Entropic optimal transport beyond product reference couplings: the Gaussian case on Euclidean space

Nikitas Georgakis; Paul Freulon; Victor Panaretos

arxiv: 2507.01709 · v2 · submitted 2025-07-02 · 🧮 math.ST · stat.ML· stat.TH

Entropic optimal transport beyond product reference couplings: the Gaussian case on Euclidean space

Paul Freulon , Nikitas Georgakis , Victor Panaretos This is my paper

Pith reviewed 2026-05-19 06:54 UTC · model grok-4.3

classification 🧮 math.ST stat.MLstat.TH

keywords entropic optimal transportGaussian reference couplingmatrix optimizationdual variablestrajectory reconstructioncontinuous-time process

0 comments

The pith

Entropic optimal transport with Gaussian reference couplings reduces to a matrix optimization problem yielding explicit primal and dual solutions.

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

The paper shows that regularizing optimal transport with entropy toward a Gaussian reference coupling instead of a product measure simplifies the problem to a matrix optimization task. This reduction works for squared Euclidean costs on Euclidean space and gives a full description of the optimal coupling along with its dual variables. The change matters for dynamic problems because Gaussian references produce transitions that fit together into a consistent continuous-time process, allowing reconstruction of trajectories from observations at only a few time points.

Core claim

For the entropic optimal transport problem with squared Euclidean cost, replacing the usual product reference coupling by a Gaussian one reduces the variational problem to a matrix optimization. The resulting matrix problem supplies complete characterizations of both the optimal primal coupling and the dual variables. This structure is motivated by the need to assemble finitely many time marginals into coherent continuous-time dynamics, a task that product references cannot accomplish without breaking consistency.

What carries the argument

Reduction of the regularized optimal transport problem to a matrix optimization problem that parameterizes the Gaussian reference coupling and yields explicit solutions for the primal and dual variables.

If this is right

The optimal coupling and dual potentials are obtained directly from the solution of the matrix problem.
Transitions induced by the Gaussian reference assemble into a coherent continuous-time process.
The framework supports reconstruction of trajectory dynamics from a finite collection of time marginals.

Where Pith is reading between the lines

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

The matrix reduction may suggest analogous finite-dimensional simplifications when other structured non-product references are used.
In statistical applications the explicit form could lead to faster algorithms for fitting models with temporal dependence.

Load-bearing premise

The reference coupling must be Gaussian and the cost must be squared Euclidean distance on Euclidean space for the reduction to a matrix problem to hold.

What would settle it

For low-dimensional Gaussian marginals, solve the matrix optimization, construct the implied coupling, and check whether it achieves the minimal value of the entropic objective compared with any other coupling.

Figures

Figures reproduced from arXiv: 2507.01709 by Nikitas Georgakis, Paul Freulon, Victor Panaretos.

**Figure 1.** Figure 1: Impact of the regularization parameter ε on the transport plan πε solution of the classic entropic optimal transport problem. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗

**Figure 2.** Figure 2: Impact of the correlation parameter ρ on the transport plan πε solution of the entropic optimal transport problem when the regularizing parameter ε is set to two. 0 2 4 6 8 10 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 W W 2 2 = 0 = 0.3 = 0.6 = 0.9 = 0.95 = 0.99 [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗

**Figure 3.** Figure 3: Evolution of the bias of the entropic transport cost [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗

**Figure 4.** Figure 4: Evolution of the bias of the entropic transport cost [PITH_FULL_IMAGE:figures/full_fig_p030_4.png] view at source ↗

read the original abstract

The Optimal Transport (OT) problem with squared Euclidean cost consists in finding a coupling between two input measures that maximizes correlation. Consequently, the optimal coupling is often singular with respect to the Lebesgue measure. Regularizing the OT problem with an entropy term yields an approximation called entropic optimal transport. Entropic penalties steer the induced coupling toward a reference measure with desired properties. For instance, when seeking a diffuse coupling, the most popular reference measures are the Lebesgue measure and the product of the two input measures. In this work, we study the case where the reference coupling is not a product, focussing on the Gaussian case as a core paradigm. We establish a reduction of such a regularised OT problem to a matrix optimization problem, enabling us to provide a complete description of the solution, both in terms of the primal variable and the dual variables. Beyond its intrinsic interest, allowing non-product references is essential in dynamic statistical settings. As a key motivation, we address the reconstruction of trajectory dynamics from finitely many time marginals where, unlike product references, Gaussian process references produce transitions that assemble into a coherent continuous-time process.

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 reduces entropic OT with non-product Gaussian references to a matrix optimization problem and supplies explicit primal-dual solutions for the squared-Euclidean case.

read the letter

The main takeaway is that this paper extends entropic optimal transport to non-product Gaussian reference couplings and reduces the problem to a matrix optimization with full primal and dual characterizations. They focus on the Gaussian case as a core example. This allows them to derive closed forms that go beyond the product-measure setups common in the field. The key is that the reference is Gaussian but not independent, which affects the entropy regularization. This extension is useful for dynamic statistical settings. When reconstructing trajectories from marginals at discrete times, you need the optimal transitions to be consistent across time. Gaussian process references achieve this coherence, while product references fall short. The paper highlights this motivation effectively. The derivations seem solid and direct from the objective. There is no apparent circularity or unsupported fitting in the reduction. A potential limitation is the dependence on squared Euclidean cost and Euclidean space. The matrix reduction exploits the specific structure of Gaussians under this cost, so broader applicability would require additional work. I would check the full paper for the explicit matrix construction to ensure it is fully justified. This paper suits readers interested in optimal transport applications to dynamics and inference. It provides concrete tools for those working on continuous-time process reconstruction. Given the clear novelty and the absence of major flaws, it merits a serious referee. I would recommend sending it for peer review.

Referee Report

2 major / 2 minor

Summary. The paper studies entropic optimal transport with squared-Euclidean cost on Euclidean space when the reference coupling is a non-product Gaussian (or Gaussian process). It reduces the regularized problem to a matrix optimization problem and supplies explicit characterizations of both the primal coupling and the dual variables. The construction is motivated by the need to reconstruct coherent continuous-time trajectory dynamics from finitely many time marginals, where product references fail to produce consistent transitions.

Significance. If the reduction and explicit solutions are correct, the work provides a concrete extension of entropic OT beyond the standard product or Lebesgue references. The matrix formulation yields a complete primal-dual description that is directly usable for Gaussian-process references, addressing a practical gap in dynamic statistical settings. The emphasis on assembling transitions into a continuous-time process is a clear strength for applications in trajectory inference.

major comments (2)

[§3, Theorem 3.2] §3, Theorem 3.2: the claimed reduction of the entropic OT objective to an unconstrained matrix optimization problem is central to the complete description of the solution; the proof sketch relies on the quadratic cost and Gaussian reference to obtain closed-form optimality conditions, but the explicit matrix construction and verification that the resulting kernel satisfies the marginal constraints for arbitrary dimensions should be expanded to confirm no hidden parameter fitting occurs.
[§4.2, Proposition 4.3] §4.2, Proposition 4.3: the dynamic consistency of the assembled transitions under the Gaussian-process reference is asserted to follow directly from the reference property, yet the argument would benefit from an explicit check that the finite-dimensional marginals remain consistent with a single underlying Gaussian process when the number of time points increases.

minor comments (2)

[§2 and §3] Notation for the reference covariance matrices is introduced in §2 but reused with different subscripts in §3 without a consolidated table; a single reference table would improve readability.
[Introduction] The abstract states that the reduction enables a 'complete description' of primal and dual variables; the introduction could more explicitly contrast this with the product-reference case to highlight the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and positive recommendation for minor revision. We address each major comment below and have prepared revisions to incorporate the suggested clarifications.

read point-by-point responses

Referee: [§3, Theorem 3.2] §3, Theorem 3.2: the claimed reduction of the entropic OT objective to an unconstrained matrix optimization problem is central to the complete description of the solution; the proof sketch relies on the quadratic cost and Gaussian reference to obtain closed-form optimality conditions, but the explicit matrix construction and verification that the resulting kernel satisfies the marginal constraints for arbitrary dimensions should be expanded to confirm no hidden parameter fitting occurs.

Authors: We appreciate the referee's suggestion to strengthen the presentation of Theorem 3.2. Upon review, we agree that expanding the proof to include the explicit matrix construction and a step-by-step verification of the marginal constraints in general dimensions will improve readability and confirm the absence of any hidden parameters. The derivation is based solely on the closed-form optimality conditions from the quadratic cost and the Gaussian reference covariance. We will include this expanded proof in the revised manuscript. revision: yes
Referee: [§4.2, Proposition 4.3] §4.2, Proposition 4.3: the dynamic consistency of the assembled transitions under the Gaussian-process reference is asserted to follow directly from the reference property, yet the argument would benefit from an explicit check that the finite-dimensional marginals remain consistent with a single underlying Gaussian process when the number of time points increases.

Authors: We thank the referee for this observation. While the consistency follows from the defining property of the Gaussian process reference (i.e., its finite-dimensional distributions being multivariate Gaussian with consistent covariances), we acknowledge that an explicit check would be beneficial for readers. In the revised version, we will add a detailed verification, perhaps in a remark following Proposition 4.3, showing that the marginals for any finite set of times are consistent with the underlying process via the Kolmogorov consistency conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's central reduction of the entropic OT problem with non-product Gaussian reference coupling to a matrix optimization problem is derived directly from the regularized objective, the squared-Euclidean cost, and the Gaussian process reference properties on Euclidean space. Explicit primal and dual solutions follow from this formulation, and the assembly of transitions into a coherent continuous-time process is a direct consequence of the reference measure choice. No load-bearing step reduces by construction to a fitted input, self-definition, or unverified self-citation chain; the argument remains self-contained under the stated modeling assumptions without circular re-expression of inputs as predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard squared-Euclidean OT setup plus the modeling decision to use a Gaussian (process) reference; no free parameters or new invented entities are declared in the abstract.

axioms (2)

domain assumption The cost function is squared Euclidean distance on Euclidean space.
Stated in the title and abstract as the setting for the Gaussian case.
domain assumption The reference coupling can be chosen as a non-product Gaussian process whose transitions assemble into a continuous-time process.
Invoked in the motivation paragraph of the abstract for the dynamic reconstruction application.

pith-pipeline@v0.9.0 · 5734 in / 1286 out tokens · 32342 ms · 2026-05-19T06:54:39.765217+00:00 · methodology

discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Sliced-Regularized Optimal Transport
stat.ML 2026-04 unverdicted novelty 7.0

SROT regularizes the OT plan toward a smoothened sliced OT plan, producing more accurate approximations to exact OT than entropic OT while also improving on the sliced OT reference.
Sliced-Regularized Optimal Transport
stat.ML 2026-04 unverdicted novelty 7.0

SROT regularizes the OT transport plan toward a sliced OT reference, yielding better approximations of exact OT than entropic OT and improving on the sliced OT plan itself.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages · cited by 1 Pith paper

[1]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar´ e. Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser, Boston, 2005

work page 2005
[2]

C. R. Baker. Joint measures and cross-covariance operators. Transactions of the American Mathe- matical Society, 186:273–289, 1973

work page 1973
[3]

Bakonyi and H

M. Bakonyi and H. J. Woerdeman. Matrix completions, moments, and sums of Hermitian squares . Princeton University Press, 2011

work page 2011
[4]

Bengtsson and K

I. Bengtsson and K. ˙Zyczkowski. Geometry of Quantum States: An Introduction to Quantum En- tanglement. Cambridge University Press, Cambridge, UK, 2006

work page 2006
[5]

R. Bhatia. Positive definite matrices. In Positive Definite Matrices. Princeton university press, 2009

work page 2009
[6]

Bhatia, T

R. Bhatia, T. Jain, and Y. Lim. On the bures-wasserstein distance between positive definite matrices. Expositiones Mathematicae, 37(2):165–191, 2019

work page 2019
[7]

S. P. Boyd and L. Vandenberghe. Convex optimization. Cambridge university press, 2004

work page 2004
[8]

Chewi, J

S. Chewi, J. Niles-Weed, and P. Rigollet. Statistical optimal transport. arXiv preprint arXiv:2407.18163, 2024

work page arXiv 2024
[9]

Chizat, P

L. Chizat, P. Roussillon, F. L´ eger, F.-X. Vialard, and G. Peyr´ e. Faster wasserstein distance estima- tion with the sinkhorn divergence. Advances in Neural Information Processing Systems , 33:2257– 2269, 2020. 30

work page 2020
[10]

J. A. Cuesta-Albertos, C. Matr´ an-Bea, and A. Tuero-Diaz. On lower bounds for the l 2-wasserstein metric in a hilbert space. Journal of Theoretical Probability, 9(2):263–283, 1996

work page 1996
[11]

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

work page 2013
[12]

Feydy, T

J. Feydy, T. S´ ejourn´ e, F.-X. Vialard, S.-i. Amari, A. Trouv´ e, and G. Peyr´ e. Interpolating between optimal transport and mmd using sinkhorn divergences. In The 22nd International Conference on Artificial Intelligence and Statistics , pages 2681–2690. PMLR, 2019

work page 2019
[13]

C. R. Givens and R. M. Shortt. A class of wasserstein metrics for probability distributions. Michigan Mathematical Journal, 31(2):231–240, 1984

work page 1984
[14]

Janati, M

H. Janati, M. Cuturi, and A. Gramfort. Debiased sinkhorn barycenters. In International Conference on Machine Learning, pages 4692–4701. PMLR, 2020

work page 2020
[15]

Janati, B

H. Janati, B. Muzellec, G. Peyr´ e, and M. Cuturi. Entropic optimal transport between unbalanced gaussian measures has a closed form. Advances in neural information processing systems, 33:10468– 10479, 2020

work page 2020
[16]

L´ eonard

C. L´ eonard. A survey of the schrodinger problem and some of its connections with optimal transport, 2013

work page 2013
[17]

J. R. Magnus and H. Neudecker. Matrix differential calculus with applications in statistics and econometrics. John Wiley & Sons, 2019

work page 2019
[18]

Mallasto, A

A. Mallasto, A. Gerolin, and H. Q. Minh. Entropy-regularized 2-Wasserstein distance between Gaussian measures. Information Geometry, 5(1):289–323, 2022

work page 2022
[19]

S. D. Marino and A. Gerolin. An optimal transport approach for the schr¨ odinger bridge problem and convergence of sinkhorn algorithm. Journal of Scientific Computing , 85(2):27, 2020

work page 2020
[20]

H. Q. Minh. Entropic regularization of wasserstein distance between infinite-dimensional gaussian measures and gaussian processes. Journal of Theoretical Probability, 36(1):201–296, 2023

work page 2023
[21]

M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information . Cambridge University Press, Cambridge, UK, 2000

work page 2000
[22]

M. Nutz. Introduction to entropic optimal transport. Lecture notes, Columbia University , 2021

work page 2021
[23]

V. M. Panaretos and Y. Zemel. An invitation to statistics in Wasserstein space . Springer Nature, 2020

work page 2020
[24]

L. Pardo. Statistical inference based on divergence measures. Chapman and Hall/CRC, 2018

work page 2018
[25]

Peyr´ e, M

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

work page 2019
[26]

Rigollet and A

P. Rigollet and A. J. Stromme. On the sample complexity of entropic optimal transport. The Annals of Statistics, 53(1):61–90, 2025

work page 2025
[27]

A. Takatsu. Wasserstein geometry of gaussian measures. Osaka J. Math. , 2011

work page 2011
[28]

C. Villani. Optimal transport: old and new , volume 338. Springer, 2009

work page 2009
[29]

C. Villani. Topics in optimal transportation , volume 58. American Mathematical Soc., 2021. 31 A Auxiliary results On the space of squared matrices Md(R), the Hilbert-Schmidt (also called Frobenius) scalar product is defined by ⟨A, B⟩HS := tr(ABT ); and reduces to ⟨A, B⟩HS = tr(AB) between symmetric matrices. Proposition A.1 (Kullback-Leibler divergence) ...

work page 2021

[1] [1]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar´ e. Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser, Boston, 2005

work page 2005

[2] [2]

C. R. Baker. Joint measures and cross-covariance operators. Transactions of the American Mathe- matical Society, 186:273–289, 1973

work page 1973

[3] [3]

Bakonyi and H

M. Bakonyi and H. J. Woerdeman. Matrix completions, moments, and sums of Hermitian squares . Princeton University Press, 2011

work page 2011

[4] [4]

Bengtsson and K

I. Bengtsson and K. ˙Zyczkowski. Geometry of Quantum States: An Introduction to Quantum En- tanglement. Cambridge University Press, Cambridge, UK, 2006

work page 2006

[5] [5]

R. Bhatia. Positive definite matrices. In Positive Definite Matrices. Princeton university press, 2009

work page 2009

[6] [6]

Bhatia, T

R. Bhatia, T. Jain, and Y. Lim. On the bures-wasserstein distance between positive definite matrices. Expositiones Mathematicae, 37(2):165–191, 2019

work page 2019

[7] [7]

S. P. Boyd and L. Vandenberghe. Convex optimization. Cambridge university press, 2004

work page 2004

[8] [8]

Chewi, J

S. Chewi, J. Niles-Weed, and P. Rigollet. Statistical optimal transport. arXiv preprint arXiv:2407.18163, 2024

work page arXiv 2024

[9] [9]

Chizat, P

L. Chizat, P. Roussillon, F. L´ eger, F.-X. Vialard, and G. Peyr´ e. Faster wasserstein distance estima- tion with the sinkhorn divergence. Advances in Neural Information Processing Systems , 33:2257– 2269, 2020. 30

work page 2020

[10] [10]

J. A. Cuesta-Albertos, C. Matr´ an-Bea, and A. Tuero-Diaz. On lower bounds for the l 2-wasserstein metric in a hilbert space. Journal of Theoretical Probability, 9(2):263–283, 1996

work page 1996

[11] [11]

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

work page 2013

[12] [12]

Feydy, T

J. Feydy, T. S´ ejourn´ e, F.-X. Vialard, S.-i. Amari, A. Trouv´ e, and G. Peyr´ e. Interpolating between optimal transport and mmd using sinkhorn divergences. In The 22nd International Conference on Artificial Intelligence and Statistics , pages 2681–2690. PMLR, 2019

work page 2019

[13] [13]

C. R. Givens and R. M. Shortt. A class of wasserstein metrics for probability distributions. Michigan Mathematical Journal, 31(2):231–240, 1984

work page 1984

[14] [14]

Janati, M

H. Janati, M. Cuturi, and A. Gramfort. Debiased sinkhorn barycenters. In International Conference on Machine Learning, pages 4692–4701. PMLR, 2020

work page 2020

[15] [15]

Janati, B

H. Janati, B. Muzellec, G. Peyr´ e, and M. Cuturi. Entropic optimal transport between unbalanced gaussian measures has a closed form. Advances in neural information processing systems, 33:10468– 10479, 2020

work page 2020

[16] [16]

L´ eonard

C. L´ eonard. A survey of the schrodinger problem and some of its connections with optimal transport, 2013

work page 2013

[17] [17]

J. R. Magnus and H. Neudecker. Matrix differential calculus with applications in statistics and econometrics. John Wiley & Sons, 2019

work page 2019

[18] [18]

Mallasto, A

A. Mallasto, A. Gerolin, and H. Q. Minh. Entropy-regularized 2-Wasserstein distance between Gaussian measures. Information Geometry, 5(1):289–323, 2022

work page 2022

[19] [19]

S. D. Marino and A. Gerolin. An optimal transport approach for the schr¨ odinger bridge problem and convergence of sinkhorn algorithm. Journal of Scientific Computing , 85(2):27, 2020

work page 2020

[20] [20]

H. Q. Minh. Entropic regularization of wasserstein distance between infinite-dimensional gaussian measures and gaussian processes. Journal of Theoretical Probability, 36(1):201–296, 2023

work page 2023

[21] [21]

M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information . Cambridge University Press, Cambridge, UK, 2000

work page 2000

[22] [22]

M. Nutz. Introduction to entropic optimal transport. Lecture notes, Columbia University , 2021

work page 2021

[23] [23]

V. M. Panaretos and Y. Zemel. An invitation to statistics in Wasserstein space . Springer Nature, 2020

work page 2020

[24] [24]

L. Pardo. Statistical inference based on divergence measures. Chapman and Hall/CRC, 2018

work page 2018

[25] [25]

Peyr´ e, M

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

work page 2019

[26] [26]

Rigollet and A

P. Rigollet and A. J. Stromme. On the sample complexity of entropic optimal transport. The Annals of Statistics, 53(1):61–90, 2025

work page 2025

[27] [27]

A. Takatsu. Wasserstein geometry of gaussian measures. Osaka J. Math. , 2011

work page 2011

[28] [28]

C. Villani. Optimal transport: old and new , volume 338. Springer, 2009

work page 2009

[29] [29]

C. Villani. Topics in optimal transportation , volume 58. American Mathematical Soc., 2021. 31 A Auxiliary results On the space of squared matrices Md(R), the Hilbert-Schmidt (also called Frobenius) scalar product is defined by ⟨A, B⟩HS := tr(ABT ); and reduces to ⟨A, B⟩HS = tr(AB) between symmetric matrices. Proposition A.1 (Kullback-Leibler divergence) ...

work page 2021