pith. sign in

arxiv: 2604.19981 · v1 · submitted 2026-04-21 · 🧮 math.OC · math.FA

Debiasing optimal transport: classical and entropic

Pierre-Cyril Aubin-Frankowski , Virginie Ehrlacher , Gabriele Todeschi This is my paper

Pith reviewed 2026-05-10 01:44 UTC · model grok-4.3

classification 🧮 math.OC math.FA
keywords optimal transportdebiasabilityentropic regularizationunbalanced optimal transportmaximum mean discrepancycost functionsminimax
0
0 comments X

The pith

A symmetric cost function is debiasable exactly when it equals the infimum over an auxiliary space of the sum of a potential evaluated at each argument.

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

The paper defines a symmetric cost as debiasable if it obeys the inequality c(x,y) at least half of c(x,x) plus half of c(y,y). It proves this property is equivalent to the cost admitting the representation c(x,y) as the infimum over z of ψ(x,z) plus ψ(y,z) for some auxiliary space and function ψ. The authors apply the characterization to costs between probability measures, showing that entropic optimal transport costs with any positive regularization parameter are debiasable when the ground cost is negative definite or the induced kernel is continuous and positive definite. The same holds for the classical case and the maximum mean discrepancy limit, with all results extending to unbalanced transport and producing new decomposition formulas.

Core claim

A symmetric cost c is debiasable if and only if there exist an auxiliary set Z and a function ψ such that c(x,y) equals the infimum over z in Z of ψ(x,z) plus ψ(y,z). For entropic costs on spaces of probability measures with regularization ε in (0, +∞], this representation holds under negative definiteness of the ground cost or continuity and positive definiteness of the induced kernel, established by a convex-nonconcave minimax argument. The characterization covers ε = 0 and ε = +∞ as well and yields decomposition formulas for entropic optimal transport that extend to the unbalanced setting.

What carries the argument

The inf-representation c(x,y) = inf_z ψ(x,z) + ψ(y,z) for auxiliary space Z and function ψ, which is shown to be equivalent to debiasability and generalizes the midpoint identity for squared geodesic distances.

If this is right

  • Entropic optimal transport costs between probability measures are debiasable whenever the ground cost is negative definite.
  • The same debiasability holds for ε greater than zero when the induced kernel is continuous and positive definite.
  • The results apply equally to unbalanced optimal transport with different total masses.
  • New decomposition formulas become available for all regimes of entropic optimal transport.

Where Pith is reading between the lines

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

  • The inf-representation supplies a constructive way to verify the inequality without checking every pair of points.
  • The decomposition formulas may simplify numerical schemes for entropic distances beyond the cases already treated.
  • Similar inf-representations could be sought for other families of regularized transport costs on different spaces.

Load-bearing premise

Every symmetric cost satisfying the debiasability inequality is assumed to admit an inf-representation over some auxiliary space, but the paper does not prove that such a representation always exists.

What would settle it

Exhibit one concrete symmetric cost function that obeys c(x,y) greater than or equal to half c(x,x) plus half c(y,y) for all pairs yet cannot be written as inf_z ψ(x,z) + ψ(y,z) for any auxiliary Z and ψ, or compute an entropic cost whose ground cost is not negative definite and verify whether the inequality holds.

read the original abstract

We study the notion of debiasability for cost functions arising in optimal transport. We call a symmetric cost function $c:\mathscr{X}\times\mathscr{X}\to\mathbb{R}\cup\{+\infty\}$ debiasable if it satisfies $c(x,y)\ge \tfrac{1}{2}c(x,x)+\tfrac{1}{2}c(y,y)$ for all $x,y\in\mathscr{X}$. Building on an equivalent characterization by an inf-representation $c(x,y)=\inf_{z\in\mathscr{Z}}\psi(x,z)+\psi(y,z)$ for some set $\mathscr{Z}$ and some function $\psi: \mathscr{X}\times \mathscr{Z} \to \mathbb{R} \cup \{+\infty\}$, interpreted as a generalization of the midpoint identity for squared geodesic distances, we investigate the debiasability of costs defined on spaces of probability measures. Our primary focus is the entropic regularization of optimal transport across different regimes of the regularization parameter $\varepsilon \in [0,+\infty]$, encompassing classical optimal transport ($\varepsilon=0$), entropic optimal transport ($\varepsilon>0$), and the Maximum Mean Discrepancy ($\varepsilon=+\infty$). For $\varepsilon \in (0,+\infty]$, we investigate sufficient conditions, such as negative definiteness of the ground cost or continuity and positive definiteness of the induced kernel, handled then via a convex-nonconcave minimax argument. All our results extend naturally to unbalanced optimal transport settings and we generalize in this way the findings of \cite{feydy2019interpolating} and \cite{sejourne2019sinkhorn}. As a byproduct, we derive novel decomposition formulas for entropic optimal transport, which may be of independent interest.

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 / 2 minor

Summary. The manuscript defines debiasability for symmetric cost functions c on X×X by the inequality c(x,y) ≥ ½c(x,x) + ½c(y,y) and provides an equivalent inf-representation characterization c(x,y) = inf_z ψ(x,z) + ψ(y,z). It studies this property for entropic OT costs across ε ∈ [0,∞], deriving sufficient conditions (negative definiteness of the ground cost or continuity/positive-definiteness of the induced kernel) for ε > 0 via a convex-nonconcave minimax argument, extends all results to unbalanced OT, generalizes prior work on interpolating and Sinkhorn divergences, and obtains new decomposition formulas.

Significance. If the minimax argument is placed on firm footing, the work supplies a unified theoretical lens on debiasing across classical, entropic, and MMD regimes together with concrete sufficient conditions and unbalanced extensions; the decomposition formulas may also be useful independently.

major comments (2)
  1. [entropic regularization for ε > 0] The sufficient conditions for debiasability when ε > 0 rest on a convex-nonconcave minimax argument (abstract and the section treating entropic regularization). Standard minimax theorems do not apply directly to nonconcave problems; the manuscript must therefore verify the additional structure (compactness of the relevant sets, continuity of the objective, or existence of saddle points) that guarantees the inf-representation holds under the stated negative-definiteness or positive-definiteness hypotheses.
  2. [characterization of debiasability] The characterization theorem asserts that a symmetric cost is debiasable if and only if it admits the inf-representation over some auxiliary space Z. The manuscript does not demonstrate that such a representation exists for every symmetric cost satisfying the defining inequality, nor does it specify restrictions on Z that would make the equivalence hold in the OT setting.
minor comments (2)
  1. [introduction] The abstract states that the results generalize the findings of Feydy et al. (2019) and Séjourné et al. (2019); a short paragraph in the introduction contrasting the new minimax conditions and unbalanced extension with those earlier works would clarify the incremental contribution.
  2. [throughout] Notation for spaces (script X, Z, etc.) and the regularization parameter ε should be checked for consistency between the abstract, definitions, and statements of the main theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments, which will help strengthen the rigor and clarity of the manuscript. We address each major comment below and will incorporate revisions accordingly.

read point-by-point responses

  1. Referee: [entropic regularization for ε > 0] The sufficient conditions for debiasability when ε > 0 rest on a convex-nonconcave minimax argument (abstract and the section treating entropic regularization). Standard minimax theorems do not apply directly to nonconcave problems; the manuscript must therefore verify the additional structure (compactness of the relevant sets, continuity of the objective, or existence of saddle points) that guarantees the inf-representation holds under the stated negative-definiteness or positive-definiteness hypotheses.

    Authors: We agree that the application of the minimax argument requires explicit verification of the requisite structure. Under the negative-definiteness hypothesis on the ground cost, the space of probability measures is compact in the weak topology, and the entropic objective is jointly continuous. For the positive-definiteness case on the induced kernel, we can restrict to a compact subset of the reproducing kernel Hilbert space. In the revised manuscript we will add a dedicated paragraph (or subsection) in the entropic-regularization section that invokes a suitable minimax theorem for convex-nonconcave problems (e.g., via compactness and lower-semicontinuity) and confirms the existence of saddle points. This will place the argument on firm footing while leaving the main results unchanged. revision: yes

  2. Referee: [characterization of debiasability] The characterization theorem asserts that a symmetric cost is debiasable if and only if it admits the inf-representation over some auxiliary space Z. The manuscript does not demonstrate that such a representation exists for every symmetric cost satisfying the defining inequality, nor does it specify restrictions on Z that would make the equivalence hold in the OT setting.

    Authors: The equivalence is stated for general symmetric costs on an arbitrary space. One direction (inf-representation implies the inequality) is immediate. For the converse, an explicit construction of Z and ψ is possible: when the cost is continuous we may take Z = X and define ψ via a suitable extension; more generally Z can be taken as the completion of X with respect to the pseudometric induced by the inequality. We acknowledge that the manuscript presents the construction only in outline. In the revision we will expand the proof of the characterization theorem to give the full construction, state the precise restrictions on Z (e.g., Z Polish when working with Borel measures in OT), and verify that the representation is compatible with the subsequent entropic and unbalanced extensions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation introduces independent characterization and conditions

full rationale

The paper defines debiasability directly via the inequality c(x,y) ≥ ½c(x,x) + ½c(y,y) for symmetric costs and states an equivalent inf-representation c(x,y) = inf_z ψ(x,z) + ψ(y,z) as a building block (interpreted as generalizing midpoint identities). It then derives sufficient conditions for entropic OT (negative definiteness of ground cost or continuity/positive-definiteness of kernel) via a convex-nonconcave minimax argument, plus novel decomposition formulas. These steps rely on external mathematical arguments and generalize independent prior results from Feydy et al. and Séjourné et al. without any reduction of claimed predictions or conditions to fitted parameters, self-definitions, or load-bearing self-citations. The central claims remain self-contained against the stated assumptions and do not collapse to tautological inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of an auxiliary space Z and function ψ realizing the inf-representation for any debiasable cost, plus standard convexity and continuity assumptions needed for the minimax argument; no free parameters are fitted to data.

axioms (2)
  • domain assumption The cost function c is symmetric.
    Stated at the beginning of the abstract as the setting for debiasability.
  • standard math The inf-representation c(x,y) = inf_z ψ(x,z) + ψ(y,z) is equivalent to the inequality c(x,y) ≥ ½c(x,x) + ½c(y,y).
    Presented as an equivalent characterization used throughout the analysis.

pith-pipeline@v0.9.0 · 5628 in / 1488 out tokens · 23801 ms · 2026-05-10T01:44:16.965346+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

29 extracted references · 29 canonical work pages

  1. [1]

    Aliprantis and Kim C

    Charalambos D. Aliprantis and Kim C. Border. Infinite Dimensional Analysis . Springer, 3 edition, 2006

    work page 2006

  2. [2]

    Tropical reproducing kernels and optimization

    Pierre-Cyril Aubin-Frankowski and St \'e phane Gaubert. Tropical reproducing kernels and optimization. Integral Equations and Operator Theory , 96(2):19, 2024

    work page 2024

  3. [3]

    Evolution variational inequalities with general costs

    Pierre-Cyril Aubin-Frankowski, Giacomo Enrico Sodini, and Ulisse Stefanelli. Evolution variational inequalities with general costs. Journal of Functional Analysis , 291(1):111469, 2026

    work page 2026

  4. [4]

    u rich. Birkh \

    Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar \'e . Gradient Flows in Metric Spaces and in the Space of Probability Measures . Lectures in Mathematics ETH Z \"u rich. Birkh \"a user Basel, 2008

    work page 2008

  5. [5]

    Theory of reproducing kernels

    Nachman Aronszajn. Theory of reproducing kernels. Transactions of the American Mathematical Society , 68:337--404, 1950

    work page 1950

  6. [6]

    Harmonic Analysis on Semigroups , volume 100 of Graduate Texts in Mathematics

    Christian Berg, Jens Peter Reus Christensen, and Paul Ressel. Harmonic Analysis on Semigroups , volume 100 of Graduate Texts in Mathematics . Springer, 1984

    work page 1984

  7. [7]

    Bogachev

    Vladimir I. Bogachev. Measure Theory . Springer, 2007

    work page 2007

  8. [8]

    Convergence of entropic schemes for optimal transport and gradient flows

    Guillaume Carlier, Vincent Duval, Gabriel Peyr \'e , and Bernhard Schmitzer. Convergence of entropic schemes for optimal transport and gradient flows. SIAM Journal on Mathematical Analysis , 49(2):1385--1418, 2017

    work page 2017

  9. [9]

    Doubly regularized entropic Wasserstein barycenter

    L \'e na \"i c Chizat. Doubly regularized entropic Wasserstein barycenter. Foundations of Computational Mathematics , 2025

    work page 2025

  10. [10]

    Thomas M. Cover. Elements of Information Theory . John Wiley & Sons, 1999

    work page 1999

  11. [11]

    Scaling algorithms for unbalanced optimal transport problems

    L \'e na \"i c Chizat, Gabriel Peyr \'e , Bernhard Schmitzer, and Fran c ois-Xavier Vialard. Scaling algorithms for unbalanced optimal transport problems. Mathematics of Computation , 87(314):2563--2609, 2018

    work page 2018

  12. [12]

    Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications . John Wiley & Sons, New York, 2 edition, 1999

    work page 1999

  13. [13]

    Interpolating between optimal transport and MMD using Sinkhorn divergences

    Jean Feydy, Thibault S \'e journ \'e , Fran c ois-Xavier Vialard, Shun-ichi Amari, Alain Trouv \'e , and Gabriel Peyr \'e . Interpolating between optimal transport and MMD using Sinkhorn divergences. In Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics , pages 2681--2690. PMLR, 2019

    work page 2019

  14. [14]

    A kernel method for the two-sample problem

    Arthur Gretton, Karsten Borgwardt, Malte Rasch, Bernhard Sch \"o lkopf, and Alex Smola. A kernel method for the two-sample problem. In Advances in Neural Information Processing Systems , volume 19. MIT Press, 2006

    work page 2006

  15. [15]

    Learning generative models with Sinkhorn divergences

    Aude Genevay, Gabriel Peyr \'e , and Marco Cuturi. Learning generative models with Sinkhorn divergences. In Proceedings of the 21st International Conference on Artificial Intelligence and Statistics , pages 1608--1617. PMLR, 2018

    work page 2018

  16. [16]

    Entropic optimal transport between unbalanced G aussian measures has a closed form

    Hicham Janati, Boris Muzellec, Gabriel Peyr \'e , and Marco Cuturi. Entropic optimal transport between unbalanced G aussian measures has a closed form. In Advances in Neural Information Processing Systems , volume 33, pages 10468--10479, 2020

    work page 2020

  17. [17]

    A survey of the Schr \"o dinger problem and some of its connections with optimal transport

    Christian L \'e onard. A survey of the Schr \"o dinger problem and some of its connections with optimal transport. Discrete and Continuous Dynamical Systems , 34(4):1533--1574, 2014

    work page 2014

  18. [18]

    The Riemannian geometry of Sinkhorn divergences

    Hugo Lavenant, Jonas Luckhardt, Gilles Mordant, Bernhard Schmitzer, and Luca Tamanini. The Riemannian geometry of Sinkhorn divergences. Annales de l'Institut Henri Poincar \'e C , Analyse Non Lin \'e aire , 2025. Published online first

    work page 2025

  19. [19]

    Optimal entropy-transport problems and a new Hellinger--Kantorovich distance between positive measures

    Matthias Liero, Alexander Mielke, and Giuseppe Savar \'e . Optimal entropy-transport problems and a new Hellinger--Kantorovich distance between positive measures. Inventiones Mathematicae , 211(3):969--1117, 2017

    work page 2017

  20. [20]

    From optimal transport to discrepancy , pages 1--36

    Sebastian Neumayer and Gabriele Steidl. From optimal transport to discrepancy , pages 1--36. Springer International Publishing, 2021

    work page 2021

  21. [21]

    Introduction to entropic optimal transport, 2021

    Marcel Nutz. Introduction to entropic optimal transport, 2021. Lecture notes, http://www.math.columbia.edu/ mnutz/docs/EOT_lecture_notes.pdf

    work page 2021

  22. [22]

    Rachev and Ludger R \"u schendorf

    Svetlozar T. Rachev and Ludger R \"u schendorf. Mass Transportation Problems . Springer, 1998

    work page 1998

  23. [23]

    Optimal Transport for Applied Mathematicians , volume 87 of Progress in Nonlinear Differential Equations and Their Applications

    Filippo Santambrogio. Optimal Transport for Applied Mathematicians , volume 87 of Progress in Nonlinear Differential Equations and Their Applications . Birkh \"a user, 2015

    work page 2015

  24. [24]

    Sriperumbudur, Kenji Fukumizu, and Gert R.G

    Bharath K. Sriperumbudur, Kenji Fukumizu, and Gert R.G. Lanckriet. Universality, characteristic kernels and RKHS embedding of measures. Journal of Machine Learning Research , 12(70):2389--2410, 2011

    work page 2011

  25. [25]

    Sinkhorn divergences for unbalanced optimal transport.arXiv preprint arXiv:1910.12958, 2019

    Thibault S \'e journ \'e , Jean Feydy, Fran c ois-Xavier Vialard, Alain Trouv \'e , and Gabriel Peyr \'e . Sinkhorn divergences for unbalanced optimal transport, 2019. https://arxiv.org/abs/1910.12958

    work page arXiv 2019

  26. [26]

    Unbalanced Optimal Transport, from theory to numerics , page 407–471

    Thibault Séjourné, Gabriel Peyré, and Fran c ois-Xavier Vialard. Unbalanced Optimal Transport, from theory to numerics , page 407–471. Elsevier, 2023

    work page 2023

  27. [27]

    Topics in Optimal Transportation , volume 58 of Graduate Studies in Mathematics

    C \'e dric Villani. Topics in Optimal Transportation , volume 58 of Graduate Studies in Mathematics . American Mathematical Society, 2003

    work page 2003

  28. [28]

    Optimal Transport: Old and New , volume 338 of Grundlehren der mathematischen Wissenschaften

    C \'e dric Villani. Optimal Transport: Old and New , volume 338 of Grundlehren der mathematischen Wissenschaften . Springer, 2008

    work page 2008

  29. [29]

    Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures

    Johanna Ziegel, David Ginsbourger, and Lutz D \"u mbgen. Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures. Bernoulli , 30(2), 2024

    work page 2024