A variational approach to regularity theory in optimal transportation

Michael Goldman (LJLL)

arxiv: 1907.05627 · v1 · pith:MKB3KKYXnew · submitted 2019-07-12 · 🧮 math.AP

A variational approach to regularity theory in optimal transportation

Michael Goldman (LJLL) This is my paper

Pith reviewed 2026-05-24 22:41 UTC · model grok-4.3

classification 🧮 math.AP

keywords optimal transportMonge-Ampère equationregularity theoryvariational methodsPoisson equationpartial regularitymatching problems

0 comments

The pith

A quantitative linearization shows the Monge-Ampère equation reduces to the Poisson equation near the identity for optimal transport maps.

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

The paper establishes a quantitative version of the fact that the Monge-Ampère equation linearizes to the Poisson equation around the identity map. This controlled approximation holds in the setting of optimal transport between suitable measures. The result supplies a variational method that yields a new proof of partial regularity for transport maps and confirms structural predictions for maps arising in matching problems.

Core claim

The linearization of the Monge-Ampère equation around the identity is the Poisson equation, and this identity holds in quantitative form for optimal transport maps.

What carries the argument

Quantitative linearization of the Monge-Ampère equation to the Poisson equation.

If this is right

A variational proof of the partial regularity theorem of Figalli and Kim.
Rigorous confirmation of the structural predictions made by Caracciolo et al. for optimal maps in matching problems.
A template for applying similar linearization arguments to other regularity questions in optimal transport.

Where Pith is reading between the lines

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

The same quantitative control could be used to derive error estimates for numerical approximations that replace the Monge-Ampère equation by its Poisson limit.
The method may extend to transport problems on manifolds where the linearization is expected to involve the Laplace-Beltrami operator.
It supplies a possible route to stability results that quantify how far a map can deviate from the identity before the Poisson approximation breaks.

Load-bearing premise

Optimal transport maps exist and possess enough regularity for the Monge-Ampère equation to be well-posed.

What would settle it

An explicit pair of measures whose optimal map stays close to the identity yet whose second variation deviates from the Poisson equation by a fixed amount.

read the original abstract

This paper describes recent results obtained in collaboration with M. Huesmann and F. Otto on the regularity of optimal transport maps. The main result is a quantitative version of the well-known fact that the linearization of the Monge-Amp{\`e}re equation around the identity is the Poisson equation. We present two applications of this result. The first one is a variational proof of the partial regularity theorem of Figalli and Kim and the second is the rigorous validation of some predictions made by Carraciolo and al. on the structure of the optimal transport maps in matching problems.

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 gives a variational route to a quantitative linearization of the Monge-Ampère equation around the identity and uses it to reprove the Figalli-Kim partial regularity result plus validate some matching predictions.

read the letter

The main contribution is a quantitative version of the standard linearization fact—that the Monge-Ampère equation around the identity becomes the Poisson equation—obtained through a variational argument with Huesmann and Otto. They then apply this to recover the Figalli-Kim partial regularity theorem variationally and to make rigorous some earlier predictions on the structure of optimal maps in matching problems. The variational control on the linearization is the piece that looks new and potentially reusable for other regularity questions in optimal transport. The matching application stands out because it turns informal heuristics into theorems rather than just re-deriving an existing result. The partial regularity proof is already known, so the paper's value sits in the method rather than in new statements. The abstract gives no sign of circularity or hidden regularity assumptions that would undermine the linearization step, and the stress-test note aligns with that. Still, without the explicit error estimates or the precise function spaces used to close the variational argument, it is difficult to gauge how much simpler or more general the approach really is compared with direct estimates in the literature. This is for people already working in optimal transport regularity or in the analysis of matching problems. A reader who cares about variational techniques for Monge-Ampère-type equations or who wants to see the matching predictions checked would find it worth reading. The work is coherent on its own terms and targets a central technical step, so it deserves referee time even if the final verdict depends on the details of the estimates.

Referee Report

0 major / 2 minor

Summary. The paper presents a variational approach, developed in collaboration with M. Huesmann and F. Otto, to obtain a quantitative linearization of the Monge-Ampère equation around the identity map (recovering the Poisson equation). This is applied to give a variational proof of the partial regularity theorem of Figalli and Kim for optimal transport maps, and to rigorously validate predictions by Carraciolo et al. on the structure of optimal transport maps in matching problems.

Significance. If the quantitative estimates hold with the claimed error control, the variational method supplies a new route to linearization estimates that avoids presupposing the regularity being proved, which could be useful for extending regularity results in optimal transport and related nonlinear PDEs. The two applications demonstrate concrete utility by recovering a known partial-regularity theorem and confirming conjectural predictions in matching problems.

minor comments (2)

The abstract refers to 'Carraciolo and al.'; this should be corrected to the standard citation form 'Carracciolo et al.'
As the manuscript describes recent joint results, the introduction would benefit from a clearer statement of the precise quantitative linearization estimate (including the functional whose second variation is controlled) before turning to the applications.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of the manuscript. The recommendation of minor revision is noted. However, the report contains no listed major comments, so there are no specific points requiring point-by-point response or revision.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper's central claim is a quantitative strengthening, via variational methods, of the standard linearization of the Monge-Ampère equation around the identity (yielding the Poisson equation). This is applied to recover the known partial regularity result of Figalli-Kim and to validate external predictions from matching problems. No quoted steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the derivation remains independent of its target outputs and is benchmarked against prior external theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5610 in / 925 out tokens · 18955 ms · 2026-05-24T22:41:27.233232+00:00 · methodology

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

?

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

the linearization of the Monge-Ampère equation around the identity is the Poisson equation … harmonic approximation result … excess improvement by tilting
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Δψ = μ − λ … ΔΦ = c in BR … ∫ |j − ρ∇Φ|² ≤ τE + CD

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

27 extracted references · 27 canonical work pages · 2 internal anchors

[1]

Ajtai, J

M. Ajtai, J. Koml´ os, and G. Tusn´ ady,On optimal matchings. , Combinatorica 4 (1984), 259–264

work page 1984
[2]

Alberti, R

G. Alberti, R. Choksi, and F. Otto, Uniform energy distribution for an isoperimetric problem with long-range interactions , J. Amer. Math. Soc. 22 (2009), no. 2, 569–605

work page 2009
[3]

Ambrosio, F

L. Ambrosio, F. Glaudo, and D. Trevisan, On the optimal map in the 2-dimensional random matching problem , 2019, p. 20

work page 2019
[4]

Ambrosio, F

L. Ambrosio, F. Stra, and D. Trevisan, A PDE approach to a 2-dimensional matching problem, Probab. Theory Related Fields 173 (2019), no. 1-2, 433–477

work page 2019
[5]

Armstrong, T

S. Armstrong, T. Kuusi, and J.-C. Mourrat, Quantitative stochastic homogenization and large-scale regularity, ArXiv e-prints (2017). 12

work page 2017
[6]

S. N. Armstrong and C. K. Smart, Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. ´Ec. Norm. Sup´ er. (4)49 (2016), no. 2, 423–481

work page 2016
[7]

Avellaneda and F.-H

M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization , Comm. Pure Appl. Math. 40 (1987), no. 6, 803–847

work page 1987
[8]

L. A. Caﬀarelli, A localization property of viscosity solutions to the Monge -Amp` ere equation and their strict convexity , Ann. of Math. (2) 131 (1990), no. 1, 129–134

work page 1990
[9]

, The regularity of mappings with a convex potential , J. Amer. Math. Soc. 5 (1992), no. 1, 99–104

work page 1992
[10]

Campanato, Propriet` a di una famiglia di spazi funzionali , Ann

S. Campanato, Propriet` a di una famiglia di spazi funzionali , Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 137–160

work page 1964
[11]

Caracciolo, C

S. Caracciolo, C. Lucibello, G. Parisi, and G. Sicuro, Scaling hypothesis for the Eu- clidean bipartite matching problem , Physical Review E 90 (2014), no. 1

work page 2014
[12]

Caracciolo and G

S. Caracciolo and G. Sicuro, Scaling hypothesis for the Euclidean bipartite matching problem. II. Correlation functions , Physical Review E 91 (2015), no. 6

work page 2015
[13]

De Philippis and A

G. De Philippis and A. Figalli, The Monge-Amp` ere equation and its link to optimal transportation, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 4, 527–580

work page 2014
[14]

, Partial regularity for optimal transport maps, Publ. Math. Inst. Hautes ´Etudes Sci. 121 (2015), 81–112

work page 2015
[15]

Figalli and Y.-H

A. Figalli and Y.-H. Kim, Partial regularity of Brenier solutions of the Monge-Amp` e re equation, Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 559–565

work page 2010
[16]

Gloria, S

A. Gloria, S. Neukamm, and F. Otto, A regularity theory for random elliptic operators , ArXiv e-prints (2014)

work page 2014
[17]

Goldman, M

M. Goldman, M. Huesmann, and F. Otto, In preparation

work page
[18]

, A large-scale regularity theory for the Monge-Ampere equat ion with rough data and application to the optimal matching problem , arXiv e-prints (2018), arXiv:1808.09250

work page internal anchor Pith review Pith/arXiv arXiv 2018
[19]

, Quantitative linearization results for the Monge-Amp` ere equation, arXiv e- prints (2019)

work page 2019
[20]

A variational proof of partial regularity for optimal transportation maps

M. Goldman and F. Otto, A variational proof of partial regularity for optimal trans - portation maps , arXiv:1704.05339 (2017)

work page internal anchor Pith review Pith/arXiv arXiv 2017
[21]

Huesmann and K.-T

M. Huesmann and K.-T. Sturm, Optimal transport from Lebesgue to Poisson , Ann. Probab. 41 (2013), no. 4, 2426–2478. 13

work page 2013
[22]

Kitagawa and R

J. Kitagawa and R. McCann, Free discontinuities in optimal transport , Arch. Ration. Mech. Anal. 232 (2019), no. 3, 1505–1541

work page 2019
[23]

Ledoux, A ﬂuctuation result in dual Sobolev norm for the optimal matc hing prob- lem, 2019

M. Ledoux, A ﬂuctuation result in dual Sobolev norm for the optimal matc hing prob- lem, 2019

work page 2019
[24]

Maggi, Sets of ﬁnite perimeter and geometric variational problems , Cambridge Studies in Advanced Mathematics, vol

F. Maggi, Sets of ﬁnite perimeter and geometric variational problems , Cambridge Studies in Advanced Mathematics, vol. 135, Cambridge University Pr ess, Cambridge, 2012, An introduction to geometric measure theory

work page 2012
[25]

Santambrogio, Optimal transport for applied mathematicians , Progress in Nonlinear Diﬀerential Equations and their Applications, vol

F. Santambrogio, Optimal transport for applied mathematicians , Progress in Nonlinear Diﬀerential Equations and their Applications, vol. 87, Birkh¨ auser/ Springer, Cham, 2015, Calculus of variations, PDEs, and modeling

work page 2015
[26]

Talagrand, Upper and lower bounds for stochastic processes: modern met hods and classical problems, vol

M. Talagrand, Upper and lower bounds for stochastic processes: modern met hods and classical problems, vol. 60, Springer Science & Business Media, 2014

work page 2014
[27]

Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol

C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. 14

work page 2003

[1] [1]

Ajtai, J

M. Ajtai, J. Koml´ os, and G. Tusn´ ady,On optimal matchings. , Combinatorica 4 (1984), 259–264

work page 1984

[2] [2]

Alberti, R

G. Alberti, R. Choksi, and F. Otto, Uniform energy distribution for an isoperimetric problem with long-range interactions , J. Amer. Math. Soc. 22 (2009), no. 2, 569–605

work page 2009

[3] [3]

Ambrosio, F

L. Ambrosio, F. Glaudo, and D. Trevisan, On the optimal map in the 2-dimensional random matching problem , 2019, p. 20

work page 2019

[4] [4]

Ambrosio, F

L. Ambrosio, F. Stra, and D. Trevisan, A PDE approach to a 2-dimensional matching problem, Probab. Theory Related Fields 173 (2019), no. 1-2, 433–477

work page 2019

[5] [5]

Armstrong, T

S. Armstrong, T. Kuusi, and J.-C. Mourrat, Quantitative stochastic homogenization and large-scale regularity, ArXiv e-prints (2017). 12

work page 2017

[6] [6]

S. N. Armstrong and C. K. Smart, Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. ´Ec. Norm. Sup´ er. (4)49 (2016), no. 2, 423–481

work page 2016

[7] [7]

Avellaneda and F.-H

M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization , Comm. Pure Appl. Math. 40 (1987), no. 6, 803–847

work page 1987

[8] [8]

L. A. Caﬀarelli, A localization property of viscosity solutions to the Monge -Amp` ere equation and their strict convexity , Ann. of Math. (2) 131 (1990), no. 1, 129–134

work page 1990

[9] [9]

, The regularity of mappings with a convex potential , J. Amer. Math. Soc. 5 (1992), no. 1, 99–104

work page 1992

[10] [10]

Campanato, Propriet` a di una famiglia di spazi funzionali , Ann

S. Campanato, Propriet` a di una famiglia di spazi funzionali , Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 137–160

work page 1964

[11] [11]

Caracciolo, C

S. Caracciolo, C. Lucibello, G. Parisi, and G. Sicuro, Scaling hypothesis for the Eu- clidean bipartite matching problem , Physical Review E 90 (2014), no. 1

work page 2014

[12] [12]

Caracciolo and G

S. Caracciolo and G. Sicuro, Scaling hypothesis for the Euclidean bipartite matching problem. II. Correlation functions , Physical Review E 91 (2015), no. 6

work page 2015

[13] [13]

De Philippis and A

G. De Philippis and A. Figalli, The Monge-Amp` ere equation and its link to optimal transportation, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 4, 527–580

work page 2014

[14] [14]

, Partial regularity for optimal transport maps, Publ. Math. Inst. Hautes ´Etudes Sci. 121 (2015), 81–112

work page 2015

[15] [15]

Figalli and Y.-H

A. Figalli and Y.-H. Kim, Partial regularity of Brenier solutions of the Monge-Amp` e re equation, Discrete Contin. Dyn. Syst. 28 (2010), no. 2, 559–565

work page 2010

[16] [16]

Gloria, S

A. Gloria, S. Neukamm, and F. Otto, A regularity theory for random elliptic operators , ArXiv e-prints (2014)

work page 2014

[17] [17]

Goldman, M

M. Goldman, M. Huesmann, and F. Otto, In preparation

work page

[18] [18]

, A large-scale regularity theory for the Monge-Ampere equat ion with rough data and application to the optimal matching problem , arXiv e-prints (2018), arXiv:1808.09250

work page internal anchor Pith review Pith/arXiv arXiv 2018

[19] [19]

, Quantitative linearization results for the Monge-Amp` ere equation, arXiv e- prints (2019)

work page 2019

[20] [20]

A variational proof of partial regularity for optimal transportation maps

M. Goldman and F. Otto, A variational proof of partial regularity for optimal trans - portation maps , arXiv:1704.05339 (2017)

work page internal anchor Pith review Pith/arXiv arXiv 2017

[21] [21]

Huesmann and K.-T

M. Huesmann and K.-T. Sturm, Optimal transport from Lebesgue to Poisson , Ann. Probab. 41 (2013), no. 4, 2426–2478. 13

work page 2013

[22] [22]

Kitagawa and R

J. Kitagawa and R. McCann, Free discontinuities in optimal transport , Arch. Ration. Mech. Anal. 232 (2019), no. 3, 1505–1541

work page 2019

[23] [23]

Ledoux, A ﬂuctuation result in dual Sobolev norm for the optimal matc hing prob- lem, 2019

M. Ledoux, A ﬂuctuation result in dual Sobolev norm for the optimal matc hing prob- lem, 2019

work page 2019

[24] [24]

Maggi, Sets of ﬁnite perimeter and geometric variational problems , Cambridge Studies in Advanced Mathematics, vol

F. Maggi, Sets of ﬁnite perimeter and geometric variational problems , Cambridge Studies in Advanced Mathematics, vol. 135, Cambridge University Pr ess, Cambridge, 2012, An introduction to geometric measure theory

work page 2012

[25] [25]

Santambrogio, Optimal transport for applied mathematicians , Progress in Nonlinear Diﬀerential Equations and their Applications, vol

F. Santambrogio, Optimal transport for applied mathematicians , Progress in Nonlinear Diﬀerential Equations and their Applications, vol. 87, Birkh¨ auser/ Springer, Cham, 2015, Calculus of variations, PDEs, and modeling

work page 2015

[26] [26]

Talagrand, Upper and lower bounds for stochastic processes: modern met hods and classical problems, vol

M. Talagrand, Upper and lower bounds for stochastic processes: modern met hods and classical problems, vol. 60, Springer Science & Business Media, 2014

work page 2014

[27] [27]

Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol

C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. 14

work page 2003