Sharp Asymptotics for Regularized Optimal Transport

Alberto Gonz\'alez-Sanz; Carlos Cardoso-Perell\'o; Marcel Nutz

REVIEW 2 major objections 3 minor 46 references

Sharp Asymptotics for Regularized Optimal Transport

T0 review · 2 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper establishes the exact small-regularization asymptotics of L^p-regularized and entropically regularized optimal transport: the gap to the unregularized cost is an explicit constant times a power of ε (with a logarithmic correction

desk verdict Genuinely new sharp asymptotics for p-ROT and EOT under weaker conditions, but the standing global C^2 uniformly elliptic Brenier assumption (A2) is heavier than advertised and is not proved or given verifiable sufficient conditions. read the letter →

arxiv 2607.18191 v1 pith:AUD23LXB submitted 2026-07-20 math.AP math.OCmath.PR

Carlos Cardoso-Perelló , Alberto González-Sanz , Marcel Nutz This is my paper

classification math.APmath.OCmath.PR MSC 49Q2260E1565K10

keywords regularizedoptimaltransportentropicL^pregularizationsharpasymptoticsBarenblattprofilesquantizationBrenierpotentialMonge–Ampèreidentity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that as the regularization strength ε shrinks to zero, the added cost caused by regularization is not merely small but has an exact, explicit leading term. For L^p-regularized transport (1

What carries the argument

The load-bearing objects are the local optimal profiles. Near the diagonal, the Bregman divergence D(x,∇φ(x′)) = φ(x)+φ*(∇φ(x′))−⟨x,∇φ(x′)⟩ behaves like (1/2)||x−x′||²_{∇²φ(x)}, so the dual problem selects a normalized local kernel: a Gaussian for entropy regularization and a Barenblatt-type power kernel for L^p regularization. The Monge–Ampère identity det∇²φ(x) = f_μ(x)/f_ν(∇φ(x)) converts these kernels into the explicit constants in the theorems. The second key piece is a quantization construction: the local kernels form only a subcoupling whose marginals are dominated by μ, and the residual marginal mass is completed by block or nearest-neighbour couplings whose cost is controlled by qua

What would settle it

Take μ=ν equal to the standard Gaussian in d=1 with p=3/2; the theorem predicts (ROT−OT)/ε^{2/(d(p−1)+2)} → C_{1,3/2} ∫ f_μ(x)^{d(p−1)/(d(p−1)+2)} dx, an explicit number. A numerical solver accurate below that scale finding a different limit or no limit would falsify the claim. Alternatively, construct μ,ν satisfying the moment assumptions whose Brenier map is only C^1 with a point singularity; if the predicted limit still holds, (A2) is not necessary, and if it fails, the assumption is essential.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the small-ε deviation of regularized optimal transport from unregularized OT obeys an explicit asymptotic law with no unknown constants: Theorem 2.1 gives lim_{ε→0+} [ROT_{ε,p}(μ,ν)−OT(μ,ν)] / ε^{2/(d(p−1)+2)} = C_{d,p} ∫ [f_ν(∇φ(x)) f_μ(x)]^{−(p−1)/(d(p−1)+2)} dμ(x), and Theorem 2.2 gives lim_{ε→0+} [EOT_ε(μ,ν)−OT(μ,ν)+(d/2)ε log(πε)] / ε = −(1/2)(Ent(μ)+Ent(ν)). The assumptions are (A1) absolute continuity with finite second moments and (A2) the Brenier potential is C² with uniform ellipticity bounds on its Hessian; for p>2 the finite case additionally assumes bounded supports with Lipschitz boundaries and two-sided density bounds. Wh

Load-bearing premise

The whole proof rests on Assumption (A2): the optimal transport (Brenier) potential φ is C² on all of R^d with Hessian bounded between two positive constants σ_m I and σ_M I; the paper states this is the standard conclusion of classical regularity theory but does not prove it from conditions on μ and ν, and if it fails the Bregman quadratic bounds, the change of variables, and the quantization estimates all collapse.

Editorial extensions

If this is right

For L^p-regularized transport with any p in (1,2], the first-order error is now known exactly under only absolute continuity and a finite (2+β)-moment, removing the compact-support and density-bound restrictions of earlier results.
The family p in (1,2) interpolates between the quadratic case (exponent 2/(d+2)) and the entropic case, whose logarithmic correction emerges as p→1+, giving a quantitative bridge between two widely used regularizers.
For p>2 the same formula holds when the marginals have bounded support with Lipschitz boundary and two-sided bounded densities; without such conditions the constant can be infinite, so the regularized cost pulls away from OT faster than the power law.
The entropic formula holds even when the entropy terms are infinite, covering marginals whose densities are unbounded or have heavy tails, with the logarithmic term governing the divergence to −∞.
Because the constants depend only on dimension p and on the densities and optimal map, the asymptotic error of regularized transport is computable from data, not merely qualitatively small.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The two-step recipe — duality produces the local kernel, quantization fills the residual marginal mass — is likely to yield sharp constants for other convex regularizers whose conjugates scale like powers, not just entropy and L^p penalties.
Lemma 2.5 shows the p→1+ limit of the L^p expansion reproduces the EOT expansion; with uniform-in-p estimates, a rigorous double limit could make entropic asymptotics a corollary of the L^p theorem, unifying the two results further.
The standing C² ellipticity assumption (A2) is stronger than many applications supply; the localization arguments suggest the theorem may survive if the Bregman quadratic approximation holds only on growing compact sets, which would widen applicability to densities with zeros or singularities.
Since K(μ,ν,p) is an explicit integral of the densities and the optimal map, the formula offers a practical calibration tool: choose ε to meet a target absolute error in the OT cost using plug-in estimates of f_μ and f_ν.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper delivers what it claims: sharp first-order (EOT) and exact-rate (p-ROT) asymptotics for regularized OT, with explicit constants. The p≠2 range is new, as is the treatment of unbounded marginals with only 2+β moments for p≤2. The quantization-based upper bound is a real technical contribution, cleaner than the existing global-correction arguments. Lower bounds via duality are solid and the consistency of the p→1 limit with the EOT result is a nice touch. Credit where due: this is a serious proof paper with explicit error estimates and no obvious circularity or over-citation.

The soft spot is exactly what the stress-test identifies: Assumption (A2). The entire proof — Bregman bounds, Monge–Ampère change of variables, local profiles, quantization constants — rests on the Brenier potential being C^2 on all of R^d with uniform ellipticity. The paper calls this “the standard conclusion of classical regularity theory,” but that is not right in this generality. Classical regularity gives interior C^{2,α} under additional convexity/boundary/density hypotheses; it does not give global uniform ellipticity for arbitrary absolutely continuous measures with finite moments. The counterexamples are simple and convincing: take μ=γ and ν=(x↦x^3)#γ on R; the Brenier potential is x^4/4, with Hessian zero at the origin. Or map a ball to an annulus — no global C^1 diffeomorphism can do that. So (A2) is a real restriction, and the abstract's “mild assumptions” oversells the scope.

That said, this is not a fatal flaw in the proofs; it is a gap between the stated assumptions and the advertised generality. The paper should either prove (A2) from explicit conditions on μ,ν, or reformulate the results as conditional on (A2) and discuss which natural pairs satisfy it. The remark about correcting [15, Cor. 3.14] is terse; showing the exact invalid step would help. The p>2 cubical construction is intricate and I did not verify every line, but I found no concrete error.

Who should read this: anyone working on regularization asymptotics or computational OT. The theorems will be cited, but the assumption question needs to be settled before the results are used as a black box. It deserves a serious referee, with the request that (A2) be clarified. I would accept it for review and ask for revision.

Referee Report

2 major / 3 minor

Summary. The paper studies the vanishing-regularization limit of L^p-regularized optimal transport (1<p<∞) and entropic optimal transport (EOT). Under assumptions (A1) (absolute continuity with finite second moments) and (A2) (global C^2 uniformly elliptic Brenier potential), with additional hypotheses (H1)-(H2) for p>2 in the finite-constant case, it proves exact asymptotic limits: (ROT_{ε,p}-OT)/ε^{2/(d(p-1)+2)} → C_{d,p}K(μ,ν,p), and (EOT_ε-OT+(d/2)ε log(πε))/ε → K_EOT(μ,ν). The proof strategy separates the local profile computation (Barenblatt for L^p, Gaussian for entropy) from the global enforcement of marginal constraints by a quantization completion. The paper also establishes a formal consistency between the L^p and entropic limits as p→1+, and analyzes finiteness of the limiting constants.

Significance. If correct, the results provide the sharp small-regularization asymptotics in a substantially broader setting than earlier work, unifying and extending the known EOT and quadratically-regularized cases. The quantization construction is a genuine methodological novelty, and the paper supplies explicit constants and falsifiable predictions. The lower bounds are clean duality estimates with controlled errors, and the upper-bound construction is nontrivial and credible. The principal weakness is the strength and unverified nature of Assumption (A2), which narrows the advertised scope; this is a significant caveat but does not undermine the conditional correctness of the derivations.

major comments (2)

[Section 2.2, Assumption (A2)] Assumption (A2) requires the Brenier potential φ to be globally C^2 with uniform ellipticity σ_m I ⪯ ∇²φ ⪯ σ_M I on all of R^d. This condition is load-bearing: Lemma 3.2 (Bregman quadratic bounds), Lemma 3.4 (change of variables/diffeomorphism), Lemma 3.5 and Theorems 3.6–3.7 (lower bounds), and Lemmas 4.3–4.6 together with the upper-bound constructions all rely on the two-sided bound on ∇²φ. The paper states that (A2) 'is the standard conclusion of the classical regularity theory', but classical Monge–Ampère regularity yields at best interior C^{2,α} estimates under additional hypotheses; it does not give global uniform ellipticity from (A1). Natural examples satisfying (A1) but not (A2) exist, e.g., μ=γ, ν=(x↦x^3)_#γ, where φ(x)=x^4/4 and ∇²φ(0)=0. The authors should either prove (A2) from verifiable conditions on μ,ν, or state the theorems explicitly as conditional on this hypothesis
[Abstract and Introduction] The paper repeatedly advertises 'mild assumptions' and emphasizes that no compactness, boundary regularity, or density bounds are required (Abstract; Section 1). While the 2+β moment condition is mild, Assumption (A2) is a global, quantitative regularity hypothesis that excludes many smooth, compactly supported pairs (e.g., a Brenier map sending a ball onto an annulus cannot be a global C^1 diffeomorphism). Thus the statements of Theorems 2.1 and 2.2 are formally correct conditional on the stated hypotheses, but the advertised scope is materially narrower than the theorems deliver. This discrepancy should be corrected in the presentation, and the conditions under which (A2) actually holds should be discussed.

minor comments (3)

[Section 3.2, proof of Lemma 3.5] The proof of Lemma 3.5 invokes 'Lemma 4.1' before that lemma is stated (it appears later, in Section 4.1.1). Consider moving the Barenblatt integral lemma to Section 3, or at least adding a forward-reference note.
[Section 5, proof of Remark 2.3] The entropy calculation for the first counterexample (the measure with density h(x)=1_{(0,e^{-1})}(x)/(x(log(1/x))^2)) is very compressed. The displayed integral is hard to follow; expanding the substitution steps would improve readability.
[Equation (4)] The constant C_{d,p} is called 'dimensional' in the Introduction, but it depends on both d and p. Suggest 'dimension- and p-dependent constant'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the asymptotics are derived from explicit dual and primal constructions against standard OT inputs.

full rationale

The paper's central claim is a sharp asymptotic limit for a well-defined regularized OT problem. The limit is not taken as an input or defined into existence: the constants C_{d,p}K(μ,ν,p) and K_EOT(μ,ν) are explicit integrals involving the Brenier map, and the proofs provide matching lower and upper bounds. The lower bound evaluates the dual objective at explicit test functions; the upper bound constructs explicit couplings from local Barenblatt/Gaussian profiles and a quantization completion, then shows the remainder is negligible at the claimed scale. The local profiles are obtained from the first-order optimality conditions of the dual (not smuggled in via citation), and the quantization lemmas are proved from elementary estimates and standard quantization bounds. Although the paper cites prior work by overlapping authors (e.g., [19] for the Bregman-divergence lemma and [15] for comparison), those cited results are elementary or standard and do not incorporate the target asymptotics; they are independent support, not circular load-bearing. The manuscript itself states that the consistency Lemma 2.5 does not imply the EOT limit from the p-ROT limit, so no circular derivation exists there. The strong assumption (A2) (global C^2 uniformly elliptic Brenier potential) is a substantial regularity hypothesis, and its advertised scope may be narrower than claimed, but this is a question of correctness/generality, not of circularity. No step reduces the predicted limit to the definition of the constant or to a self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper has zero invented entities and zero fitted free parameters — the constants B_{d,p}, C_{d,p}, K(μ,ν,p) are explicit formulas in the input data. The entire load rests on the analytical assumptions (A2) (global C² uniform ellipticity of the Brenier potential), the (2+β)-moment condition for the first regime, and (H1)-(H2) for the second.

assumptions (5)

domain assumption Brenier potential φ for (μ,ν) is C² on all of R^d with uniform ellipticity 0<σ_m≤λ_min(∇²φ), λ_max(∇²φ)≤σ_M<∞ (Assumption A2).
The paper explicitly states this as an assumption in §2.2 and uses it throughout (Bregman bounds Lemma 3.2, Monge–Ampère Lemma 3.4, Taylor localizations, quantization cells). The paper remarks it is 'the standard conclusion' of regularity theory, but does not prove it or derive it from explicit conditions on μ,ν.
domain assumption Push-forward formula T=∇φ and Monge–Ampère identity f_μ = f_ν∘T det∇²φ (equation (9)).
Invoked in §3.2 (Lemma 3.4) and throughout §4. It follows from (A1)-(A2) and classical theory, but it is a load-bearing fact about the relation between the marginals and the transport map.
domain assumption For EOT and p≤2, μ,ν have finite (2+β)-moments, which gives L¹ integrability of density powers via the inequality in Remark 2.3.
Used in Lemma 4.3, the proof of Theorem 2.1(i) (s≥0 and f_μ^s ∈ L¹), and the EOT entropy estimates in §4.4.
domain assumption For p>2 with finite K, (H1) bounded interiors with Lipschitz boundaries and (H2) densities bounded above and below.
Explicitly assumed in Theorem 2.1(ii) and used in Lemma 4.4/4.5 (uniform cell densities, boundary layers) and in §4.3.
standard math Standard results quoted without proof: Fenchel–Rockafellar weak duality (Prop. 3.1), Graf–Luschgy quantization bound (Cor. 6.7), Folland's Lebesgue differentiation theorem, beta-function identities.
The duality and quantization estimates are external standard results; the paper marks the proof of Prop. 3.1 as 'standard and hence omitted.'

how reviews work

0 comments

Cite this review

Pith. "Pith review of Sharp Asymptotics for Regularized Optimal Transport." pith.science (2026). https://pith.science/paper/AUD23LXB

@misc{pith2026260718191,
  author       = {Pith},
  title        = {Pith review of: Sharp Asymptotics for Regularized Optimal Transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AUD23LXB}},
  note         = {Machine review of arXiv:2607.18191}
}

abstract

We study the small-regularization limit for $L^p$-regularized optimal transport with $1<p<\infty$ and for entropically regularized optimal transport (EOT). The exact first-order (respectively, second-order) asymptotics are determined explicitly under mild assumptions on the source and target measures. Our work generalizes the existing results for quadratic and entropic regularization, and connects them by a natural interpolation via $p\in(1,2)$. We derive all these asymptotics in a unified manner by a novel approach that separates the local computation of the optimal profile from the global enforcement of the marginal constraints: convex duality leads to Gaussian profiles for entropy and Barenblatt profiles for $L^p$-regularization, while a quantization construction turns these local profiles into couplings.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

46 extracted references · 11 linked inside Pith

[1]

Balakrishnan, T

S. Balakrishnan, T. Manole, and L. Wasserman. Statistical inference for optimal transport maps: Recent advances and perspectives.arXiv:2506.19025, 2025

arXiv 2025
[2]

Bayraktar, S

E. Bayraktar, S. Eckstein, and X. Zhang. Stability and sample complexity of divergence regu- larized optimal transport.Bernoulli, 31(1):213–239, 2025

2025
[3]

Blondel, V

M. Blondel, V. Seguy, and A. Rolet. Smooth and sparse optimal transport. In A. Storkey and F. Perez-Cruz, editors,Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, volume 84 ofProceedings of Machine Learning Research, pages 880–
[4]

G. Carlier. On the linear convergence of the multi-marginal Sinkhorn algorithm.SIAM J. Optim., 32(2):786–794, 2022

2022
[5]

Carlier, V

G. Carlier, V. Duval, G. Peyré, and B. Schmitzer. Convergence of entropic schemes for optimal transport and gradient flows.SIAM J. Math. Anal., 49(2):1385–1418, 2017

2017
[6]

Carlier, P

G. Carlier, P. Pegon, and L. Tamanini. Convergence rate of general entropic optimal transport costs.Calc. Var. Partial Differential Equations, 62(4):Paper No. 116, 28, 2023

2023
[7]

Y. Chen, T. T. Georgiou, and M. Pavon. On the relation between optimal transport and Schrödinger bridges: a stochastic control viewpoint.J. Optim. Theory Appl., 169(2):671–691, 2016

2016
[8]

Chewi, J

S. Chewi, J. Niles-Weed, and P. Rigollet.Statistical optimal transport, volume 2364 ofLecture Notes in Mathematics. Springer, Cham, 2025

2025

Show all 46 references

[9]

Chizat, A

L. Chizat, A. Delalande, and T. Vaškevičius. Sharper exponential convergence rates for Sinkhorn’s algorithm in continuous settings.Math. Program., 215(1-2):809–858, 2026

2026
[10]

Conforti, A

G. Conforti, A. Durmus, and G. Greco. Quantitative contraction rates for Sinkhorn algorithm: beyond bounded costs and compact marginals.Preprint arXiv:2304.04451v1, 2023. 48

2023 arXiv
[11]

Conforti and L

G. Conforti and L. Tamanini. A formula for the time derivative of the entropic cost and appli- cations.J. Funct. Anal., 280(11):108964, 2021

2021
[12]

M. Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport.Advances in neural information processing systems, 26, 2013

2013
[13]

del Barrio, A

E. del Barrio, A. González-Sanz, J.-M. Loubes, and D. Rodríguez-Vítores. Distributional limit theory for optimal transport.arXiv:2505.19104, 2025

2025 arXiv
[14]

Eckstein

S. Eckstein. Hilbert’s projective metric for functions of bounded growth and exponential con- vergence of Sinkhorn’s algorithm.Probab. Theory Related Fields, 193(1-2):585–621, 2025

2025
[15]

Eckstein and M

S. Eckstein and M. Nutz. Convergence rates for regularized optimal transport via quantization. Math. Oper. Res., 49(2):1223–1240, 2024

2024
[16]

Essid and J

M. Essid and J. Solomon. Quadratically regularized optimal transport on graphs.SIAM J. Sci. Comput., 40(4):A1961–A1986, 2018

2018
[17]

G. B. Folland.Real Analysis: Modern Techniques and Their Applications. John Wiley & Sons, 2nd edition, 1999

1999
[18]

Franklin and J

J. Franklin and J. Lorenz. On the scaling of multidimensional matrices.Linear Algebra Appl., 114/115:717–735, 1989

1989
[19]

Garriz-Molina, A

A. Garriz-Molina, A. González-Sanz, and G. Mordant. Infinitesimal behavior of quadrat- ically regularized optimal transport and its relation with the porous medium equation. arXiv:2407.21528, 2024

2024 arXiv
[20]

Genevay, L

A. Genevay, L. Chizat, F. Bach, M. Cuturi, and G. Peyré. Sample complexity of Sinkhorn divergences. InThe 22nd International Conference on Artificial Intelligence and Statistics, pages 1574–1583. PMLR, 2019

2019
[21]

Ghosal and M

P. Ghosal and M. Nutz. On the convergence rate of Sinkhorn’s algorithm.Mathematics of Operations Research, 51(2):1080–1096, 2026

2026
[22]

González-Sanz, E

A. González-Sanz, E. del Barrio, and M. Nutz. Sample complexity of quadratically regularized optimal transport.arXiv:2511.09807, 2025

2025
[23]

González-Sanz, S

A. González-Sanz, S. Eckstein, and M. Nutz. Sparse regularized optimal transport without curse of dimensionality.Preprint arXiv:2505.04721, 2025

2025 arXiv
[24]

González-Sanz, R

A. González-Sanz, R. S. Gvalani, and L. Koch. Sharp local sparsity of regularized optimal transport.arXiv:2604.00843, 2026

2026 arXiv
[25]

González-Sanz, M

A. González-Sanz, M. Nutz, and A. Riveros Valdevenito. Linear convergence of gradient descent for quadratically regularized optimal transport.Preprint arXiv:2509.08547, 2025

2025 arXiv
[26]

González-Sanz, M

A. González-Sanz, M. Nutz, and A. Riveros Valdevenito. Polyak–Łojasiewicz inequal- ity for quadratically regularized optimal transport.SIAM J. Optim., forthcoming, 2026. arXiv:2605.27175

2026 arXiv
[27]

González-Sanz, M

A. González-Sanz, M. Nutz, and A. Stromme. Finite-sample bounds for regularized optimal transport.Preprint arXiv:2606.25947, 2026

2026 arXiv
[28]

González-Sanz and M

A. González-Sanz and M. Nutz. Sparsity of quadratically regularized optimal transport: Scalar case.SIAM J. Math. Anal., forthcoming.arXiv:2410.03353
[29]

Graf and H

S. Graf and H. Luschgy.Foundations of quantization for probability distributions, volume 1730 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 2000

2000
[30]

Han and J

R. Han and J. Wiesel. Sample complexity for divergence regularized optimal transport with radial cost.arXiv:2510.05685, 2026

2026
[31]

C. Léonard. From the Schrödinger problem to the Monge-Kantorovich problem.J. Funct. Anal., 49 262(4):1879–1920, 2012

1920
[32]

C. Léonard. A survey of the Schrödinger problem and some of its connections with optimal transport.Discrete Contin. Dyn. Syst., 34(4):1533–1574, 2014

2014
[33]

Lorenz, P

D. Lorenz, P. Manns, and C. Meyer. Quadratically regularized optimal transport.Appl. Math. Optim., 83(3):1919–1949, 2021

1919
[34]

Mena and J

G. Mena and J. Niles-Weed. Statistical bounds for entropic optimal transport: sample complex- ity and the central limit theorem. InAdvances in Neural Information Processing Systems 32, pages 4541–4551. 2019

2019
[35]

Muzellec, R

B. Muzellec, R. Nock, G. Patrini, and F. Nielsen. Tsallis regularized optimal transport and ecological inference.Proceedings of the AAAI Conference on Artificial Intelligence, 31(1), 2017

2017
[36]

Nutz.Introduction to Entropic Optimal Transport

M. Nutz.Introduction to Entropic Optimal Transport. Lecture notes, Columbia University, 2021.https://www.math.columbia.edu/~mnutz/docs/EOT_lecture_notes.pdf

2021
[37]

M. Nutz. Quadratically regularized optimal transport: existence and multiplicity of potentials. SIAM J. Math. Anal., 57(3):2622–2649, 2025

2025
[38]

Nutz and C

M. Nutz and C. Zhong. Entropic regularization of Monge’s problem.Preprint arXiv:2604.21578, 2026

2026 arXiv
[39]

S. Pal. On the difference between entropic cost and the optimal transport cost.Ann. Appl. Probab., 34(1B):1003–1028, 2024

2024
[40]

Peyré and M

G. Peyré and M. Cuturi. Computational optimal transport: With applications to data science. Foundations and Trends in Machine Learning, 11(5-6):355–607, 2019

2019
[41]

Schmitzer

B. Schmitzer. Stabilized sparse scaling algorithms for entropy regularized transport problems. SIAM J. Sci. Comput., 41(3):A1443–A1481, 2019

2019
[42]

Schrödinger

E. Schrödinger. Über die Umkehrung der Naturgesetze. Sitzungsberichte Preuss. Akad. Wiss. Akad. Wiss., Berlin. Phys. Math., 144:144–153, 1931

1931
[43]

Sinkhorn

R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Statist., 35:876–879, 1964

1964
[44]

Villani.Optimal Transport: Old and New

C. Villani.Optimal Transport: Old and New. Springer-Verlag Berlin Heidelberg, 2008

2008
[45]

Wiesel and X

J. Wiesel and X. Xu. Sparsity of quadratically regularized optimal transport: Bounds on con- centration and bias.SIAM J. Math. Anal., 57(6):6498–6521, 2025

2025
[46]

Zhang, G

S. Zhang, G. Mordant, T. Matsumoto, and G. Schiebinger. Manifold learning with sparse regularised optimal transport.arXiv preprint arXiv:2307.09816, 2023. 50

2023 arXiv

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Balakrishnan, T

S. Balakrishnan, T. Manole, and L. Wasserman. Statistical inference for optimal transport maps: Recent advances and perspectives.arXiv:2506.19025, 2025

arXiv 2025

[2] [2]

Bayraktar, S

E. Bayraktar, S. Eckstein, and X. Zhang. Stability and sample complexity of divergence regu- larized optimal transport.Bernoulli, 31(1):213–239, 2025

2025

[3] [3]

Blondel, V

M. Blondel, V. Seguy, and A. Rolet. Smooth and sparse optimal transport. In A. Storkey and F. Perez-Cruz, editors,Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, volume 84 ofProceedings of Machine Learning Research, pages 880–

[4] [4]

G. Carlier. On the linear convergence of the multi-marginal Sinkhorn algorithm.SIAM J. Optim., 32(2):786–794, 2022

2022

[5] [5]

Carlier, V

G. Carlier, V. Duval, G. Peyré, and B. Schmitzer. Convergence of entropic schemes for optimal transport and gradient flows.SIAM J. Math. Anal., 49(2):1385–1418, 2017

2017

[6] [6]

Carlier, P

G. Carlier, P. Pegon, and L. Tamanini. Convergence rate of general entropic optimal transport costs.Calc. Var. Partial Differential Equations, 62(4):Paper No. 116, 28, 2023

2023

[7] [7]

Y. Chen, T. T. Georgiou, and M. Pavon. On the relation between optimal transport and Schrödinger bridges: a stochastic control viewpoint.J. Optim. Theory Appl., 169(2):671–691, 2016

2016

[8] [8]

Chewi, J

S. Chewi, J. Niles-Weed, and P. Rigollet.Statistical optimal transport, volume 2364 ofLecture Notes in Mathematics. Springer, Cham, 2025

2025