Data-to-Energy Stochastic Dynamics
Authors: Kirill Tamogashev
Nikolay Malkin
Venue: ICLR 2026 Poster (2026)
Keywords: Schrödinger bridge
Bayesian posterior inference
stochastic differential equations
Iterative Proportional Fitting
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: The paper presents a novel algorithm for modelling data-to-energy Schrödinger bridges.

Abstract: The Schrödinger 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 enable the inference of such
  dynamics only for cases where samples from both distributions are available.
  In this paper, we propose the first general method for modelling
  Schrödinger 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 Schrödinger 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.

Published as a conference paper at ICLR 2026
DATA-TO-ENERO STOCHASTIC DYNAMICS
Kirill Tamogashey University of Edinburgh
Nikolay Malkin University of Edinburgh, CIFAR Fellow rmalkin@ed. ac.uk uk
k, t tamogashevvssss..om ed. ac. uk
ABSTRACT
The Schrödinger 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 enable the inference of such dynamics only for cases where samples from both distributions general method for modelling Schrödinger 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 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-ent eentaty--eeery IPF on synthetic problems, finding that it can successfully learn transports between multimodal distributions. As secondary consequence of our reinforcement learning formulation, which assumes fixed time discretisation scheme for the dynamics, we find that existing data-to-data Schrödinger bridge algorithms can be substantially improved by learning the diffusion coefficient of the dynamics. Finally, apply the newly developed algorithm to the problem of sampling posterior distributions in latent spaces generative models, thus creating data-free image-to-ing to-image translation method.
available. In this paper, we propose the first
INTRODUCTION
Two modern approaches to generative modelling that have paved the way for scalable and efficient generation of high-fidelity images (Dhariwal & Nichol, 2021; Rombach et al., 2021), videos (Polyak et al., 2024), audio (Chen et al., 2021 aa; Kong et al., 2021) and text (Nie et al., 2025; Sahoo et al., 2025) diffusion models and flow matching. Diffusion models SSohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2021b) assume a noising stochastic process that transforms data into a tractable noise distribution and use score-based techniques to learn its reverse process, which transforms noise into data. Flow matching (Liu et al., 2023; Albergo et al., 2025; Lipman et al., 2023; Tong et al., 2024a) learns deeeddependent deterministic dynamics that give transportation map between two arbitrary distributions. Both approaches can be seen as special cases of the more general problem of learning stochastic dynamics between arbitrary distributions.
The problem of inferring an optimal stochastic bridge between two distributions is called the Schrödinger bridge (SB) problem. Initially proposed in Schrödinger (1931; 1932), it has recently been studied using various machine learning techniques (Huang et al., 2021; Vargas et al., 2021; Chen al., 2021b; Stromme, 2023; Shi al., 2023; Tong et al., 2024b). One computational approach the Schrödinger bridge problem is the iterative proportional fitting (IPF) algorithm (Fortet, 1940; Vargas et al., 2021; De Bortoli et al., 2021), which maintains pair of processes in forward and reverse time and iteratively updates them by solving half-bridge problems (see $2). Upon convergence, the two processes become time reversals of each other and solve the SB problem. Notably, the typical training of diffusion models with a fixed noising process that transforms data distribution into a Gaussian by construction is a degenerate case of IPF that converges in a single step. However, existing variants of IPF work only in setting where samples from both marginal distributions

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Published as a conference paper at ICLR 2026
available and thus cannot be used model bridges when one (or both) of the marginal distributions is given as an unnormalised density, without access to data samples.
We propose to extend the IPF algorithm to the case where one both marginal distributions are given by unnormalised densities or energy functions: p(x) e e- /z. Z z e -((x) dx, where & can be queried, but Z is unknown. Our proposed data-to-energy (or energy-to-energy) IPF generalises recently developed techniques for training diffusion models to sample from a distribution given by an unnormalised density (Zhang & Chen, 2022; Vargas al., 2023; 2024; Berner et al., 2024; Albergo & Vanden- Eijnden, 2025; Blessing et al., 2025a, inter alia). In particular, we build upon off-policy reinforcement learning losses and stabilisation techniques for diffusion samplers (Richter et al., 2020; Richter & Berner, 2024; Lahlou et al., 2023; Sendera et al., 2024; Gritsaev et al., 2025) to propose an efficient training for the IPF steps the data-to-energy setting. Our algorithm the first general method for inferring data-to-energe and energy-to-energe Schrödinger bridges.
Wielding the newly proposed algorithm, we make three contributions:
(1) We show that the proposed data-to-energy and energy-to to-eyytoo-erryy IPF algorithms successfully learn stochastic bridges with low transport cost between synthetic datasets and densities, performing par with the transports learnt by data-to-data IPF using samples from a ground-truth oracle. (2) As a secondary contribution, we show that - as a consequence of the time discretisation used in our reinforcement learning formulation existing data-to-data IPF algorithms can be improved by learning the diffusion coefficient of the dynamics, in addition to the drift, generalising the results of Gritsaev al. (2025) for diffusion samplers to the more general SB setting.
(3) We apply the data-to-energy IPF algorithm to the problem of translating prior distributions to posteriors in latent spaces of generative models, generalising the outsourced diffusion sampling of Venkatraman et al. (2025) to yield a scalable data-free image-to-image translation method.
2 DATA-TO-DATA SCHRÖDINGER BRIDGES
2.1 ITERATIVE PROPORTIONAL FITTING FOR DATA-TO-DATA SB
Setting: The SB problem and its connection to optimal transport. We present some background on the SB problem; see Léonard (2014); De Bortoli et al. (2021) for relevant and more detailed overviews. Let Po and pı be two given distributions over the space Rd, assumed to be absolutely continuous (thus used interchangeably with their density functions) and of finite variance. Let Q be a reference process on the time interval [O, I] taking values in Rd (usually an Ornstein- Omnttin-UUllnbeekk Uhlenbeck process, such as the Wiener process). The Schrödinger bridge problem can be formalised as:
PT argmin (KL (P, || Qr) s.t. (NO)aP; = po. (T))aP, = P1).
where the minimisation is taken over all processes P, whose marginals at times / = 0 and = 1, written PT and (F,) equal po and Pl. respectively. The solution P; is stochastic dynamical system that transports Po to p1 that is, bridge - that is the closest to O in KL divergence. If the reference process Q is given by an Itô stochastic differential equation (SDE)
dx, Free (X,,t) !) dr + or dW, Xo qo-
then, under mild conditions (see Léonard (2014)) the solution to (1) exists, is unique, and also takes the form of an SDE:
P, dx, = F(X,,)) di + 0, dW,, Xo ~ Po-
with the same diffusion coefficient. The KL then takes the form of a dynamic transport cost (Berner et al., 2026)
= + Ex,-F, /'' (X,, - -F(,)) dr,
20²
reducing the problem (1) to one of inferring the drift function F minimising the cost (2). This representation makes clear that as o, - O, the SB problem approaches the dynamic optimal transport problem between Po and Pı with squared-euclidean cost (see Tong al. (2024a)). For o, 0. the joint marginal distribution of F; over X,, X, is an entropy optimal transport, a key observation in the derivation of the SB algorithm in Tong et al. (2024b).
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Iterative proportional fitting. Computationally, the SB problem be solved using the iterative proportional fitting (IPF) algorithm (Fortet, 1940; Vargas et al., 2021; De Bortoli et al., 2021). IPF
defines a recursion initialised at P =
F argmin [L |I F7)) (TO)AIP, ]
F Po 1/0
(3a)
nll argmin F,
KL(P, |I s.t. = P1
Pi @ 111
F
(3b)
where each step is the solution to a half-bridge problem, which pins the previous iterate at one of the marginals po or P1 (here Fn *oo denotes the conditional process given X6 and Fatl the conditional process given X1). IPF for Schrödinger bridges is thus dynamic generalisation of the Sinkhorn algorithm (Sinkhorn, 1964), which computes entropic optimal transport by iteratively renormalising cost matrix over rows and columns. It can be shown (De Bortoli et al., 2021) that the iterates F and """ converge to the same stochastic process and that this process solves the SB problem (1).
r|1
7 Qr is represented as an SDE, then the iterates defined (3b) and (3a) can be represented as forward-tin time and reverse-time SDEs initialised at po and Pt. respectively. We can thus introduce SDEs with neurally parametrised drifts (Tzen & Raginsky, 2019):
Fr dx, eX, dr dW,
(4a)
Fr dx, = F (X, di dw,.
(4b)
where coincides with the diffusion coefficient of the reference process, and perform the iterations (3) as optimisation problems over On and 4n- (We henceforth occasionally drop the subscript from the parameters, with the understanding that (3a) optimises y fn+11 under a fixed 0, and (3b)
optimises = 6 under a fixed fnn+1.)
Data-to-data IPF in a time discretisation.
To approximately perform the optimisations involved IPF with respect to the parameters 0 and y,, we discretise the SDEs (4) representing F,, and P, over K steps using the Muler-Maruyama scheme with Ar x. The continuous time processes are thus approximated by discrete-ting time Markov chains, i.e., joint distributions over discrete- time trajectories = (Ao. x1), having the factorisation:
po(r(x))
Euler-Manyyam step with step size Ar
K-1
P)(()=po(x)) k=0
P Oo (xRA)) o (a N (KAA 7 (xRAA kAr)A, kAI)A,, Sir
(5a)
(r(())
(T))=)((()))
(o (xRne). (x(a-1I) = (JRAe (AI)A, Ar) (Sb)
reverse-time Fuler Maruyama step with step size At
As proposed in Vargas et al. (2021), the two IPF optimisation problems in (3) can be approximately solved by maximum likelihood. On the level of the discretised processes, this amounts to the following recurrence:
Pn+1 argmaa B Xp po. Ban (TiXo)
log Po (x xxar)
(6a)
k=
0x+1 argmax B - T- F (T|x)
log P (x xeAC).
(6b)
k=0
The optimisation problem in (6a) (resp. (6b)) requires taking a sample from the marginal distribution Xo ~ Po (resp. X1 ~ P1), rolling out a trajectory in forward time from Po o (resp. in reverse time from w) initialised at the sample, and maximising the log-likelihood of this trajectory in the opposite direction, i.e., under Foo (resp. under Po). This procedure essentially requires samples from Po (resp. from P1) to be available.
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Algorithm 1: Data-to-Data IPF
Algorithm 2: Data-to-Energy IPF
for π O...., Pmax do
Initialise buffer 8 =
/* Backward IPF step (6a)
*/
for ...... Pmax do
while not converged do
/ Backward IPF step (6a)
*/
20 po
while not converged do
XAr. XAr. using (5a)
.1) p xm)
Xn pu. - | |xx)
Gradient step on yp with
Gradient step on ip with
Vlog P 7 xx)
Vlog P o xx)
Update 2 with samples x1
1+ Forward IPF step (6b)
/* Forward IFF step (8)
while not converged do
7
while not converged do if on-policy then
P1
== (xo.... .co.....N. .I.......-Ar.-3
(T(
xo Po. T 1
x1) using (5b)
10
else
Gradient step on with
X1 -1 B, 7()) ( (x]) 7(2) ...... - ((N) ~ Po(T 1x (
11
v log P (T xo)
10 return 0, 9
Gradient step 0 with
VVr, log [o( + Ex((
return 0, 4
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.
We call this iterative algorithm the aigliikeihoodd method and give an algorithmic presentation in Algorithm 1. It contrasts with the method proposed in De Bortoli et al. (2021), which uses slightly different discretisation scheme, although the two coincide in the continuous-time limit (K - oo).
al., 2015; Ho et al., 2020; Song et al., 2021b) can be seen as a special case of the IPF algorithm that converges in a single step. If po data distribution and #1 is Gaussian, and Q is noising process that transports any source distribution to Pı by construction, then Q2 initialised at is already bridge between and p1-
Diffusion models special case. Diffusion models
Thus the first iteration of IPF learning F by maximum-likelihon training (6a) on trajectories sampled from Po @ Q lo already yields bridge between Po and p1. so all subsequent iterations are
redundant and P) solves the SB problem. In practice, training F as a neural SDE proceeds by score matching, which is simply Buo-Blackwellisee estimate of the IPF maximum mokimum-likellhood objective (Song et al., 2021a).
2.2 DISCRETISATION ALLOWS FLEXIBLE KERNELS
The majority of existing work (Chen al., 2021b; Vargas et al., 2021: De Bortoli et al., 2021; Shi et al., 2023) trains only the drift functions F 8. F v or related objects using objectives similar to (6). In contrast to that, we propose to train not only the drift, but also diffusion coefficients of both processes, by replacing the variances 2 in (5) by learnt functions d2 (xz, kAr) and 5² (x2, kAr). We expect this to correct for the effect time discretisation error, inspired by the results for diffusion samplers in Gritsaev et al. (2025).
The optimisation problems in (6) can then be solved with respect to the parameters both the drift and diffusion coefficients. We compare this approach to those that do not lear the variance in Table 1.
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3 DATA-FREE SCHRÖDINGER BRIDGES
3.1 IPF FOR TEEERYY SB
We now consider the setting where samples from pe are available, but p1 is given by an unnormalised density (x( = e (1(x) /Z and Z- f=fe (x) -- dx is unknown. In this case, the odd-numbered IPF steps (3a) can be performed (via (6a)), but the even-numbered steps (3b) cannot be done using (6b), as they require samples from P1- Instead, we need an objective that would fit P as forward-time SDE matching @ Inal without samples from pt.
In the time discretisation (5), the IPF step (3b) for Fp+l requires enforcing that for every xo. P (T xo) (T (x)) over all trajectories = (xo.Xar...... starting at xo-
To approximately enforce this proportionality, we introduce a source-conditional variant of the second-moment, or log-variance, loss used for training diffusion samplers (Richter al. (2020); see Berner et al. (2026) for an overview of related losses and their consistency properties in the continuous- time limit). The idea to minimise the variance of the log-ratio of the two sides of the proportionality over trajectories t sharing the same Xo:
Po(T xo) Liv(xo. 0) = Var log (x(x)) (T (T
Polxnai 1(K-1)N) +
E +
(7) where the variance is taken over some full-support distribution p pm (- xо) over trajectories. (The normalising constant Z does not affect the variance, nor do need to know the marginal of the process being learnt at time 0.) The empirical variance over a batch of N trajectories sampled from (r xx) is an unbiased estimate of (7) and can be used in the loss. (The training policy below takes N non-i.i.d trajectories, thus p train (. xo) can be distribution over batches of r.)
Var log
1=1 D (x xKS))
Because this proportionality must hold for every Xo, we must also average (7) over Xp (xp), where po
train some training distribution over xo. Thus the full objective for the forward IPF step is
Po(r() xо) (T 1x(1))
+ Eia
L(A) B xy 20-pp
mail (wo) Br .....,(N) ((N) piinn (-(o))
Var;
log
(8)
The choice of (xa) and p trann xo) is very important and we discuss it in the next subsection.
Comparison with diffusion samplers. The objective (7) is conditional variant of the log- varianee, or VarGrad, loss (Richter et al., 2020) previously used for diffusion srfuesio--based samplers of unnormalised
target densities. In that setting but not in ours, since the marginal of L) Fnntl time 0 is not po x|]
unless IPF has converged - the density of the process being learnt at time 0 known to be (x(x)). Thus (x() can be placed in the numerator of the loss in (7) and variance can be taken over both xo and r. An alternative objective would fit the density at xo. yielding a variant of the related trajectory balance (TB) loss for diffusion samplers (Malkin et al., 2022; Lahlou et al., 2023).
Yet another alternative would avoid IPF altogether and simply perform joint optimisation of both and & using a VarGrad- or TB-like objective to enforce that (o P (T Ax) = ( x1) over all trajectories Such bridge sampling approach is taken in Blessing et al. (2025a;b); Gritsaev et al. (2025). However, this approach would yield a bridge that is not necessarily the solution to the SB problem (1), since the KI. the reference process is not minimised. In future work, it would be inter- esting to compare our IPF approach with those that regularise bridge sampling losses by the cost (2).
3.2 OFF-POLICY TRAINING METHODS
The objective (8) leaves room for the choice of training distributions p train (xp) and p train xo), which can vary over the course of training. In this way, training with this loss is a form of off-policy reinforcement learning, a connection that has been elucidated and exploited for improved training of diffusion samplers in Sendera et al. (2024).
naive choice would take przin Po and p ir | (x)) = PT for complex, high-dimensional distributions on-policy training is insufficient, modes that not
[T] x²) (on-policy training). However,
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discovered by the sampler are very unlikely to be explored. To facilitate training adapt practices from diffusion sampling literature to guide the sampling process towards the areas of high density of the target distribution pt. In the following paragraphs, we discuss such techniques.
Replay buffer. We keep replay buffer of final samples x1 from the process 7 ( xo). During training, to obtain Xo, we sample x1 from the buffer, then sample a reverse trajectory to obtain xo for training: Xo p- xx). As the model trains and begins to better approximate pl. the buffer becomes populated with samples that are probable under p1. Thus, the buffer helps the sampler focus on the relevant regions of the space and retain information about previously discovered modes.
Reverse trajectories. To obtain the batch of trajectories r starting at Xo. we use both the reverse trajectory used to produce xp (see above) and a batch of N - trajectories drawn on-policy from P'o(T xo) to form a batch of trajectories sharing their initial point. The reuse of the backward trajectories allows the algorithm to learn on trajectories that reach high-density regions of P1- However, using only reverse trajectories prevents the model from sufficiently exploring the whole space. Therefore, careful tuning is needed to strike a perfect balance between exploration and exploitation. We use N = for all our experiments.
Langevin updates. Following the method of Sendera et al. (2024) for diffusion samplers, we periodically update the buffer using a few steps unadjusted Langevin on the density Pt to correct for the sampler's 's imperfect fit to the target.
Mixing on- on-policy and policy training. In the training policy, we use a mixture of initial points xo and trajectories sampled on-policy and those sampled using the langevin-updateer buffer with reverse trajectory reuse, as described above. The frequency using the buffer is called the off-policy ratio, and we ablate different off-policy ratios in Table 3. For most of the experiments use constant off- off-policy ratio 0.8.
We provide details of all off aff-policy methods in Appendix D, and these methods are ablated in $5.4.
3.3 ITERATIVE PROPORTIONAL. FITTING WITH DATA-TO-ENEAA STEPS
The full IPF algorithm for data-to-energy SB alternates between backward steps that train yp to convergence using the muximum-likelihood objective (6a) and forward steps that train @ to conver- gence using (8). The backward step trained using samples from po and forward trajectories from PT xo), while the forward step trained using trajectories obtained using the off-policy methods described above. The complete algorithm is summarised in Algorithm 2. We reuse the model weights and preserve the buffer state from the previous IPF step, although we randomly reinitialise a fraction of samples stored in buffer for the outsourced experiments.
Energy-to-eners generalisation. The data-to-energy IPF algorithm can easily be generalised to the case where samples from neither Po nor Pı are available, but both are given by unnormalised densities (x(x) = (x) /7, In this case, both the backward and forward IPF steps must be performed using the variance- hasee loss (7), with appropriate choices of training distributions (such as keeping separate replay buffers for both marginals). We call this energy-to-enens SB and show preliminary results in $5.2.
Evaluation metrics. We consider three metrics for evaluating the approximate solutions to the SB problem yielded by our data-to -neeryyy IPF algorithm. Because SB is constrained optimisation problem, is necessary to measure both the constraint satisfaction (i.e., that the solution is a transport from Po to P1) and the cost (divergence from the reference process). To this end measure ELBO, path KI., and Wasserstein distance to oracle samples from the target distribution (when available); Appendix B for details.
4 OUTSOURCED SAMPLING WITH SCHRÖDINGER BRIDGES
We describe how our algorithm for solving data-to-energy Schrödinger bridges can be applied the problem of Bayesian posterior sampling under a pretrained generative model prior p(x) by pulling the sampling problem back its latent space.

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Table I: Comparison of data-to-data IPF methods. Bold indicates the best-performing method.
Distributions
Gauss <>
GMM
Gauss
Two Moons
Two Moons <> GMM
Algorithm J Metric
Path KL (L)
())
Path KL (1)
WF(()
Path KL (1)
DSBM-IMF (Shi et al., 2023)
0.046
1.004-0117
0.061 =003
1.813-090
0,043+0012
1.893.0,00
DSBM-IMF+ (Shi et al., 2023)
0. 0.060
0.909=4.26
0.049
1.848=0285
0.038
18894
[SF]'M (Tong et al., 2024b)
2.041
2.295
0.053
1.32
0.057
2.823
IPF-b
DSB mean (De Bortoli et al., 2021)
0.093+00.06
5.886244400
0.1 0.111
6.398 9999
0.078
5.520a3.165
DSB score (De Bortoli et al., 2021)
0.052a0.018
5.645=2.474
=.171
14.34628.776
0,066220035
5.231
SDE (Chen
2021b)
0.037:a.010
2.088=1.238
0.033.00004
2.2622a0:268
0.025:0.01
3.915
I.L. fixed var. = Vargas et al. (2021)
0.037:00.01
2.507 00007.0.666 366
0.033
2.351
0.031
3.710
I.I. learnt var. (ours)
0.04
2.840-0668
0.022
4.288 1886
0.023.n0913
3.938.0.526
Consider posterior of the form p(x y) p(x)r(x, y), where p(x) is prior over the data space (e.g. images) and r(x. y) is a constraint function that encodes the conditional information about the sample x (e.g., class likelihood match to text prompt). If the pretrained generative model is expressed as a deterministic function f of a random noise variable ~ p(z), Venkatraman et al. (2025) proposed to sample the posterior pulled back to the noise space, with density p(z y) x p(z)r(f(zz) y), using a diffusion sampler. If z is distributed with this density, then samples / f(z) follow the desired posterior distribution p(x y) in data space. Such a method was successfully applied in the latent spaces of various models types, such as GANs, continuous normalising flows
Instead of using a diffusion sampler, we propose to model a Schrödinger bridge between the dis- tributions p(z) and p(z | y). Since neither the normalising constant nor samples from the latter distribution are available, we use the data-to-energy algorithm described in $3.1. Modelling a Schrödinger bridge instead of simply diffusion sampler has the advantage of transporting prior samples to nearby posterior samples in latent space, which is expected to preserve semantic content that is not constrained by y; we show this empirically in $5.3.
Metries for outsourced stochastic transport. In order to evaluate the performance of our method on the stochastic optimal transport task in the latent space, compute path KL in the latent space, as well as the L static transport cost between prior samples from (x())) and the pushed-forward samples from the approximated posterior (T xx) (o)x)) For image tasks with classifier reward, in order to evaluate the quality of generated images with respect to the target posterior, we use the mean log- cog-conntraint value and FID (Heusel et al., 2017). The latter is computed between images decoded from latents sampled from the trained SB model and images of the target class(es), which are not available to the model during training.
EXPERIMENTS
5.1 BENEFITS OF TRAINABLE VARIANCE
In order to show the benefits of trainable vari- ance in dineeiisretised processes, we present the comparisons of existing data-to-data SB
elergy
methods with our proposed algorithms, includ-
0.05
ing both learnt-variance and fixed-variance al- ternatives. The data-to-data experiments are conducted on several synthetic 2-dimensional benchmarks (following Shi al. (2023)): Gauss
004
000
0.0
0301
+ GMM, Gauss + Two Moons, Two Moons + GMM (where Gauss is an isotropic Gaus- sian and GMM is a mixture of eight Gaussians). We compare with Shi et al. (2023) (DSBM and DSBM++), De Bortoli et al. (2021) (DSB), which also uses IPF for training, Chen et al. (2021b) (SDE), which uses a continuous time version of IPF, and Tong et al. (2024b) ([SF)2M) which relies minibatch approximation to entropic optimal transport. All experiments use K 20 discretisation steps. The results in Table I clearly show the benefits of training the variances, despite the discrete-time processes not being consistent with underlying continuous-time process.
ot
Nancor of sege
Figure 2: 1W2 and Path KL depending on the num- ber of discretization steps for Gauss + GMM.
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We further investigate the effect of learnt variance varying numbers of discretisation steps. We find that learnt variance allows for more accurate modelling in both totta-tt-eeegyy and data-to-date settings when the number of steps is small. Results are shown in Fig. (in numerical form in Table 4).
For all 2-dimensional experiments use dX, d2,,, 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 detailed experiment configuration in Appendix D.
5.2 DATA-TO-ENERO AND ENERGY-TO-ENERGY SCHRÖDINGER BRIDGE f,
To prove the viability of data-to-energy Schrödinger bridge training we compare the bridge between Gaussian and GMM distributions trained using data-to-data and data-to-energy IPF
f
IPF step 7 IPF step
versions (where the data-to-energy version has no access to the data samples that the data-to-data algorithm sees). Fig. 3 shows the resulting IPF trajectories and Table 4 shows that the Lata-to-energy model is comparable to one trained with data samples.
IPT step 20
IPF step 20 IPT step 7 HPF step
5.3 OUTSOURCED SCHRÖDINGER BRIDGE
Finally, show the scalability of our method by running data-to-energy Schrödinger bridge algorithm in the latent space of generative model.
We use the StyleGAN (Karras et al., 2020; 2021) and SN-GAN (Miyato al., 2018) generators trained on CIFAR-10 (Krizhevsky, 2009), and VAE (Kingma Welling, 2014; Rezende et al., 2014) generator trained on MNIST (LeCun et al., 1998). We train the bridge model between the la- tent space prior p()). which in our case is always a Gaussian distribution, and a reward-reweiihhted prior of the form r y(( p(z). The reward function r(x. y) is classifier that returns the probability of the object belonging
step
IPT step
the class y (i.e., the probability that x is, for example, a boat for CIFAR-10 10 or the digit 5 for MNIST).
IPF step
5t
MNIST experiments are conducted in two setups: (a) the reward function returns the probability that x is even or odd (b) the reward function returns the probability that 5. For CIFAR-10 experiments we use StyleGAN (Karras et al., 2020; 2021) gen-
Figure 3: Comparison of learnt processes at vari- ous IPF iterations for data-to-data (top), data-to- energy (middle) and energy-to svergy-to-energe to-energy (bottom) settings. For the energy-to-enery setting we use two different mixtures of Gaussian distributions. erator with latent dimension 512 and SN-GAN (Miyato et al., 2018) generator with latent dimension 128. We use a pretrained classifier model as a reward function. Samples are shown in Fig. 4; more CIFAR-10 10 results can be found in Appendix
In Table 2 we compute FID between samples of the target class posterior obtained from the trained SB and images belonging to the target class the dataset; we use the same metric for a set of ground truth posterior samples obtained by rejection sampling. Remarkably, the transported samples tend to have lower FID than samples from the true distribution.
These results reveal benefit of training Schrödinger bridge model, as opposed to diffusion sampler or an arbitrary stochastic mapping. can be seen in Fig. that images already belonging to the target class change little, while those belonging other classes maintain features that unrelated to the target class: the background and global structure are preserved. This suggests that style transfer for higher-dimensiee images be a promising application of our method.
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SN-GAN
StyleGAN
Car
Cat
Dog
Horse
Horse
Truck
Prior
Posterior
Prior
Posterior
Prior
Posterior
Prior
Posterior
Prior
Posterior
Prior
Posterior
Prior
Fives
Prior
Even
Prior
Odd
9
9
5
5 6
6
8
7
0
7
7
4 0
4
8
6
7
7
5
9
7
9
7
0
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1
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5
9
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5
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0
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0
a
2
8
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5
à
9
5
3
9
3
5
5
2
2
0
6 4 2
0
0
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r
T
0
0
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o
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5 5 4
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f
6
9 9
9
5
Figure 4: Outsourced Schrödinger bridge on MNIST and CIFAR-10.
The bridge preserves style
features (thickness, background colour, orientation) while transforming digits the target class.
Table 2: ELBO (T) values
SN-GAN
StyleGAN
and FID (L) scores compar- ing CIFAR-10 samples from the posterior under a GAN
Algorithm J Class
Dog
Horse
Horse
Truck
Same class
10.371
17.950
15.034
12.876
12.876
9.289
prior and classifier to data samples of the target class. In this setting diffusion sampler performed worse than other methods, therefore, FID scores are computed only for "Dog' and 'Car' classes.
Rejection sampling
31.334
42.242
43.691
35.019
97.855
76.403
Langevin
25.296
33.619
37.665
27.601
85.611
68.678
Diffusion sampler
83.940
60.512
Outsourced SB
22.312
40.489
37.287
33.021
58.988
55.346
ELBO
Outsourced SB
-6.737
-9,356
-6.087
-5.961
-21.784
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Table 3: Ablation of off-policy reinforcement learning techniques on the SN-GAN outsourced sampling problem. Bold indicates the best result, underlined indicates second best. The model is trained amortise sampling from one class (dogs).
Algorithm J Metric
ELBO (1)
Path KL. ())
xx)x
mean log-reward (1)
on-policy
-190.920
1506.407
10.949
-0.233
buffer + Langevin
188.351
622.895
10.957
-0.125
-188.554
383.514
10.130
-0.286
+ reuse backward trajectory
-188.149
206.094
10.046
-0.657
annealed off-policy ratio
-188.355
244.270
11.386
-0.149
smaller off- oollpol ratio
-188.620
668.23
11.027
-0.131
5.4 ABLATIONS OF OFF-POLICY TECHNIQUES
In order to verify our design choices for the high-dimensional experiments we provide detailed ablation of the various off off-policy reinforcement learning tricks described in $3.2. The results are given in Table 3. We use a simple on-policy setup baseline. We find that saving samples in replay buffer (buffer) to use them for sampling backward trajectories and applying Langevin update to the buffer samples (buffer Langevin) significantly improves the Path KL metrics, therefore yielding a bridge closer to the optimal one. Moreover, we show that reusing backward trajectories for the computation of loss also improves Path KL, giving the best model based on this metric. However, reusing backward trajectories negatively impacts the mean log-reward metric, possibly because it prevents mode collapse. To balance between modelling modes and achieving low transport cost, we explore the possibilities of both setting a smaller off-ppolccy ratio fraction of off-policy trajectories and annealing the off-policy ratio throughout the training; the latter sometimes produces improvements. All ablation experiments are conducted using the SN-GAN generator and VGG 13 classifier on the CIFAR-10 dataset. All models are trained to amortise sampling from one class (Dogs) and are trained with the same seed.
5.5 COMPARISON WITH ANALYTICAL SOLUTION
To validate the proposed SB algorithm, we apply to the problem of finding the SB between two Gaussians. In this case, the problem has an analytical solution (Bunne et al., 2023). For our experiments, we train SB between N (). and N ([:). 3.0 1.2 )). We compare the data-to-energy and data- data SB algo- rithms described in this paper, as well as DSBM (De Bortoli et al., 2021), to the SB computed analyt- ically. For comparison, we use the Wasserstein-2 distance computed between samples obtained from the SB algorithm and analytical solution, for each time step of the trajectory. The results are in Fig. 5.
Figure 5: SB algorithms' comparison in W/ to the analytical solution.
1.2 1.0
6 CONCLUSION
This paper shows the potential of training data-to-energy Schrödinger bridges in time discretisation with learnable drift and variance for the forward and backward processes. We showed that, despite its complexity, our method can be successfully scaled. Future work should focus more scaling the data-to- data-to-energy Schrödinger bridges to higher dimensions, as well as arbitrary prior distributions, which would significantly improve the versatility of the proposed algorithms. Morcover, the trained samplers are prone to mode collapse, therefore. future work should investigate techniques to further improve mode coverage. One more promising direction would be to amortise over the distribution of conditions in image generation problems, instead of learning a model for each specific condition. Finally, we excited explore various domains in which the proposed algorithms can be applied. Interesting areas include text-conditional image reward fine-tuning of diffusion models and discrete Schrödinger bridge problems (recently explored in Carter al. (2026)).
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ACKNOWLEDGEMENTS
The authors thank Sanghyeok Choi for comments on the manuscript.
This work enabled by the computational resources of the Edinburgh International Data Facility (EIDF) with funding from the Edinburgh Generative Al Laboratory (GAIL).
NM acknowledges support from the CIFAR Learning in Machines and Brains programme.
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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 solid theoretical background and scalable applied algorithms. The problem is concerned with finding the optimal transportation map which minimises given transportation cost. Originally, the problem was proposed by Monge (Monge, 1781). ). In 20th century this problem was reformulated and generalised by Kantorovich in series of works including Kantorovich & Rubinshtein (1958); Kantorovich (1960). Since then the problem has been rigorously studied, refer to Peyré et al. (2019); Villani (2008) for detailed presentation of theory. In a discrete case optimal transport can be solved using Sinkhorn's algorithm (Sinkhorn, 1964; Peyré et al., 2019). Recent works have applied optimal transport to series of machine learning problems, including image-to-ing to-iaag translation (Korotin al., 2022), voice conversion (Asadulaev et al., 2024), and super-resolution (Gazdieva et al., 2025).
Schrödinger bridges. The Schrödinger bridge problem concerned with finding stochastic optimal transport dynamics between two distributions. The problem was originally proposed by Schrödinger (Schrödinger, 1931; 1932). The Schrödinger bridge problem can be seen as a regularised version of dynamic optimal transport (Léonard, 2014) and has interesting connections to optimal control theory (Chen et al., 2021c). Computationally, the problem can be solved using Iterative Proportional Fitting (IPF) algorithm (Fortet, 1940; Deming & Stephan, 1940; Sinkhorn, 1964). De Bortoli et al. (2021): Vargas et al. (2021) proposed the scalable formulation of this method that allows to compute Schrödinger bridge between pair of distributions given by unbiased samples. Chen al. (2021b) proposes a continuous-time variant of IPF. Methods distinct from IPF have also been proposed (Shi al., 2023; Tong al., 2024b); all of them assume access to samples from the target distribution for an unbiased objective.
Diffusion samplers. Data-to-energy Schrödinger bridge is related the problem of sampling from an unnormalised density. Diffusion samplers (Zhang & Chen, 2022; Vargas et al., 2023; Richter & Berner, 2024; Berner et al., 2024; Blessing et al., 2025a) represent one of the approaches that solve this problem. Some methods use off-policy reinforcement learning techniques (Lahlou et al., 2023; Sendera al., 2024) amortise sampling from intractable density. The theoretical connection among various objectives was established in Berner et al. (2026).
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 work shows that sampling can be efficiently conducted the latent space of generator, where the density landscape is smoother.
B METRICS FOR DATA-TO-ENT SB
First, if both samples from Po and the density of Po are available, we evaluate the quality of the learnt P. a as a sampler of #ı using the evidence lower bound:
(T (xTxx)exxp(EE(mm
ELBO = E
xp-polanp po log
log Z,
Po(T xo)Po(xo)
which equals the true log Z log (x) dv if and only if the processes P. o and T coincide.
Second, if samples from p1 are available (even not available to the learner during training), we report the 2- Wassersteii distance WW between batches of true samples from p (x1) and samples obtained from the learnt model Po xp), which measures the discrepancy between the target and modelled marginals.
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Third, to approximate the cost, compute the path KL in discrete time:
KL Fo Po @ 1o) KL (po(xo) @ p or xo) || po q( xxo
(9)
P (T xxo
No- Pu. po(
log
q(T xx))
(10)
K-1
B B Po- Po((:))
KL p (x xxx 11 xx))
(11)
where (T)) is the time discretisation of the reference process Qr. The estimator using transition KLs in (1I) can be seen to Rao-Blacde Rao-Blackwelllsedd (lower-variance) variant of the estimator (10), and we use it because the KL can be computed analytically (as all transition kernels are Gaussian). It can be shown (Appendix C) that this path KL equivalent to path energy as used in Shi al. (2023).
C RELATION BETWEEN PATH KL AND PATH ENERGY
In this section we explain the relation between path KL and path energy (Shi et al., 2023). Assuming that the transition kernels are given by:
( xx) = N (xL + volx,, kAr)Ar, x
(12)
(k xx) = N (xA.
(13)
where v) r) is learnt drift, 2² is constant and is the same for both Prjo and Orjo- time-discrete path KL can be written in the following form:
KL po(xx) @ Po(T xo) po(xo) @ q(T 1x0)
(14)
p xo)
K-1
Polxee 1xt)
Σ
= E x po.-poo-oo/aa
log
= E xx0-p.7--oott
log
(15)
q0 xx)
q(* xx)
k=O
ve (x kdr)
(X k2r)11 Ar)
B Po. -- (
(sR+) xk)
(16)
20
Given that X1+1 va(x kAr)& o NIEE, N(O. for 0 s k s K and the Ex, are independent, the path KI. can be finally written as:
KL (po(xo) @ (T xo) po(xo) @ q(( (x0
(17)
( k2r)ll M
I/ [I 1)]
= =B P0
Ar
Ar-0 20 m
20
(18)
The limit is justified by the Girsanov theorem (Särkkä & 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 to-a-t experiments are conducted under the unified setup. For the neural network we use an MLP with 3 hidden layers and 64 neurons in each layer; each layer is followed by LayerNorm(Ba. et al., 2016) and SiLU activation function. All neural networks are trained using AdamW optimiser with learning rate 0.0008. Sampling is done in 20 steps with mmx 0.2 and d: = 0.01. We train forward and backward models for 4000 steps at each IPF iteration and we train each model for 20 IPF iterations. M M is trained using 160, 000 steps. The metrics are computed using 10, 000 samples from the target distributions and 10, 000 samples obtained from the learnt forward process. All the metrics averaged over 5 seeds (42, 43, 44, 45, 46).
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D.2 2D DATA-TO-ENERGY EXPERIMENTS
For the data-to-energy experiments we use the same neural networks as for data-to-data experiments. We use 20 steps for sampling with /max = 0.8 and dr 0.04. Neural networks are optimised with AdamW optimiser with learning rate 0.0005, When Langevin update is used, we update buffer samples every 500 steps during training of the forward process. Langevin used with the step size 0.01 and we do 50 updates each time. We use 2 trajectories from each xo for the computation of the VarGrad loss. All the metries are averaged over 5 seeds (42, 43, 44, 45, 46).
D.3 2D ENERGY-TO-ENERG EXPERIMENTS
We provide a description of energy-to-eners experiment shown in Fig. 3. We use GMM with modes for the distribution Po and GMM with 8 modes for distribution p1- For both distributions we rely exclusively on the corresponding log-densities. We do not use samples from either po or Pt. We keep replay buffers for both densities. We initialise both replay buffers with Gaussian noise the beginning of training. Since samples are unavailable, we use the objective Equation (7) to learn the forward process and a similar objective:
(T x1)
K
( (kki) + Eo(xo) (19)
Liv(xi, Φ) = Var log (T and xn)p)()))
= Var
log Pxxesee (x(--1)s))
to learn the backward process.
D.4 OUTSOURCED SCHRÖDINGER BRIDGE EXPERIMENTS
Experiments on MNIST. For MNIST experiments, we use custom VAE with 3 layers in both the decoder and encoder. We use custom MNIST classifier as reward model, which consists of MLP layers. Each layer is followed by the ReLU activation function, except for the last one, which uses sigmoid. We train the forward and backward networks for 5, 000 steps during each IPF iteration, with 20 IPF iterations in total. All the networks are trained with Adam V optimiser using learning rate 0.0008. We do not use Langevin updates for this experiment, relying only on replay buffer. We use the same MLPs as 2D data-to-energy experiment to parameterise the backward and forward drift and variance.
Experiments on CIFAR-10 with SN-GAN and StyleGAN. For the CIFAR-10 experiments, we use MLP with 3 hidden layers, which has 256 hidden units for the SN-GAN experiments and 512 for the StyleGAN experiments. We train forward network for 500 steps and backward network for 100 steps during each IPF iteration, for total of 300 IPF iterations. All the networks trained with AdamW optimiser using learning rate 0.0005. Langevin updates are made every 500 iterations during the training of the forward network. We run Langevin for 500 steps with initial step size of 0.01 and anneal step size to 0.001 during the updates.
We use 20 steps for sampling with dr 0.04 for the StyleGAN experiments and 0,005 for SN-GAN experiments. We use Wiener process, dX, = v2dW, as the reference process. All the main experiments are conducted with replay buffer and Langevin updates, with off-policy ratio of 0.8, and the backward trajectories are reused for computing VarGrad loss.
For the reward model we use VGG (Simonyan & Zisserman, 2015) classifier pretrained on CIFAR-10 10. The weights are taken from https: : / / i thul. com/co/////ooooo use VGG-13 for SN-GAN experiments and VGG-19 for StyleGAN experiments.
We
For the rejection sampling (ground truth), the FID score is computed between 6, 000 images sampled proportional to the probabilities obtained from classifier and 6, 000 images from the CIFAR-10 dataset. For the outsourced SB the FID score is computed between 6, 000 samples from the learnt model and 6, 000 real CIFAR-10 samples. All scores computed only the images of a specific class. All other metrics (path KL, mean log-reward, Lxp. ELBO) are computed using a batch size of 512.
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Table SB metrics for varying the number of time discretisation steps in data-to-data and data-to-
energy setting with both learnt and fixed variance (Gauss + GMM).
Number of steps
5
10
K
20
40
Algorithm J Metric
W
Path KI.
W
Path KI.
W
Path KT.
Wi
Path KL.
var.
0.026-000s
4.s
3.127-098
0.042.nmxx
2.840-ams
3.070.0m0
Data-to-data fixed var.
0.066-am05
0,988=0m
0.038
1,825=00ms
0,037
2.507
0.023eamo
2.739
Data-to-energy learnt var
0.023-ame
41745-0
0.022-ams
3.162-030
0.028-om1
3.337-um7
0.020-amm
2.838-0000
Data-to-encrgy fixed
0.063
1.152-0
0,050
2.047=0.23
0.035
3.069
0.021=ama
3.360en
E ADDITIONAL RESULTS
In addition to Fig. 2 we also provide the metrics for the ablation in Table 4. We use the same architecture and hyperparameters as data-to-dad experiments and vary only the number of sampling steps. For the data-to- teta-tto-neryyy rums use replay buffer with Langevin updates, the off- off-policy ratio set to 0.8 and the backward trajectories are not reused for the loss computation. W2 is computed using 10, 000 ground truth samples and samples from 7 xx))p((x)). Path KL is also computed on 10, 000 samples.
VISUAL EXAMPLES FOR OUTSOURCED SB
F.1 UNCURATED EXAMPLES
Prior
Posterior
Figure 6: Uncurated examples of outsourced SB with SN-GAN for the class cars.
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Prior
Posterior
Figure 7: Uncurated examples of outsourced SB with SN-GAN for the class cats.
Prior
Posterior
Figure 8: Uncurated examples of outsourced SB with SN-GAN for the class dogs.
Prior
Posterior
Figure 9: Uncurated examples of outsourced SB with SN-GAN for the class horses.
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Published as a conference paper at ICLR 2026
Prior
Posterior
Figure 10: Uncurated examples of outsourced SB with StyleGAN for the class horses.
Prior
Posterior
Figure 11: Uncurated examples of outsourced SB with StyleGAN for the class trucks.
Posterior (Cats)
Posterior (Dogs)
Figure 12: Uncurated examples of outsourced SB with SN-GAN between classes cats and dogs.
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Published as a conference paper at ICLR 2026
Posterior (Cars)
Posterior (Trucks)
Figure 13: Uncurated examples of outsourced SB with SN-GAN between classes cars and trucks.
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Reviews:
[{"review_id": "iZb768uv3y", "reviewer": "Reviewer_kkJj", "summary": "The authors propose a modification of the Iterative Proportional Fitting (IPF) procedure to estimate a Schrödinger bridge between two distributions. Their approach does not require samples from either distribution, but instead relies on their unnormalized densities. At each step, they sample from the current approximation and add these samples to a replay buffer, similar to how it is done in reinforcement learning settings. They also explore the use of a learned diffusion scale coefficient in their framework and argue that this modification can improve any existing IPF algorithm. Finally, the authors demonstrate the performance of their proposed algorithm using synthetic toy datasets of Gaussian mixtures, as well as reward-driven energy estimations in the latent spaces of small generative models (GANs and Normalizing Flows), supporting potential applications to Style Transfer tasks.", "questions": "* Please try to examine and explain the findings regarding the FID discrepancy between Schrödinger bridge and rejection sampling in more detail.\n* The paper could benefit from including more challenging distribution pairs, even in the toy setting, such as the widely adopted moons and swiss roll datasets, as well as higher-dimensional Gaussian mixtures.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The idea is novel and natural, combining established concepts from Schrödinger bridges and entropy-regularized reinforcement learning. The choice of experiments to support the plausibility of the algorithm is adequate, though limited to toy examples.\n* The theoretical foundations are well established, with an extensive literature review and a thorough background section on stochastic optimal transport and diffusion-based sampling. The appendix, which explains the metrics, is also well written.\n* The paper is clearly written and easy to follow, although the experiment section is somewhat compressed in favor of the background overview.\n* The paper demonstrates how the method could be applied to Style Transfer, but its main significance, in my view, lies in the thorough examination of the mathematical foundations.", "weaknesses": "* There are many unexplored applications for this method that would be more interesting to see in the paper. In particular, areas where unnormalized density is the primary source of information and data samples are very limited or nearly nonexistent—such as transferring molecular structures where energy or reward is given by physical simulation, or transferring grasping poses between different robot embodiments given a 3D object description. These examples would be better testbeds for the method’s applicability than outsourced image generation, as in the latter case it is difficult to distinguish between failure modes of latent-to-image generators and the method itself. Additionally, the source distribution in image generation is typically chosen to be Gaussian, which is again too simple and narrow for a comprehensive experimental overview. The authors do, however, mention in the conclusion that such experiments could be explored in future work.\n* The FID discrepancy between rejection sampling and Schrödinger bridge samples is not well motivated and feels somewhat misleading, especially given the claim that the target distribution learned by the generative model is \"ground truth.\" In reality, this may simply be a poor approximation.\n* There are good analytical estimations in the literature for Schrödinger bridges between Gaussian mixtures, and for some reason, I did not find a measurement of the discrepancy between the learned bridge and the analytical approximation in these toy experiments.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HeLRhFLbVG", "reviewer": "Reviewer_Rvjv", "summary": "This paper proposes a generalization of the Iterative Proportional Fitting (IPF) procedure for Schrödinger Bridge (SB) estimation to settings where one or both marginals are given only by unnormalized densities (energy functions) rather than by data samples. The proposed data-to-energy IPF leverages a variance-based (VarGrad) objective previously used in diffusion samplers for unnormalized target densities. In addition, the paper shows that learning both drift and diffusion coefficients improves discretized SB optimization compared to drift-only models. Extensive experiments demonstrate the method on synthetic 2D datasets (Gaussian, GMM, Two-Moons) and on latent-space posterior sampling tasks (StyleGAN, SN-GAN, VAE). The approach is further extended to an energy-to-energy SB variant, demonstrating fully sample-free transport between energy-based densities.", "questions": "- In Fig. 3, why does the energy-to-energy experiment use a different data setup (GMM-to-GMM), whereas the data-to-data and data-to-energy experiments use Gaussian-to-GMM?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper introduces a generalization of IPF to the data-free setting, allowing SB learning even when only energy functions are available. This is an important theoretical and practical advance, broadening the applicability of SB to posterior inference and energy-based models.\n- The authors provide extensive ablations on trainable variance and other off-policy reinforcement learning techniques, such as replay buffers, Langevin updates, and off-policy ratio.\n- The inclusion of learnable variance meaningfully improves discrete SB optimization accuracy, especially at low discretization levels.\n- The latent-space posterior sampling experiments demonstrate the practical utility of the proposed method.", "weaknesses": "- The convergence guarantee of the proposed data-to-energy IPF (Eq. 7) is not established. While classical IPF has convergence results under certain regularity assumptions [1], it remains unclear whether those extend to the variance-based updates and off-policy sampling. A discussion of theoretical convergence would strengthen the contribution.\n- The paper lacks comparison with a closely related concurrent work, the Adjoint Schrödinger Bridge Sampler [2], which also addresses SB estimation when one marginal is given as an unnormalized density. A direct comparison or discussion of methodological differences would clarify novelty.\n- In Table 1, the Path KL metric is reported as “larger = better.” However, by definition, the SB objective minimizes KL (P ‖ Q), suggesting that smaller Path KL values should correspond to more optimal bridges. If the smaller Path KL is better, then the reported results appear to indicate a tradeoff: the trainable-variance model improves Wasserstein distance but worsens Path KL.  In this case, the justification for employing trainable variance under the tradeoff is required. \n\n[1] De Bortoli, Valentin, et al. \"Diffusion schrödinger bridge with applications to score-based generative modeling.\" NeurIPS 2019.    \n[2] Liu, Guan-Horng, et al. \"Adjoint Schr\\\" odinger Bridge Sampler.\" NeurIPS 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tRL58SklFR", "reviewer": "Reviewer_k5Aj", "summary": "This paper studies the problem of data-to-energy generation on synthetic datasets, aiming to learn mappings between data-driven and energy-based distributions. To achieve this, the authors employ an Iterative Proportional Fitting (IPF) procedure, which is essentially a variant of the Sinkhorn algorithm, to bridge the two distributions through alternating updates. The method is evaluated on several synthetic problems, including posterior sampling in the latent space of generative models, and compared with standard data-to-data approaches.", "questions": "- How exactly is Langevin noise incorporated into the proposed method? Is it applied only at the final iteration, or added throughout the IPF trajectory?\n\n- Does the method still converge or perform meaningfully without Langevin noise, or is it a necessary component for stability?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper presents an interesting conceptual perspective by framing the connection between data-driven and energy-driven distributions as a transport problem.\n\n- The manuscript is clearly written and easy to follow, with a straightforward presentation of the core algorithmic idea.", "weaknesses": "While the proposed idea is conceptually interesting, I have several concerns regarding its scalability, novelty, and experimental rigor.\n\n- **Computational inefficiency.** The proposed method employs standard IPF training, which is known to be computationally demanding. It requires simulating and storing both forward and backward trajectories, leading to significant memory and runtime costs. Moreover, due to the log-variance computation in Eq. (8), the algorithm must maintain multiple trajectories for each source sample $x_0\\sim p_0$, further amplifying the computational overhead. This raises serious questions about the method’s scalability to high-dimensional or large-scale settings.\n\n- **Limited comparison with related methods.** The paper does not include comparisons with existing and more scalable energy-to-energy [2] and data-to-energy [3] approaches. These frameworks perform only forward simulations, efficiently handle trajectory storage, and are generally better suited for practical applications. Although [3] may be considered concurrent work, it was publicly available nearly three months before the submission deadline and should at least be discussed for completeness.\n\n- **Questionable experimental design.** The experimental evaluation is restricted to self-curated synthetic datasets and lacks standardized benchmarks, making it difficult to assess general applicability.\n\n(1) On synthetic datasets, it would be more informative to include samplers capable of energy-to-energy or data-to-energy methods, as these are the most directly comparable baselines.\n\n(2) The posterior sampling from latent formulated as $r(f(x), y) p_0(x)$ appears somewhat artificial and lacks a clear grounding in established probabilistic models. In contrast, a more canonical formulation sampling from energy-tilted data distributions $e^{r(x)/\\alpha} p_{data}$ is well studied in the literature. For estimating $p_{data}$, the authors could have leveraged normalizing flows [4], continuous normalizing flows [5], or VAE/diffusion-based ELBO approximations that provide explicit or approximate log-likelihoods. Furthermore, comparison against rejection sampling seems too weak; many stronger training-based or training-free posterior sampling methods could serve as more relevant baselines.\n\n(3) Evaluating the approach on more established and quantitatively meaningful benchmark tasks is required. At a minimum, incorporating comparative analyses against existing samplers on standard sampling problems would provide a more rigorous and convincing validation of the proposed framework.\n\nOverall, the absence of experiments on well-established benchmarks and the lack of comparisons with efficient samplers substantially weaken the empirical claims.\n\n- **Limited methodological novelty.** From a theoretical standpoint, the proposed formulation largely adapts the framework of [1] to the case where the energy function is defined on only one marginal. While this adaptation is logically consistent, it does not appear to introduce a fundamentally new methodological contribution beyond prior work.\n\nReferences\n\n[1] From Discrete-Time Policies to Continuous-Time Diffusion Samplers.\n\n[2] Sequential Controlled Langevin Dynamics.\n\n[3] Adjoint Schrödinger Bridge Sampler.\n\n[4] Glow: Generative Flow with Invertible 1×1 Convolutions.\n\n[5] Normalizing Flows are Capable Generative Models.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "KJaLyEH9M5", "reviewer": "Reviewer_oxUm", "summary": "This paper proposes a method for modeling SB when distributions are unnormalized, without accessing samples. This is generally called \"data-free,\" and the proposed data-to-energy IPF procedure attempts to successfully learn transportation between multimodal distributions. The authors validate the approach in various circumstances, including high-dimensional image-to-image translation.", "questions": "* I assume that the training time and memory requirement of Data-to-Energy IPF in synthetic (<50 dim) settings are good, but that it will suffer from the curse of dimensionality for image data (>1000 dim). Could you please give us the training time and memory requirements of standard and Data-to-Energy IPF procedures for various dimensions?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "* This method is (partially) data-free using an energy-based interpretation; thus, it is likely to find considerable applications in restricted domains.\n* The proposed scheme (Eqs. (7-8)) appears to be sound.\n* This method can be efficiently outsourced, reaching sufficient scalability for image-to-image translation by using a pretrained latent model (possibly GANs).", "weaknesses": "* Limited scalability. The promise of scalability heavily relies on projected and cleaned modalities of embeddings. In other words, the model cannot be a head-first approach to solve an unknown problem, since this requires an unconstrained pretrained generative model.\n* Limited comparison. Meaningful comparison studies are only present in synthetic datasets, with mixed results. I can find results from the references indicating that DSB variants show exceptional FID scores, often surpassing those of GANs. The authors are encouraged to report these scores, or acknowledge that their outsourced techniques share limitations on performance with pretrained GANs.\n* Contribution. I understand the proposed method as a better-executed IPF with enhanced alignment with the model space. That being said, the manuscript does not successfully demonstrate the benefits of data-to-energy (or energy-to-energy) IPF, and the suggested numerical results contain a lot of overlapping regions, limiting the significance of the methodology.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)