Pith. sign in

REVIEW 4 major objections 5 minor 53 references

Bridge matching that uses the potential gradient outperforms the prior scheme

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 19:18 UTC pith:VNLNES7P

load-bearing objection Real derivations, honest experiments, and a genuine but unquantified gap between the ideal IMF extension and the implemented plug-in algorithm; worth a careful referee. the 4 major comments →

arxiv 2607.16987 v1 pith:VNLNES7P submitted 2026-07-18 stat.ML cs.LG

Twisted Schr\"odinger Bridge Matching

Maxence Noble , Marie Scheid , Yazid Janati , Eric Moulines , Alain Durmus This is my paper
classification stat.ML cs.LG
keywords Schrödinger bridgebridge matchingFeynman–Kac transformstochastic optimal controldiffusion modelstrajectory inferencecontrol variatesoptimal transport
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tackles the generalized Schrödinger bridge problem, where trajectories are penalized by a time-dependent potential in addition to the usual kinetic-energy cost. The authors show that this problem is equivalent to finding a bridge with respect to a twisted Brownian reference, formed by Feynman–Kac reweighting of Brownian motion. They then derive TSBM, a bridge-matching algorithm that rigorously implements the Iterative Markovian Fitting scheme for this twisted reference, producing a regression loss that depends explicitly on the gradient of the potential and reduces exactly to the DSBM objective when the potential is zero. The paper argues that this provides a theoretically grounded alternative to the existing GSBM method, with empirical evidence on crowd navigation and sparse single-cell trajectory inference.

Core claim

Treating the generalized Schrödinger bridge as a twisted Brownian bridge problem, the authors establish that the reciprocal projection can be learned by a variational reverse-KL problem with a tractable loss, and that the Markovian projection's optimal drift is a conditional expectation of the Brownian bridge velocity plus a potential-gradient correction integrated over the remaining trajectory. This yields computable bridge-matching losses (forward and backward) whose zero-potential limit is exactly the DSBM objective in continuous time and the DDPM objective in discrete time. They further introduce learnable control variates that leave the optimal drift unchanged while reducing estimator v

What carries the argument

The central object is the twisted bridge Q^{V,σ}_{|0,T}, the law of Brownian motion conditioned on endpoints under a Feynman–Kac weight exp(−∫V/σ²). Because its drift is intractable, TSBM approximates it through a variational family of Gaussian stochastic interpolants minimizing a reverse KL, then plugs that approximation into the Markovian projection losses, whose regression targets are the Brownian bridge velocity (x_T−x_t)/(T−t) corrected by integrals of −∇V along the future (forward) or past (backward). Learnable control variates, built from zero-conditional-mean score identities, subtract variance without biasing the learned drift.

Load-bearing premise

The practical algorithm replaces the intractable twisted bridge with a variational approximation and provides no control on the approximation gap, so if that gap is large the learned drift targets the wrong path measure.

What would settle it

Solve the twisted bridge in closed form for a quadratic potential (which is known to be solvable), run TSBM on that problem, and measure the KL divergence between the learned variational bridge and the true twisted bridge; if the gap does not shrink with more variational capacity, the claim of recovering the true twisted-SB solution is not supported.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • TSBM recovers the DSBM objective when the potential vanishes, making it a strict generalization of existing diffusion bridge matching; in discrete time it recovers the DDPM loss.
  • The Markovian projection loss uses the exact conditional velocity field of the twisted bridge, so it does not inherit the variational approximation error into the drift regression, unlike GSBM's loss.
  • The bias–variance decomposition of the control-variate losses means optimizing over both the drift and the control variate provably minimizes the original TSBM loss plus a variance term, preserving the same optimal drift.
  • Empirically, TSBM consistently improves the estimated reciprocal projection and achieves lower Schrödinger-bridge objective values across crowd navigation and single-cell settings, at the cost of somewhat worse terminal marginal feasibility.

Where Pith is reading between the lines

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

  • The variational-gap assumption is unquantified: if the gap between the true twisted bridge and the variational stochastic interpolant is large, the Markovian projection targets a different path measure, and the claimed recovery of the iterative fixed point no longer holds; a bound or diagnostic would settle this.
  • Because the TSBM losses are unchanged as σ→0, the method should yield a twisted analogue of rectified flow; testing in the deterministic zero-noise limit could extend the framework to ordinary transport maps.
  • The trajectory-based control variates for general (t,s) pairs may transfer to other regression-against-conditional-expectation objectives beyond Schrödinger bridges, such as general stochastic optimal control matching.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper considers the generalized Schrödinger bridge problem in which the reference path measure is a twisted Brownian motion, i.e., a Feynman–Kac reweighting of Brownian motion by a time-dependent potential V. Building on the Iterative Markovian Fitting (IMF) framework and its special case DSBM, the authors derive an extension they call Twisted Schrödinger Bridge Matching (TSBM). The main theoretical contributions are: (i) a variational reformulation of the reciprocal projection as a reverse-KL problem with tractable objectives that involve the potential (Eqs. (16)–(17)); (ii) a characterization of the Markovian projection in continuous time giving explicit drift formulas that depend on ∇V (Proposition 2, Eqs. (18)–(19)) and corresponding regression losses (Eqs. (20)–(21)); (iii) a control-variate scheme that preserves the optimal drift while reducing variance (Propositions 3, 11, 12); and (iv) a discrete-time variant D-TSBM that reduces to the DDPM objective in the zero-potential case. The paper presents experiments on crowd navigation and single-cell trajectory inference, reporting that TSBM often achieves lower SB cost than GSBM, at the price of worse terminal-marginal feasibility in several settings.

Significance. If the theoretical claims are correct, TSBM is a valuable contribution: it provides the first bridge-matching method that explicitly connects generalized Schrödinger bridge problems to the IMF scheme, with losses that exactly recover DSBM/DDPM in the zero-potential limit. The control-variate construction with a bias–variance decomposition is original and likely to be useful beyond this setting. The paper is also strong on the empirical side, with careful ablations, bidirectional training, and experiments across dimensions. However, the central claim of a “rigorous extension” currently outstrips what is proved for the implemented algorithm, because the intractable twisted bridge is replaced by a Gaussian variational approximation with no control on the induced bias. Several supporting results are also stated with omitted proofs or under explicit informal assumptions. These issues are fixable but require either additional analysis or a more careful scoping of the claims.

major comments (4)
  1. [§3.1–3.2, §F.2, Algorithm 3] The Markovian projection losses (20)–(21) are derived under the hypothesis Π ∈ R(Q_{V,σ}) (Proposition 2). In the implementation, however, Π is replaced by Π_{ψ⋆} = P_{0,T} Q_{ψ⋆|0,T}, where Q_{ψ⋆|0,T} is obtained by the reverse-KL problem (15) over Gaussian stochastic interpolants. For a non-quadratic potential V, Corollary 7 shows that the true twisted bridge Q_{V,σ|0,T} has nonlinear drift and non-Gaussian marginals, so the variational family cannot represent it. No estimate of KL(Q_{ψ⋆}‖Q_{V,σ}) or of the resulting bias in the Markovian projection is provided. Consequently, the learned drift may solve a different Schrödinger bridge problem, and the central claim that TSBM rigorously extends IMF to problem (5) is not supported for the implemented algorithm. The authors acknowledge the limitation by mentioning Gaussian mixtures as future work, but the gap affects the main theoretical c
  2. [§C.3, proof of Proposition 10] The proof of Proposition 10 — the result that underpins the forward and backward Markovian projection losses (20)–(21) — states: 'The structure of the proof closely follows that of [47, Proposition 2] and relies on analogous technical assumptions, which we omit here for readability.' Since this proposition is load-bearing for the paper's central derivation, the omitted technical assumptions (integrability of V, growth conditions on ∇V, interchange of differentiation and expectation, etc.) and the full argument cannot simply be delegated to a reference. Please provide a complete proof or a precise statement of the assumptions under which the drift formulas (18)–(19) hold.
  3. [§D.1, Proposition 16 and §E.2] Proposition 16 is explicitly labeled 'Informal version,' and the formal version (Proposition 26) assumes — without proof — that P_N converges to some P_∞ in the weak topology as N→∞. This convergence hypothesis is an axiom that is not established; it is essential for the claim that D-TSBM converges to the continuous-time TSB solution and for the connection to the DDPM loss. Please state conditions on V and the discretization under which the weak convergence holds, or weaken the claim accordingly.
  4. [§G.1 (annealing and boundary substitution)] The practical TSBM objective deviates from the theoretically analyzed loss (20) in two important ways: the ∇V contribution is annealed by a factor ϵ_k that ramps up from 0 to 0.9, and the TSBM regression target is replaced by the GSBM target whenever t/T ≤ 0.1 or t/T ≥ 0.9. These modifications are reported only in the appendix, yet they mean that the loss actually optimized in the crowd-navigation experiments is not the loss derived in Proposition 2. This is not a minor implementation detail: it changes the objective and may be partly responsible for the observed feasibility–optimality trade-off. Please state these heuristics in the main text and discuss their effect on the validity of the theoretical guarantees.
minor comments (5)
  1. [§5.3 heading] The heading 'Singe-cell inference under sparse observations' contains a typo; it should read 'Single-cell.'
  2. [Table 1] The metric reported as 'forward KL-based metric E_{π0,1}[KL(Q^σ_{t|0,1}‖Q^{ψ⋆}_{t|0,1})]' is not explicitly defined in the main text; the caption should state how the KL between Gaussian bridges is computed and what 'averaged over 8 metric evaluations' means.
  3. [§F.2, Eq. (76)–(77)] The transition kernels are written with a factor exp(−σ²/2 J^ψ) that appears dimensionally unusual; please verify the exponent and the variance term, and state the dependence on γ^ψ more explicitly.
  4. [§D.1, Proposition 16] Numbering a proposition 'Informal version' is nonstandard; consider moving this to a remark and keeping the formal statement as the main proposition.
  5. [Acknowledgments] The acknowledgments thank 'MD' for financial support, but no author with those initials appears in the author list; please correct the attribution.

Circularity Check

0 steps flagged

No significant circularity: TSBM's losses are derived from KL projections and Girsanov/h-transform identities, not from fitted parameters renamed as predictions.

full rationale

The derivation chain is self-contained. The generalized SB problem is reformulated as a twisted reference measure via the Feynman–Kac density (4), and the equivalence to the SOC problem (5) is obtained by Girsanov (Section C.1). Proposition 8 shows, by the KL chain rule and Girsanov, that the reciprocal-projection loss (16)/(36) is exactly proportional to E_{P_{0,T}}[KL(Q^ψ_{|0,T} || Q^{V,σ}_{|0,T})], so the variational objective is the target itself, not a fitted input later relabeled as a prediction. Proposition 10 (main-text Proposition 2) derives the Markovian-projection drifts (40)–(41) from Doob h-transform/Tweedie identities, and the losses (20)–(21) are regression reformulations of the same conditional expectations. The control-variate result (Proposition 12) gives a bias–variance decomposition while leaving the optimal drift invariant, so no fitted quantity is renamed as a prediction. The practical replacement of the intractable twisted bridge by the Gaussian spline bridge Q^{ψ⋆}_{|0,T} (Eq. (15), Section F.2) is an approximation with an acknowledged, uncontrolled gap; this is a correctness/robustness limitation, not a circular step, since the method would be exact in the zero-gap limit and no equation is defined in terms of its own output. The only author-overlapping citation is [48] (Durmus is a co-author), used for 'exponential convergence of IMF'; it is background motivation and is not used to derive the TSBM updates, so it is not load-bearing. The paper also evaluates against external benchmarks (Brownian-bridge recovery, crowd navigation, single-cell data), providing independent falsifiable checks. No prediction in the paper reduces to its input by construction.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 0 invented entities

No new physical entities are introduced. The 'twisted' reference measure is a standard Feynman–Kac transform. The main free parameters are experimental hyperparameters; the derivation itself depends on standard mathematical tools plus a few unstated regularity assumptions.

free parameters (6)
  • Entropic regularization σ = 2 (crowd nav), 0.1 (single-cell)
    Controls the noise level in the reference Brownian motion; chosen per experiment, not derived.
  • Warmup fraction ϵ = 0.9
    Anneals the gradient contribution in the first Markovian projection; hand-set for crowd navigation tasks.
  • Boundary swap thresholds = t/T ≤ 0.1 or ≥ 0.9
    Switches to GSBM target near boundaries to stabilize training; a post-hoc modification.
  • Number of spline control points = 15 (mean), 30 (std)
    Capacity of the variational bridge family; chosen by hand.
  • Control variate polynomial degree K = 4
    Complexity of the α parameterization; larger degrees did not improve performance.
  • Single-cell cost hyperparameters (h, kNN, β) = h=0.1/1/7, kNN=5, β=20/100
    Tuned per dimension to define the state cost; affect experimental results.
axioms (5)
  • standard math Girsanov theorem and Doob h-transform for deriving KL losses
    Used throughout Section 3 and C.
  • domain assumption ZV < ∞ and V ∈ C^{0,1} so that QV,σ is a well-defined probability measure
    Assumed in Proposition 4 and throughout.
  • ad hoc to paper Sufficient integrability of α-based control variate terms
    The proof of Proposition 3 states 'we assume throughout that the quantities involved are sufficiently integrable' without explicit conditions.
  • ad hoc to paper Convergence of PN to P∞ in weak topology for the discrete-time limit
    Stated as an assumption in Proposition 16 (informal version); the formal proof relies on it.
  • domain assumption The variational family of Gaussian stochastic interpolants can approximate QV,σ|0,T with negligible gap
    Required for the practical algorithm to be unbiased; no quantitative guarantee is given.

pith-pipeline@v1.3.0-alltime-deepseek · 49901 in / 8483 out tokens · 73823 ms · 2026-08-01T19:18:58.446673+00:00 · methodology

0 comments
read the original abstract

Over the past few years, diffusion-based Schr\"odinger bridge models have been proposed to approximate optimal transport dynamics between two prescribed boundary distributions, with successful applications to generative modeling. More precisely, these methods aim to estimate a path measure whose initial and terminal marginals match the two boundary distributions, while minimizing the Kullback-Leibler divergence with respect to a reference Markov process. In this work, we consider the generalized Schr\"odinger bridge problem, in which the reference process is a twisted Brownian motion, that is, a Feynman-Kac transform of a Brownian motion induced by a time-dependent differentiable potential. Building on the Iterative Markovian Fitting (IMF) paradigm, and in particular on its special case Diffusion Schr\"odinger Bridge Matching (DSBM), which corresponds to the zero potential case, we introduce Twisted Schr\"odinger Bridge Matching (TSBM), a diffusion-based method designed to handle both continuous- and discrete-time potentials. Unlike previous approaches, TSBM provides a rigorous extension of the IMF scheme to the generalized Schr\"odinger bridge problem. This derivation leads to a new bridge-matching loss that depends explicitly on the gradient of the potential and recovers the DSBM objective when the potential vanishes, yielding improved performance. We further introduce trajectory-based variance-reduction techniques that substantially stabilize optimization and may be useful beyond the present setting. Finally, we empirically demonstrate the benefits of TSBM for trajectory inference across increasingly high-dimensional settings, including crowd navigation and single-cell data. Code available at https://github.com/maxencenoble/twisted-sb-matching.

Figures

Figures reproduced from arXiv: 2607.16987 by Alain Durmus, Eric Moulines, Marie Scheid, Maxence Noble, Yazid Janati.

Figure 1
Figure 1. Figure 1: GSBM and TSBM learn￾ing error of the Brownian bridge variance over time (d = 2, σ = 2). 3Toy experiments in the discrete-time setting, absent from the experimental study of [29], are reported in Section G.3. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Learned TSB dynamics for Stunnel (top), Gmm (middle), Vneck (bottom) with [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of metrics for TSBM and GSBM on 2D-Stunnel with [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Snapshots of learned TSBM dynamics for 2D single-cell ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Learned TSB dynamics in the single-observation discrete-time setting. The intermediate observation distribution at t1 = 0.5 is shown in grey. Top: forward time direction. Bottom: backward time direction. Multi-observation setting. We next consider a more challenging setting with two intermediate observation sets at times t1 = 0.4 and t2 = 1.6, with horizon T = 2, source distribution µ = N((−10, −5), 0.2Id)… view at source ↗
Figure 6
Figure 6. Figure 6: Learned TSB dynamics in the multi-observation discrete-time setting. The intermediate observation distributions at t1 = 0.4 and t2 = 1.6 are respectively shown in pink and grey. Top: forward time direction. Bottom: backward time direction [PITH_FULL_IMAGE:figures/full_fig_p041_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Effect of control variate on the learned TSBM dynamics in the multi-observation discrete-time setting (only 1 outer iteration is performed). The intermediate observation distributions at t1 = 0.4 and t2 = 1.6 are respectively shown in pink and grey. Top: forward time direction. Bottom: backward time direction. (Left): Without control variate. (Right): With control variate. 41 [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 8
Figure 8. Figure 8: Backward learned TSB dynamics for Stunnel (top), Gmm (middle), Vneck (bottom) with d = 2. This is complementary to [PITH_FULL_IMAGE:figures/full_fig_p044_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Evolution of backward metrics for TSBM and GSBM on 2D-Stunnel with σ = 2. The vertical dotted line marks the end of bidirectional pretraining or, equivalently, the first unidirectional outer iteration. This is complementary to [PITH_FULL_IMAGE:figures/full_fig_p044_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Evolution of forward and backward metrics for TSBM and GSBM on 2D-Gmm with σ = 2. The vertical dotted line marks the end of bidirectional pretraining or, equivalently, the first unidirectional outer iteration. 50 100 Vneck 2D - Optimality (forward) 10 2 10 1 10 0 Vneck 2D - Feasibility (forward) 0 2 4 6 8 10 12 Training epochs (x10 3 ) 50 100 Vneck 2D - Optimality (backward) 0 2 4 6 8 10 12 Training epoch… view at source ↗
Figure 11
Figure 11. Figure 11: Evolution of forward and backward metrics for TSBM and GSBM on 2D-Vneck with σ = 2. The vertical dotted line marks the end of bidirectional pretraining or, equivalently, the first unidirectional outer iteration [PITH_FULL_IMAGE:figures/full_fig_p045_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Learned TSB dynamics for Stunnel (top), Gmm (middle), Vneck (bottom) with [PITH_FULL_IMAGE:figures/full_fig_p046_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Learned TSBM dynamics for Stunnel (top), Gmm (middle), Vneck (bottom) with [PITH_FULL_IMAGE:figures/full_fig_p046_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: 2D Single-cell data splitted by snapshot times : endpoint samples (day 0, day 24) and [PITH_FULL_IMAGE:figures/full_fig_p047_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Learned TSB dynamics for 2D single-cell data for [PITH_FULL_IMAGE:figures/full_fig_p049_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Snapshots of learned GSBM dynamics for 2D single-cell ( [PITH_FULL_IMAGE:figures/full_fig_p050_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Snapshots of learned GSBM dynamics for 2D single-cell ( [PITH_FULL_IMAGE:figures/full_fig_p050_17.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

53 extracted references · 6 linked inside Pith

  1. [1]

    Stochastic interpolants: A unifying framework for flows and diffusions.Journal of Machine Learning Research, 26(209): 1–80, 2025

    Michael Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.Journal of Machine Learning Research, 26(209): 1–80, 2025

    2025

  2. [2]

    A computational fluid mechanics solution to the monge-kantorovich mass transfer problem.Numerische Mathematik, 84(3):375–393, 2000

    Jean-David Benamou and Yann Brenier. A computational fluid mechanics solution to the monge-kantorovich mass transfer problem.Numerische Mathematik, 84(3):375–393, 2000

    2000

  3. [3]

    Bridge matching sampler: Scalable sampling via generalized fixed-point diffusion matching

    Denis Blessing, Lorenz Richter, Julius Berner, Egor Malitskiy, and Gerhard Neumann. Bridge matching sampler: Scalable sampling via generalized fixed-point diffusion matching. In International Conference on Machine Learning, 2026

    2026

  4. [4]

    Diffusion schrödinger bridge with applications to score-based generative modeling.Advances in Neural Information Processing Systems, pages 17695–17709, 2021

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

    2021

  5. [5]

    Schrodinger bridge flow for unpaired data translation

    Valentin De Bortoli, Iryna Korshunova, Andriy Mnih, and Arnaud Doucet. Schrodinger bridge flow for unpaired data translation. InAdvances in Neural Information Processing Systems, 2024

    2024

  6. [6]

    Lev M Bregman. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming.USSR computational mathematics and mathematical physics, 7(3):200–217, 1967

    1967

  7. [7]

    Ricky T. Q. Chen and Yaron Lipman. Flow matching on general geometries. InInternational Conference on Learning Representations, 2024

    2024

  8. [8]

    Likelihood training of Schrödinger bridge using forward-backward SDEs theory.International Conference on Learning Representations, 2022

    Tianrong Chen, Guan-Horng Liu, and Evangelos A Theodorou. Likelihood training of Schrödinger bridge using forward-backward SDEs theory.International Conference on Learning Representations, 2022

    2022

  9. [9]

    Deep momentum multi-marginal schrödinger bridge.Advances in Neural Information Processing Systems, pages 57058–57086, 2023

    Tianrong Chen, Guan-Horng Liu, Molei Tao, and Evangelos Theodorou. Deep momentum multi-marginal schrödinger bridge.Advances in Neural Information Processing Systems, pages 57058–57086, 2023

    2023

  10. [10]

    Nonequilibrium markov processes conditioned on large deviations

    Raphaël Chetrite and Hugo Touchette. Nonequilibrium markov processes conditioned on large deviations. InAnnales Henri Poincaré, volume 16, pages 2005–2057. Springer, 2015

    2005

  11. [11]

    On the stability of interacting processes with applications to filtering and genetic algorithms

    Pierre Del Moral and Alice Guionnet. On the stability of interacting processes with applications to filtering and genetic algorithms. InAnnales de l’Institut Henri Poincaré (B) Probability and Statistics, volume 37, pages 155–194. Elsevier, 2001

    2001

  12. [12]

    The monte-carlo method for filtering with discrete-time observations.Probability Theory and Related Fields, 120(3):346–368, 2001

    Pierre Del Moral, Jean Jacod, and Philip Protter. The monte-carlo method for filtering with discrete-time observations.Probability Theory and Related Fields, 120(3):346–368, 2001. 11

    2001

  13. [13]

    Stochastic optimal control matching.Advances in Neural Information Processing Systems, pages 112459–112504, 2024

    Carles Domingo i Enrich, Jiequn Han, Brandon Amos, Joan Bruna, and Ricky TQ Chen. Stochastic optimal control matching.Advances in Neural Information Processing Systems, pages 112459–112504, 2024

    2024

  14. [14]

    Doob’s lagrangian: A sample-efficient variational approach to transition path sampling.Advances in Neural Information Processing Systems, pages 65791–65822, 2024

    Yuanqi Du, Michael Plainer, Rob Brekelmans, Chenru Duan, Frank Noe, Carla P Gomes, Alan Aspuru-Guzik, and Kirill Neklyudov. Doob’s lagrangian: A sample-efficient variational approach to transition path sampling.Advances in Neural Information Processing Systems, pages 65791–65822, 2024

    2024

  15. [15]

    Interpolating between optimal transport and mmd using sinkhorn divergences

    Jean Feydy, Thibault Séjourné, François-Xavier Vialard, Shun-ichi Amari, Alain Trouve, and Gabriel Peyré. Interpolating between optimal transport and mmd using sinkhorn divergences. InInternational Conference on Artificial Intelligence and Statistics, pages 2681–2690, 2019

    2019

  16. [16]

    Stochastic differential equations for the non linear filtering problem.Osaka J

    M Fujisaki, G Kallianpur, and H Kunita. Stochastic differential equations for the non linear filtering problem.Osaka J. Math, 9:19–40, 1972

    1972

  17. [17]

    Adversarial schrödinger bridge matching.Advances in Neural Information Processing Systems, pages 89612–89651, 2024

    Nikita Gushchin, Daniil Selikhanovych, Sergei Kholkin, Evgeny Burnaev, and Aleksandr Korotin. Adversarial schrödinger bridge matching.Advances in Neural Information Processing Systems, pages 89612–89651, 2024

    2024

  18. [18]

    Controlled sequen- tial monte carlo.The Annals of Statistics, 48(5):2904–2929, 2020

    Jeremy Heng, Adrian N Bishop, George Deligiannidis, and Arnaud Doucet. Controlled sequen- tial monte carlo.The Annals of Statistics, 48(5):2904–2929, 2020

    2020

  19. [19]

    Denoising diffusion probabilistic models.Advances in Neural Information Processing Systems, pages 6840–6851, 2020

    Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models.Advances in Neural Information Processing Systems, pages 6840–6851, 2020

    2020

  20. [20]

    Schrödinger bridge matching for tree-structured costs and entropic wasserstein barycentres

    Samuel Howard, Peter Potaptchik, and George Deligiannidis. Schrödinger bridge matching for tree-structured costs and entropic wasserstein barycentres. InAdvances in Neural Information Processing Systems, 2025

    2025

  21. [21]

    Striebel

    G Kallianpur and C. Striebel. Estimation of stochastic systems: Arbitrary system process with additive white noise observation errors.The Annals of Mathematical Statistics, 39(3):785–801, 1968

    1968

  22. [22]

    Bronstein, Joey Bose, and Francesco Di Giovanni

    Kacper Kapusniak, Peter Potaptchik, Teodora Reu, Leo Zhang, Alexander Tong, Michael M. Bronstein, Joey Bose, and Francesco Di Giovanni. Metric flow matching for smooth interpola- tions on the data manifold. InAdvances in Neural Information Processing Systems, 2024

    2024

  23. [23]

    Variational diffusion models

    Diederik Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational diffusion models. Advances in Neural Information Processing Systems, pages 21696–21707, 2021

    2021

  24. [24]

    Stochastic derivatives and generalized h-transforms of markov processes

    Christian Léonard. Stochastic derivatives and generalized h-transforms of markov processes. arXiv preprint arXiv:1102.3172, 2011

    Pith/arXiv arXiv 2011

  25. [25]

    Some properties of path measures.Séminaire de Probabilités XLVI, pages 207–230, 2014

    Christian Léonard. Some properties of path measures.Séminaire de Probabilités XLVI, pages 207–230, 2014

    2014

  26. [26]

    A survey of the Schrödinger problem and some of its connections with optimal transport.Discrete & Continuous Dynamical Systems-A, 34(4):1533–1574, 2014

    Christian Léonard. A survey of the Schrödinger problem and some of its connections with optimal transport.Discrete & Continuous Dynamical Systems-A, 34(4):1533–1574, 2014

    2014

  27. [27]

    Flow matching for generative modeling.International Conference on Learning Representations, 2023

    Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling.International Conference on Learning Representations, 2023

    2023

  28. [28]

    Deep generalized schrödinger bridge.Advances in Neural Information Processing Systems, pages 9374–9388, 2022

    Guan-Horng Liu, Tianrong Chen, Oswin So, and Evangelos Theodorou. Deep generalized schrödinger bridge.Advances in Neural Information Processing Systems, pages 9374–9388, 2022

    2022

  29. [29]

    Guan-Horng Liu, Yaron Lipman, Maximilian Nickel, Brian Karrer, Evangelos Theodorou, and Ricky T. Q. Chen. Generalized schrödinger bridge matching. InInternational Conference on Learning Representations, 2024

    2024

  30. [30]

    Rectified flow: A marginal preserving approach to optimal transport.arXiv preprint arXiv:2209.14577, 2022

    Qiang Liu. Rectified flow: A marginal preserving approach to optimal transport.arXiv preprint arXiv:2209.14577, 2022. 12

    Pith/arXiv arXiv 2022

  31. [31]

    Simplifying, stabilizing and scaling continuous-time consistency models

    Cheng Lu and Yang Song. Simplifying, stabilizing and scaling continuous-time consistency models. InInternational Conference on Learning Representations, 2025

    2025

  32. [32]

    Strong stochastic flow maps.arXiv preprint arXiv:2606.01086, 2026

    Sam McCallum, Zander W Blasingame, Timothy Herschell, Niklas Rindtorff, Alexander Tong, and James Foster. Strong stochastic flow maps.arXiv preprint arXiv:2606.01086, 2026

    Pith/arXiv arXiv 2026

  33. [33]

    Moon, David van Dijk, Zheng Wang, Scott Gigante, Daniel B

    Kevin R. Moon, David van Dijk, Zheng Wang, Scott Gigante, Daniel B. Burkhardt, William S. Chen, Kristina Yim, Antonia van den Elzen, Matthew J. Hirn, Ronald R. Coifman, Natalia B. Ivanova, Guy Wolf, and Smita Krishnaswamy. Visualizing structure and transitions for biological data exploration.bioRxiv, page 120378, 2019

    2019

  34. [34]

    Action matching: Learning stochastic dynamics from samples

    Kirill Neklyudov, Rob Brekelmans, Daniel Severo, and Alireza Makhzani. Action matching: Learning stochastic dynamics from samples. InInternational Conference on Machine Learning, pages 25858–25889. PMLR, 2023

    2023

  35. [35]

    A computational framework for solving Wasserstein lagrangian flows

    Kirill Neklyudov, Rob Brekelmans, Alexander Tong, Lazar Atanackovic, Qiang Liu, and Alireza Makhzani. A computational framework for solving Wasserstein lagrangian flows. In International Conference on Machine Learning, pages 37461–37485. PMLR, 2024

    2024

  36. [36]

    Tree-based diffu- sion Schrödinger bridge with applications to Wasserstein barycenters.Advances in Neural Information Processing Systems, 36:55193–55236, 2023

    Maxence Noble, Valentin De Bortoli, Arnaud Doucet, and Alain Durmus. Tree-based diffu- sion Schrödinger bridge with applications to Wasserstein barycenters.Advances in Neural Information Processing Systems, 36:55193–55236, 2023

    2023

  37. [37]

    Nikolas Nüsken and Lorenz Richter. Solving high-dimensional hamilton–jacobi–bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space.Partial differential equations and applications, 2(4):48, 2021

    2021

  38. [38]

    Itô maps for any-step sdes.arXiv preprint arXiv:2606.11156, 2026

    Zhengkai Pan, Peter Potaptchik, Wenxi Yao, Michael S Albergo, and Jakiw Pidstrigach. Itô maps for any-step sdes.arXiv preprint arXiv:2606.11156, 2026

    Pith/arXiv arXiv 2026

  39. [39]

    Diffusion bridge mixture transports, schrödinger bridge problems and generative modeling.Journal of Machine Learning Research, 24(374):1–51, 2023

    Stefano Peluchetti. Diffusion bridge mixture transports, schrödinger bridge problems and generative modeling.Journal of Machine Learning Research, 24(374):1–51, 2023

    2023

  40. [40]

    Non-denoising forward-time diffusions.arXiv preprint arXiv:2312.14589, 2023

    Stefano Peluchetti. Non-denoising forward-time diffusions.arXiv preprint arXiv:2312.14589, 2023

    Pith/arXiv arXiv 2023

  41. [41]

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

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

    2019

  42. [42]

    Aram-Alexandre Pooladian, Carles Domingo-Enrich, Ricky T. Q. Chen, and Brandon Amos. Neural optimal transport with lagrangian costs. InUncertainty in Artificial Intelligence. PMLR, 2024

    2024

  43. [43]

    Cambridge university press, 2000

    L Chris G Rogers and David Williams.Diffusions, Markov processes, and martingales: Itô calculus, volume 2. Cambridge university press, 2000

    2000

  44. [44]

    Springer Science & Business Media, 2009

    Bernard Roynette and Marc Yor.Penalising brownian paths, volume 1969. Springer Science & Business Media, 2009

    1969

  45. [45]

    Align your flow: Scaling continuous-time flow map distillation

    Amirmojtaba Sabour, Sanja Fidler, and Karsten Kreis. Align your flow: Scaling continuous-time flow map distillation. InAdvances in Neural Information Processing Systems, 2025

    2025

  46. [46]

    Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique.Annales de l’Institut Henri Poincaré, 2(4):269–310, 1932

    Erwin Schrödinger. Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique.Annales de l’Institut Henri Poincaré, 2(4):269–310, 1932

    1932

  47. [47]

    Diffusion schrödinger bridge matching.Advances in Neural Information Processing Systems, 2023

    Yuyang Shi, Valentin De Bortoli, Andrew Campbell, and Arnaud Doucet. Diffusion schrödinger bridge matching.Advances in Neural Information Processing Systems, 2023

    2023

  48. [48]

    Exponential convergence guarantees for iterative markovian fitting

    Marta Gentiloni Silveri, Giovanni Conforti, and Alain Durmus. Exponential convergence guarantees for iterative markovian fitting. InAdvances in Neural Information Processing Systems, 2025

    2025

  49. [49]

    Score-based generative modeling through stochastic differential equations

    Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021. 13

    2021

  50. [50]

    Nonlinear filtering formulas for discrete-time observations

    Yoshiki Takeuchi and Hajime Akashi. Nonlinear filtering formulas for discrete-time observations. SIAM Journal on Control and Optimization, 19(2):244–261, 1981

    1981

  51. [51]

    Transport with support: Data-conditional diffusion bridges.Transactions on Machine Learning Research, 2023

    Ella Tamir, Martin Trapp, and Arno Solin. Transport with support: Data-conditional diffusion bridges.Transactions on Machine Learning Research, 2023

    2023

  52. [52]

    Schrödinger bridge with quadratic state cost is exactly solvable.arXiv preprint arXiv:2406.00503, 2024

    Alexis MH Teter, Wenqing Wang, and Abhishek Halder. Schrödinger bridge with quadratic state cost is exactly solvable.arXiv preprint arXiv:2406.00503, 2024

    Pith/arXiv arXiv 2024

  53. [53]

    − Z T t T−s T−t ∇Vs(Xs)ds|X t =x t,X T =x T # . Using the same reparameterization argument forZ t(xt), we further obtain ∇xt logZ t(xt) = 1 σ2 EQV,σ |t

    Alexander Tong, Jessie Huang, Guy Wolf, David Van Dijk, and Smita Krishnaswamy. Trajecto- rynet: A dynamic optimal transport network for modeling cellular dynamics. InInternational Conference on Machine Learning, pages 9526–9536. PMLR, 2020. 14 Organization of the supplementary The appendix is organized as follows. Section A summarizes the notation used t...

    2020