Optimal transport of signed measures: existence, uniqueness and fractal structure

Bwo'Nyahre Baidi Barthelemy; Houpa Danga Duplex Elvies; Kouakep Tchaptche Yannick

arxiv: 2605.01523 · v2 · submitted 2026-05-02 · 🧮 math.AP

Optimal transport of signed measures: existence, uniqueness and fractal structure

Bwo'Nyahre Baidi Barthelemy , Kouakep Tchaptche Yannick , Houpa Danga Duplex Elvies This is my paper

Pith reviewed 2026-05-12 02:48 UTC · model grok-4.3

classification 🧮 math.AP

keywords optimal transportsigned measuresMonge-Ampère equationfractal setsHausdorff dimensionJordan decompositionexistence and uniqueness

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The pith

Optimal transport maps exist and are unique for signed measures, preserving the Hausdorff dimension of their fractal supports.

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

This paper extends the classical theory of optimal transport from probability measures to signed measures that may have fractal singular parts. It proves that under certain regularity assumptions, there is a unique optimal transport map for a cost function that treats same-sign and opposite-sign transports differently using a penalty term. The solution is characterized by coupled Monge-Ampère equations derived from a double Legendre transform. This matters because many real-world measures, like differences in densities or charges, are signed and can have irregular fractal structures, so a theory that handles them directly opens the door to transporting such data without splitting into positive and negative parts artificially.

Core claim

Under suitable regularity and structural assumptions (H1-H5), there exists a unique optimal transport map T for signed measures using a cost that distinguishes same-sign and opposite-sign transports via a positional penalty lambda. This map is characterized by coupled Monge-Ampère equations and a double Legendre transform system. Furthermore, the optimal transport preserves the Hausdorff dimension and Ahlfors regularity of fractal sets in the measures.

What carries the argument

The adaptive regularization technique combined with the Jordan decomposition of signed measures, which enables the construction of the optimal map T while respecting sign and fractal properties.

Load-bearing premise

The five unspecified regularity and structural assumptions H1-H5 on the measures, cost function, and supports hold, and the adaptive regularization technique does not introduce artifacts that alter the signed or fractal nature.

What would settle it

Construct a specific signed measure with fractal support satisfying H1-H5 but where the computed transport map changes the Hausdorff dimension of the support or where multiple optimal maps exist.

read the original abstract

This paper develops a comprehensive theory of optimal transport for signed (real) measures on Rd. Extending the classical Brenier theorem, we consider Jordan decompositions of measures with possibly fractal singular parts. Under suitable regularity and structural assumptions (H1-H5), we prove existence and uniqueness of an optimal transport map T for a cost that distinguishes same-sign and opposite sign transports via a positional penalty lambda. We derive coupled Monge-Ampere equations and a double Legendre transform system characterizing the solution. Moreover, we show that the optimal transport preserves Hausdorff dimension and Ahlfors regularity of fractal sets. The proof relies on a novel adaptive regularization technique that respects the signed and fractal nature of the measures.

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.

Desk Editor's Note private letter to a colleague

The paper pushes optimal transport to signed measures with fractal parts via a sign-aware cost and adaptive regularization, but the dimension-preservation claim lacks explicit convergence controls.

read the letter

The main takeaway is that this work tries to extend the Brenier theorem to signed measures whose singular parts can be fractal, using a cost that adds a positional penalty lambda for opposite-sign transports. They prove existence and uniqueness of the map under assumptions H1-H5, derive coupled Monge-Ampère equations plus a double Legendre transform system, and claim the transport preserves Hausdorff dimension and Ahlfors regularity of those fractal sets. The proof route is a new adaptive regularization meant to keep the signed and fractal character intact. That combination of ideas is the genuinely new piece; most prior optimal transport work stays with positive measures or smooth densities. The setup is coherent on its own terms and the authors engage the classical literature without obvious circularity. The regularization technique is a reasonable attempt to handle the technical obstacles that signed data and fractals create. The soft spot is exactly where the stress-test note flags it: the limit behavior of the adaptive regularization is not analyzed in enough detail to guarantee that Hausdorff dimension and Ahlfors regularity survive. If the parameter choice smooths positive and negative parts at different rates, the preservation can fail even when the transport map itself exists. The five assumptions H1-H5 are stated but their interaction with the regularization step is not spelled out with explicit error bounds or convergence rates, so the strongest claim rests on the weakest link. This is for people already working in optimal transport or geometric measure theory who need to handle signed or singular data. A reader looking for modeling tools in complex systems might skim it for the cost construction and the coupled equations. The paper is coherent enough and the claims specific enough that it deserves a serious referee rather than a desk reject; the technical gaps are fixable in revision if the authors supply the missing limit analysis. I would send it to review but ask the referees to focus on the regularization convergence.

Referee Report

2 major / 2 minor

Summary. The paper extends the classical Brenier theorem to optimal transport of signed measures on R^d, including Jordan decompositions with possibly fractal singular parts. Under regularity and structural assumptions (H1-H5), it proves existence and uniqueness of an optimal transport map T for a cost distinguishing same-sign and opposite-sign transports via a positional penalty lambda. The solution is characterized by coupled Monge-Ampère equations and a double Legendre transform system. It further claims that the transport preserves Hausdorff dimension and Ahlfors regularity of fractal sets, with the proof relying on a novel adaptive regularization technique that respects the signed and fractal nature of the measures.

Significance. If the central claims hold, particularly the dimension preservation result, this would constitute a notable extension of optimal transport theory to signed measures with singular fractal supports. The coupled equations and double Legendre transform provide a new characterization that could enable further analysis in PDEs and geometric measure theory. The adaptive regularization approach, if shown to converge without artifacts, represents a technical innovation with potential applications in fields handling signed densities or singular data.

major comments (2)

[adaptive regularization and fractal preservation proof] The section describing the adaptive regularization technique and its convergence: the claim that this technique preserves Hausdorff dimension and Ahlfors regularity of fractal singular parts rests on the regularization converging to the original signed measure without altering the support's dimension. No explicit limit analysis is provided showing that the local density-dependent parameter choice does not smooth positive and negative parts differently or introduce new singularities under H1-H5; this is load-bearing for the fractal preservation result.
[characterization via coupled equations] The derivation of the coupled Monge-Ampère equations and double Legendre transform system (under H1-H5): while the abstract states these characterize the solution, the manuscript provides no details on error controls, verification steps, or how the sign-distinguishing cost and lambda interact with the equations to ensure uniqueness; this undercuts assessment of the existence/uniqueness claims.

minor comments (2)

[Abstract] The abstract introduces lambda as a 'positional penalty' without specifying its functional form, admissible range, or dependence on the measures.
[Introduction or assumptions section] Assumptions H1-H5 are referenced but their precise statements (e.g., on supports, regularity of densities, or cost properties) should be stated explicitly early in the paper for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will revise the manuscript accordingly to provide the requested details and analysis.

read point-by-point responses

Referee: [adaptive regularization and fractal preservation proof] The section describing the adaptive regularization technique and its convergence: the claim that this technique preserves Hausdorff dimension and Ahlfors regularity of fractal singular parts rests on the regularization converging to the original signed measure without altering the support's dimension. No explicit limit analysis is provided showing that the local density-dependent parameter choice does not smooth positive and negative parts differently or introduce new singularities under H1-H5; this is load-bearing for the fractal preservation result.

Authors: We agree that the convergence analysis requires more explicit treatment to fully support the dimension-preservation claim. In the revised version we will add a dedicated subsection with a rigorous limit analysis. Under H1-H5 we will show that the locally adaptive parameter converges in total variation to the original signed measure, that the positive and negative parts are regularized symmetrically with respect to their supports, and that no new singularities are created; the argument relies on uniform bounds on the density-dependent mollification radii together with the Ahlfors-regularity assumption to control the Hausdorff measure of the approximated supports. revision: yes
Referee: [characterization via coupled equations] The derivation of the coupled Monge-Ampère equations and double Legendre transform system (under H1-H5): while the abstract states these characterize the solution, the manuscript provides no details on error controls, verification steps, or how the sign-distinguishing cost and lambda interact with the equations to ensure uniqueness; this undercuts assessment of the existence/uniqueness claims.

Authors: We acknowledge that the derivation section would benefit from additional intermediate steps. In the revision we will expand the relevant section to include (i) a priori error estimates between the regularized and limiting problems, (ii) verification that the double Legendre transform system is satisfied in the viscosity sense, and (iii) a uniqueness argument showing that the sign-distinguishing cost together with the positional penalty lambda forces the optimal map to be the unique solution of the coupled Monge-Ampère system under H1-H5, via a monotonicity argument on the subgradient. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends classical results independently

full rationale

The paper extends the classical Brenier theorem to signed measures using a sign-distinguishing cost with positional penalty lambda, derives coupled Monge-Ampere equations and double Legendre transform under assumptions (H1-H5), and claims preservation of Hausdorff dimension and Ahlfors regularity via a novel adaptive regularization technique. No quoted equations or steps in the abstract reduce the central existence/uniqueness or fractal-preservation results to the inputs by construction, fitted parameters renamed as predictions, or load-bearing self-citation chains. The regularization is presented as a proof tool respecting signed/fractal structure without evidence of tautological reduction. This is the common case of a self-contained extension of prior independent theorems.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The theory depends on five domain assumptions H1-H5 whose details are not provided, the free parameter lambda in the cost, and standard mathematical tools like Jordan decomposition; no invented entities are introduced.

free parameters (1)

lambda
Positional penalty parameter used in the cost to distinguish same-sign and opposite-sign transports.

axioms (2)

domain assumption H1-H5: suitable regularity and structural assumptions on measures and cost
Invoked to guarantee existence, uniqueness, and fractal preservation; specifics not given in abstract.
standard math Jordan decomposition applies to signed measures
Standard background result used to split measures into positive and negative parts.

pith-pipeline@v0.9.0 · 5424 in / 1359 out tokens · 73621 ms · 2026-05-12T02:48:58.658500+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Under five structural assumptions (H1–H5), we prove existence and uniqueness of an optimal transport map T for a total cost C encoding a sign-dependent positional penalty λ. ... coupled Monge–Ampère equations ... double Legendre transform system
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The optimal map is shown to preserve Hausdorff dimension and Ahlfors regularity of every fractal component.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

novel adaptive regularization operator Rn ... kernel K(ds)_h whose normalization is exact by construction

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 1 internal anchor

[1]

Brenier, Y. (1991). Polar factorization and monotone rearrangement of vector-valued func- tions.Comm. Pure Appl. Math., 44(4), 375–417

work page 1991
[2]

Caffarelli, L. A. (1992). The regularity of mappings with a convex potential.J. Amer. Math. Soc., 5(1), 99–104

work page 1992
[3]

Ciosmak, K. J. (2021). Optimal transport for vector measures.Calc. Var., 60, 230

work page 2021
[4]

Cohn, D. L. (2013).Measure Theory, 2nd ed. Birkhäuser

work page 2013
[5]

& Semmes, S

David, G. & Semmes, S. (1993).Analysis of and on Uniformly Rectifiable Sets. AMS

work page 1993
[6]

De Masi, L., Marchese, A., & Massaccesi, A. (2025). Variational models of robust optimal transport. arXiv:2511.05146

work page arXiv 2025
[7]

& Shenfeld, Y

De Philippis, G. & Shenfeld, Y. (2024). Optimal transport maps, majorization, and log- subharmonic measures. arXiv:2411.12109

work page internal anchor Pith review arXiv 2024
[8]

(2003).Fractal Geometry, 2nd ed

Falconer, K. (2003).Fractal Geometry, 2nd ed. Wiley

work page 2003
[9]

Kantorovich, L. (1942). On the transfer of masses.Dokl. Akad. Nauk SSSR, 37(2), 227–229

work page 1942
[10]

Mainini, E. (2012). A description of transport cost for signed measures.J. Math. Sci., 181, 837–855

work page 2012
[11]

Monge, G. (1781). Mémoire sur la théorie des déblais et des remblais.Histoire de l’Académie Royale des Sciences de Paris

work page
[12]

& Cuturi, M

Peyré, G. & Cuturi, M. (2019). Computational optimal transport.Found. Trends Mach. Learn., 11(5–6), 355–607

work page 2019
[13]

Piccoli, B., Rossi, F., & Tournus, M. (2023). A Wasserstein norm for signed measures.Com- mun. Math. Sci., 21(1), 247–290

work page 2023
[14]

Rockafellar, R. T. (1970).Convex Analysis. Princeton University Press

work page 1970
[15]

(2009).Optimal Transport: Old and New

Villani, C. (2009).Optimal Transport: Old and New. Springer. 7

work page 2009

[1] [1]

Brenier, Y. (1991). Polar factorization and monotone rearrangement of vector-valued func- tions.Comm. Pure Appl. Math., 44(4), 375–417

work page 1991

[2] [2]

Caffarelli, L. A. (1992). The regularity of mappings with a convex potential.J. Amer. Math. Soc., 5(1), 99–104

work page 1992

[3] [3]

Ciosmak, K. J. (2021). Optimal transport for vector measures.Calc. Var., 60, 230

work page 2021

[4] [4]

Cohn, D. L. (2013).Measure Theory, 2nd ed. Birkhäuser

work page 2013

[5] [5]

& Semmes, S

David, G. & Semmes, S. (1993).Analysis of and on Uniformly Rectifiable Sets. AMS

work page 1993

[6] [6]

De Masi, L., Marchese, A., & Massaccesi, A. (2025). Variational models of robust optimal transport. arXiv:2511.05146

work page arXiv 2025

[7] [7]

& Shenfeld, Y

De Philippis, G. & Shenfeld, Y. (2024). Optimal transport maps, majorization, and log- subharmonic measures. arXiv:2411.12109

work page internal anchor Pith review arXiv 2024

[8] [8]

(2003).Fractal Geometry, 2nd ed

Falconer, K. (2003).Fractal Geometry, 2nd ed. Wiley

work page 2003

[9] [9]

Kantorovich, L. (1942). On the transfer of masses.Dokl. Akad. Nauk SSSR, 37(2), 227–229

work page 1942

[10] [10]

Mainini, E. (2012). A description of transport cost for signed measures.J. Math. Sci., 181, 837–855

work page 2012

[11] [11]

Monge, G. (1781). Mémoire sur la théorie des déblais et des remblais.Histoire de l’Académie Royale des Sciences de Paris

work page

[12] [12]

& Cuturi, M

Peyré, G. & Cuturi, M. (2019). Computational optimal transport.Found. Trends Mach. Learn., 11(5–6), 355–607

work page 2019

[13] [13]

Piccoli, B., Rossi, F., & Tournus, M. (2023). A Wasserstein norm for signed measures.Com- mun. Math. Sci., 21(1), 247–290

work page 2023

[14] [14]

Rockafellar, R. T. (1970).Convex Analysis. Princeton University Press

work page 1970

[15] [15]

(2009).Optimal Transport: Old and New

Villani, C. (2009).Optimal Transport: Old and New. Springer. 7

work page 2009