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arxiv: 2606.20710 · v1 · pith:JEYKJETUnew · submitted 2026-06-16 · 🧮 math.NA · cs.NA· math.PR

Variable Exponent Wasserstein Spaces: Stability of Entropy Convexity and Modified R\'enyi Entropy

Ambroise Soglo , Cyrille Comb\'et\'e , Koffi Wilfrid Hou\'edanou , L\'eonard Todjihound\'e This is my paper

Pith reviewed 2026-06-26 23:57 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.PR
keywords variable exponent Wasserstein spaceentropy convexitymodified Renyi entropyBakry-Emery tensorLog-Sobolev inequalityTalagrand inequalityoptimal transport
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The pith

Entropy remains convex along geodesics in Wasserstein spaces with small spatial changes to the transport exponent.

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

The paper extends the Lott-Villani theorem by equipping the space of probability measures with a distance Wp built from the Lagrangian |v| raised to p(x) = 2 + ε(x), where ε is a small function. It proves that Boltzmann entropy satisfies convexity with constant K minus C times the supremum norm of ε along the resulting geodesics. A modified Rényi entropy is constructed to cancel an extra logarithmic term that appears in the second-order expansion of Wp squared. This adjustment produces an exact match between the convexity deficit and the Bakry-Émery tensor. The same construction yields versions of the logarithmic Sobolev and Talagrand inequalities that survive the perturbation.

Core claim

In the Wasserstein space equipped with the variable-exponent distance Wp derived from |v|^{p(x)}, the entropy satisfies (K - C‖ε‖∞)-convexity along Wp-geodesics. A modified Rényi entropy compensates the log divergence in Wp² expansions, yielding a sharp equivalence where the Bakry-Émery tensor determines the effective curvature.

What carries the argument

The modified Rényi entropy that exactly cancels the logarithmic correction arising in the second-order expansion of Wp squared.

If this is right

  • Perturbed logarithmic Sobolev inequalities hold in the variable-exponent Wasserstein space.
  • Perturbed Talagrand inequalities continue to be valid under the same perturbation.
  • The functional inequalities remain robust when the transport exponent varies spatially by a small amount.
  • The Lott-Villani theorem on entropy convexity generalizes to this class of spatially varying metrics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The stability result suggests that other curvature-dimension conditions in optimal transport may tolerate small metric perturbations.
  • The same logarithmic compensation technique could be tested on time-dependent or random exponents.
  • Numerical verification on low-dimensional manifolds would give concrete bounds on the constant C appearing in the convexity deficit.
  • The construction may connect to variable-exponent Sobolev inequalities arising in nonlinear PDE analysis.

Load-bearing premise

The perturbation ε must be small enough in supremum norm that Wp defines a genuine metric and the logarithmic term in its squared expansion remains controllable.

What would settle it

An explicit calculation on a compact Riemannian manifold with a chosen ε whose supremum norm exceeds a threshold, showing that either the convexity constant drops below zero or the modified Rényi entropy no longer matches the Bakry-Émery tensor.

Figures

Figures reproduced from arXiv: 2606.20710 by Ambroise Soglo, Cyrille Comb\'et\'e, Koffi Wilfrid Hou\'edanou, L\'eonard Todjihound\'e.

Figure 1
Figure 1. Figure 1: Boltzmann entropy (shifted by log(2π)) along W2 and Wp(θ) geodesics on S 1 . Both curves are computed analytically to order O(α 2 ) with α = 0.5, ε = 0.1. The upward shift of the Wp(·) curve is consistent with K − C∥ε∥∞ = 0 − C · 0.1. 10.2 Remark on the torus T 2 A similar analytical computation can be performed on T 2 = S 1 × S 1 with product metric (also flat, K = 0), using tensor products of the S 1 den… view at source ↗
read the original abstract

We study the Wasserstein space $\mathcal{P}(M)$ equipped with a distance $\Wp$ constructed from the Lagrangian $L(x,v)=|v|^{p(x)}$ where $p(x)=2+\varepsilon(x)$ with $\varepsilon$ small. Building on the fundamental work of Lott and Villani on the $K$-geodesic convexity of the Boltzmann entropy in $(\mathcal{P}(M),\Wb)$, we establish a generalized inequality showing that the entropy remains $\left(K - C\|\varepsilon\|_\infty\right)$-convex along $\Wp$-geodesics. We then introduce a modified R\'enyi entropy that exactly compensates the logarithmic divergence that appears in the expansions of $\Wp^2$, obtaining thus a sharp equivalence that reaveals the Bakry-\'Emery tensor as the effective curvature in the variable exponent setting. As applications, we derive perturbed versions of the Log-Sobolev and Talagrand inequalities in variable exponent Wasserstein spaces, showing that these fundamental functional inequalities are robust under small perturbations of the transport exponent. This work generalizes the Lott-Villani theorem and its consequences (\emph{J. Lott and C. Villani, Ann. of Math. \textbf{169} (2009), 903-991}) to situations where the transport metric varies spatially.

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.

Referee Report

2 major / 1 minor

Summary. The paper generalizes the Lott-Villani theorem on K-geodesic convexity of Boltzmann entropy in Wasserstein space to variable-exponent metrics Wp induced by the Lagrangian L(x,v)=|v|^{p(x)} with p(x)=2+ε(x) for small ε. It establishes (K-C‖ε‖∞)-convexity of entropy along Wp-geodesics and introduces a modified Rényi entropy that compensates the logarithmic divergence in the Wp² expansion, yielding a sharp equivalence identifying the Bakry-Émery tensor as effective curvature. Perturbed Log-Sobolev and Talagrand inequalities are derived as applications, showing robustness under small spatial perturbations of the exponent.

Significance. If the small-ε regime and expansion control are rigorously established, the result meaningfully extends curvature-dimension conditions and functional inequalities from constant-exponent to spatially varying transport costs. It directly builds on the Lott-Villani framework and demonstrates stability of key inequalities, which could be relevant for modeling inhomogeneous media or variable-metric optimal transport problems.

major comments (2)
  1. [Abstract (central construction) and the section deriving the modified Rényi entropy] The (K-C‖ε‖∞)-convexity and the modified-Rényi equivalence both rest on the claim that the second-order expansion of Wp² contains only a controllable logarithmic correction when ‖ε‖∞ is small. The manuscript must explicitly state the threshold on ‖ε‖∞ that guarantees Wp is a metric and that spatial variation of ε introduces no non-logarithmic error terms in the second variation of geodesics; without this, the compensation performed by the modified Rényi entropy is incomplete and the Bakry-Émery identification fails.
  2. [The section establishing the perturbed convexity inequality] The second-variation analysis for Wp-geodesics is asserted to carry over from the constant-p case once ‖ε‖∞ is small, but no explicit error estimates or verification of the small-ε regime are provided. If non-uniform terms arise from the x-dependence in L(x,v), the claimed sharp equivalence to the Bakry-Émery tensor does not hold; a concrete bound on the remainder in the expansion of Wp² is required.
minor comments (1)
  1. [Abstract] Typo in abstract: 'reaveals' should read 'reveals'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for your thorough and constructive review. We agree that explicit statements on the threshold for ‖ε‖∞ and concrete remainder bounds are needed to make the small-ε regime fully rigorous. We address each major comment below and will incorporate the requested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract (central construction) and the section deriving the modified Rényi entropy] The (K-C‖ε‖∞)-convexity and the modified-Rényi equivalence both rest on the claim that the second-order expansion of Wp² contains only a controllable logarithmic correction when ‖ε‖∞ is small. The manuscript must explicitly state the threshold on ‖ε‖∞ that guarantees Wp is a metric and that spatial variation of ε introduces no non-logarithmic error terms in the second variation of geodesics; without this, the compensation performed by the modified Rényi entropy is incomplete and the Bakry-Émery identification fails.

    Authors: We agree that an explicit threshold must be stated. The current manuscript assumes ε sufficiently small for p(x) > 1 and for Wp to be a metric, but does not record the precise bound. In revision we will add the condition ‖ε‖∞ < 1/2 (ensuring 1 < p(x) < 3 uniformly), under which the second-order expansion of Wp² admits only a controllable logarithmic correction with no non-logarithmic remainder arising from the x-dependence of ε. This statement will appear in the abstract and be proved in the section on the modified Rényi entropy, thereby completing the compensation argument and the identification with the Bakry-Émery tensor. revision: yes

  2. Referee: [The section establishing the perturbed convexity inequality] The second-variation analysis for Wp-geodesics is asserted to carry over from the constant-p case once ‖ε‖∞ is small, but no explicit error estimates or verification of the small-ε regime are provided. If non-uniform terms arise from the x-dependence in L(x,v), the claimed sharp equivalence to the Bakry-Émery tensor does not hold; a concrete bound on the remainder in the expansion of Wp² is required.

    Authors: We accept that explicit remainder estimates were omitted. In the revised manuscript we will derive a concrete bound showing that the error term in the second variation of Wp² is at most C‖ε‖∞ (log(1/‖ε‖∞) + 1) along geodesics, under the threshold ‖ε‖∞ < 1/2. This bound confirms that no non-uniform terms beyond the logarithmic correction appear, so the second-variation analysis carries over from the constant-p case and the sharp equivalence to the Bakry-Émery tensor remains valid. The estimate will be inserted in the section on perturbed convexity. revision: yes

Circularity Check

1 steps flagged

Modified Rényi entropy defined to exactly compensate log divergence in Wp², yielding sharp equivalence by construction

specific steps
  1. self definitional [Abstract]
    "We then introduce a modified Rényi entropy that exactly compensates the logarithmic divergence that appears in the expansions of Wp², obtaining thus a sharp equivalence that reaveals the Bakry-Émery tensor as the effective curvature in the variable exponent setting."

    The modified Rényi entropy is introduced precisely to cancel the log divergence in the Wp² expansion; the claimed sharp equivalence to the Bakry-Émery tensor therefore follows immediately from this definitional choice rather than from a separate derivation.

full rationale

The paper's central claim of a sharp equivalence revealing the Bakry-Émery tensor as effective curvature rests on introducing a modified Rényi entropy chosen specifically to cancel the logarithmic term arising in the Wp² expansion. This step reduces the equivalence to the definition of the compensator rather than an independent derivation. The (K - C‖ε‖∞)-convexity perturbs the external Lott-Villani theorem and does not itself reduce by construction, so overall circularity is partial rather than total. No other load-bearing steps (self-citations, fitted parameters, or imported uniqueness) are exhibited in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; free parameters, axioms, and invented entities cannot be audited in detail. The construction implicitly assumes the Lagrangian L(x,v)=|v|^{p(x)} defines a valid distance for small ε and that the second variation of Wp² admits a controllable logarithmic term.

pith-pipeline@v0.9.1-grok · 5799 in / 1240 out tokens · 25086 ms · 2026-06-26T23:57:06.855776+00:00 · methodology

discussion (0)

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Reference graph

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