Data-to-Energy Stochastic Dynamics

Kirill Tamogashev; Nikolay Malkin

REVIEW 4 major objections 4 minor 14 references

Data-to-Energy Stochastic Dynamics

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper proposes the first general algorithm for Schrödinger bridge problems in which one or both endpoints are known only as unnormalised densities, extending iterative proportional fitting with a variance-minimising loss and demonstrat

desk verdict A genuinely new data-to-energy Schrödinger bridge algorithm that is likely the first of its kind, but the forward IPF step is heuristic and the paper's own tables contradict the secondary learned-variance claim. read the letter →

arxiv 2509.26364 v2 pith:XQHXV5QW submitted 2025-09-30 cs.LG

Kirill Tamogashev , Nikolay Malkin This is my paper

classification cs.LG

keywords Schrödingerbridgeiterativeproportionalfittingunnormaliseddensitysamplingdiffusionsamplersoff-policyreinforcementlearningstochasticoptimaltransportlearnedcoefficientlatent-spaceimagetranslation

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 proposes the first general algorithm for solving Schrödinger bridge problems when one or both endpoint distributions are known only through an unnormalised density—a function you can evaluate but cannot sample from. Existing bridge algorithms require samples from both endpoints; this one replaces the sample-based half-bridge updates with a variance-minimisation loss borrowed from diffusion samplers, so the iterative proportional fitting recursion can run data-free. The authors show on synthetic problems that the resulting bridges match sample-based ones, and apply it to posterior sampling in latent spaces of generative models, yielding image-to-image translation where semantic content is preserved. A secondary claim is that learning the diffusion coefficient as well as the drift improves existing data-to-data bridge algorithms at coarse time discretisations.

What carries the argument

The central mechanism is the iterative proportional fitting (IPF) recursion, which alternates half-bridge projections onto the two endpoint marginals. In the data-to-energy setting the forward projection is carried out by a source-conditional log-variance (VarGrad-type) loss: over a batch of trajectories sharing an initial point x0, it minimises the variance of log p_theta(tau|x0) - log p_phi(tau|x1) - log p1(x1), a quantity whose normalising constant cancels. Off-policy training distributions—a replay buffer of terminal samples, Langevin refreshes, and reuse of reverse trajectories—supply the x0 locations and trajectory batches that make the loss tractable in high dimensions.

What would settle it

For a two-dimensional Gauss-to-GMM pair with a known analytic or high-accuracy sample-based reference bridge, run data-to-energy IPF to convergence, then evaluate the time-0 marginal of the forward process against p0 and compare the joint (x0,x1) law against the entropic-optimal-transport coupling. Significant deviation in the time-0 marginal or the coupling would show that the conditional variance loss alone does not identify the bridge without an additional marginal constraint.

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Extended reading notes

Core claim

The paper claims that the Schrödinger bridge between a sample-available distribution p0 and an energy-defined distribution p1 can be computed by running IPF where the forward half-bridge is solved with a source-conditional log-variance loss rather than maximum likelihood. The loss minimises the variance, over trajectories sharing the same x0, of log p_theta(tau|x0) - log p_phi(tau|x1) - log p1(x1); because the normalising constant of p1 cancels, no samples from p1 are needed. The paper further claims that learning the diffusion coefficients of the discretised processes—not only the drifts—systematically improves existing data-to-data bridge algorithms, and that applying the resulting data-to

Load-bearing premise

The forward IPF step is trained by minimising the variance of a trajectory log-ratio conditional on x0, which only approximately enforces p_theta(tau|x0) proportional to p1(x1)p_phi(tau|x1) and does not by itself constrain the time-0 marginal of the learned process to equal p0; if the off-policy training distribution over x0 misses regions of p0, the fixed point of the recursion need not be the Schrödinger bridge.

Editorial extensions

If this is right

Bridge solvers can now target Boltzmann distributions, posteriors, and other energy models without a sample oracle, the main regime in scientific computing.
Learning the diffusion coefficient as well as the drift yields better bridges under coarse discretisation, so existing sample-based bridge pipelines can be improved by the same change.
Posterior sampling in latent spaces becomes a stochastic transport problem; class-conditional translation preserves unconstrained semantic structure such as background and digit style.
Energy-to-energy bridges, with no samples from either endpoint, are feasible when both sides are equipped with replay buffers.
Data-to-energy IPF reduces to diffusion sampler training when the forward process is already a bridge, connecting the new method to the diffusion-sampling literature.

Reading between the lines

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

The authors left implicit that the same variance loss could handle constraints expressed as arbitrary reward-shaped weights, not just probabilities, since only the log-weight appears in the objective.
The off-policy buffer acts as an implicit importance-sampling distribution; a natural extension would be to reweight buffer samples by importance weights or anneal the off-policy ratio dynamically, which the ablation suggests trades off mode coverage and transport cost.
The paper does not directly report the time-0 marginal of the learned forward process; a sharper fixed-point test would compare it to p0, since this is the constraint the variance loss only approximately enforces.

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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

First, the good news: this paper does something new. Existing SB algorithms need samples from both endpoints; the authors replace one endpoint with an unnormalized density and run IPF with a conditional VarGrad loss in the forward step and a replay-buffer/off-policy scheme to obtain x0. The synthetic experiments are plausible, the image-to-image translation in latent space is a nice application, and code is provided.

The main soft spot is the theory, or rather the lack of it. Equation (8) minimizes, for each x0, the variance of the log ratio over trajectories. That enforces proportionality of p_θ(τ|x0) to p_1(x1)p_φ(τ|x1), but the normalizing constant is free to vary with x0. The time-1 marginal of the implied forward process will be p1 only if, at the fixed point, the backward step has made the two processes exact reversals and the training distribution over x0 covers the relevant support. That fixed-point consistency might hold—the alternating structure could work it out—but the paper doesn't give the argument, and the off-policy x0 distribution (from buffer plus reverse rollouts) has no guarantee of covering p0. The authors say the loss 'approximately enforces' the constraint, which is honest, but for a paper claiming the first general method in this setting, this needs either a proof or a much clearer statement of what is being approximated.

Second, the secondary claim about learning the diffusion coefficient is not supported by the numbers. In Table 1, learned variance improves W2 on two of three tasks but makes path KL worse in all three. Table 4 is similar. The text says the tables 'clearly show' benefits—they don't.

The image experiments are suggestive but under-benchmarked: no comparison to a diffusion sampler (e.g., the outsourced diffusion sampling they generalize) or to bridge-sampling alternatives, and FID below the rejection-sampling ground truth is suspicious enough to need discussion.

Bottom line: the core idea is novel and the paper is worth a serious referee, but revisions need to address the fixed-point question and tone down the variance claim. I'd bring it to a reading group to kick around the IPF issue.

Referee Report

4 major / 4 minor

Summary. The paper proposes an extension of iterative proportional fitting (IPF) for Schrödinger bridge problems in which one or both endpoint marginals are available only as unnormalised densities (data-to-energy and energy-to-energy settings). The forward half-bridge is trained with a conditional log-variance (VarGrad) loss, Eq. (7)–(8), using off-policy replay buffers, reverse-trajectory reuse, and Langevin updates; the backward half-bridge is trained by maximum likelihood as in data-to-data IPF. The authors also train the diffusion coefficients of the forward/backward processes, claiming this improves discrete-time data-to-data IPF, and they apply the method to latent-space posterior sampling of GAN/VAE priors for image translation. The main positive results are on 2D synthetic benchmarks and on FID/reward metrics for CIFAR-10 and MNIST latent-space experiments. The central claim is that this is the first general method for data-to-energy and energy-to-energy Schrödinger bridges.

Significance. If the algorithm were proven to converge to the Schrödinger bridge, it would fill a genuine gap: existing IPF and Schrödinger-bridge matching methods require samples from both marginals, while many applications (Bayesian posteriors, reward-weighted priors) provide only unnormalised densities. The off-policy formulation is a plausible and potentially useful way to adapt diffusion-sampler losses to the bridge setting, and the authors have provided code. The empirical evaluation uses external metrics (W2 against oracle samples, FID against real images), which reduces circularity. However, the central convergence claim is not established: the forward IPF step is only heuristically justified, and the reported experiments do not demonstrate that the fixed point satisfies the p0 marginal constraint that defines the Schrödinger bridge. The secondary claim about learning diffusion coefficients is also only weakly supported by the tables. The paper is therefore more of a promising empirical proposal than a validated general method at this stage.

major comments (4)

[§3.1, Eq. (8) and Algorithm 2] The forward IPF step is trained by minimising Eq. (8) over x0 drawn from p_train^0. In practice p_train^0 is a mixture of on-policy samples x0~p0 and off-policy samples obtained by reverse rollouts from a replay buffer. The paper states only that this loss 'approximately enforce[s] this proportionality' and that the choice of p_train^0 is 'very important'; no argument is given that the fixed point satisfies the p0 marginal constraint. This is load-bearing because the backward step in Algorithm 2 (line 4) conditions on x0~p0 and will query the forward conditional at those x0. For the energy-to-energy variant (§D.3), there is no on-policy p0 component and buffers are initialised with Gaussian noise, so coverage of p0 is even less certain. Please provide a support/convergence analysis, or modify the algorithm so that p_train^0 provably covers p0.
[§3.1, Eq. (7)–(8)] The variance in Eq. (7) is defined over a full-support distribution p_train(τ|x0), but the implementation uses N=2 non-i.i.d. trajectories: one reverse roll-out from the buffer-conditioned backward process and one (or more) forward on-policy roll-outs. The empirical variance of a non-exchangeable batch is not an unbiased estimator of the variance of either marginal trajectory distribution, and at a global optimum it need not force p_θ(τ|x0) ∝ p_φ(τ|x1)p1(x1) for draws from the two different proposal mechanisms. This undermines the theoretical link between Eq. (8) and the half-bridge constraint. Please either use i.i.d. batches or provide a separate consistency argument for the non-i.i.d. estimator.
[§5.1, Tables 1 and 4] The claim that learning the diffusion coefficient improves existing data-to-data IPF is not supported by the reported numbers. In Table 4, data-to-data learnt-variance has higher Path KL than fixed-variance at K=5, 10, 20 and 40, and in Table 1 it is worse or mixed on the Gauss↔GMM and Gauss↔Two Moons tasks. The statement that the results 'clearly show the benefits of training the variances' is therefore an overclaim. Please re-analyse the results or temper the contribution to specific settings (e.g., small numbers of steps and W2 metric).
[§5.3, Table 2 and Fig. 4] The image-translation experiments contain no baselines: there is no comparison to the outsourced diffusion sampling method of Venkatraman et al. (2025) that the paper generalises, nor to a non-Schrödinger stochastic transport baseline. Consequently the claim that modelling a Schrödinger bridge is beneficial 'as opposed to a diffusion sampler' is unsupported. Additionally, the model's FID is lower than rejection-sampling FID for several classes, which is surprising and suggests either a bias in the classifier reward, mode collapse, or an artefact of the FID computation; this should be explained before the image results can be interpreted.

minor comments (4)

[Appendix C, Eq. (17)] The Euler-Maruyama increment in Eq. (17) writes the noise term as σΔt ξ_k; it should be σ√Δt ξ_k under the convention used in Eq. (5a). This affects the intermediate algebra of the path-KL derivation, though the final limit is plausible.
[§5.1] Typos: '2-dimentional', 'leant variance', 'undelying'. Appendix headings contain 'outsoursed'.
[§5.2] The comparison between data-to-data and data-to-energy IPF in Fig. 3 is visual only; reporting quantitative metrics with error bars for the energy-to-energy setting would strengthen the claim of 'comparable' performance.
[§3.3 and Algorithm 2] The notation for the forward and backward IPF steps is inconsistent: Eq. (3a)/(3b) are labelled with arrows but are later referred to as 'backward' and 'forward' respectively. Please align the terminology.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; the central data-to-energy IPF construction is independent of its inputs and is validated on external benchmarks, though it uses several self-cited techniques as building blocks.

full rationale

The paper's central contribution is a new conditional log-variance loss (Eq. 7-8) applied to the standard IPF recursion. The loss is designed to enforce the half-bridge proportionality p_theta(tau|x0) proportional to p1(x1) p_phi(tau|x1); it is not fitted to the reported evaluation quantities, and the off-policy buffer supplies the appropriate time-0 marginal of the backward conditional (p1(x1) times the backward conditional process) for training the forward half-bridge. The lack of a formal convergence proof for the off-policy alternating scheme is a correctness and robustness concern, not a circular reduction of the result to its inputs. Headline results are measured with external, non-circular metrics: W2 against ground-truth samples, FID against real images, ELBO, and path KL against the reference process. Self-citations (Gritsaev et al. 2025, Sendera et al. 2024, Venkatraman et al. 2025, Berner et al. 2025) are used as modular building blocks rather than as evidence for the central claim; the paper provides its own comparisons, ablations, and synthetic benchmarks. No equation reduces to another by construction, and no fitted parameter is renamed as a prediction. The 'first general method' claim is a novelty assertion, not a circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The main assumptions are the standard SB existence theory, the queryability of the energy, and two ad hoc approximations: the finite-batch variance loss in place of an exact half-bridge projection, and the use of learned variances in a fixed Euler-Maruyama discretisation. The free parameters are tuning choices for the off-policy training distribution, not physical constants.

free parameters (3)

off-policy ratio = 0.8
Controls the mixture of buffer-based and on-policy trajectories in Eq. (8); ablated in Table 3 and set to 0.8 for main experiments.
VarGrad trajectory batch size N = 2
Number of trajectories sharing x0 used in the variance estimate (8); set for all experiments with no analysis of bias or variance.
Langevin update schedule = 50-500 steps; step size 0.01 annealed to 0.001
Moves replay-buffer points toward p1 during training; hyperparameters differ across experiments and are chosen by hand.

assumptions (4)

standard math The Schrödinger bridge problem has a unique solution that is an SDE with the same diffusion coefficient as the reference process.
Invoked in §2.1 via Léonard (2014); needed for the IPF recursion to terminate at an SDE bridge.
domain assumption p0 and p1 are absolutely continuous with finite variance, and E1 is evaluable with an unknown normalising constant Z.
Used throughout §2.1 and §3.1; if the energy is not smooth or only a noisy oracle is available, the Langevin update in §3.2 and the log-ratio loss are not well-defined.
ad hoc to paper For the forward IPF step it is sufficient to enforce p_theta(tau|x0) proportional to p_phi(tau|x1) exp(-E1(x1)) for each x0 via a finite-batch variance loss (Eq. 7), and this fixed point satisfies the SB constraints.
Stated in §3.1 as 'To approximately enforce this proportionality'; no proof that the p0 marginal constraint is preserved or that N=2 trajectories suffice.
ad hoc to paper Euler-Maruyama discretisation with learned per-step variances yields processes that remain close to continuous-time SB solutions.
Assumed in §2.2; used to justify learning diffusion coefficients; not established, and Table 1/4 give mixed evidence.

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Cite this review

Pith. "Pith review of Data-to-Energy Stochastic Dynamics." pith.science (2026). https://pith.science/paper/XQHXV5QW

@misc{pith2026250926364,
  author       = {Pith},
  title        = {Pith review of: Data-to-Energy Stochastic Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XQHXV5QW}},
  note         = {Machine review of arXiv:2509.26364}
}

read the original abstract

The Schr\"odinger bridge problem is concerned with finding a stochastic dynamical system bridging two marginal distributions that minimises a certain transportation cost. This problem, which represents a generalisation of optimal transport to the stochastic case, has received attention due to its connections to diffusion models and flow matching, as well as its applications in the natural sciences. However, all existing algorithms allow to infer such dynamics only for cases where samples from both distributions are available. In this paper, we propose the first general method for modelling Schr\"odinger bridges when one (or both) distributions are given by their unnormalised densities, with no access to data samples. Our algorithm relies on a generalisation of the iterative proportional fitting (IPF) procedure to the data-free case, inspired by recent developments in off-policy reinforcement learning for training of diffusion samplers. We demonstrate the efficacy of the proposed data-to-energy IPF on synthetic problems, finding that it can successfully learn transports between multimodal distributions. As a secondary consequence of our reinforcement learning formulation, which assumes a fixed time discretisation scheme for the dynamics, we find that existing data-to-data Schr\"odinger bridge algorithms can be substantially improved by learning the diffusion coefficient of the dynamics. Finally, we apply the newly developed algorithm to the problem of sampling posterior distributions in latent spaces of generative models, thus creating a data-free image-to-image translation method. Code: https://github.com/mmacosha/d2e-stochastic-dynamics

Figures

Figures reproduced from arXiv: 2509.26364 by the authors.

**Figure 1.** Left: Algorithm for data-to-data IPF. Right: Algorithm for data-to-energy IPF, showing the replay buffer with backward trajectory reuse (§3.1), with differences highlighted in red. redundant and ←− P 1 𝑡 solves the SB problem. In practice, training ←− P 1 𝑡 as a neural SDE proceeds by score matching, which is simply a Rao-Blackwellised estimate of the IPF maximum-likelihood objective (Song et al., 2021a). 2.2 DISCRE… view at source ↗

**Figure 2.** W2 2 and Path KL depending on the number of discretization steps for Gauss ↔ GMM. For all 2-dimensional experiments we use d𝑋𝑡 = √ 2 d𝑊𝑡 as the reference process, training is done using 4000 steps for both backward and forward processes and 20 IPF steps. We use the same neural network architecture for all 2- dimensional experiments. We provide a detailed experiment configuration in §D. 5.2 DATA-TO-ENERGY AND ENERGY… view at source ↗

**Figure 3.** Comparison of learnt processes at various IPF iterations for data-to-data (top), data-toenergy (middle) and energy-to-energy (bottom) settings. For the energy-to-energy setting we use two different mixtures of Gaussian distributions. MNIST experiments are conducted in two setups: (a) reward function returns the probability that 𝑥 is even or odd (b) reward function returns the probability that 𝑥 = 5. For CIFAR-10… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Outsourced Schrodinger bridge on MNIST and CIFAR-10. The bridge preserves style ¨ features (thickness, background colour, orientation) while tranforming digits to the target class [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Curated examples of outsoursed SB with SN-GAN and StyleGAN generators. [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: Uncurated examples of outsoursed SB with SN-GAN for the class [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: Uncurated examples of outsoursed SB with SN-GAN for the class [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

**Figure 8.** Figure 8: Uncurated examples of outsoursed SB with SN-GAN for the class [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

**Figure 9.** Figure 9: Uncurated examples of outsoursed SB with SN-GAN for the class [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

**Figure 10.** Figure 10: Uncurated examples of outsoursed SB with StyleGAN for the class [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

**Figure 11.** Figure 11: Uncurated examples of outsoursed SB with StyleGAN for the class [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

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Reference graph

Works this paper leans on

14 extracted references · 7 linked inside Pith

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A RELATED WORKS In this section we establish links between our method and other research directions in the literature

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where𝑞(𝜏) is the time discretisation of the reference process Q𝑡

"𝐾−1∑︁ 𝑘=0 KL(− →𝑝𝜃(𝑥′ 𝑘+1|𝑥 𝑘)∥𝑞(𝑥 ′ 𝑘+1|𝑥 𝑘)) # (11) 14 Preprint. where𝑞(𝜏) is the time discretisation of the reference process Q𝑡. The estimator using transition KLs in (11) can be seen to be a Rao-Blackwellised (lower-variance) variant of the estimator in (10), and we use it because the KL can be computed analytically (as all transition kernels are Ga...

2023
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This yields the path energy used in Shi et al

∫ ∥𝑣𝜃(𝑥𝑡,𝑡)∥ 2d𝑡 (18) The limit is justified by the Girsanov theorem (S¨arkk¨a & Solin, 2019). This yields the path energy used in Shi et al. (2023). D EXPERIMENT DETAILS D.1 DATA-TO-DATA EXPERIMENTS All data-to-data experiments are conducted under the unified setup. For neural network we use an MLP with 3 hidden layers and 64 neurons in each layer, each ...

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

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Stochastic interpolants: A unifying framework for flows and diffusions.arXiv preprint arXiv:2303.08797,

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From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster training.arXiv preprint arXiv:2501.06148,

Julius Berner, Lorenz Richter, Marcin Sendera, Jarrid Rector-Brooks, and Nikolay Malkin. From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster training.arXiv preprint arXiv:2501.06148,

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Timofei Gritsaev, Nikita Morozov, Kirill Tamogashev, Daniil Tiapkin, Sergey Samsonov, Alexey Naumov, Dmitry Vetrov, and Nikolay Malkin

10 Preprint. Timofei Gritsaev, Nikita Morozov, Kirill Tamogashev, Daniil Tiapkin, Sergey Samsonov, Alexey Naumov, Dmitry Vetrov, and Nikolay Malkin. Adaptive destruction processes for diffusion samplers.arXiv preprint arXiv:2506.01541,

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Improving and generalizing flow-based generative models with minibatch optimal transport.Transactions on Machine Learning Research, 2024a

Alexander Tong, Kilian FATRAS, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector- Brooks, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport.Transactions on Machine Learning Research, 2024a. Alexander Tong, Nikolay Malkin, Kilian Fatras, Lazar Atanackovic, Yanlei Zhang, Guillau...

arXiv 1905

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A RELATED WORKS In this section we establish links between our method and other research directions in the literature

13 Preprint. A RELATED WORKS In this section we establish links between our method and other research directions in the literature. Optimal transport.Optimal transport is a well-established area of research with a solid theoretical background and scalable applied algorithms. The problem is concerned with finding the optimal transportation map which minimi...

1958

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The theoretical connection among various objectives was established in Berner et al

to amortise sampling from intractable density. The theoretical connection among various objectives was established in Berner et al. (2025). Outsourced sampling.The concept of outsourced diffusion sampling – modelling continuous- time dynamics in latent space for posterior inference under pretrained priors – was proposed in Venkatraman et al. (2025). The w...

2025

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where𝑞(𝜏) is the time discretisation of the reference process Q𝑡

"𝐾−1∑︁ 𝑘=0 KL(− →𝑝𝜃(𝑥′ 𝑘+1|𝑥 𝑘)∥𝑞(𝑥 ′ 𝑘+1|𝑥 𝑘)) # (11) 14 Preprint. where𝑞(𝜏) is the time discretisation of the reference process Q𝑡. The estimator using transition KLs in (11) can be seen to be a Rao-Blackwellised (lower-variance) variant of the estimator in (10), and we use it because the KL can be computed analytically (as all transition kernels are Ga...

2023

[8] [14]

This yields the path energy used in Shi et al

∫ ∥𝑣𝜃(𝑥𝑡,𝑡)∥ 2d𝑡 (18) The limit is justified by the Girsanov theorem (S¨arkk¨a & Solin, 2019). This yields the path energy used in Shi et al. (2023). D EXPERIMENT DETAILS D.1 DATA-TO-DATA EXPERIMENTS All data-to-data experiments are conducted under the unified setup. For neural network we use an MLP with 3 hidden layers and 64 neurons in each layer, each ...

2019