fitted)
Regularization strength; appears in the density and in the stability constant. Not fit to data.

assumptions (3)

domain assumption Maximum correlation functional T(·,μ) is convex along Wasserstein geodesics (Prop. 3.3 of [17])
Used to prove strong geodesic convexity of the regularized functional and the stability inequality.
domain assumption Lower semicontinuity of entropy and T with respect to narrow convergence under moment bounds (Props. 2.1 and 3.1 of [17])
Needed to pass from minimizing sequences to a minimizer in Proposition 2.1.
standard math Standard optimal transport duality: T admits primal/dual representation and equality (e.g. [16])
Defines T and justifies the characterization of minimizers via potentials u.

invented entities (1)

α-Gaussian regularized moment measure and moment map
purpose: Generalizes the moment measure equation so that a solution exists for every μ with finite first moment, and enables quantitative stability.
New mathematical object introduced in this paper; no external empirical handle, but internally well-defined.

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Cite this review

Pith. "Pith review of Regularized Moment Measures." pith.science (2026). https://pith.science/paper/RPVQJPSH

@misc{pith2026250613218,
  author       = {Pith},
  title        = {Pith review of: Regularized Moment Measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RPVQJPSH}},
  note         = {Machine review of arXiv:2506.13218}
}

abstract

In the work "Dealing with moment measures via entropy and optimal transport", Santambrogio provided an optimal transport approach to study existence of solutions for the moment measure equation, that is: given $\mu$, find $u$ such that $ (\nabla u)_{\sharp}e^{-u}=\mu$. In particular he proves that $u$ satisfies the previous equation if and only if $e^{-u}$ is the minimizer of an entropy and a transport cost. Here we study a modified minimization problem, in which we add a strongly convex regularization depending on a positive $\alpha$ and we link its solutions to a modified moment measure equation $(\nabla u)_{\sharp}e^{-u-\frac{\alpha}{2} \|x\|^2}= \mu$. Exploiting the regularization term, we study the stability of the minimizers.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Quantitative stability for the Brascamp-Lieb inequality and moment measures
math.FA 2025-11 conditional novelty 7.0 of 10

For any convex potential, the L1 distance from a function to the Brascamp-Lieb optimizer manifold is controlled by the square root of its deficit, with a dimension-only constant independent of the potential.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

[17]

Santambrogio

F. Santambrogio. Dealing with moment measures via entropy and optimal transport.Journal of Functional Analysis, 271(2):418–436, 2016. Email address: delalande.alex@gmail.com Università di Genova, Dipartimento di Matematica, MaLGa, Genova, Italy Email address: sara.farinelli@edu.unige.it

work page 2016
[1]

K. J. Böröczky and A. De. Stability of the Prékopa-Leindler inequality for log-concave functions. Advances in Mathematics, 386:107810, 2021

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K. J. Böröczky, A. Figalli, and J. PG Ramos. A quantitative stability result for the Prékopa- Leindler inequality for arbitrary measurable functions.Annales de l’Institut Henri Poincaré C, 41(3):565–614, 2023

work page 2023
[3]

Cordero-Erausquin and B

D. Cordero-Erausquin and B. Klartag. Moment measures. Journal of Functional Analysis, 268(12):3834–3866, 2015

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De Philippis, A.R

G. De Philippis, A.R. Mészáros, F. Santambrogio, and B. Velichkov. Bv estimates in optimal transportation and applications.Archive for Rational Mechanics and Analysis, 219:829–860, 2016

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M. Fathi. Stein kernels and moment maps.The Annals of Probability, 47(4):2172–2185, 2019

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Fathi and D

M. Fathi and D. Mikulincer. Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and stein kernels.Annali Scuola Normale Superiore - Classe di Scienze, 23(3):1417–1445, 2022

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Figalli and J

A. Figalli and J. PG Ramos. Improved stability versions of the Prékopa-Leindler inequality, 2024

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H. Khanh and F. Santambrogio. q-moment measures and applications: A new approach via optimal transport. Journal of Convex Analysis, 28(4):1033–1052, 2021. REGULARIZED MOMENT MEASURES 13

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [17]

Santambrogio

F. Santambrogio. Dealing with moment measures via entropy and optimal transport.Journal of Functional Analysis, 271(2):418–436, 2016. Email address: delalande.alex@gmail.com Università di Genova, Dipartimento di Matematica, MaLGa, Genova, Italy Email address: sara.farinelli@edu.unige.it

work page 2016

[2] [1]

K. J. Böröczky and A. De. Stability of the Prékopa-Leindler inequality for log-concave functions. Advances in Mathematics, 386:107810, 2021

work page 2021

[3] [2]

K. J. Böröczky, A. Figalli, and J. PG Ramos. A quantitative stability result for the Prékopa- Leindler inequality for arbitrary measurable functions.Annales de l’Institut Henri Poincaré C, 41(3):565–614, 2023

work page 2023

[4] [3]

Cordero-Erausquin and B

D. Cordero-Erausquin and B. Klartag. Moment measures. Journal of Functional Analysis, 268(12):3834–3866, 2015

work page 2015

[5] [4]

De Philippis, A.R

G. De Philippis, A.R. Mészáros, F. Santambrogio, and B. Velichkov. Bv estimates in optimal transportation and applications.Archive for Rational Mechanics and Analysis, 219:829–860, 2016

work page 2016

[6] [5]

M. Fathi. Stein kernels and moment maps.The Annals of Probability, 47(4):2172–2185, 2019

work page 2019

[7] [6]

Fathi and D

M. Fathi and D. Mikulincer. Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and stein kernels.Annali Scuola Normale Superiore - Classe di Scienze, 23(3):1417–1445, 2022

work page 2022

[8] [7]

Figalli and J

A. Figalli and J. PG Ramos. Improved stability versions of the Prékopa-Leindler inequality, 2024

work page 2024