Convergence Rate of the Solution of Multi-marginal Schrodinger Bridge Problem with Marginal Constraints from SDEs

Geoffrey Schiebinger; Rentian Yao; Young--Heon Kim

REVIEW 2 major objections 5 minor 24 references

Convergence Rate of the Solution of Multi-marginal Schrodinger Bridge Problem with Marginal Constraints from SDEs

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper proves that multi-marginal Schrödinger bridges built from SDE marginals converge to the true path measure at rate $O(m^{-1})$ in KL divergence, extending known results to time-dependent drift.

desk verdict Genuine extension to time-dependent drift, but the main theorem is not proven under Assumption A because C2(Ψ) can be infinite; repairable and worth reviewing. read the letter →

arxiv 2507.09151 v1 pith:PMEOBARZ submitted 2025-07-12 math.PR math.STstat.TH

Rentian Yao , Young--Heon Kim , Geoffrey Schiebinger This is my paper · ORCID

classification math.PRmath.STstat.TH MSC 60J6060H1049Q22

keywords multi-marginalSchrödingerbridgeKLdivergenceconvergencerateSDEmarginalstime-dependentdriftGirsanovtheoremMarkovpropertytrajectoryinference

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

This paper establishes a quantitative convergence rate for the multi-marginal Schrödinger bridge problem when the prescribed marginals are pulled from a diffusion process with a time-dependent drift. The main theorem says that on the flat torus, if the potential $\Psi$ is smooth enough and the time points are equally spaced, the KL divergence between the true SDE path measure and the bridge constrained at $m$ time points is $O(m^{-1})$. For general time grids the error is controlled by the largest spacing $\Delta_m$. This matters because trajectory inference and related problems observe snapshots of a process whose drift changes over time, and the result guarantees that the finitely constrained bridge is a controlled approximation to the underlying SDE. The proof is built from a two-marginal estimate of order $\varepsilon^2$, summed over consecutive intervals via the Markov property of the bridge.

What carries the argument

The central object is the multi-marginal Schrödinger bridge $\widehat{R}^{T_m}$, the unique minimizer of KL divergence against reversible Brownian motion subject to prescribed marginals at times $t_0,\dots,t_m$. Three mechanisms carry the proof. First, the Markov property of the bridge, combined with the chain rule for KL, decomposes the total divergence into one two-marginal term per interval. Second, Girsanov's theorem identifies the KL divergence between the SDE law and the Wiener measure as half the integrated expected squared drift, which anchors the two-marginal comparison. Third, the variational formulation of the Schrödinger bridge provides an admissible velocity field built from the SDE drift and $\nabla\log\rho_t$; comparing it with the optimal field yields the $\varepsilon^2$ bound through an exponential expectation controlled by constants $C_1$ and $C_2$ that depend only on $\Psi$ and $\sup|\nabla\log\rho_t|$.

What would settle it

Take $X=\mathbb{T}$, $\tau=1$, $\Psi=0$, and $\rho_0(x)\propto\sin^2(x/2)$ so that $\nabla\log\rho_0$ is unbounded while Assumption A holds; numerically solve the two-marginal problem (1.4) for small $\varepsilon$ and estimate $\mathrm{KL}(R^*_{[0,\varepsilon]}\|\widehat{R}_{[0,\varepsilon]})$. If the decay is slower than $\varepsilon^2$, the finiteness of $\sup|\nabla\log\rho_t|$ is genuinely needed; if it still decays like $\varepsilon^2$, the stated condition is stronger than necessary.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a time-dependent drift does not break the convergence rate obtained for time-independent drift: the total error accumulates only through the largest gap between constraint times. Theorem 1.1 states $$\mathrm{KL}(R^* \parallel \widehat{R}^{T_m}) \le \left(\frac{3C_1}{2}+\sqrt{\frac{5C_1}{2}C_2}\right)\Delta_m,$$ and hence with equal spacing $\Delta_m=T/m$ the rate is $O(m^{-1})$, with constants independent of $m$. The engine is the $m=1$ result, Theorem 1.2, which gives the same kind of bound on a short interval of length $\varepsilon$ as $O(\varepsilon^2)$. Because the multi-marginal bridge is Markov, the full KL divergence decomposes into a sum of two-marginal terms, so $m$ intervals of length $O(1/m)$ produce $m\cdot O(m^{-2})=O(m^{-1})$.

Load-bearing premise

The proof needs $\sup_{t,x}|\nabla\log\rho_t(x)|$ to be finite because this quantity defines the constant $C_2$, but Assumption A only requires the initial law to have finite KL divergence against Lebesgue measure, so the theorem is not established for all initial densities the assumptions nominally allow.

Editorial extensions

If this is right

Doubling the number of equally spaced snapshots halves the KL error between the bridge and the true SDE path measure.
For nonuniform grids, only the largest gap matters: shrinking the maximum spacing $\Delta_m$ shrinks the error linearly, regardless of how the remaining points are placed.
The two-marginal estimate gives interval-halving a factor-four improvement: a bridge on a window of length $\varepsilon$ has KL error $O(\varepsilon^2)$.
Time-dependent drift, common in developmental biology, is covered by the same convergence guarantee as the time-independent case.

Reading between the lines

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

Beyond the paper: the constant $C_2$ involves $\sup|\nabla\log\rho_t|$, which is not implied by Assumption A; the sharp condition for the $O(m^{-1})$ rate may be a gradient or log-Sobolev bound on the marginal densities rather than mere finite KL of $\rho_0$.
Beyond the paper: because the KL divergence decomposes by intervals, the rate should transfer to approximate solvers: any method that computes each two-marginal bridge to within $\delta_m$ error would inherit an overall error of $O(\Delta_m + m\delta_m)$, indicating where computational effort should be spent.
Beyond the paper: on noncompact spaces such as $\mathbb{R}^d$, an analogous $\varepsilon^2$ estimate should hold when the drift is dissipative and the marginal densities satisfy uniform gradient bounds; the torus assumption mainly supplies compactness for existence and finite constants.

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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

Dear colleague,

Two things to know about this paper: the contribution is real, and the main theorem as stated is not proven under its own assumptions. The authors extend Agarwal et al.'s convergence result for multi-marginal Schrödinger bridges to time-dependent SDE drift, which is a genuine step forward for trajectory inference. The decomposition into independent two-marginal bridges via the Markov property (their (2.1)) is clean, and the Girsanov and variational arguments are mostly sound.

The soft spot is Step 3.2. The constant C2(Ψ) is defined as sup over [0,1]×X of |∇log ρ_t|, and the text claims this is finite by compactness. That only follows if ρ_t is bounded away from zero uniformly. Assumption A gives ρ0 ∈ P^r_2 with finite KL, which allows smooth densities with zeros (e.g., (1−cos x)^2 on T^1). For such ρ0, the heat-evolved density has |∇log ρ_t| ~ t^{−1/2} near the zero, so the sup over any interval starting at 0 is infinite. Worse, if one patches the proof by taking the sup over [δ,1], the constant scales like δ^{−1/2}, and applying this to each interval of length T/m gives a per-interval error O(m^{−3/2}), summing to O(m^{−1/2}) rather than O(m^{−1}). So the claimed uniform constant independent of m is not established, even for smooth initial densities with zeros.

This is repairable: add an assumption that ρ0 is C^2 and bounded below, or that sup |∇log ρ_t| is finite, and the theorem becomes true with constants depending on that bound. As it stands, the theorem overreaches its assumptions. Minor issues: the text mixes Φ and Ψ in several places, cites a nonexistent Theorem 1.3, and says C2 depends only on Ψ when it also depends on ρ0.

The result is likely correct under a slightly stronger hypothesis, and the paper should go to peer review. A referee should ask for the C2 assumption to be fixed and the notation cleaned up. The audience is researchers in Schrödinger bridges, optimal transport, and trajectory inference; they will care about the explicit rate even if the proof needs amendment.

Referee Report

2 major / 5 minor

Summary. The paper studies the multi-marginal Schrödinger bridge problem (MSB) when the marginal constraints are the time-marginals of an SDE with a constant diffusion coefficient and a time-dependent drift. The main result, Theorem 1.1, claims that on the flat torus, under Assumption A and Ψ∈C^{2,4}, the KL divergence between the SDE path measure R* and the solution of the MSB with m time-point marginals is bounded by a constant times Δ_m, the maximal gap between consecutive time points, and hence is O(m^{-1}) for equally spaced points. The proof first decomposes the KL divergence over adjacent intervals using the Markov property of Schrödinger bridges (Section 2.1), then proves a two-marginal estimate of order ε² (Theorem 1.2) via Girsanov's theorem and the variational formulation of the Schrödinger bridge (Section 2.2). The paper frames this as an extension of prior work by Agarwal et al. to time-dependent drifts.

Significance. If the main theorem is valid, the result provides a quantitative convergence rate for multi-marginal Schrödinger bridge approximations to SDE path measures, a problem motivated by trajectory inference and finite-sample analysis. The proof strategy—decomposing the problem into two-marginal bridges and combining Girsanov estimates with the Benamou–Brenier-type variational formula—is natural and mostly self-contained. The paper also benefits from stating the result in terms of an explicit bound with constants independent of m, which is a concrete and useful form. However, the proof as written contains a load-bearing gap concerning the uniform finiteness of a constant C₂(Ψ) defined through the log-density gradient of the SDE marginals; this gap prevents the stated theorem from being established under the current assumptions.

major comments (2)

[Section 2.2, Step 3.2] The constant C2(Ψ) is defined as sup_{t∈[0,1],x∈X} max{|∇Ψ(t,x)|, |∇ log ρ(t,x)|} and asserted to be finite by compactness. This assertion is not a consequence of Assumption A: a density ρ0 satisfying KL(ρ0∥λ_X)<∞ may vanish at a point, in which case ∇ log ρ0 is unbounded. For example, on T^1 take ρ0(x)∝(1−cos x)^2; then ∇ log ρ0(x) behaves like 2/x near x=0. Even for t>0, the heat-evolved density ρ_t has sup_x |∇ log ρ_t(x)| ∼ t^{-1/2} as t→0, so no finite C2 exists for any interval starting at t=0. Since this constant enters the bound (1.5) of Theorem 1.2 and, through Theorem 1.2, the main bound of Theorem 1.1, the proof does not establish the stated results under Assumption A. Moreover, even when ρ_t is smooth and bounded below, C2 depends on the lower bound of ρ0, so the claimed dependence of the constants on only T, τ, and Ψ is not supported by the argument.
[Section 2.1] The proof of Theorem 1.1 applies the two-marginal estimate of Theorem 1.2 to every interval [t_{j-1}, t_j], including the first one with t0=0. The issue with C2(Ψ) is not cosmetic: if one attempts to fix it by taking the supremum over [δ,1] with δ=T/m, the resulting constant grows with m. The per-interval contribution is then ε² C2 ∼ (T/m)² (m/T)^{1/2} for intervals with a singular initial density, and the sum over m intervals is O(m^{-1/2}) rather than O(m^{-1}) in the argument as written. Thus the claimed O(m^{-1}) rate with a constant independent of m is not recovered by the current proof; a uniform-in-m control of the log-density gradient, or an alternative estimate avoiding the supremum of |∇ log ρ|, is needed.

minor comments (5)

[Section 2.2, Step 2] In the display after 'By Girsanov’s Theorem', the drift is written as ∇Φ(t, Z_t); this should be ∇Ψ(t, Z_t) to match the SDE (1.2).
[Section 2.1] The sentence 'Moreover, we have from Theorem 1.3 that bR^T_m|_{[t_{j-1},t_j]} = ...' refers to a nonexistent Theorem 1.3; it should refer to Proposition 1.3 or to the Markov property of Schrödinger bridges cited in the preceding paragraph.
[Section 2.1, final display] The assignment C1 = C'_1 T and C2 = C'_2 √T is algebraically inconsistent with the preceding bound involving Σ_j (t_j−t_{j-1})² ≤ Δ_m T; using C2 = C'_2 T (or equivalently adjusting C1) makes the display correct.
[Section 2.2, Steps 3.1–3.2] The constants C1(Ψ) and C2(Ψ) are defined over [0,1]×X, implicitly assuming ε ≤ 1; for ε > 1 the constants should be defined over [0,ε] or the theorem should be restricted to small ε.
[Section 2.2, Step 3.2] The admissible pair is written as '(µ_t, v_t) = (ρεt, ε∇Ψ(εt, ·) − ε 2 ∇ log ρεt)'; the expression 'ε 2' is ambiguous and should be written as ε/2.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the O(m^{-1}) rate is derived from explicit Girsanov, Ito, and variational bounds with constants obtained analytically, not fitted; the self-citations used in the proof (e.g., [13] for the Markov property) are dually cited to independent sources, and the flagged gaps (finiteness of C2(Ψ), phantom 'Theorem 1.3' reference) are correctness issues, not circularity.

full rationale

The derivation chain of Theorem 1.1 is self-contained as an analytic argument and does not reduce to its own inputs. The two-marginal result (Theorem 1.2) is proved in four steps: Step 1 splits KL(R* ∥ bR[0,ε]) using Equation (10) of [14] and the identity bR[0,ε](·|Z0,Zε) = W^{ρ0}_{[0,ε]}(·|Z0,Zε), cited to the independent survey [15]; Step 2 computes KL(R*_{[0,ε]} ∥ W^{ρ0}_{[0,ε]}) exactly by Girsanov's theorem; Step 3 bounds the residual term by Ito's formula and the variational formulation of the Schrödinger bridge (Proposition 1.6, cited to [3], an independent work), with constants C1(Ψ) and C2(Ψ) defined as explicit analytic suprema over Ψ and the SDE marginals ρ_t. The admissible-pair estimate (2.5) is a genuine use of the infimum in (1.7), so the ε² rate is a derived consequence of Ψ ∈ C^{2,4} differentiability and not an assumed conclusion; no parameter is fitted to the target quantity KL(R* ∥ bR^T_m). Section 2.1 then decomposes KL(R* ∥ bR^T_m) into a sum of subinterval two-marginal KL divergences using the Markov property of the multi-marginal bridge, cited as 'Proposition D.1 in [13] or Lemma 3.4 in [2]'; since [2] is an independent source, the self-citation [13] is not the sole load-bearing support. Two concerns are flagged for correctness rather than circularity. (i) In Step 3.2 the paper defines C2(Ψ) = sup_{t∈[0,1],x∈X} max{|∇Ψ(t,x)|, |∇log ρ(t,x)|} and asserts this is finite 'due to the compactness of [0,1]×X'; finiteness of sup|∇log ρ_t| is not implied by Assumption A (ρ0 ∈ P^r_2 with KL(ρ0∥λ_X) < ∞ allows smooth densities with zeros, for which the sup is infinite near t=0), so the constants of Theorem 1.1 are not established for all ρ0 admitted by Assumption A, and a patched proof would likely degrade the rate. (ii) In Section 2.1 the subinterval identity 'bRTm_{[tj−1,tj]} = argmin_{R_{tj}=ρtj} KL(R ∥ W^τ_{[tj−1,tj]})' is attributed to 'Theorem 1.3', which does not exist in the paper (Proposition 1.3 is a different statement), leaving this load-bearing step under-referenced or mis-referenced. Neither concern is a circular step: no equation in the proof restates KL(R* ∥ bR^T_m) as its own bound, and no result from the authors' prior work is invoked as an unverified premise that forces the theorem's conclusion.

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

The proof relies on standard stochastic analysis tools plus one unstated regularity assumption on the log-density gradient. No free parameters are fitted to data, and no new physical or mathematical entities are introduced.

assumptions (5)

standard math Girsanov theorem for the SDE density with time-dependent drift
Used in Step 2 and Step 3 of the proof of Theorem 1.2 to compute KL divergences and Radon-Nikodym derivatives.
domain assumption Markov property of the multi-marginal Schrödinger bridge
Invoked in Section 2.1 to decompose the path-space KL divergence into a sum over adjacent time intervals; cited to [13] and [2].
domain assumption Variational formulation of the two-marginal Schrödinger bridge
Used in Step 3.2 via Proposition 1.6, taken from [3], to bound the action of the scaled bridge.
domain assumption Existence of potential functions for the two-marginal bridge
Used in Step 1 to write the bridge density as e^{(φ+ψ)/ε} pε ρ0 ρε; standard Schrödinger bridge theory cited to [14, 20].
ad hoc to paper Uniform bound on |∇ log ρ_t| over [0,T] × T^d
The proof defines C2(Ψ) as this supremum in Step 3.2 and uses it to bound the admissible velocity field. This bound is not derived from Assumption A and can be infinite for initial densities with vanishing mass.

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Cite this review

Pith. "Pith review of Convergence Rate of the Solution of Multi-marginal Schrodinger Bridge Problem with Marginal Constraints from SDEs." pith.science (2026). https://pith.science/paper/PMEOBARZ

@misc{pith2026250709151,
  author       = {Pith},
  title        = {Pith review of: Convergence Rate of the Solution of Multi-marginal Schrodinger Bridge Problem with Marginal Constraints from SDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PMEOBARZ}},
  note         = {Machine review of arXiv:2507.09151}
}

abstract

In this paper, we investigate the multi-marginal Schrodinger bridge (MSB) problem whose marginal constraints are marginal distributions of a stochastic differential equation (SDE) with a constant diffusion coefficient, and with time dependent drift term. As the number $m$ of marginal constraints increases, we prove that the solution of the corresponding MSB problem converges to the law of the solution of the SDE at the rate of $O(m^{-1})$, in the sense of KL divergence. Our result extends the work of~\cite{agarwal2024iterated} to the case where the drift of the underlying stochastic process is time-dependent.

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Works this paper leans on

24 extracted references · 14 canonical work pages

[1]

Iterated schr¨ odinger bridge approximation to wasserstein gradient flows

Medha Agarwal, Zaid Harchaoui, Garrett Mulcahy, and Soumik Pal. Iterated schr¨ odinger bridge approximation to wasserstein gradient flows. arXiv preprint arXiv:2406.10823 , 2024

arXiv 2024
[2]

An entropy minimiza- tion approach to second-order variational mean-field games

Jean-David Benamou, Guillaume Carlier, Simone Di Marino, and Luca Nenna. An entropy minimiza- tion approach to second-order variational mean-field games. Mathematical Models and Methods in Applied Sciences, 29(08):1553–1583, 2019

work page 2019
[3]

Stochastic control liaisons: Richard sinkhorn meets gaspard monge on a schr¨ odinger bridge.Siam Review, 63(2):249–313, 2021

Yongxin Chen, Tryphon T Georgiou, and Michele Pavon. Stochastic control liaisons: Richard sinkhorn meets gaspard monge on a schr¨ odinger bridge.Siam Review, 63(2):249–313, 2021. 12

work page 2021
[4]

Trajectory inference via mean-field langevin in path space

L´ ena ¨ ıc Chizat, Stephen Zhang, Matthieu Heitz, and Geoffrey Schiebinger. Trajectory inference via mean-field langevin in path space. Advances in Neural Information Processing Systems , 35:16731– 16742, 2022

work page 2022
[5]

Sinkhorn distances: Lightspeed computation of optimal transport

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

2013
[6]

Diffusion schr¨ odinger bridge with applications to score-based generative modeling

Valentin De Bortoli, James Thornton, Jeremy Heng, and Arnaud Doucet. Diffusion schr¨ odinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 34:17695–17709, 2021

work page 2021
[7]

Reflected schr¨ odinger bridge for constrained generative modeling.arXiv preprint arXiv:2401.03228 , 2024

Wei Deng, Yu Chen, Nicole Tianjiao Yang, Hengrong Du, Qi Feng, and Ricky TQ Chen. Reflected schr¨ odinger bridge for constrained generative modeling.arXiv preprint arXiv:2401.03228 , 2024

arXiv 2024
[8]

Single-cell reconstruction of developmental trajectories during zebrafish embryogenesis

Jeffrey A Farrell, Yiqun Wang, Samantha J Riesenfeld, Karthik Shekhar, Aviv Regev, and Alexan- der F Schier. Single-cell reconstruction of developmental trajectories during zebrafish embryogenesis. Science, 360(6392):eaar3131, 2018

work page 2018

Show all 24 references

[9]

Private continuous-time synthetic trajectory generation via mean-field langevin dynamics

Anming Gu, Edward Chien, and Kristjan Greenewald. Private continuous-time synthetic trajectory generation via mean-field langevin dynamics. arXiv preprint arXiv:2506.12203 , 2025

2025 arXiv
[10]

Trajectory inference with smooth schr \” odinger bridges

Wanli Hong, Yuliang Shi, and Jonathan Niles-Weed. Trajectory inference with smooth schr \” odinger bridges. arXiv preprint arXiv:2503.00530 , 2025

2025 arXiv
[11]

The sinkhorn–knopp algorithm: convergence and applications

Philip A Knight. The sinkhorn–knopp algorithm: convergence and applications. SIAM Journal on Matrix Analysis and Applications , 30(1):261–275, 2008

2008
[12]

Statistical inference for ergodic diffusion processes

Yury A Kutoyants. Statistical inference for ergodic diffusion processes . Springer Science & Business Media, 2013

2013
[13]

Toward a mathematical theory of trajectory inference

Hugo Lavenant, Stephen Zhang, Young-Heon Kim, and Geoffrey Schiebinger. Toward a mathematical theory of trajectory inference. The Annals of Applied Probability , 34(1A):428–500, 2024

2024
[14]

From the schr¨ odinger problem to the monge–kantorovich problem

Christian L´ eonard. From the schr¨ odinger problem to the monge–kantorovich problem. Journal of Functional Analysis, 262(4):1879–1920, 2012

1920
[15]

A survey of the schr¨ odinger problem and some of its connections with optimal transport

Christian L´ eonard. A survey of the schr¨ odinger problem and some of its connections with optimal transport. arXiv preprint arXiv:1308.0215 , 2013

2013 arXiv
[16]

Multimarginal schr¨ odinger barycenter

Pengtao Li and Xiaohui Chen. Multimarginal schr¨ odinger barycenter. arXiv preprint arXiv:2502.02726, 2025

2025
[17]

Generalized schr¨ odinger bridge matching.arXiv preprint arXiv:2310.02233 , 2023

Guan-Horng Liu, Yaron Lipman, Maximilian Nickel, Brian Karrer, Evangelos A Theodorou, and Ricky TQ Chen. Generalized schr¨ odinger bridge matching.arXiv preprint arXiv:2310.02233 , 2023

2023 arXiv
[18]

Schr¨ odinger bridges with multimarginal constraints

Abdulwahab Mohamed, Alberto Chiarini, and Oliver Tse. Schr¨ odinger bridges with multimarginal constraints. 2021

2021
[19]

The entropic optimal (self-) transport problem: Limit distributions for decreasing regularization with application to score function estimation

Gilles Mordant. The entropic optimal (self-) transport problem: Limit distributions for decreasing regularization with application to score function estimation. arXiv preprint arXiv:2412.12007 , 2024. 13

2024 arXiv
[20]

Introduction to entropic optimal transport

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

2021
[21]

Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming

Geoffrey Schiebinger, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subramanian, Aryeh Solomon, Joshua Gould, Siyan Liu, Stacie Lin, Peter Berube, et al. Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming. Cell, 176(...

2019
[22]

Deep generative learning via schr¨ odinger bridge

Gefei Wang, Yuling Jiao, Qian Xu, Yang Wang, and Can Yang. Deep generative learning via schr¨ odinger bridge. In International conference on machine learning , pages 10794–10804. PMLR, 2021

2021
[23]

Topological schr¨ odinger bridge matching

Maosheng Yang. Topological schr¨ odinger bridge matching. arXiv preprint arXiv:2504.04799 , 2025

2025 arXiv
[24]

Learning density evolution from snapshot data

Rentian Yao, Atsushi Nitanda, Xiaohui Chen, and Yun Yang. Learning density evolution from snapshot data. arXiv preprint arXiv:2502.17738 , 2025. 14

2025 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Iterated schr¨ odinger bridge approximation to wasserstein gradient flows

Medha Agarwal, Zaid Harchaoui, Garrett Mulcahy, and Soumik Pal. Iterated schr¨ odinger bridge approximation to wasserstein gradient flows. arXiv preprint arXiv:2406.10823 , 2024

arXiv 2024

[2] [2]

An entropy minimiza- tion approach to second-order variational mean-field games

Jean-David Benamou, Guillaume Carlier, Simone Di Marino, and Luca Nenna. An entropy minimiza- tion approach to second-order variational mean-field games. Mathematical Models and Methods in Applied Sciences, 29(08):1553–1583, 2019

work page 2019

[3] [3]

Stochastic control liaisons: Richard sinkhorn meets gaspard monge on a schr¨ odinger bridge.Siam Review, 63(2):249–313, 2021

Yongxin Chen, Tryphon T Georgiou, and Michele Pavon. Stochastic control liaisons: Richard sinkhorn meets gaspard monge on a schr¨ odinger bridge.Siam Review, 63(2):249–313, 2021. 12

work page 2021

[4] [4]

Trajectory inference via mean-field langevin in path space

L´ ena ¨ ıc Chizat, Stephen Zhang, Matthieu Heitz, and Geoffrey Schiebinger. Trajectory inference via mean-field langevin in path space. Advances in Neural Information Processing Systems , 35:16731– 16742, 2022

work page 2022

[5] [5]

Sinkhorn distances: Lightspeed computation of optimal transport

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

2013

[6] [6]

Diffusion schr¨ odinger bridge with applications to score-based generative modeling

Valentin De Bortoli, James Thornton, Jeremy Heng, and Arnaud Doucet. Diffusion schr¨ odinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 34:17695–17709, 2021

work page 2021

[7] [7]

Reflected schr¨ odinger bridge for constrained generative modeling.arXiv preprint arXiv:2401.03228 , 2024

Wei Deng, Yu Chen, Nicole Tianjiao Yang, Hengrong Du, Qi Feng, and Ricky TQ Chen. Reflected schr¨ odinger bridge for constrained generative modeling.arXiv preprint arXiv:2401.03228 , 2024

arXiv 2024

[8] [8]

Single-cell reconstruction of developmental trajectories during zebrafish embryogenesis

Jeffrey A Farrell, Yiqun Wang, Samantha J Riesenfeld, Karthik Shekhar, Aviv Regev, and Alexan- der F Schier. Single-cell reconstruction of developmental trajectories during zebrafish embryogenesis. Science, 360(6392):eaar3131, 2018

work page 2018