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Rigidity and Quantitative Stability of the Sliced Wasserstein Deficit

T0 review · 0 major / 3 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read The sliced Wasserstein deficit is zero exactly when the Brenier map between the measures is homothetic affine.

desk verdict The paper gives a precise rigidity result for when the sliced Wasserstein deficit vanishes and introduces a new SPK constant for stability, which looks like a genuine addition worth checking in detail. read the letter →

arxiv 2605.25453 v2 pith:SPVLDZAO submitted 2026-05-25 math.MG math.FAmath.PRmath.SP

Bang-Xian Han This is my paper · ORCID
classification math.MGmath.FAmath.PRmath.SP
keywords slicedWassersteindistanceBreniermaprigidityoptimaltransportquantitativestabilityPoincaréinequalitygradientfields
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 examines the nonnegative difference between the squared Wasserstein distance of order two and d times the squared sliced Wasserstein distance. It proves this difference vanishes if and only if the gradient of the convex potential pushing one measure forward to the other is an affine map of the form lambda x plus a constant vector. The result requires the source measure to be absolutely continuous with respect to Lebesgue measure. The authors then introduce a sliced Poincaré-Korn constant that controls how much the deficit can be bounded away from zero when the map stays close to affine, and they compute the constant explicitly for Gaussians while showing that standard curvature or Poincaré assumptions are insufficient to guarantee positivity of the constant.

What carries the argument

The sliced Wasserstein deficit D(μ,ν) = (1/d)W₂²(μ,ν) - SW₂²(μ,ν), whose vanishing is equivalent to the Brenier map being homothetic affine.

What would settle it

A pair of measures in dimension two or higher, with the source absolutely continuous, whose Brenier map is known to be nonlinear, yet whose sliced Wasserstein deficit is exactly zero, would disprove the rigidity statement.

Watch

Extended reading notes

Core claim

We prove that D(μ,ν)=0 if and only if the Brenier map T=∇φ from μ to ν is homothetic affine, T(x)=λx+b μ-a.e., for some λ≥0 and b∈R^d. For quantitative stability, we introduce the sliced Poincaré-Korn constant κ_SPK(μ) as a spectral gap of an averaged ridge-projection quadratic form on gradient fields modulo the family {λx+b}. Whenever this constant is positive, we prove a stability estimate for the sliced Wasserstein deficit, up to a one-dimensional Lipschitz scale for the projected monotone transports. We obtain the sharp SPK constant for the Gaussian measures, and establish positive SPK bounds for bounded perturbations of the Gaussian and compact classes of gradient fields for fixed sourc

Load-bearing premise

The source measure must be absolutely continuous with respect to Lebesgue measure so that a unique Brenier map exists.

Editorial extensions

If this is right

  • The inequality SW₂²(μ,ν) ≤ (1/d)W₂²(μ,ν) becomes equality precisely when the transport map is homothetic affine.
  • A positive sliced Poincaré-Korn constant yields a quantitative bound relating the size of the deficit to the distance of the Brenier map from the affine class, up to one-dimensional Lipschitz factors.
  • The sliced Poincaré-Korn constant is positive and sharp for centered Gaussians and remains positive under small bounded perturbations of the Gaussian.
  • Anisotropic Gaussians demonstrate that neither Bakry-Émery curvature bounds nor standard Poincaré inequalities are sufficient to guarantee a positive sliced Poincaré-Korn constant.

Reading between the lines

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

  • The rigidity result may be used to characterize when high-dimensional measures differ only by an affine transformation through projection-based distances.
  • Numerical approximation of the sliced Poincaré-Korn constant for a given source measure could provide practical error controls when replacing Wasserstein distances by their sliced versions in computations.
  • The obstruction examples suggest that any attempt to derive sliced Poincaré-Korn inequalities from curvature must incorporate dimension-dependent or anisotropy-dependent corrections.
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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

0 major / 3 minor

Summary. The paper proves that the sliced Wasserstein deficit D(μ,ν) := (1/d)W₂²(μ,ν) − SW₂²(μ,ν) vanishes if and only if the Brenier map T = ∇φ from μ to ν is homothetic affine (T(x) = λx + b μ-a.e.) for μ ≪ ℒ^d in dimension d ≥ 2. It introduces the sliced Poincaré–Korn constant κ_SPK(μ) as a spectral gap on gradient fields modulo affine homotheties and derives quantitative stability estimates whenever this constant is positive. Explicit sharp values are obtained for Gaussian measures, positive bounds are established for bounded perturbations of Gaussians and compact classes of gradient fields, and anisotropic Gaussians are shown to obstruct both Bakry–Émery curvature and standard Poincaré inequalities from implying a global SPK inequality.

Significance. The rigidity characterization clarifies the equality case in the elementary comparison between sliced and full Wasserstein distances, which underpins the use of sliced Wasserstein distances as computationally tractable proxies in statistics and machine learning. The SPK constant supplies a new, explicitly computable stability modulus with sharp Gaussian examples; the obstruction result for anisotropic Gaussians is a useful negative result that delineates the reach of curvature-based methods.

minor comments (3)
  1. Abstract, line 3: 'an new spectral gap' should read 'a new spectral gap'.
  2. The stability statement is phrased 'up to a one-dimensional Lipschitz scale for the projected monotone transports'; a precise statement of the dependence on this scale (or a reference to the relevant proposition) would improve readability.
  3. The paper invokes μ ≪ ℒ^d to guarantee uniqueness of the Brenier map; a brief reminder of the classical theorem (e.g., Brenier or McCann) in the setup section would help readers who are not optimal-transport specialists.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the paper, the positive significance assessment, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper establishes a rigidity theorem stating that the sliced Wasserstein deficit D(μ,ν) vanishes if and only if the Brenier map is homothetic affine, under the standard assumption μ ≪ ℒ^d that guarantees existence and uniqueness of the map via classical optimal transport theory. The SPK constant is introduced as a new spectral gap quantity on gradient fields modulo affine homotheties and used to derive stability estimates when positive, with explicit computations for Gaussians. No load-bearing steps reduce by construction to fitted inputs, self-definitions, or unverified self-citations; the derivation chain relies on standard properties of Wasserstein distances and Brenier maps without circular reduction to the paper's own inputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 1 invented entities

The paper relies on standard results from optimal transport theory and introduces one new entity without external evidence.

assumptions (2)
  • standard math Existence and uniqueness of the Brenier map T=∇φ when the source measure is absolutely continuous with respect to Lebesgue measure
    Invoked in the setup and the statement of the rigidity result for measures in P2(R^d), d≥2.
  • standard math The inequality SW_2^2(μ,ν) ≤ (1/d) W_2^2(μ,ν) that defines the non-negative deficit
    Stated as the elementary comparison that motivates the deficit.
invented entities (1)
  • sliced Poincaré-Korn (SPK) constant κ_SPK(μ)
    purpose: To serve as a spectral gap that enables quantitative stability estimates for the sliced Wasserstein deficit
    Newly defined as an averaged ridge-projection quadratic form on gradient fields modulo affine maps; no independent evidence outside the paper is provided.

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Pith. "Pith review of Rigidity and Quantitative Stability of the Sliced Wasserstein Deficit." pith.science (2026). https://pith.science/paper/SPVLDZAO

@misc{pith2026260525453,
  author       = {Pith},
  title        = {Pith review of: Rigidity and Quantitative Stability of the Sliced Wasserstein Deficit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SPVLDZAO}},
  note         = {Machine review of arXiv:2605.25453}
}
abstract

The sliced Wasserstein distance $SW_2(\mu,\nu)$ compares high-dimensional probability measures by averaging one-dimensional optimal transport distances over linear projections. Although sliced Wasserstein distances are now standard computational tools in statistics, imaging, and machine learning, the rigidity behind the elementary comparison \[ SW_2^2(\mu,\nu)\leq \frac1d W_2^2(\mu,\nu) \] has not been systematically studied. Let $\mu,\nu\in\mathcal P_2(\mathbb R^d)$, $d\ge2$, with $\mu\ll\mathcal L^d$, and define the sliced Wasserstein deficit by \[ {\mathrm D}(\mu,\nu):=\frac1d W_2^2(\mu,\nu)-SW_2^2(\mu,\nu)\geq 0. \] We prove that ${\mathrm D}(\mu,\nu)=0$ if and only if the Brenier map $T=\nabla\varphi$ from $\mu$ to $\nu$ is homothetic affine, \[ T(x)=\lambda x+b \qquad \mu\text{-a.e.}, \] for some $\lambda\ge0$ and $b\in \mathbb R^d$. For quantitative stability, we introduce the sliced Poincar\'e--Korn (SPK) constant $\kappa_{\mathrm{SPK}}(\mu)$, defined as an new spectral gap of an averaged ridge-projection quadratic form on gradient fields modulo the family $\{\lambda x+b\}$. Whenever this constant is positive, we prove a stability estimate for the sliced Wasserstein deficit, up to a one-dimensional Lipschitz scale for the projected monotone transports. We obtain the sharp SPK constant for the Gaussian measures as the most important example, and establish positive SPK bounds for bounded perturbations of the Gaussian and compact classes of gradient fields for fixed source measures. Finally, we show that anisotropic Gaussians give a sharp obstruction: neither a Bakry--\'Emery lower curvature bound nor a usual Poincar\'e inequality alone can imply a global sliced Poincar\'e--Korn inequality.

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