pith. sign in

arxiv: 1907.07900 · v1 · pith:I6I5UWXEnew · submitted 2019-07-18 · 🧮 math.AP · math-ph· math.MP· math.OC

Multi-marginal Entropy-Transport with repulsive cost

Augusto Gerolin , Anna Kausamo , Tapio Rajala This is my paper

Pith reviewed 2026-05-24 19:53 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPmath.OC
keywords multi-marginal optimal transportentropy-transportGamma-convergencerepulsive costKantorovich dualitymetric spaces
0
0 comments X

The pith

Entropy-transport functionals with repulsive costs admit minimizers in metric spaces and Gamma-converge to multi-marginal optimal transport.

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

The paper studies the entropy-transport functional when paired with repulsive cost functions in a multi-marginal setting. It supplies conditions that guarantee the existence of a minimizer within a broad class of metric spaces. The same conditions allow the authors to prove that the entropy-transport functional Gamma-converges to the corresponding unregularized multi-marginal optimal transport problem with repulsive cost. They further establish an entropy-regularized form of Kantorovich duality for these functionals.

Core claim

Under sufficient conditions on the repulsive cost, the entropy-transport functional possesses a minimizer in a class of metric spaces. As the entropy parameter tends to zero, the functional Gamma-converges to the multi-marginal optimal transport problem with the same repulsive cost. The entropy-regularized version of Kantorovich duality holds for the functional.

What carries the argument

The entropy-transport functional with repulsive cost functions, which augments a multi-marginal transport cost by an entropy term.

If this is right

  • Minimizers of the regularized entropy-transport problem exist under the stated conditions on the cost.
  • Solutions to the unregularized multi-marginal problem can be recovered as limits of the regularized minimizers.
  • Kantorovich duality applies directly to the entropy-regularized problem.

Where Pith is reading between the lines

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

  • The Gamma-convergence result supplies a variational justification for using entropy regularization as an approximation scheme for repulsive multi-marginal problems.
  • The duality statement may permit dual formulations that simplify numerical computation of the regularized problem in concrete metric spaces.

Load-bearing premise

The cost functions must satisfy a repulsive property that enables both existence of minimizers and the Gamma-convergence statement.

What would settle it

A metric space and repulsive cost for which no minimizer of the entropy-transport functional exists would falsify the existence claim.

Figures

Figures reproduced from arXiv: 1907.07900 by Anna Kausamo, Augusto Gerolin, Tapio Rajala.

Figure 1
Figure 1. Figure 1: The dependence of the minimizer of the entropic-transport problem (1.1) on the entropic parameter ε for the one-dimensional Coulomb cost, N = 2 and ρ ∼ N(0, 5). The pictures show part of the support of the optimal coupling γε around the origin. From the left to right: ε = 104 , 10−2 , 10−3 , 10−4 , 10−5 . Organization of the paper. In Section 2 we introduce the setting and study sufficient conditions for t… view at source ↗
read the original abstract

In this paper we study theoretical properties of the entropy-transport functional with repulsive cost functions. We provide sufficient conditions for the existence of a minimizer in a class of metric spaces and prove the $\Gamma$-convergence of the entropy-transport functional to a multi-marginal optimal transport problem with a repulsive cost. We also prove the entropy-regularized version of the Kantorovich duality.

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

0 major / 3 minor

Summary. The manuscript studies the entropy-transport functional with repulsive cost functions. It provides sufficient conditions for existence of minimizers in a class of metric spaces, proves Γ-convergence of the entropy-transport functional to the multi-marginal optimal transport problem with repulsive cost, and establishes the entropy-regularized Kantorovich duality.

Significance. If the results hold, the work supplies a useful extension of entropy-regularized multi-marginal transport theory to repulsive costs in metric spaces. The Γ-convergence statement supplies a rigorous justification for approximation schemes, while the duality result extends classical Kantorovich theory to the regularized setting. These contributions are of moderate significance for researchers working on multi-marginal problems and numerical optimal transport.

minor comments (3)
  1. [Section 2] §2 (or wherever the repulsive property is defined): the precise statement of the repulsive condition on the cost should be isolated as a numbered assumption or definition to make the hypotheses of Theorems 3.1, 4.2, and 5.3 immediately verifiable.
  2. [Section 4] The Γ-convergence proof (likely §4) relies on tightness arguments; a short remark on whether the metric-space assumptions guarantee uniform integrability of the entropy terms would clarify the passage to the limit.
  3. [Throughout] Notation for the multi-marginal entropy functional should be introduced once and used consistently; occasional switches between E_ε and F_ε (if present) slow reading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of our manuscript on the entropy-transport functional with repulsive costs and for recommending minor revision. The assessment that the work provides a useful extension of entropy-regularized multi-marginal transport theory, along with rigorous justification via Gamma-convergence and duality, is appreciated. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper consists of existence proofs, Gamma-convergence arguments, and duality results for an entropy-transport functional under explicitly stated sufficient conditions (including a repulsive cost property) in metric spaces. These are standard mathematical derivations in optimal transport; no parameters are fitted to data, no quantities are renamed as predictions, and no load-bearing steps reduce by definition or self-citation to the target conclusions. The repulsive property is an input assumption, not derived from the results. The derivation chain is self-contained against external benchmarks in analysis and optimal transport theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no information on free parameters, background axioms, or new entities; all three lists are therefore empty.

pith-pipeline@v0.9.0 · 5584 in / 1069 out tokens · 18683 ms · 2026-05-24T19:53:32.053095+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 3 internal anchors

  1. [1]

    Agueh and G

    M. Agueh and G. Carlier , Barycenters in the Wasserstein space , SIAM J. on Mathematical Analysis, 43 (2011), pp. 904–924

    work page 2011

  2. [2]

    Splitting Methods in Communication and Imaging, Science and Engineering

    J.-D. Benamou, G. Carlier, and L. Nenna , A numerical method to solve optimal transport problems with Coulomb cost, to appear as a chapter in “Splitting Methods in Communication and Imaging, Science and Engineering”, Springer, (2015)

    work page 2015

  3. [3]

    , Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm , arXiv preprint arXiv:1710.08234, (2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  4. [4]

    Buttazzo, T

    G. Buttazzo, T. Champion, and L. De Pascale , Continuity and estimates for multimarginal optimal transportation problems with singular costs , Applied Mathematics & Optimization, (2016), pp. 1–16

    work page 2016

  5. [5]

    Buttazzo, L

    G. Buttazzo, L. De Pascale, and P. Gori-Giorgi, Optimal-transport formulation of electronic Density Functional Theory, Physical Review A, 85 (2012), p. 062502

    work page 2012

  6. [6]

    Carlier, V

    G. Carlier, V. Duval, G. Peyr ´e, and B. Schmitzer , Convergence of entropic schemes for optimal transport and gradient flows , SIAM J. Math. Anal., 49 (2017), pp. 1385–1418

    work page 2017

  7. [7]

    Cotar, G

    C. Cotar, G. Friesecke, and C. Kl¨uppelberg, Density Functional Theory and optimal transportation with Coulomb cost , Comm. Pure Appl. Math., 66 (2013), pp. 548–599

    work page 2013

  8. [8]

    , Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg- Kohn functional, Archive for Rational Mechanics and Analysis, 228 (2018), pp. 891–922

    work page 2018

  9. [9]

    Cuturi , Sinkhorn distances: Lightspeed computation of optimal transport , in Advances in neural information processing systems, 2013, pp

    M. Cuturi , Sinkhorn distances: Lightspeed computation of optimal transport , in Advances in neural information processing systems, 2013, pp. 2292–2300

    work page 2013

  10. [10]

    Di Marino, L

    S. Di Marino, L. De Pascale, and M. Colombo , Multimarginal optimal transport maps for 1- dimensional repulsive costs , Canadian Journal of Mathematics – Journal Canadien des Math´ ematiques, 67 (2015), pp. 350–368

    work page 2015

  11. [11]

    Di Marino and A

    S. Di Marino and A. Gerolin, An optimal transportation approach for the Schr¨ odinger Bridge problem and convergence of the Sinkhorn algorithm , in preparation

    work page

  12. [12]

    Di Marino, A

    S. Di Marino, A. Gerolin, and L. Nenna , Optimal transport theory for repulsive costs , Topological Optimization and Optimal Transport: In the Applied Sciences, 17 (2017)

    work page 2017

  13. [13]

    Ekeland and R

    I. Ekeland and R. Temam, Analyse convexe et problemes variationnelles, Dunod Gauthier-Villars, Paris, (1974)

    work page 1974

  14. [14]

    Flamary and N

    R. Flamary and N. Courty, POT Python Optimal Transport library , 2017

    work page 2017

  15. [15]

    Gerolin, A

    A. Gerolin, A. Kausamo, and T. Rajala , Duality theory for multimarginal optimal transport with repulsive costs in metric spaces , ESAIM Control Optim. Calc. Var., (to appear), (2018)

    work page 2018

  16. [16]

    Second order differentiation formula on compact $RCD^*(K,N)$ spaces

    N. Gigli and L. Tamanini , Second order differentiation formula on compact RCD∗(K, N) spaces, arXiv:1701.03932, (2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  17. [17]

    Gozlan and C

    N. Gozlan and C. L´eonard, Transport inequalities. a survey, Markov Processes and Related Fields, 16 (2010), pp. 635–736

    work page 2010

  18. [18]

    H. G. Kellerer, Duality theorems for marginal problems , Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und verwandte Gebiete, 67 (1984)

    work page 1984

  19. [19]

    L´eonard, From the Schr¨ odinger problem to the Monge–Kantorovich problem, Journal of Functional Analysis, 262 (2012), pp

    C. L´eonard, From the Schr¨ odinger problem to the Monge–Kantorovich problem, Journal of Functional Analysis, 262 (2012), pp. 1879–1920

    work page 2012

  20. [20]

    , A survey of the Schr¨ odinger problem and some of its connections with optimal transport , Discrete Contin. Dyn. Syst., (2014), pp. 1533–1574

    work page 2014

  21. [21]

    E. H. Lieb, Density functionals for Coulomb systems , in Inequalities, Springer, 2002, pp. 269–303

    work page 2002

  22. [22]

    Mikami, Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes , Probability theory and related fields, 129 (2004), pp

    T. Mikami, Monge’s problem with a quadratic cost by the zero-noise limit of h-path processes , Probability theory and related fields, 129 (2004), pp. 245–260

    work page 2004

  23. [23]

    Nenna, Numerical Methods for Multi-Marginal Opimal Transportation , PhD thesis, Universit´ e Paris- Dauphine, 2016

    L. Nenna, Numerical Methods for Multi-Marginal Opimal Transportation , PhD thesis, Universit´ e Paris- Dauphine, 2016

    work page 2016

  24. [24]

    Peyr´e and M

    G. Peyr´e and M. Cuturi , Computational Optimal Transport, vol. 11, Now Publishers, Inc., 2019

    work page 2019

  25. [25]

    Schr¨odinger, ¨Uber die umkehrung der naturgesetze , Verlag Akademie der wissenschaften in kommis- sion bei Walter de Gruyter u

    E. Schr¨odinger, ¨Uber die umkehrung der naturgesetze , Verlag Akademie der wissenschaften in kommis- sion bei Walter de Gruyter u. Company, 1931

    work page 1931

  26. [26]

    The strictly-correlated electron functional for spherically symmetric systems revisited

    M. Seidl, S. Di Marino, A. Gerolin, K. Giesbertz, L. Nenna, and P. Gori-Giorgi , The Strictly- Correlated Electron functional for spherically symmetric systems revisited , Accepted in Physical Review A (available in arXiv:1702.05022), (2016)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  27. [27]

    Sturm, On the geometry of metric measure spaces , Acta Math., 196 (2006), pp

    K.-T. Sturm, On the geometry of metric measure spaces , Acta Math., 196 (2006), pp. 65–131. 20 AUGUSTO GEROLIN, ANNA KAUSAMO, AND TAPIO RAJALA Vrije Universiteit Amsterdam, Department of Theoretical Chemistry, FEW, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands E-mail address: augustogerolin@gmail.com University of Jyvaskyla, Department of Mathemati...

    work page 2006