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Variational Schr\"odinger Momentum Diffusion

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that momentum Schrödinger bridges can be trained without forward simulation by replacing the forward score with an adaptively tuned linear variational score, and proves that the generated samples converge to the data.

desk verdict Promising simulation-free momentum diffusion with a real gap in the adaptive transition formulas; the theoretical claim is not supported as written, but the algorithm merits a serious revision. read the letter →

arxiv 2501.16675 v1 pith:XLXZDYNS submitted 2025-01-28 stat.ML cs.LG

Kevin Rojas , Yixin Tan , Molei Tao , Yuriy Nevmyvaka , Wei Deng This is my paper · ORCID
classification stat.MLcs.LG MSC 60J6068T07
keywords Schrödingerbridgemomentumdiffusionvariationalscoresimulation-freegenerativemodelcriticallydampedLangevinstochasticapproximationoptimaltransportscore-basedmodeling
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 tries to make momentum Schrödinger bridge (mSB) generative models trainable without simulating forward trajectories. It replaces the intractable forward score with a linearized variational score matrix and updates that matrix adaptively during training so the transport plan improves while the backward score is learned simulation-free. The central theoretical claim is that, under regularity assumptions, samples generated by the resulting algorithm converge in total variation to the data distribution, with an error made of forward-process convergence, discretization, and score-estimation terms. The method also uses a critical-damping transform that reduces the two variational scores to one. If correct, this gives momentum diffusion a training cost comparable to standard diffusion while preserving some optimal-transport benefit and removing warm-up initialization.

What carries the argument

The mechanism is the linear forward SDE $d\vec a_t = -\tfrac12\beta D_t \vec a_t\,dt + g\,d\vec w_t$, where $D_t = \begin{pmatrix}0&-1\\1&\gamma\end{pmatrix}\otimes I_d - 2\gamma A_{a,t}$ and the matrix $A_{a,t}$ is the variational score. Because this is a linear Gaussian process, the transition is available in closed form through the mean $\mu_{t|0}=e^{-\frac12\beta[D]_t}x_0$ and a covariance given by a differential Lyapunov equation, which is what removes forward simulation. The other load-bearing identity is the critical-damping transform $A_{v,t}=\tfrac12-\tfrac1\gamma\sqrt{R(1-2\gamma A_{x,t})}$, which collapses the two score blocks into one trainable block. The stochastic-approximation update of $A_{a,t}$ from backward samples is what makes the plan data-adaptive and is the subject of the convergence theorems.

What would settle it

Take a fixed non-commuting schedule of variational scores $A_{a,t}$, run the forward SDE (5) with a fine numerical integrator, and compare the empirical distribution of $\vec a_T$ with the closed-form Gaussian $\mathcal N(\mu_{T|0},\Sigma_{T|0})$ from Eqs. (7) and (8); a total-variation or Wasserstein gap that does not shrink with the integrator step size would show that the simulation-free sampler is not exact for time-varying scores.

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Extended reading notes

Core claim

The paper's central claim is that the hard optimal-transport problem of the momentum Schrödinger bridge can be replaced, without losing too much transport benefit, by a search over generalized Ornstein–Uhlenbeck processes with linear drift corrections. Within this restricted class the forward marginals are Gaussian and can be sampled in closed form, so the backward score can be trained from cached mean and covariance rather than from simulated forward paths. The paper then proves, in Theorems 1 through 3, that if the variational score matrix $A_{a,t}$ is updated by stochastic approximation on the transport cost, it converges to a locally optimal equilibrium $A^\star_{a,t}$, and the total variation distance between the generated samples and the data is bounded by $\sqrt{\mathrm{KL}(p_{\mathrm{data}}\|\mu_x)+\mathrm{FI}(p_{\mathrm{data}}\|\mu_x)}\,e^{-T}$ plus discretization, score-estimation, and adaptation error. The critical-damping relation ties the two score components together, so only one matrix needs to be trained.

Load-bearing premise

The load-bearing premise is that the time-varying drift matrix $D_t$ can be exponentiated as if it were constant, giving exact simulation-free Gaussian transitions; if the adaptively optimized matrices $D_s$ and $D_t$ do not commute, this replacement is only approximate and the cached samples used to train the score are not exact draws from the stated forward SDE.

Editorial extensions

If this is right

  • Forward training cost drops to a cache-and-sample procedure: draw from $\mathcal N(\mu_{t|0},\Sigma_{t|0})$, Cholesky factor once, and regress the score; no forward SDE simulation is needed.
  • The transport plan is no longer fixed: as $A_{a,t}$ adapts, the effective damping and frequency of each dimension become data-dependent, which is how stretched anisotropic shapes can be generated while keeping $\beta$ moderate.
  • Underdamped schedules with $R<1$ inherit the faster mixing of kinetic Langevin dynamics, and the paper's experiments show comparable or better sample quality than CLD and ULD at earlier training stages.
  • The same machinery transfers to real-world tasks: conditional time-series forecasting and unconditional CIFAR-10 generation run with a standard U-Net, a second-order Heun discretization, and no warm-up initialization from a pretrained score model.

Reading between the lines

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

  • Editorial inference: the theory's forward-process term $\sqrt{\mathrm{KL}(p_{\mathrm{data}}\|\mu_x)+\mathrm{FI}(p_{\mathrm{data}}\|\mu_x)}$ is finite only when the data is not too far from the Gaussian invariant measure $\mu_x$; for heavy-tailed or highly multimodal data, the practical guarantee would rest almost entirely on the score-estimation and adaptation terms.
  • Editorial inference: if the non-commutativity of $D_s$ and $D_t$ is material, the simulation-free cache is an approximate sampler rather than an exact one; a testable extension is to replace the matrix exponential with a time-ordered exponential or a piecewise-constant schedule and measure the change in sample quality.
  • Editorial inference: the critical-damping transform restricts the search to a one-dimensional family of score matrices; a natural ablation is to train both blocks freely on a small anisotropic problem and compare transport cost against the constrained version, which would reveal how much optimality the stabilization costs.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes Variational Schrödinger Momentum Diffusion (VSMD), a simulation-free generative model that approximates the forward score of a momentum Schrödinger bridge by linear variational score matrices, yielding a Gaussian transition for the augmented (position, velocity) process. The neural backward score is trained against these cached Gaussian transitions, while the variational score matrices are adapted by stochastic approximation to improve transport; a critical-damping transform reduces the two score matrices to one. Experiments cover anisotropic synthetic data, multivariate time-series forecasting, and CIFAR-10 image generation.

Significance. If the theoretical claims held, VSMD would be a useful step toward scalable momentum Schrödinger bridges with transport-optimized, simulation-free training. The empirical evaluation is reasonably broad, the damping-transform reduction is elegant, and the paper honestly compares with CLD, ULD, and the overdamped VSDM baseline. However, the central convergence theorem rests on a Gaussian-transition formula whose exactness requires commutativity conditions that are neither stated nor satisfied by the adaptive, time-varying variational scores; the proposed time-varying extension in Remark 1 is mathematically incorrect. The theoretical contribution is therefore conditional on a substantial repair of the forward-transition argument.

major comments (4)
  1. [§4.2, Eqs. (7)–(8)] The closed-form formulas for the mean and covariance are valid for a linear SDE with time-dependent drift only when the drift matrices commute at different times or when the drift is constant. In Algorithm 1, A_a,t is updated at every time step and every stochastic-approximation stage, so D_t varies with t and generally does not commute with D_s. Eq. (7) replaces the time-ordered exponential with an ordinary matrix exponential, and Eq. (8) does the same for the augmented Lyapunov transition. Consequently, the cached samples a_t|a_0 drawn in Algorithm 1 are not exact draws from the forward SDE (5), the score loss (11) does not match the conditional score of the stated forward process, and the backward drift in Eq. (12) is not the exact reverse-time drift. This invalidates the link between the training objective and the fixed-generation bound in Theorem 1 for the adaptive schedule actually used.
  2. [§B.1, Remark 1] The proposed extension of Theorem 1 to time-varying A_a,t is not valid. For fixed T, replacing D_t in Eq. (5) by its time average D̄ = (1/T)∫₀ᵀ D_s ds does not produce the same distribution at time T unless the family {D_t} commutes. The mean of the true process is the time-ordered exponential, and the covariance is the solution of the non-autonomous Lyapunov equation, not the constant-drift covariance generated by D̄. Therefore the 'convergence of forward process' term in Theorem 1 and Theorem 3 does not follow for the adaptively time-varying variational scores.
  3. [§B.2–§B.3, Theorems 2–3] Theorems 2 and 3 are presented as restatements of theorems in Deng et al. (2024b). As restatements, they inherit the assumptions of that paper, but Theorem 3 depends on Theorem 1, whose time-varying extension is invalid as shown above. The first term of the bound in Theorem 3 also presumes convergence of the forward process to an invariant measure μ_x; for a non-autonomous D_t, no such unique invariant measure is established, so that term is not meaningful in the adaptive setting. The convergence proof needs to be re-derived for non-autonomous D_t, or the statement must be restricted to a commuting or time-invariant family of variational scores.
  4. [§4.3, Eq. (13)] The transport objective in Eq. (13) is defined using the learned backward score ←zᶿ_t rather than the true Schrödinger bridge potential. Thus the 'transport-optimized equilibrium' A⋆_a,t is a local optimum of a variational objective that depends on the current neural score. Theorem 2, even if its assumptions were verified, would establish convergence to this coupled equilibrium, not that the forward process solves the momentum Schrödinger bridge optimal transport problem. The paper's phrasing in §4.3 overstates the transport-optimality guarantee, and Assumption A2's local strong convexity is assumed for this coupled objective without verification.
minor comments (5)
  1. [§5.2] In the sentence 'We compare against CLD, and VDSM using the same architecture', the acronym VDSM should be VSDM.
  2. [Eq. (7)] The notation on the right-hand side of Eq. (7) writes x0, but the mean evolution is for the augmented variable a_t = (x_t, v_t); it should be a0, or the block structure should be made explicit.
  3. [Eq. (14)] The damping transform contains a matrix square root; the paper should specify the principal matrix square root and state the conditions under which the resulting A_v,t remains symmetric and positive definite as required by Assumption A1.
  4. [Assumption A2] The Hessian inequality mI ≼ ∂²L/∂A²(A) ≼ MI is written as a matrix inequality, but A is a matrix and the Hessian is a fourth-order tensor; a precise definition in terms of an operator norm on matrices would be needed.
  5. [Appendix C, Table 3] The FID comparison groups methods under 'OT' and 'SGMs' without clarifying the protocol or reporting confidence intervals; the comparison is informal and should be labeled as such.

Circularity Check

1 steps flagged · score 6.0 of 10

The headline adaptive-convergence theorem is explicitly imported as a restatement/natural extension of a same-author prior paper, making the theoretical core load-bearing self-citation; the empirical benchmarks remain independent.

  1. self citation load bearing [Appendix B.2-B.3 (Lemma 2, Theorems 2-3)]
    "The following is a restatement of Lemma 2 in Deng et al. (2024b)... The next result is a restatement of Theorem 2 in Deng et al. (2024b)... The following is a natural extension of Theorem 3 in Deng et al. (2024b) since both algorithms follow from the framework of multivariate diffusion... In particular, the stochastic approximation part is standard and inherited from Deng et al. (2024b)."

    These labels cover the exact chain that produces the paper's central theoretical claim. Lemma 2 supplies the local stability needed for the stochastic approximation iterates; Theorem 2 is explicitly a 'restatement' of a theorem in Deng et al. (2024b), whose authors include Deng, Tan, and Nevmyvaka; Theorem 3, the adaptive TV bound advertised in the abstract, is introduced as a 'natural extension' of that same paper's Theorem 3. No in-paper proof of the momentum/multivariate version is given; the proof sketch instead says the stochastic approximation part is 'standard and inherited from Deng et al. (2024b)' and otherwise refers to Ma et al. (2021) and Chen et al. (2023b). Since Deng et al.

full rationale

I did not find construction-level circularity in the main simulation-free training loop. The forward samples in Eqs. (7)-(8) are Gaussian draws from a stated linear SDE and are used to train the score via loss (11); any mismatch is charged in the A5 error term epsilon_score. The transport objective (13) does use the learned backward score z_theta to define the variational-score loss, but Theorem 3 explicitly carries epsilon_score and sqrt(eta_k) terms, so this is an acknowledged coupled/adaptive loop rather than an identity. The damping transform (14) is derived from the standard critical-damping condition, not from the target distribution. The genuine circularity is the theoretical scaffold: Lemma 2, Theorem 2, and Theorem 3 are presented, respectively, as a 'restatement', a 'restatement', and a 'natural extension' of theorems in Deng et al. (2024b), a paper with overlapping authorship, with the proof of the multivariate Lyapunov extension delegated to external citations. Thus the paper's headline convergence statement is inherited from a same-author citation chain, which is load-bearing. Separately, Remark 1's time-averaged-drift argument and the use of ordinary matrix exponentials in Eqs. (7)-(8) for noncommuting D_t are a correctness gap (the exact solution would be a time-ordered exponential), but that is not a circularity under the hard rules and does not enter the score beyond confirming that the imported theory is not independently re-proved here.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central algorithm depends on the Gaussian-transition formulas, on hyperparameters beta, gamma, R and step sizes, on the learned variational score matrices, and on assumptions A1-A6. The most fragile unstated premise is that the time-varying drift matrices commute so that matrix-exponential updates are exact; the local strong convexity of the transport loss and the bounded score estimation error are also unverified.

free parameters (5)
  • Diffusion scale beta = 5 or 10 on toy data; VPSDE schedule with beta_max=10 for VSDM-10(VP)
    Chosen by hand per experiment to balance anisotropic accuracy and straightness; central to the trade-off shown in Figures 3-4.
  • Damping ratio R = 0.7 (VSULD toy), 0.9 (CIFAR-10)
    Tuned manually; the paper states under-damping can be much faster than critical damping based on observation, and R controls the damping transform in Eq. (14).
  • Friction coefficient gamma = Not reported systematically
    Defines CLD/ULD dynamics; convergence effects are cited from prior work, but the values used in experiments are not listed.
  • Score and SA step sizes = Forward score lr 3e-4, backward score lr 3e-6, decay .99 in image experiments
    Required by Assumption A6; values are heuristics rather than derived quantities.
  • Variational score matrices A_x,t and A_v,t = Learned per time step
    The core adaptive parameters, optimized by stochastic approximation against a loss that uses the learned backward score; they define the transport plan and the damping relation Eq. (14).
assumptions (7)
  • ad hoc to paper D_s commutes with D_r (or D_t is constant or diagonal) for the exponential formulas in Eqs. (7)-(8)
    Needed for the matrix-exponential transition to be the exact solution of the time-varying linear SDE; not stated in the main text and generally false for adaptive A_{a,t}.
  • domain assumption A1: I - 2 gamma A_x,t and I - 2 A_v,t are symmetric positive-definite and the operator norm of D is O(1)
    Ensures an invariant Gaussian measure exists and the process is stable; unverified in experiments.
  • ad hoc to paper A2: local strong convexity of the transport loss in A
    Needed for Theorem 2 via the mean-field ODE; not checked and hard to verify.
  • domain assumption A3: score functions are Lipschitz in input and in A
    Standard smoothness borrowed from Chen et al. (2023b); plausible but not verified.
  • domain assumption A5: score estimation error bounded by eps_score
    The convergence bounds depend on this unverified bound; in practice the score network is finite and biased by the linear approximation.
  • standard math Exponential convergence of the linear forward process to its invariant Gaussian with rate c, via the Lyapunov function of Ma et al. (2021)
    Borrowed from Ma et al. (2021) and Chen et al. (2023b); the proof in Appendix B.1 refers to their appendices.
  • ad hoc to paper The linearized gOU class contains a process whose KL is a local optimum of the momentum Schrödinger bridge
    Section 4.3 states the solution bP acts as a local optimum without proof.

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Cite this review

Pith. "Pith review of Variational Schr\"odinger Momentum Diffusion." pith.science (2026). https://pith.science/paper/XLXZDYNS

@misc{pith2026250116675,
  author       = {Pith},
  title        = {Pith review of: Variational Schr\"odinger Momentum Diffusion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XLXZDYNS}},
  note         = {Machine review of arXiv:2501.16675}
}
read the original abstract

The momentum Schr\"odinger Bridge (mSB) has emerged as a leading method for accelerating generative diffusion processes and reducing transport costs. However, the lack of simulation-free properties inevitably results in high training costs and affects scalability. To obtain a trade-off between transport properties and scalability, we introduce variational Schr\"odinger momentum diffusion (VSMD), which employs linearized forward score functions (variational scores) to eliminate the dependence on simulated forward trajectories. Our approach leverages a multivariate diffusion process with adaptively transport-optimized variational scores. Additionally, we apply a critical-damping transform to stabilize training by removing the need for score estimations for both velocity and samples. Theoretically, we prove the convergence of samples generated with optimal variational scores and momentum diffusion. Empirical results demonstrate that VSMD efficiently generates anisotropic shapes while maintaining transport efficacy, outperforming overdamped alternatives, and avoiding complex denoising processes. Our approach also scales effectively to real-world data, achieving competitive results in time series and image generation.

Figures

Figures reproduced from arXiv: 2501.16675 by the authors.

Figure 1
Figure 1. Comparison with existing methodologies and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. CLD-5 (left two) v.s. VSULD-5 (right two) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Sample quality evaluation. The damping ratios for ULD and VSULD are both fixed to 0.7. (e.g., the X-axis of the spiral dataset). Specifically, evaluating the straightness metric as suggested in Deng et al. (2024b), we observe in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (15 more)
Figure 5
Figure 5. Figure 5: Overdamped versus underdamped models. 5.2 Time Series Forecasting We demonstrate our models ability in a real world multivariate probabilistic forecasting. Given a sequence x1:N = {(ti , xi)} N i=1 where ti ∈ R is a time variable and xi ∈ R d . Our goal is to predict t…
Figure 7
Figure 7. Figure 7: Unconditional generated samples using VS [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 6
Figure 6. Figure 6: Sample forecasts of our method in the solar [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 8
Figure 8. Figure 8: Sample forecasts of VSCLD in the electricity dataset [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Sample forecasts of VSCLD in the solar dataset [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Sample forecasts of VSCLD in the exchange rate dataset [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Sample forecasts of VSULD in the electricity dataset [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: Sample forecasts of VSULD in the solar dataset [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: Sample forecasts of VSULD in the exchange rate dataset [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: Sample forecasts of VSDM in the electricity dataset [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: Sample forecasts of VSDM in the solar dataset [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
Figure 16
Figure 16. Figure 16: Sample forecasts of VSDM in the exchange rate dataset [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]
Figure 17
Figure 17. Figure 17: Sample forecasts of CLD in the electricity dataset [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]
Figure 18
Figure 18. Figure 18: Sample forecasts of CLD in the solar dataset [PITH_FULL_IMAGE:figures/full_fig_p020_18.png]
Figure 19
Figure 19. Figure 19: Sample forecasts of CLD in the exchange rate dataset [PITH_FULL_IMAGE:figures/full_fig_p020_19.png]

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

74 extracted references · 67 canonical work pages · cited by 1 Pith paper

  1. [1]

    S., Bof, N

    Albergo, M. S., Bof, N. M., and Vanden-Eijnden, E. (2023). Stochastic Interpolants: A Unifying Framework for Flows and Diffusions . arXiv:2303.08797v1 , pages 1--48

    arXiv 2023

  2. [2]

    Albergo, M. S. and Vanden-Eijnden, E. (2023). Building Normalizing Flows with Stochastic Interpolants . In International Conference on Learning Representation (ICLR)

    work page 2023

  3. [3]

    Anderson, B. D. (1982). Reverse-time Diffusion Equation Models . Stochastic Processes and Their Applications , 12(3):313--326

    work page 1982

  4. [4]

    F., Ang, M

    Ansari, A. F., Ang, M. L., and Soh, H. (2020). Refining Deep Generative Models via Discriminator Gradient Flow . In International Conference on Learning Representations

    work page 2020

  5. [5]

    Bartosh, G., Vetrov, D., and Naesseth, C. A. (2024). Neural Flow Diffusion Models: Learnable Forward Process for Improved Diffusion Modelling . In Advances in Neural Information Processing Systems (NeurIPS)

    work page 2024

  6. [6]

    Benveniste, A., M\'etivier, M., and Priouret, P. (1990). Adaptive A lgorithms and S tochastic A pproximations . Berlin: Springer

    work page 1990

  7. [7]

    Bunne, C., Hsieh, Y.-P., Cuturi, m., and Krause, A. (2023). The Schr\"odinger Bridge between Gaussian Measures has a Closed Form . In AISTATS

    work page 2023

  8. [8]

    Caluya, K. F. and Halder, A. (2022). Wasserstein Proximal Algorithms for the Schr\" o dinger Bridge Problem: Density Control with Nonlinear Drift . IEEE Transactions on Automatic Control , 67(3):1163--1178

    work page 2022

Show all 74 references
  1. [9]

    Chen, C., Ding, N., and Carin, L. (2015). On the C onvergence of S tochastic G radient MCMC A lgorithms with H igh-order I ntegrators. In Advances in Neural Information Processing Systems (NeurIPS) , pages 2278--2286

    2015

  2. [10]

    Chen, H., Lee, H., and Lu, J. (2023a). Improved Analysis of Score-based Generative Modeling: User-friendly Bounds under Minimal Smoothness Assumptions . In International Conference on Machine Learning , pages 4735--4763

    2023

  3. [11]

    Chen, R. T. Q., Rubanova, Y., Bettencourt, J., and Duvenaud, D. (2018). Neural Ordinary Differential Equations . In Advances in Neural Information Processing Systems (NeurIPS)

    2018

  4. [12]

    Chen, S., Chewi, S., Li, J., Li, Y., Salim, A., and Zhang, A. R. (2023b). Sampling is as Easy as Learning the Score: Theory for Diffusion Models with Minimal Data Assumptions . International Conference on Learning Representation (ICLR)

    2023

  5. [13]

    A., Susskind, J., and Zhai, S

    Chen, T., Gu, J., Dinh, L., Theodorou, E. A., Susskind, J., and Zhai, S. (2024). Generative Modeling with Phase Stochastic Bridges . In International Conference on Learning Representation (ICLR)

    2024

  6. [14]

    Chen, T., Liu, G.-H., Tao, M., and Theodorou, E. A. (2023c). Deep Momentum Multi-Marginal Schr\"odinger Bridge . In Advances in Neural Information Processing Systems (NeurIPS)

    2023

  7. [15]

    Chen, T., Liu, G.-H., and Theodorou, E. A. (2022). Likelihood Training of Schr\" o dinger Bridge using Forward-Backward SDEs Theory . In International Conference on Learning Representation (ICLR)

    2022

  8. [16]

    Chen, Y., Deng, W., Fang, S., Li, F., Yang, N., Zhang, Y., Rasul, K., Zhe, S., Schneider, A., and Nevmyvaka, Y. (2023d). Provably Convergent Schr\" o dinger Bridge with Applications to Probabilistic Time Series Imputation . In International Conference on Machine Learning (ICML)

    2023

  9. [17]

    and Georgiou, T

    Chen, Y. and Georgiou, T. (2016). Stochastic Bridges of Linear Systems . IEEE Transactions on Automatic Control , 61(2)

    2016

  10. [18]

    T., and Pavon, M

    Chen, Y., Georgiou, T. T., and Pavon, M. (2021). Stochastic Control Liaisons: Richard Sinkhorn Meets Gaspard Monge on a Schr\" o dinger Bridge . SIAM Review , 63(2):249--313

    2021

  11. [19]

    S., Bartlett, P

    Cheng, X., Chatterji, N. S., Bartlett, P. L., and Jordan, M. I. (2017). Underdamped Langevin MCMC: A Non-Asymptotic Analysis . In Proc. of Conference on Learning Theory (COLT)

    2017

  12. [20]

    Dalalyan, A. S. and Karagulyan, A. G. (2019). User-friendly G uarantees for the L angevin M onte C arlo with I naccurate G radient. Stochastic Processes and their Applications , 129:12:5278--5311

    2019

  13. [21]

    Dalalyan, A. S. and Riou-Durand, L. (2020). On Sampling from a Log-concave Density using Kinetic Langevin Diffusions . Bernoulli , 26(3):1956--1988

    2020

  14. [22]

    De Bortoli, V., Korshunova, I., Mnih, A., and Doucet, A. (2024). Schr\"odinger Bridge Flow for Unpaired Data Translation . In Advances in Neural Information Processing Systems (NeurIPS)

    2024

  15. [23]

    De Bortoli, V., Thornton, J., Heng, J., and Doucet, A. (2021). Diffusion Schr\" o dinger Bridge with Applications to Score-Based Generative Modeling . In Advances in Neural Information Processing Systems (NeurIPS)

    2021

  16. [24]

    T., Du, H., Feng, Q., and Chen, R

    Deng, W., Chen, Y., Yang, N. T., Du, H., Feng, Q., and Chen, R. T. Q. (2024a). Reflected Schr\"odinger Bridge for Constrained Generative Modeling . In Proc. of the Conference on Uncertainty in Artificial Intelligence (UAI)

    2024

  17. [25]

    Deng, W., Luo, W., Tan, Y., Bilo s , M., Chen, Y., Nevmyvaka, Y., and Chen, R. T. Q. (2024b). Variational Schr\"odinger Diffusion Models . In Proc. of the International Conference on Machine Learning (ICML)

    2024

  18. [26]

    and Nichol, A

    Dhariwal, P. and Nichol, A. (2022). Diffusion Models Beat GANs on Image Synthesis . In Advances in Neural Information Processing Systems (NeurIPS)

    2022

  19. [27]

    Dockhorn, T., Vahdat, A., and Kreis, K. (2022). Score-Based Generative Modeling with Critically-Damped Langevin Diffusion . In Proc. of the International Conference on Learning Representation (ICLR)

    2022

  20. [28]

    Eyring, L., Klein, D., Uscidda, T., Palla, G., Kilbertus, N., Akata, Z., and Theis, F. (2024). Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translation . In ICLR

    2024

  21. [29]

    Gushchin, N., Kolesov, A., Korotin, A., Vetrov, D., and Burnaev, E. (2023). Entropic Neural Optimal Transport via Diffusion Processes . In Advances in Neural Information Processing Systems (NeurIPS)

    2023

  22. [30]

    Hairer, E., Lubich, C., and Wanner, G. (2006). Geometric Numerical Integration . Springer, Berlin, 2nd edition

    2006

  23. [31]

    P., Poole, B., Norouzi, M., Fleet, D

    Ho, J., Chan, W., Saharia, C., Whang, J., Gao, R., Gritsenko, A., Kingma, D. P., Poole, B., Norouzi, M., Fleet, D. J., and Salimans, T. (2022). Imagen Video: High Definition Video Generation with Diffusion Models . In arXiv:2210.02303

    2022 arXiv

  24. [32]

    Ho, J., Jain, A., and Abbeel, P. (2020). Denoising Diffusion Probabilistic Models . In Advances in Neural Information Processing Systems (NeurIPS)

    2020

  25. [33]

    Karras, T., Aittala, M., Aila, T., and Laine, S. (2022). Elucidating the Design Space of Diffusion-Based Generative Models . In Advances in Neural Information Processing Systems (NeurIPS)

    2022

  26. [34]

    Kim, B., Kwon, G., Kim, K., and Ye, J. C. (2024). Unpaired Image-to-Image Translation via Neural Schr\"odinger Bridge . In Proc. of the International Conference on Learning Representation (ICLR)

    2024

  27. [35]

    P., Salimans, T., Poole, B., and Ho, J

    Kingma, D. P., Salimans, T., Poole, B., and Ho, J. (2021). Variational Diffusion Models . ArXiv , abs/2107.00630

    2021 arXiv

  28. [36]

    Kong, Z., Ping, W., Huang, J., Zhao, K., and Catanzaro, B. (2021). DiffWave: A Versatile Diffusion Model for Audio Synthesis . In Proc. of the International Conference on Learning Representation (ICLR)

    2021

  29. [37]

    Korotin, A., Gushchin, N., and Burnaev, E. (2024). Light Schrödinger Bridge . In Proc. of the International Conference on Learning Representation (ICLR)

    2024

  30. [38]

    and Santambrogio, F

    Lavenant, H. and Santambrogio, F. (2022). The Flow Map of the Fokker–Planck Equation Does Not Provide Optimal Transport . Applied Mathematics Letters , 133

    2022

  31. [39]

    Lee, H., Lu, J., and Tan, Y. (2022). Convergence for Score-based Generative Modeling with Polynomial Complexity . Advances in Neural Information Processing Systems (NeurIPS)

    2022

  32. [40]

    and Matthews, C

    Leimkuhler, B. and Matthews, C. (2013). Rational construction of stochastic numerical methods for molecular sampling. Applied Mathematics Research eXpress , 2013(1):34--56

    2013

  33. [41]

    Lipman, Y., Chen, R. T. Q., Ben-Hamu, H., Nickel, M., and Le, M. (2023). Flow Matching for Generative Modeling . In Proc. of the International Conference on Learning Representation (ICLR)

    2023

  34. [42]

    Liu, G.-H., Chen, T., So, O., and Theodorou, E. A. (2022). Deep Generalized Schr\"odinger Bridge . In Advances in Neural Information Processing Systems (NeurIPS)

    2022

  35. [43]

    Liu, Q. (2022). Rectified Flow: A Marginal Preserving Approach to Optimal Transport . arXiv:2209.14577

    2022 arXiv

  36. [44]

    Liu, X., Gong, C., and Liu, Q. (2023). Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow . In ICLR

    2023

  37. [45]

    Lu, C., Zhou, Y., Bao, F., Chen, J., Li, C., and Zhu, J. (2022). DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps . In Advances in Neural Information Processing Systems (NeurIPS)

    2022

  38. [46]

    and Yong, J

    Ma, J. and Yong, J. (2007). Forward-Backward Stochastic Differential Equations and their Applications . Springer

    2007

  39. [47]

    S., Cheng, X., Flammarion, N., Bartlett, P

    Ma, Y.-A., Chatterji, N. S., Cheng, X., Flammarion, N., Bartlett, P. L., and Jordan, M. I. (2021). Is there an analog of nesterov acceleration for gradient-based mcmc? Bernoulli , 27(3)

    2021

  40. [48]

    and Smith, A

    Mangoubi, O. and Smith, A. (2021). Mixing of Hamiltonian Monte Carlo on Strongly Log-concave Distributions: Continuous Dynamics . The Annals of Applied Probability , 31(5):2019--2045

    2021

  41. [49]

    and Vishnoi, N

    Mangoubi, O. and Vishnoi, N. K. (2018). Dimensionally T ight R unning T ime B ounds for S econd-order H amiltonian M onte C arlo. In Advances in Neural Information Processing Systems (NeurIPS)

    2018

  42. [50]

    McCall, M. W. (2011). Classical Mechanics: From Newton to Einstein: A Modern Introduction (Second Edition) . John Wiley & Sons

    2011

  43. [51]

    McCann, R. J. (1997). A Convexity Principle for Interacting Gases . Advances in mathematics , 128(1):153–179

    1997

  44. [52]

    Neal, R. M. (2012). MCMC using Hamiltonian dynamics. In Handbook of Markov Chain M onte C arlo , volume 54, pages 113--162

    2012

  45. [53]

    Neklyudov, K., Brekelmans, R., Tong, A., Atanackovic, L., Liu, Q., and Makhzani, A. (2024). A Computational Framework for Solving Wasserstein Lagrangian Flows . In Proc. of the International Conference on Machine Learning (ICML)

    2024

  46. [54]

    G., and Trigila, G

    Pavon, M., Tabak, E. G., and Trigila, G. (2021). The Data-driven Schr\" o dinger Bridge . Communications on Pure and Applied Mathematics , 74:1545--1573

    2021

  47. [55]

    Peluchetti, S. (2023). Diffusion Bridge Mixture Transports, Schr\"odinger Bridge Problems and Generative Modeling . Journal of Machine Learning Research

    2023

  48. [56]

    Pooladian, A.-A., Ben-Hamu, H., Domingo-Enrich, C., Amos, B., Lipman, Y., and Chen, R. T. Q. (2023). Multisample Flow Matching: Straightening Flows with Minibatch Couplings . In Proc. of the International Conference on Machine Learning (ICML)

    2023

  49. [57]

    Ramesh, A., Dhariwal, P., Nichol, A., Chu, C., and Chen, M. (2022). Hierarchical Text-Conditional Image Generation with CLIP Latents . In arXiv:2204.06125v1

    2022 arXiv

  50. [58]

    Rasul, K., Seward, C., Schuster, I., and Vollgraf, R. (2021). Autoregressive Denoising Diffusion Models for Multivariate Probabilistic Time Series Forecasting . In International Conference on Machine Learning

    2021

  51. [59]

    and Monro, S

    Robbins, H. and Monro, S. (1951). A S tochastic A pproximation M ethod. Annals of Mathematical Statistics , 22:400--407

    1951

  52. [60]

    and Ho, J

    Salimans, T. and Ho, J. (2022). Progressive Distillation for Fast Sampling of Diffusion Models . In ICLR

    2022

  53. [61]

    and Solin, A

    S\"arkk\"a, S. and Solin, A. (2019). Applied Stochastic Differential Equations . Cambridge University Press

    2019

  54. [62]

    Shi, Y., De Bortoli, V., Campbell, A., and Doucet, A. (2023). Diffusion Schr\" o dinger Bridge Matching . In Advances in Neural Information Processing Systems (NeurIPS)

    2023

  55. [63]

    Singhal, R., Goldstein, M., and Ranganath, R. (2023). Where to Diffuse, How to Diffuse, and How to Get Back: Automated Learning for Multivariate Diffusions . In Proc. of the International Conference on Learning Representation (ICLR)

    2023

  56. [64]

    R., Pariset, M., Hsieh, Y.-P., Martinez, M

    Somnath, V. R., Pariset, M., Hsieh, Y.-P., Martinez, M. R., Krause, A., and Bunne, C. (2023). Aligned Diffusion Schr\"odinger Bridges . In Conference on Uncertainty in Artificial Intelligence

    2023

  57. [65]

    P., Kumar, A., Ermon, S., and Poole, B

    Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., and Poole, B. (2021). Score-Based Generative Modeling through Stochastic Differential Equations . In International Conference on Learning Representation (ICLR)

    2021

  58. [66]

    Tanaka, A. (2019). Discriminator Optimal Transport . In Neural Information Processing Systems

    2019

  59. [67]

    Tong, A., Malkin, N., Huguet, G., Zhang, Y., Rector-Brooks, J., Fatras, K., Wolf, G., and Bengio, Y. (2024). Improving and Generalizing Flow-based Generative Models with Minibatch Optimal Transport . Transactions on Machine Learning Research

    2024

  60. [68]

    Tuckerman, M. E. (2010). Statistical Mechanics: Theory and Molecular Simulation . Oxford University Press, New York

    2010

  61. [69]

    Vahdat, A., Kreis, K., and Kautz, J. (2021). Score-based Generative Modeling in Latent Space . Advances in Neural Information Processing Systems , 34:11287--11302

    2021

  62. [70]

    Vargas, F., Thodoroff, P., Lamacraft, A., and Lawrence, N. (2021). Solving Schr \"o dinger Bridges via Maximum Likelihood . Entropy , 23(9):1134

    2021

  63. [71]

    Wang, G., Jiao, Y., Xu, Q., Wang, Y., and Yang, C. (2021). Deep Generative Learning via Schr\" o dinger Bridge . In International Conference on Machine Learning (ICML)

    2021

  64. [72]

    @esa (Ref

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.