RNE: plug-and-play diffusion inference-time control and energy-based training
Authors: Jiajun He
José Miguel Hernández-Lobato
Yuanqi Du
Francisco Vargas
Venue: ICLR 2026 Poster (2026)
Keywords: diffusion generative models
SDE
SMC
sequential monte carlo
Primary area: generative models

Abstract: Diffusion models generate data by removing noise gradually, which corresponds to the time-reversal of a noising process.
However, access to only the denoising kernels is often insufficient.
In many applications, we need the knowledge of the marginal densities along the generation trajectory, which enables tasks such as inference-time control.
To address this gap, in this paper, we introduce the Radon-Nikodym Estimator (RNE).
Based on the concept of the density ratio between path distributions, it reveals a fundamental connection between marginal densities and transition kernels, providing a flexible plug-and-play framework that unifies (1) diffusion density estimation, (2) inference-time control, and (3) energy-based diffusion training under a single perspective.
Experiments demonstrated that RNE delivers strong results in inference-time control applications, such as annealing and model composition, with promising inference-time scaling performance, and achieves simple yet efficient regularisation for training energy-based diffusion models.
Additionally, our proposed RNE is modality-agnostic and applicable not only to continuous diffusion models but also to their discrete diffusion counterparts.

Published as a conference paper at ICLR 2026
RNE: PLUG-AND-PLAY DIFFUSION INFERENCE-TIME CONTROL AND ENERGY-BASED TRAINING
Jiajun He
José Miguel Hernández-Lobato University of Cambridge jmh2 3 30cmm. ac uu
University of Cambridge jh23830 cam ac uk
Yuanqi Du!
Francisco Vargas!
Cornell University
Xaira Therapeutics vargfrane gmail com
yuangiduecs.com sonngiisuess.coeee..... wow.ell.......oooo
ABSTRACT
Diffusion models generate data by removing noise gradually, which corresponds the time-reversal of noising process. However, access only the denoising kernels is often insufficient. In many applications, we need the knowledge of the marginal densities along the generation trajectory, which enables tasks such as inference-time control. To address this gap, in this paper, we introduce the RADON- NIKODYM ESTIMATOR (RNE). Based on the concept of the density ratio between path distributions, it reveals a fundamental connection between marginal densities and transition kernels, providing a flexible plug-and-plas framework that unifies (I) diffusion density estimation, (2) inference-time control, and (3) energy-basee diffusion training under a single perspective. Experiments demonstrate that RNE delivers strong results in inference-time control applications, such annealing and model composition, with promising inference-tin timee scaling performance, and achieves simple yet efficient regularisation for training energy-based diffusion models. Additionally, our proposed RNE is modality-agnostic and applicable not only continuous diffusion models but also to their discrete diffusion counterparts.
INTRODUCTION AND BACKGROUND
Diffusion models (Ho et al., 2020; Song et al., 2021 are class of flexible generative models that excel in generating high-quality samples from complex data distributions, and have had broad impact across a wide range of applications from image (Rombach et al., 2022; Karras et al., 2022), video (Ho et al., 2022) and text (Austin et al., 2021) generation, designing novel proteins (Watson et al., 2023), materials (Zeni et al., 2025) and capturing transient structures in chemical reactions (Duan et al., 2023). Diffusion models rely on a pair of forward and backwards stochastic differential equations (Eqs. (1) and (2)) to transport between data distribution and a tractable prior distribution, commonly selected as Gaussian. They then parametrise the score function V log Pt with a time-dependent network, and when trained to optimality, Eqs. (1) and (2) will be the time-reversal of each other. This allows us to generate high-quality samples by simulating the backward SDE in Eq. (2) starting from Gaussian. Equivalent diffusion models is the more flexible stochastic interpolants (A (Albergo et al., 2023; Ma et al., 2024; Gao et al., 2025) parametrisation of diffusion models (Eqs. (3) and (4)).
DM Characterisation Fixed 6;
SI Characterisation Ve,
1 2
ax, fiX.jat e,dW? ax (J(()) - (X.)) di s,åw,
(I)
v=- 2 Vloge
d.x, -
v.(X)) + (lokp((X)) di G.W,
(3)
V1,00
(2)
d...
v.(XX) 4 (lomm(()) di T.
(4)
In Eq. (1), fa typically chosen a linear function. For example, fe(ax) - x for vaiance- preserving process (Ho et al., 2020; Song et al., 2021a) with an extra hyperparameter 3r, or ft = 0 for variance-exploding process (Song al., 2021a; Karras et al., 2022). Note that one can always
I Last authors.

---

Published as a conference paper at ICLR 2026
recover the classical DM characterisation from its SI perspective by setting e <- oz. This unifying parametrisation allows us to establish theoretical connections to existing work and extend our methodology to models trained via flow matching (Albergo et al., 2023; Lipman et al., 2022).
Strietly generating samples that resemble the overall data distribution may lack practical applications. Fortunately, the progressive generation process of diffusion models naturally allows us to apply more flexible probabilistic inference, unlocking a variety of approaches and applications. For instance, in diffusion posterior sampling and inference-time steering (Dhariwal & Nichol, 2021; Ho & Salimans, 2022; Song et al., 2023a; Chung al., 2023; Song et al., 2023b; Trippe al., 2023; Rozet et al., 2024; Schneuing et al., 2024; Kong et al., 2025), the goal is to generate samples that satisfy specific constraints or exhibit desired attributes. In diffusion model composition (Liu et al., 2022: Du et al., 2023; Ajay et al., 2023; Biggs et al., 2024; Skreta et al., 2024; Thornton et al., 2025), multiple diffusion models are combined to produce samples with richer attributes. Also, in sampling tasks, diffusion models can be used to accelerate standard algorithms such as annealed importance sampling or parallel tempering (Doucet al., 2022; Chen et al., 2024; Zhang et al., 2025).
While heuristic methods such as guidance due to ad hoc design choices. By contrast, probabilistic inference techniques offer a principled approach to climinating such bias and can lead more reliable performance. central requirement for applying these techniques is to evaluate or approximate the sample density under pretrained diffusion model along the generation trajectory. One classic approach reformulates the diffusion process as a probability flow ODE (PF-ODE, Song et al., 2021b) and applies the instantaneous change-of-variables formula (Chen et al., 2018); however, this is computationally prohibitive, as requires calculating the divergence of the score network at every denoising step. To address this difficulty, some works have developed sequential Monte Carlo (SMC) algorithms based on twisting functions Feynman Kac formulations, which bypass the need for explicit density evaluation (Wu ct al., 2023; Skreta et al., 2025; Singhal et al., 2025). Other approaches introduce diffusion density estimators leveraging the Feynman- Kac formula or Itô's lemma (Huang al., 2021; Premkumar, 2024; Karczewski et al., 2024; Skreta et al., 2024). Alternatively, one can directly train energy- parametrised diffusion models (Du et al., 2023; Phillips al., 2024; Thornton et al., 2025; Zhang et al., 2025), which provide explicit access to the unnormalised marginals along the process.
be effective for these tasks, they often introduce bias
Our contributions. Despite the above advances, these methods remain disparate in their scope. The connections between these approaches remain unclear, and many depend on specialised designs, ESTIMA-
which can limit their applicability. In this paper, we close this gap with TOR (RNE), unified, flexible, and plug-and-play framework that enables density estimation, SMC weight computation for inference-time control, and better training of energy-based diffusion.
For inference-time control, RNE can compute SMC weights for any sampling process without re-deriving the formula. enabling a wide variety of options, such as Chung et al. (2023), Song et al. (2023b), and Singhal et al. (2025). This opens up broader design spaces and offers better inference-time scaling performances. For energy-based training. RNE yields a simple yet effective regulariser, significantly improving the learned energy with negligible computational overhead.
RNE generalises and unifies a wide range of established methods such as the twisted diffusion sampler (Wu et al., 2023), Feynman-Kac steering (Singhal et al., 2025). Feynman-Kac corrector (Skreta et al., 2025), guidance corrector (Lee et al., 2025), Itô density estimator (Karczewski et al., 2024; Skreta et al., 2024), Feynman Peynman-Kac density estimator (Huang et al., 2021; Premkumar, 2024), and Bookker-Plancle regulariser (Plainer et al., 2025) that may appear distinct at first glance.
RNE is not restricted to Gaussian diffusion. It applies broadly to any generative model that admits a pair of dynamics that are rime-reversal. This includes stochastic interpolants and bridge models (Shi et al., 2023; Peluchetti, 2023; Albergo et al., 2023), as well as more processes in other modalities such as continuous-time Markov chains (CTMC, Lou et al., 2023; Shi et al., 2024).
2 METHODS
In many applications of diffusion models, including inference-time steering or model composition, we need access to the marginal density Pt at time step t of the diffusion process. Unfortunately, this is generally intractable for score- based diffusion model. Instead, in most cases, it is easy to access the transition kernels (c.g., denoising or noising kernels) of the diffusion model. Therefore, a natural question is: can we connect the transition kernely of an SDE with its marginal densities?
2

---

Published as a conference paper at ICLR 2026
time-reversl
resunppl
and
ACXK in B6g (11)
Pu
denolning dirertion
(a) Gadon-Nikodyy Estimator (RNE).
(b) Inference- time control with RNC.
Fig 1: Conceptual illustration of our proposed approach. (a) RNE leverages the fact that RND between time-reversal processes is 1 to calculate marginal densities. (b) RNC applies RNE to calculate importance weights for inference time control.
Before considering diffusion models in the form of Eqs. (1) and (2), it is helpful to consider their discrete counterparts. The transition kernels in discrete time are defined by the conditional densities Pain+1 and Pn+ Therefore, the question are sceking to answer is: densities with marginal densities? In fact, this is precisely what Bayes' rule states:
connect conditional
(5) Under a Bayesian interpretation, the forward and backward transition kernels play the roles of the likelihood and the Bayesian posterior. Ordinarily, one's goal is infer the posterior, but here-assume both kernels are available-we can directly form their ratio to compute the ratio of marginals.
Pu|n+11 (X,, X ((((+))))))) = Po- (Xn+1 (X.), v(Xn, Xat1)
similar conclusion also exists in continuous time through the concept of time-reversal. Specifically, considering the SDE evolving from p, to py
dx, pt (X,) + GdW, ~ Pr.
(6)
given regularity on μr, one can define its time-reversal (Anderson, 1982; Nelson, 1967):
dX, (x.))dt + e,ddW,
X Pr'
where v = 14 - E log Pt. and pr is the law for Xt. The forward and backward diffusion processes in Eqs. (1) and (2) (or equivalently Eqs. (3) and (4)) exemplify this time-reversal pairing.
key observation is that while the processes in Eqs. (6) and (7) evolve opposite directions, they induce the same probability measure over the path space. Therefore, their Radon Nikoo-N derivative (informally, the "density ratio") is always 1. Let PM and i be the path measures of Eqs. (6) and (7) respectively. We have dFz/dP (Y,,rr)) 1, where Y|r.-1 is the solution any Itô process within the time-horizon [T, T'), with diffusion coefficient 6. The expression is merely the definition of time reversal. However, when discretising the SDEs with, e.g., the Fuler-Maruyama integrator, we easily how this definition connects marginals with transition kemels.. Concretely, first split the time horizon with N time steps = t < to <... < IN = r'. Then at each step, with Atn [6++1 Ln/. the forward and backward processes are defined
Phtiin (Ximer (x,,) = N(( 1Xe, + Mr, (Xi,) At o7 At,!), P (X 1) = N (XemlXen11 Mh.. (X 1 Al Ei Ain!).
(8)
(9)
After discretising, dF dFH/dtP
= becomes P=(Y-) MI Ph ( (Yn))
Ro 1.
OC) IIAT Ph + (Y,,, (
This approximation exact as N Formally, we define the quantity as follows:
Definition 2.1. Consider the forward and backward SDEs in Eqs. (6) and (7). Let Y be the solution to an
arbitrary process with the same diffusion coefficient. Following Eqs. (8) and (9)'s discretisation, we define
I-==11 Puin+ (Y,,, (X,,))
(Yr.r))) = lim
(10)
N
Potiin (Y,, 11 (Y..)
3

---

Published as a conference paper at ICLR 2026
With this definition, we obtain the following identity for the μ and processes satisfying time-reversal.
Pr P+(Y+)/p= R2 (r,r1))
(11)
We note that the limit in Eq. (10) can be formalised (Berner et al., 2025) thus Rij can be expressed as
((rr))))))) exp (L-
1
aY. L. 7h -..dY. + 2 L-
1 ? (11.a1l=-1u.1))))))
7" t dY,
(12)
For more detailed discussion on this expression, we defer to Vargas et al. (2023b, eq. 14-15). For readers not familiar with Itō integrals, we highlight that our approach can be fully understood and implemented as Eq. (10) with finite N and simple Gaussian kernels. That said, the connection to its continuous-time time counterpart will enable us to design better estimators.
In summary, as conceptually shown in Fig. la, for the denoising process of a pretrained diffusion model, we can always pair with its time-reversal, up to training error. By exploiting the fact that the Radon-Nikodym derivative between any diffusion process and its time-reversal is identically one, obtain a simple and intuitive formula that ties together marginal densities and transition kernels. We call this identity the RADON-NIKODYM ESTIMATOR (RNE).
direct application of this relation is for density estimation: Gaussian distribution. Interestingly, when writing R in continuous-time as Eq. (12), we will recover the estimator with density-augmented SDE (Karczewski et al., 2024, Theorem 1), and equivalently, in their concurrent work, Itō density estimator (Skreta al., 2024, Theorem 1). We will discuss this connection and application in more detail in Appendix C.I. In the following sections, we will mainly focus on demonstrating RNE for inference-time time control and energy- based training.
when 7' = 1,
pr is tractable, typically as
2.1
RNE FOR INFERANCE-TIME CONTROL
RNE provides a flexible and plug-and and-play approach for calculating the weight of Sequential Monte Carlo (SMC) for inference-time control. Given a pretrained diffusion model which samples from distribution Po (or two pretrained diffusion models for Ph and ))) we may want to generate samples from a new target qu without retraining the model(s). This includes, but not limited to: (1) annealing: 3. 4o po: (2) reward- swward-illtingppsseeror sampling: do Po exp(r) with reward/likelihodd and (3) classifier-free guidance (a = 1-3)
3) or model product (a = B): qu x (P6 (())((())))
One naive approach to generate samples from do is importance sampling (IS). Specifically, we can estimate the density po of generated samples from the pretrained diffusion model, and then calculate the importance weight as go/po, Po. However, when po and go differ significantly, the importance weight will have a large variance, rendering this approach infeasible in practice. Therefore, we consider applying importance resampling along the sampling process, essentially forming the Sequential Monte Carlo (SMC) algorithm. In the following, we first describe SMC, and then introduce RNE to calculate the importance weights, which we refer as the Radon-Nikodym Corrector (RNC).
2.1.1 SEQUENTIAL MONTE CARLO
Conceptually, SMC distributes the burden of importance sampling across the entire path, thereby reducing variance at each step. To apply SMC in this setting, we first define a sequence of intermediate target distributions corresponding to each time step. Next, we specify a proposal process from which samples are drawn. Since particles generated by the proposal may not align perfectly with the intermediate targets, we apply importance resampling to progressively realign the particles with the intended sequence of targets. To apply this procedure, we introduce three key components:
1. backward sampling ("proposal"") process: dX, = (X.) di + e, dW:
(13)
a forward auxiliary ("target") process: dY, = by bx(Y))tt + e.dW;
(14)
3. intermediate target marginal densities: Tt-
Note that the sampling and target processes are not the same; they are generally not required to be rive-reversals. We hence denote the process Eq. (13) by X and that in Eq. (14) by Y. In fact, we have a large flexibility in designing these components. We will discuss the choices of the backward and forward processes in Section 1.3.3. For the intermediate marginal, we can heuristically choose:
8. anneal:g, pe: reward-tilting:g & Pt exp(rr); CFG & product:q, (pl (pl )))))) (p(
(15)
'The limit in Eq. (10) can be understood in almost sure sense.
4

---

Published as a conference paper at ICLR 2026
where Ty is intermediate reward which can be heuristically crafted (Wu et al., 2023), or be reward model trained on the corresponding noisy data (Kong et al., 2025).
We now consider how to apply these components. Assume we have M particles {.x[m)) ~ qr at time r', now we consider how to obtain particles following !r at time 7 (T < r²). We first evolve the particles along the backward sampling process in Eq. (13) to time step resulting in M trajectories (xr(mr) (o,1) )- However, these trajectories will not follow the marginal density 9- at time 7. Therefore, we need to resample to ensure asymptotically unbiased samples from gr, as explained in the following.
Let ' be the path measure of Eq. (13) in & E [T,7]] with initial density at T' as gr', and let Qo be the path measure of Eq. (14) in t [T, with initial density at T as 9r, Here we slightly abuse the notion for simplicity: we should understand T as [r,r'1 1 the path measure starting from gr at time 7' and ends at time т, and also understand @ similarly. The importance weight of [[rrr1 is then
"|r (X (n.rg) = (X(x((()))))) = qr (x+))/- [R] (X [n))
(16)
where Rg are defined in Eq. (10). Note that while Eq. (16) calculates the weight over path space, we verify that this yields correct importance weight for the marginal gr. as shown in Appendix H.1.
We then perform self- normalised importance resampling: first, we normalise the importance weights " (x(m)
(m)
and sample M indices from the Categorical distribution defined with (rae))
E Whr ( (
these weights (mm) } ~ Categorical (w ................................ (n())). We return the particles corresponding to the resampled indices. This ensures the samples at time r follow the desired target 9, as M + co. We repeat this pipeline until reaching go and this process is conceptually illustrated in Fig. 1b.
(
2.1.2 CALCULATING IMPORTANCE WEIGHTS WITH RNC
Now, we consider how calculate the importance weight in Eq. (16). The term RG is simply a ratio of products of Gaussian densities when discretised, the only unknown term is the ratio between two
marginals 4- (m) ) / (x(")). Fortunately, as we define the intermediate target qe by modifying the marginal pt of the pre-trained diffusion model as exemplified in Eq. (15), we can express this unknown ratio using the pre-trained model's 's marginals. Precisely, plugging in the RNE in Eq. (10):
pr (Y=)/ Ppr (Yrr) (Yr..))) we obtain the following results:
RN Corrector (RNC). Consider pair of tüme-reversal forward and backward SDEs in Eqs. (6) and (7) with drifts fr and vs (or two pairs: (a (t) and y( & (2))). The SMC weight in Eq. (16) is given by
Anneal:
W|rse'1 oc (X [R]] (X.r.)))]
(17)
exp(r: (X,))
Reward:
W x exp(rr (Xrr)) (X(rr)) ((
(18)
CFG Product:
Wypse/1 x Riyl)) (Xpw)))
Re (X (X
[R (X [)
(19)
In summary, when performing SMC with RNC, we start from a pair of time-reversal forward and backward processes, which provides an estimate for the marginal density ratio. We then choose the sampling process, target process and intermediate marginal defined in Eqs. (13) to (15), and calculate the SMC weight as above. Importantly, all components in the importance weight can be approximated by Gaussian kernels 2, Note that, in practice, we only need calculate log w for each denoising step with negligible computation cost.
2.1.3 DESIGN CHOICES OF THE SAMPLING AND TARGET PROCESS
Notably, the RN Corrector works for any choice of drifts a and b, in Eqs. (13) and (14). We now examine two specific scenarios for designing the sampling and target processes:
In the first scenario, suppose we have access to the perfect diffusion mhael-thatt is, we know the exact forms of both 14, and v4, and can therefore evaluate the forward and backward kernels p
As concrete example, let's inspect the anneal IS weights in Eq. (17): B
N-1 Phin+ (Xe, (Xe+1))
NIN- Phttln (Xentz (Xen)
(20)
Pottin Xe,,t (Xea)
I P (Xem (Xentz)
5

---

Published as a conference paper at ICLR 2026
and Pi]n+1 as N oc. In this setting, we are free to choose any sampling and target processes, and this formulation supports flexible applications including annealing, reward-tilting product.
if the diffusion model is imperfectly trained, we only have access the denoising drift is parame- terised by the imperfect score network'. Hence, we no longer get access to its time-reversal term with drift pt. In this case, we can set the target process's drift b, cancel this unknown term. However, this formulation is limited to reward- reward-tilting as we will explain later.
Perfect diffusion model: flexible design choices Assume perfect diffusion model where the noising and denoising process with forward and backward drift jr and 14 are time-reversals. In this case, we enjoy great flexibility in choosing the sampling and target process. The only approximation error then arises from the time discretisation, which disappears as the discretisation steps N -> oc.
In Appendix C.3, we list some heuristic choices of the sampling and target processes for anneal, reward-tilting and produce cases. Note that they are not exhaustive-any suitable heuristics can be used without altering the core SMC algorithm implementation. We present an example pseudocode in Appendix A to highlight this Macro-like property of RNC.
Interestingly, the expressions in Eqs. (17) to (19) recover FKC (Skreta et al., 2025) as special cases for certain choices of a and 6. We discuss this connection in Appendix C.4. FKC derives its weights via the Feynman-Kac PDE and then designs the sampling process to cancel the costly divergence term. Therefore, FKC has restrictive design choices. By contrast, our RNC features higher flexibility in selecting these processes, yet still incurs extra computational overhead, allowing to heuristically select a process pair that may reduce variance (Jarzynski, 1997; Neal, 2001). Moreover, FKC requires deriving the weight formula for each task (anneal, product, etc), while RNC provides macro-style mplug-and-play recipe for computing importance weights.
Imperfect diffusion model: choice for cancellation (reward-tilting) So far, we assumed a perfect diffusion model, which gives us access to an SDE and its reversal. However, in practice, we typically encounter model imperfection and -iscrretetiitnnnn errors when calculating the marginal density ratio Eq. (11). Consequently, the resampled X, will not follow y- exactly, even as the number of samples M - oc. Fortunately, the reward-tilting case, we can still obtain exact importance weights, despite the discretisation and score estimation errors:
Proposition 2.2 (Exact SMC weight for reward-tilting with imperfect diffusion model).
exp(r= I Palm+1 (Xt,, (Xen41)
(21)
exp(r> I Palm+1 (Xt, (Xen4:)
We provide a detailed derivation and explain why it is only applicable to reward-tilting in Ap- pendix C.5. This formulation recovers the Twisted Diffusion Sampler (TDS, Wu et al., 2023, e1), and follow-up works such as Dou & Song (2024) and Feynman-Kac Steering (Singhal et al., 2025).
In summary, RNC allows to compute the importance-samplng weights for SMC without requiring explicit knowledge of the marginal density, thereby providing great flexibility in designing inference- time control while maintaining plug-and-play algorithm.
2.2 RNE FOR REGULARISING ENERGY-BASED DIFFUSION MODEL
Another application of RNE is to improve the training of energy- based diffusion models. These models have variety of applications in machine-learing force fields (Arts et al., 2023), free-energy estimation (Máté et al., 2024), neural sampler (Phillips et al., 2024; Zhang et al., 2025) and model composition (Du al., 2023), among others. Concretely, we aim train a diffusion model whose network outputs a scalar energy. However, the denoising score matching objective (Vincent, 2011) suffers from "blindness" issue (Zhang al., 2022), leading to inaccurate energy estimates.
RNE offers a natural way to enhance the accuracy of the energy-based diffusion model. Specifically, in addition to the standard DSM, we introduce the following regularisation to enforce Eq. (1I).
R Eag(X sg (X((())))) + log p. (X+') log Pr (X+))l
(22)
'The normal perspective is that we define the noising process but do not know the perfect reversal process. However, we can also interpret this case we know the denoising pracess defined via the learned network, while not having access its exact time-reversal along the noising direction.

---

Published as a conference paper at ICLR 2026
where log py and log (X+, are given by the energy parrgy-arrettriseedd diffusion model and [T, r'] is randomly selected time horizon. sq represents stop- sop-gradient. In practice, we can select a small time increment At, and apply this regularisation between randomly selected adjacent time steps t and t + At. We then calculate Ry; using a single forward and a single backward kernel. As discussed in Appendix E. and proved in Appendix H.5, this regularisation is equivalent the regularisation derived from the Fokker-Planck equation in continuous time (Plainer et al., 2025). However, our approach does not require computing or estimating the divergence, providing more efficient alternative.
2.3 RNE FOR CTMC
RNE conceptually only requires pair of dynamics that are time reversals of each other; hence it can be applied to other modalities, such as continuous-time Markov chains (CTMCs). All the results discussed above remain valid; the only difference is that R is now defined in terms of the rate matrices. We provide further details on CTMC-RNE in Appendix D.
3 STABILISING RNE WITH REFERENCE AND CONVERGENCE ANALYSIS
So far, we define Eq. (10), and apply this concept for inference time control and energy-based training. However. in practice, calculating R by directly discretising the forward and backwards SDE as in Eq. (10) can lead to instability and larger accumulated error. We provide an intuition behind this instability in Appendix G. 1. At a high level, this issue arises because, at each discretisation step, the variances of the forward and backward kernels are misaligned.
To address this issue, we introduce an analytical reference (Vargas et al., 2023a): consider an SDE with linear drift é whose initial state To is Gaussian, In this case, the marginal density मe remains Gaussian at all times, and one can derive the exact time-reversal drift vh Øz - - log n.. Let oi and : be the path measures of this analytical pair. We can rewrite Eq. (10) as follows:
(X+r)) = P+ dF
pr p. (Y7') dPP (x = (Y+) dp 3
di
PY,(Y,)) dui (M
dPu (X (XT,,1))
(23)
Phin+1 (Yr, (
II Pu+iln (Yengs (Y,))
(24)
P (Yr,,) (
II Pn+1ja (Y,,,1 (V,..)
By introducing the reference process ΠT, we obtain the Radon-Nikodym derivative path measures along the same direction, ensuring the variance of transition kernels is aligned after discretisation. In this work, we choose No to be Gaussian for simplicity. However, is not the only option-we can also use a Gaussian mixture adaptive to data. Using reference similar to the data distribution may offer more accurate results, as observed by Noble et al. (2024) in the context of neural samplers. Additionally, we highlight that the reference process only involves calculating Gaussian kernels without any extra network evaluation, and hence has almost no computational overhead in practice.
We also note that direct Buler-Maruyant discretisation (as described in Eq. (57) in the appendix) of the continuous RNE in Eq. (12) does not have such instabilities, providing a competitive practical alternative. We include a detailed discussion in Appendix G.2. However, using our proposed reference still offers more accurate results, which we empirically verify in Fig. 18 in Appendix G.2.
We now present our reference-based RNE's s convergence rate. Let's consider the case where we use the diffusion model defined in Eqs. (1) and (2), where μ4 = fi is a linear function, and 1/2 = 14 uu-ofV V log Pt is the backward drift. We choose a reference process whose forward drift Φb f4, and the initial marginal to be Gaussian. Let's denote the drift of this time-reversal as the = #4 - log m. In this case, have the following non-asymptotic guarantee:
Proposition 3.1. Consider the case where μ4 is linear forward drift. as in diffusion models. We choose an analytic reference by setting its drift as pt and To to be Gaussian, and denote its time-reversal drift as 1/t- Assuming Y has bounded I.P moments, follows that:
log R2 (Y(r.))) - log R (Krrii)lzaa - (VVAt)
(25)
11 denotes the L norm, Y is the Euler-Maruyame, discretisation Y,,, and log R.N (P(r.rg) is our x+()+) 11 P" Ghint (8s,, (8 IT-= pi Patlla (P (
discretised RNE estimator RN (Yp.rt) (Y, () I PM Painz (Y,, (oae)) ) II Po (Y Vr,.""
7

---

Published as a conference paper at ICLR 2026
Tab 1: Inference-time annealing on ALDP. *SMC will reduce sample diversity, which pre- dominantly influences W2. Therefore, W₂ for "anneal score" should be directly compared against SMC methods. Instead, energy and dis- tance TVD are less sensitive to sample diversity and are more comparable.
Energy TVD
Distance TVD
Sample
2
1x10
10-3
RNC
6
2x
Fig 3: Inference-time scaling on ALDP.
Metric
Energy TV())
Distance TV())
Sample Wa(L)
SMC)
0.023
0,173
0003
0.289
(+- 1.0.= ()
0.17
0.282
RNC (4 0.600 0.4)
0.034
0.011
0,253
(a) Ground truth (b) DSM
(c) RNE reg.
Fig 4: Learned density 2D GMM.
0.4
2300
0.00
0.2
0.2
1250
Learned energy
R4: 0.846
R 0.876
0090000
0000000
2000000
-120
Fig 2: Energy TVD (left), sample W2 (middle), and accumulated weight variance (right) by dif- ferent pairs of (Ca, Cb) for annealing on ALDP.
-3000
-200
-150 Ground truth
-175
-150
Ground
(a) DSM
(b) Dual SM
(c) RNE reg.
Tab 2: Quality of samples obtained by running denoising process (denoted DM) and rumning MCMC on learned energy at / O.
Fig 5: Learned energy vs. GT on 100D GMM,
Training method
Sample Method
Sample W/2
DM
0.1811
DSM
MCMC
0,9472
DM
0.1809
(a) Ground truth
(b) DSM
(c) RNE reg.
RNE Reg
MCMC
0.1836
Fig 6: Samples by MCMC on learned energy.
We further derive error bound on the importance weights, considering both the discretisation error and the error in the score network:
Proposition 3.2. (Discrete time and upproximate score bound) Following (Lee el al., 2023; Chen et al., 2022), assume || p, (Y=) - Escores then
|| log w exeet (M(T.r9) log we
RNC (Yrr=)))l=> S Eescore + P'VAL
(26)
where E, P' are positive constants, wo (aNC((Prrr is the discrete-time weight estimated by RN (Mp (rr")) with the learned score sG, wexace (%) (brrt) is the exact SMC weight.
We prove these results with more detailed discussion in Appendix G.3.
4 EXPERIMENTS
In this section, we conduct comprehensive experiments to evaluate our approach for both inference- time control and energy-based training. Please refer to Appendix for more details on the experiments. Inference-time annealing We evaluate our proposed method for inference-time annealing on a small molecule, alanine dipeptide (ALDP). We train the model on Thigh 800K and anneal it to Tow 300/. Since SMC typically suffers from low diversity, we use a batch size of 500 and collect 50 batches calculate the metrics. We compare RNC against FKC (Skreta al., 2025) and, for reference, a baseline that merely rescales the score without SMC correction. For RNC, we apply Eq. (17) to calculate the importance weights, and select the sampling and target process heuristically as Eq. (34), where we control the drift a, and b, via hyperparameter Xς and A We evaluate two different choices of sampling and target processes: (1) A - /Tow >e By Proposition C.3, this is theoretically identical to FKC. (2) we further introduce free parameters co Ca. X T/TTigh/ 'Toow Cb.
and c₁: Af = TTTThihhhT
We report in Tab. 1 the Wasserstein-2 (W2) distance between the ground-truth and generated samples, alongside the total variation distance (TVD) computed on both energy and interatomie-distance histograms. Figure 2 shows sweep the coefficients (ca, cb). When c. 1, c 0. RNC achieves a similar performance to FKC, which echoes Proposition C.3. More importantly, by selecting different (Ca, cb), RNC attains higher flexibility and enhanced performance compared FKC.
8

---

Published as a conference paper at ICLR 2026
We also compute the variance of the importance weights accumulated over the entire sampling trajectory for various (c),Cb). As shown in Figure 2 (right), choices with ca + c within = 0.2 tend to minimise the variance, which also correlates with high sample quality. However, while the pattern for variance persists across different targets, the optimal balance of ca and cb for achieving the highest sample quality can be task-dependent. For example, for unimodal targets with sharp peak, lower ESS reduces diversity but can actually be ad advantageous in ensuring the sample is closer to the peak. Conversely, for multimodal targets, maintaining a higher ESS preserves greater diversity and helps prevent mode collapse. To illustrate this trade-off, we include inference-time annealing results and additional analysis for the Lennard-Jones (LJ) system and Mixture-of Gfaasssia in Appendix F.3.
Inference-time product: multi-target structure-baser small-molecule ligand design Following Skreta et al. (2025), we evaluate RNC for model product with multi-targed structure-based small- molecule ligand design. For a detailed introduction to the background of this task, please refer (1) (2) where Po (!) and po to Appendix 1.2. In summary, we consider sampling from go x (Po Po represents the diffusion model's outcome conditional on two protein targets. In our experiments, we set 8 = 2 following the optimal hyperparameter used by Skreta et al. (2025). We compare our RNC method with FKC (Skreta et al., 2025) and baseline, "Sum score", which samples from the denoising process by directly summing the scores conditioned on each target without SMC.
For RNC, similar to the annealing case, we can choose any sampling and target processes. Here, we heuristically select the options in Eq. (36), where we set X; = X, = -E7 Ca and X = x/2 = = Cb. In Tab. 4, we present the result obtained with cu = 1, c. = 0. which is theoretically equivalent to FKC in continuous time. We observe that these results achieve similar performance. up to the inherent stochasticity in the generation process. We also report the performance obtained with C2 1.0, Go 0.2. As shown in Tab. 4, both FKC and RNC variants are significantly better than the heuristic score summation, at the price of lower diversity. Furthermore, RNC offers higher flexibility in choosing the sampling and target process, providing a visible gain over FKC, particularly with more ligands that have better docking scores than both reference ligands.
stitching and reward- -tiliing for maze navigation One advantage of RNC that it can be intu- itively and seamlessly extended to tasks that require more flexible controls. Here, we consider task by stitching together diffusion models trained on short trajectories. Formally, letting the short trajectories qo x
Flexible controls:
Tab 3: Success rate of trajectory stitch- ing by guidance without SMC and with RNC across 5 tasks.
navigation
fask
task 2
task 3
task 4
task 5
WO SMC
0.501
0.669
0.585
0.288
0.714
follow Pu. we aim to sample [x(1), 1............................. X(()
RNC
1.000
1,000
1.000
1.000
1.000
exp(r([X('), x() II Po(("), where the reward r is defined impose first trajectory starts from the initial position and the last trajectory ends at the target, and consecutive trajectories are connected. Despite the complexity of this task, RNC can still provide SMC weights for it without changing the general formula. We use the poi ntma ze- um-utm---t..... st. tichhv dataset from Park et al. (2024), and consider 5 different pairs of initial points and targets. We include more experimental details in Appendix I.3. We visualise the unstitched trajectories and the samples generated by SMC in Fig. 7, using different colours to represent different short trajectories. We also report the success rate in Tab. 3. For comparison, we include results without SMC, where we only use guidance from the gradient of the reward. We observe that RNC increases the success rate to 100%, demonstrating the flexibility and practical applicability of our method.
Inference-time scaling RNC allows us to increase the number of particles during inference to obtain better performance. We showcase this property with the annealing experiments on alanine dipeptide (ALDP). Specifically, we follow the setting we used in Tab. 1, with c. = 0.6, c, 0.4, and evaluate the performance scaling with different batch sizes: 100, 500, 1000, 2000, 5000. For comparison, we also report the corresponding FKC results. Fig. 3 shows the sample quality scaling with different numbers of particles. RNC not only features better sample quality but also presents better scaling properties, especially for sample diversity, as reflected by the sample W2 distance.
Training energy based models We train energy- based diffusion with RNE regularisation on both the Gaussian mixture and the alanine dipeptide (ALDP). In Fig. 4, we visualise the learned density for the standard denoising score matching (DSM) and for DSM with RNE regularisation on 2D GMM. We obtain the density value by exponentiating and normalising the learned negative energy at t = 0. The unregularised DSM fails capture the target accurately, whereas RNE regularisation enables

---

Published as a conference paper at ICLR 2026
(a) Tilting with RNC. Prompt from left to right: (1) A yellow ambu
lance, (2) An orange balloon with red spots, (3) blue jeep.
Fig 7: Visualisation of stitched tra- jectory for maze navigation. We also show examples of unstitched short trajectories in the upper-left corner.
(h) Generation without SMC.
Fig 8: Visualisation of prompt-reward-titing on masked dis- crete diffusion on ImageNet-256.
Tab 4: Multi-target SBDD performances. Better than known denotes the percentage of generated ligands with lower docking scores than both of two ground truth reference ligands. P1 and P2 are docking scores for two pockets Div. assesses the pairwise difference over molecular fingerprints.
Val. & Uniq. denotes the percentage of ligands that are both valid and unique. Qual. denotes the percentage of ligands that have good physicochemical properties.
Bener than known. (t)
(P: *P)()
mattPi,
Fl (I)
Fy top-1 (1)
Dov (1)
Val. & Uniq (
Qual. (1)
Sun sanre
-94llaa
0.861
a.
0.334a
FKC
82 52.57
-8 a50
-10 1000
0.8:4a
0.326aa.u
0.182
RNC RNC
0.280
81 186
a8.122
-0.000
-10075.1
0.823
0.042
81 1ilue
-9.585
-10
O 0.836
950
0.223
the model to recover the energy at t 0 relatively exactly. In Fig. 5, we evaluate 100D GMM, and also include dual score matching (Guth et al., 2025) as a baseline. We can see both RNE and dual score matching significantly improve the accuracy of the learned energy.
approach
To assess its potential as a conservative machine-learning force field (MLFF), we train energy-based diffusion models ALDP samples at 300K, then run MCMC on the learned energy at t = 0 and visualise the resulting Ramachandran plot in Fig. 6, where the RNE-regularised model closely reproduces the ground-truth distribution. Importantly, throughout training. only use samples, without accessing the energies or scores. From Tab. 2, the RNE regularisation does not noticeably influence the quality of the diffusion model itself, showcasing our RNE's flexibility and applicability.
RNE regularisation can also be applied to bridge mod- els such as stochastic interpolants (Albergo et al., 2023). Learning an accurate energy path can improve the accuracy of free-energy estimation via thermodynamic integration (TI, Kirkwood, 1935; Máté et al., 2025). We demonstrate this by estimating the solvation free energy of the alanine dipeptide, using the dataset and systems described in He et al. (2025a). Further details on the background and setup are provided in Appendix L66. We report the values estimated without and with RNE in Tab. 5, where RNE regularisation substantially improves the accuracy of the results.
Tab 5: ALDP solvation free energy esti- mated with thermodynamic integration.
TI RNE
TI RNE (Ours)
Reference Value
(Máté et al., 2025)
29.43 0.01
27.30 + 0.45
29.28 0.04
RNE for CTMC We also verify RNE for CTMC. More precisely, we consider CFG debiasing plus tilting the generation with ImageReward (Xu et al., 2023) with a prompt. We take the MaskGIT (Chang al., 2022) pretrained on ImageNet-256 by Besnier et al. (2025). MaskGIT defines a (latent) mask image model that predicts the conditional distributions of masked positions given a masked sample. Therefore, similar to Ren al. (2025), we can turn MaskGIT into (latent) masked discrete diffusion model by introducing a stochastic masking schedule following Shi et al. (2024). For more details, please refer to Appendix I.4. We visualise the results in Fig. As shown, RNE achieves strong alignment between the generated images and the target prompts, demonstrating both the effectiveness of RNE on CTMC and its scalability to larger image- genge-ggnratiion settings.
CONCLUSION
In this paper, we introduce the RADON-NIKODYM ESTIMATOR (RNE). It leverages the fact that for any diffusion process consider, we can pair it with its time-reversal, and the Radon-Nikodym derivative of the forward path measure with respect to the reverse path measure is always equal to one. This principle lets us decouple marginal densities from transition kernels, yielding highly flexible and plug-and-plat method for density estimation, inference- control and energy- based training, for diffusion models across modalities. We discuss its limitations in Appendix B.
10

---

Published as a conference paper at ICLR 2026
LLM USAGE DISCLOSURE
LLM was used at the sentence level to correct grammar.
ACKNOWLEDGEMENTS
We acknowledge Alexander Denker, Julius Berner, and Kirill Neklyudov for their insightful feedback and discussion on our manuscript. We acknowledge the helpful discussion with Michael Plainer, which drew our attention to the importance of energy-based diffusion models. We acknowledge Tony Ou Yang for the helpful discussion on anergy-based training, which inspired us to analyse the equiva- Ience between ENE-regularisation and Fokker- Plancr regularisation. We also acknowledge Zijing Ou for pointing out several typos in our manuscript. JH acknowledges support from the University of Cambridge Harding Distinguished Postgraduate Scholars Programme. JMHL acknowledges support from a Turing Al Fellowship under grant EP/V023756/1.
REFERENCES
Anurag Ajay, Seungwook Han, Yilun Du, Shuang Li, Abhi Gupta, Tommi Jaakkola, Josh Tenenbaum, Leslie Kaelbling, Akash Srivastava, and Pulkit Agrawal. Compositional foundation models for hierarchical planning. Advances in Neural Information Processing Systems, 36:22304-22325, 2023.
Michael S Albergo and Eric Vanden-Eijnden Nets: non-equilibrium transport sampler. arXiv preprint arXivv2410.02711, 2024.
Michael S Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions. arXiv preprint wwXiv:2303308797, 2023.
Brian DO Anderson. Reverse-time diffusion equation models. Stochastic Processes and their Applications, 12(3):313-322 1982.
Marloes Arts, Victor Garcia Satorras, Chin-Wei Huang, Daniel Zugner, Marco Federici, Cecilia Clementi, Frank Noé, Robert Pinsler, and Rianne van den Berg. Two for one: Diffusion models and force fields for soarse-graines molecular dynamics. Journal of Chemical Theory and Computation, 19(18):6151-615) 2023.
Jacob Austin, Daniel D Johnson, Jonathan Ho, Daniel Tarlow, and Rianne Van Den Berg. Structured denoising diffusion models in discrete state-spaces. Advances in neural information processing systems, 34:1798 1-17993, 2021.
Fan Bao, Chongxuan Li, Jun Zhu, and Bo Zhang, Analytic-dpm: an analytic estimate of the optimal reverse variance in diffusion probabilistic models. arXiv preprint arXiv:2201.066. 06500, 2022.
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:2501.06/4. 2025.
Victor Besnier, Mickael Chen, David Hurych, Eduardo Valle, and Mathieu Cord. Halton scheduler for masked generative image transformer. arXiv preprint arXiv:2503. /7076, 2025.
Benjamin Biggs, Arjun Seshadri, Yang Zou, Achin Jain, Aditya Golatkar, Yusheng Xie, Alessandro Achille, Ashwin Swaminathan, and Stefano Soatto. Diffusion soup: Model merging for text-to- image diffusion models. In European Conference on Computer Vision, pp. 257-274. Springer, 2024.
David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians. Journal of the American statistical Association. 112(518):859-888 877, 2017.
Tom L Blundell. Structure-based drug design. Nature, (84(6604) 1996.
Maria Laura Bolognesi and Andrea Cavalli. Multitarget drug discovery and polypharmacology. 2016.

---

Published as a conference paper at ICLR 2026
Andrew Campbell, Joe Benton, Valentin De Bortoli, Thomas Rainforth, George Deligiannidis, and Arnaud Doucet. A continuous time frumework for discrete denoising models. Advances in Neural Information Processing Systems, 35:28266- 28279, 2022.
Andrew Campbell, Jason Yim, Regina Barzilay, Tom Rainforth, and Tommi Jaakkola. Generative flows on discrete state-spaces: Enabling multimodal flows with applications to protein co-design. arXiv preprint urXiv:2402.0445 04997, 2024.
Huiwen Chang, Han Zhang, Lu Jiang, Ce Liu, and William Freeman. Maskgit: Masked generative image transformer. In Proceedings of the IEEE/CVF conference computer vision and pattern recogmition, pp. 1 11315-1 1325, 2022.
Haoxuan Chen, Yinuo Ren, Martin Rengiang Min, Lexing Ying, and Zachary Izzo. Solving inverse problems via diffusionn based priors: An approximation ensemble sampling approach. arXiv preprint arXiv:2506.0398 03979, 2025.
Junhua Chen, Lorenz Richter, Julius Bemnr,, Denis Blessing, Gerhard Neumann, and Anima Anand- kumar. Sequential controlled langevin diffusions. arXiv preprint arXiv:2412.07081, 2024.
Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances neural information processing systems, 31, 2018.
Sitan Chen, Sinho Chewi, Jerry Li, Yuanzhi Li, Adil Salim, and Anru R Zhang. Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions. arXiv preprint arXiv:2209. 2022.
John Chodera, Andrea Rizzi, Levi Naden, Kyle Beauchamp, Patrick Grinaway, Mike Henry, Iván Pulido, Josh Fass, Alex Wade, Gregory A. Ross, Andreas Kraemer, Hannah Bruce Macdonald, jaimergp. Bas Rustenburg, David W.H. Swenson, Ivy Zhang, Dominic Rufa, Andy Simmonett, Mark J. Williamson, hb0402, Jake Fennick, Sander Roet, Benjamin Ries, Ian Kenney, Irfan Alibay, Richard Gowers, and SimonBoothroyd. choderalab/opennmtoo 0.24.1, January 2025. URL pettss//doi.ozzz 10. 2ataa///eoodo..o 14782825.
Hyungjin Chung, Jeongsol Kim, Michael Thompson Mccann, Mare Louis Klasky, and Jong Chul Ye. Diffusion posterior sampling for general noisy inverse problems. In The Eleventh International Conference on Learning Representations, 2023.
Valentin De Bortoli, Michael Hutchinson, Peter Wimsberger, and Arnaud Doucet. Target score matching. arXiv preprint arXiv:2402.08667, 2024.
Alexander Denker, Francisco Vargas, Shreyas Padhy, Kieran Didi, Simon Mathis, Riccardo Barbano, Vincent Dutordoir, Emile Mathieu, Urszula Julia Komorowska, and Pictro Lio. Deft: Efficient fine-tuning of diffusion models by learning the generalised fi-transform. Advances in Neural Information Processing Systems, 37:19636- 19682, 2024.
Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 01:8780-8794 2021.
Carles Domingo-Entin Ennich. Michal Drozdzal, Brian Karrer, and Ricky TQ Chen. Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control. arXiv preprint arXiv:2409.0885 00861, 2024.
Zehao Dou and Yang Song. Diffusion posterior sampling for linear inverse problem solving: A filtering perspective. In The Twelfth International Conference on Learning Representations, 2024.
Arnaud Doucet, Will Grathwohl, Alexander G Matthess, and Heiko Strathmann. Score-based diffusion meets annealed importance sampling. Advances in Neural Information Processing 221444, 2022.
Systems,
Yilun Du, Conor Durkan, Robin Strudel, Joshua B Tenenbaum. Sander Dieleman, Rob Fergus, Jascha Sohl-Dickstein, Arnaud Doucet, and Will Sussman Grathwohl. Reduce, reuse, recycle: Composi- tional generation with energy-based based diffusion models and meme. In International conference on machine learning, pp. 8519- PMLR, 2023.
12

---

Published as a conference paper at ICLR 2026
Chenru Duan, Yuanqi Du, Haojun Jia, and Heather Kulik. Accurate transition state generation with an object-aware equivariant elementary reaction diffusion model. Nature computational science, (12):1045-1055, 2023.
Peter Eastman, Raimondas Galvelis, Raúl P Peláez, Charlles RA Abreu, Stephen Farr, Emilio Gallicchio, Anton Gorenko, Michael M Henry, Frank Hu, Jing Huang, et al. Openmm 8: molecular dynamics simulation with machine learning potentials. The Journal of Physical Chemistry B, 128 (1):109-116, 2023.
Jerome Eberhardt, Diogo
Andreas F Tillack, and Stefano Forli. Autodock vina 1.2. New docking methods, expanded force field, and python bindings. Journal of chemical information and modeling, 61()):381133888 2021.
Ruiqi Gao, Emiel Hoogeboom, Jonathan Heek, Valentin De Bortoli, Kevin Patrick Murphy, and Tim Salimans. Diffusion models and gaussian flow matching: Two sides of the same coin. In The Fourth Blogpost Track at ICLR 2025, 2025. URL htttht:///penreeee
/ttp///ppeeeeeeeeeeeee/eeeeeeeee net/ forum?id did=
C8Xyg9wy0s.
Jiaqi Guan, Wesley Wei Qian, Xingang Peng, Yufeng Su, Jian Peng, and Jianzhu Ma. 3d equivariant diffusion for target-aware molecule generation and affinity prediction. In The Eleventh International Conference on Learning Representations, 2023.
Florentin Guth, Zahra Kadkhodaie, and Eero P Simoncelli. Learning normalized image densities via dual score matching. arXiv preprint 2500:05500003310, 2025.
István Gyöngy and Miklós Rásonyi. A note on euler approximations for sdes with hölder continuous diffusion coefficients. Stochastic processes and their applications, 121(10):2189- 2200, 2011. URL 80304414911001443
rette com/
Jiajun He, Yuanqi Du, Francisco Vargas, Yuanqing Wang, Carla P Gomes, José Miguel Hernández- Lobato, and Eric Vanden-Bijnden. Feat: Free energy estimators with adaptive transport. arXiv preprint arXiv:2504. 11.50044.111.... 2025a.
Jiajun He, Yuanqi Du Francisco Vargas, Dinghuai Zhang, Shreyas Padhy, RuiKang Ou Yang, Carla Gomes, and José Miguel Hernández-Lobato. No trick, no treat: Pursuits and challenges towards simulation-free training of neural samplers. arXiv preprint arXiv:2502.0666: 2025b.
Jiajun He, Paul Jeha, Peter Potaptchik, Leo Zhang, José Miguel Hernández-Lobato, Yuanqi Du, Saifuddin Syed, and Francisco Vargas. Crepe: Controlling diffusion with replica exchange. arXiv preprint arXiv:2509.23265 2025c.
Jonathan Ho and Tim Salimans. Classifier-free diffusion guidance. arXiv preprint arXiv:2207.12598. 2022.
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-6851, 2020.
Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David J Fleet. Video diffusion models. Advances in Neural Information Processing Systems, 15:8633-8646 2022.
Peter Holderrieth, Michael S Albergo, and Tommi Jaakkola. Leaps: A discrete neural sampler via locally equivariant networks. arXiv preprint arXiv:2502.10843, 2025.
Emiel Hoogeboom, Victor Garcia Satorras, Clément Vignac, and Max Welling. Equivariant diffusion for molecule generation in 3d. In International conference machine learning, pp. 8867-8887. PMLR, 2022.
Chin-Wer Wei Huang, Jae Hyun Lim, and Aaron C Courville. A variational perspective diffusion-basee generative models and score matching. Advances in Neural Information Processing Systems, 34: 22863-22874 2021.
13

---

Published as a conference paper at ICLR 2026
Christopher Jarzynski. Nonequilibrium equality for free energy differences. Physical Review Letters, 78(14):2690, 1997.
Rafal Karczewski, Markus Heinonen, and Vikas Garg. Diffusion models as cartoonists! the curious case of high density regions. arXiv preprint arXiv:2411.01293, 2024.
Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion- based generative models. Advances in neural information processing systems, 35:26565-26577,
2022.
Diederik Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational diffusion models. Advances in neural information processing systems, 34:21696-21707, 2021.
Diederik Kingma, Max Welling, et al. Auto-encoding variational bayes, 2013.
John G. Kirkwood. Statistical mechanics of fluid mixtures. The Journal of Chemical Physics, 3(5): 300-313, 05 1935. ISSN 0021-9606. doi: 10.1063/1.17496577 URL sttpp:///oot orgg 1.63/1..749655
Peter Kloeden, Eckhard Platen, Peter Kloeden, and Eckhard Platen. Stochastic differential equations. Springer, 1992.
Lingkai Kong, Yuanqi Du, Wenhao Mu, Kirill Neklyudov, Valentin De Bortoli, Dongxia Wu, Haorui Wang, Aaron Ferber, Yian Ma, Carla P Gomes, et Diffusion models as constrained samplers for optimization with unknown constraints. In The 28th International Conference Artificial Intelligence and Statistics, 2025.
Cheuk Kit Lee, Paul Jeha, Jes Frellsen, Pietro Lio, Michael Samuel Albergo, and Francisco Var- gas. Debiasing guidance for discrete diffusion with sequential monte carlo. arXiv preprint arXiv:2502.06079 2025.
Holden Lee, Jianfeng Lu, and Yixin Tan. Convergence of score-based generative modeling for general data distributions. In International Conference Algorithmic Learning Theory, pp. 946 985. PMLR, 2023.
Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamn Maximilian Nickel, and Matt Le. Flow matching for generative modeling. arXiv preprint arXiv:2210.02747, 2022.
Nan Liu, Shuang Li, Yilun Du, Antonio Torralba, and Joshua B Tenenbaum. Compositional visual generation with composable diffusion models. In European Conference on Computer Vision, pp- 423-439. Springer, 2022.
Aaron Lou, Chenlin Meng, and Stefano Ermon. Discrete diffusion language modeling by estimating the ratios of the data distribution. 2023.
Yunhao Luo, Utkarsh A Mishra, Yilun Du and Danfci Xu. Generative trajectory stitching through diffusion composition. arXiv preprint arXiv:2503.05/5: 2025.
Nanye Ma, Mark Goldstein, Michael S Albergo, Nicholas M Boffi, Eric Munden-Eijndesd and Saining Xie. Sit: Exploring flow and diffusion-baser generative models with scalable interpolant transformers. In European Conference on Computer Vision, pp. 23 40. Springer, 2024.
Bálint Máté, Francois Fleuret, and Tristan Bereau. Neural thermodynamic integration: Free energies from energy-bare based diffusion models. The Journal of Physical Chemistry Letters, 13(45) 11404, 2024.
Bálint Máté, Francois Fleuret, and Tristan Bereau. Solvation free energies from neural thermodynamic integration. The Journal of Chemical Physics, 162(12), 2025.
Isambi S Mbalawata and Simo Särkkä. Moment conditions for convergence of particle filters with unbounded importance weights. Signal Processing, 11:1333-333 2016.
Laurence Illing Midgley, Vincent Stimper, Gregor NC Simm, Bernhard Schölkopf, and José Miguel Hernández-Lobar Flow annealed importance sampling bootstrap. arXiv preprint arXiv:2208. 01893, 2022.
14

---

Published as a conference paper at ICLR 2026
Radford M Neal. Annealed importance sampling, Statistics and computing, 11:125- 139, 2001.
Edward Nelson. Dynamical Theories of Brownian Motion. Princeton University Press, 1967. ISBN 9780691079509.
Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In International conference on machine learning, pp. 8162-8171. PMLR, 2021.
Maxence Noble, Louis Grenioux, Marylou Gabrié, and Alain Oliviero Durmus. Learned reference- based diffusion sampling for multi-modal distributions. arXiv preprint arXiv:2410. 19449, 2024.
Jingyang Ou, Shen Nie, Kaiwen Xue, Fengqi Zhu, Jiacheng Sun, Zhenguo Li, and Chongxuan Li. Your absorbing discrete diffusion secretly models the conditional distributions clean data. arXiv preprint ar arXiv:2406.0377 2024a.
Zijing Ou, Mingtian Zhang, Andi Zhang, Tim Xiao, Yingzhen Li, and David Barber, Improv- ing probabilistic diffuuson models with optimal diagonal covariance matching. arXiv preprint arXiv:2406. 10808, 2024b.
Seohong Park, Kevin Frans, Benjamin Eysenbach, and Sergey Levine. Ogbench: Benchmarking offline goal-conditioned rl. arXiv preprint arXiv:2410.2009) 2024.
Stefano Peluchetti. Diffusion bridge mixture transports, schrödinger bridge problems and generative modeling. Journal of Machine Learning Research, 24(374):1-51, 2023.
Angus Phillips, Hai-Dang Dau, Michael John Hutchinson, Valentin De Bortoli, George Deligiannidis, and Arnaud Doucet. Particle denoising diffusion sampler. arXiv preprint arXiv:240..06320, 2024.
Michael Plainer, Hao Wu, Leon Klein, Stephan Günnemann, and Frank Noé. Consistent sam- pling and simulation: Molecular dynamics with energy- diffusion models. arXiv preprint arXiv:2506.17139, 2025.
Akhil Premkumar. Diffusion density estimators. arXiv preprint arXiv:2410.0698 2024.
Yinuo Ren, Haoxuan Chen, Yuchen Zhu, Wei Guo, Yongxin Chen, Grant M Rotskoff, Molei Tao, and Lexing Ying. Fast solvers for discrete diffusion models: Theory and applications of high-order algorithms. arXiv preprint arXiv:2502.00234, 2025.
Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. High- resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF confer- ence on computer vision and pattern recognition, pp. 10684-10695, 2022.
Francois Rozet, Gérôme Andry, Francois Lanusse, and Gilles Louppe. Learning diffusion priors from observations by expectation maximization. Advances in Neural Information Processing Systems, 37:87647-87682, 2024.
Arne Schneuing, Charles Harris, Yuanqi Du, Kieran Didi, Arian Jamasb, Ilia Igashov, Weitao Du, Carla Gomes, Tom L Blundell, Pietro Lio, et al. Structure-based drug design with equivariant diffusion models. Nature Computational Science, 4(12):899-909, 2024.
Jiaxin Shi, Kehang Han, Zhe Wang, Arnand Doucet, and Michalis Titsias. Simplified and generalized masked diffusion for discrete data. Advances neural information processing systems, 37: 103131-103167, 2024.
Yuyang Shi, Valentin De Bortoli, Andrew Campbell, and Arnaud Doucet. Diffusion schr\"" odinger bridge matching, arXiv preprint arXiv:2303. 16852, 2023.
Michael R Shirts and John D Chodera. Statistically optimal analysis of samples from multiple cquilibrium states. The Journal of chemical physics, 129(12), 2008.
Raghav Singhal, Zachary Horvitz, Ryan Teehan, Mengye Ren, Zhou Yu, Kathleen McKeown, and Rajesh Ranganath. A general framework for inference-time scaling and steering of diffusion models. arXiv preprint 2rX 2025.
15

---

Published as a conference paper at ICLR 2026
Marta Skreta, Lazar Atanackovic, Avishek Joey Bose, Alexander Tong, and Kirill Neklyudov. The su- perposition of diffusion models using the it), o density estimator. arXiv preprint arXiv:2412.17762, 2024.
Marta Skreta, Tara Akhound-Sadegh. Viktor Ohanesian, Roberto Bondesan, Alán Aspuru-Guzin
Arnaud Doucet, Rob Brekelmans, Alexander Tong, and Kirill Neklyudov. Feynman-kac correctors in diffusion: Annealing, guidance, and product of experts. arXiv preprint arXiv:2503.02819 2025.
Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. In Interna-
tional Conference on Learning Representations, 2021a.
Jiaming Song, Arash Vahdat, Morteza Mardani, and Jan Kautz. Pscudoinverse-guided diffusion models for inverse problems. ICLR, 2023a.
Jiaming Song, Qinsheng Zhang, Hongxu Yin, Morteza Mardani, Ming- Yu Liu, Jan Kautz, Yongxin Chen, and Arash Vahdat. Loss-guided diffusion models for plug-and-play controllable generation. In International Conference on Machine Learning, pp. 32483-32498. PMLR, 2023b.
Yang Song, Jascha Sohl-Dickstein, Diederik Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021b.
Charlie B Tan, Avishek Joey Bose, Chen Lin, Leon Klein, Michael M Bronstein, and Alexander Tong. Scalable equilibrium sampling with sequential boltzmann generators. arXiv preprint arXiv:2502. 18462, 2025.
James Thornton, Louis Béthune, Ruixiang Zhang, Arwen Bradley, Preetum Nakkiran, and Shuangfei Zhai. Composition and control with distilled energy diffusion models and sequential monte carlo. arXiv preprint arXiv:2502. 12786, 2025.
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, pp. 1-34, 2024.
Brian L. Trippe, Jason Yim, Doug Tischer, David Baker, Tamara Broderick, Regina Barzilay, and Tommi S Jaakkola. Diffusion probabilistic modeling of protein backbones 3d for the motif- scaffolding problem. In The Eleventh International Conference on Learning Representations, 2023.
Masatoshi Uchara, Yulai Zhao, Chenyu Wang, Xiner Li, Aviv Regev, Sergey Levine, and Tommaso Biancalani. Inference-tinn time alignment in diffusion models with reward-guided generation: Tutorial and review. arXiv preprint a5Xiv:2501 20X0:25501.09685. 2025.
Suriyanarayanan Vaikuntanathan and Christopher Jarzynski. Escorted free energy simulations: Improving convergence by reducing dissipation. Physical Review Letlers, 100611) 2008
Francisco Vargas, Will Grathwohl, and Arnaud Doucet. Denoising diffusion samplers. arXiv preprint arXiv:2302. 13334, 2023a.
Francisco Vargas, Shreyas Padhy, Denis Blessing, and Nikolas Nüsken. Transport meets variational inference: Controlled monte carlo diffusions. arXiv preprint arXiv:2307.010550 2023b.
Pascal Vincent. A connection between score matching and denoising autoencoders. Neural Computa- tion, 23(7):1661-161) 2011. doi: 16.1162/ 00142.
Lingle Wang, Yujie Wu, Yuging Deng, Byungchan Kim, Levi Pierce, Goran Krilov, Dmitry Lupyan, Shaughnessy Robinson, Markus Dahlgren, Jeremy Greenwood, et al. Accurate and reliable prediction of relative ligand binding potency in prospective drug discovery by way of a modern free-energy calculation protocol and force field. Journal of the American Chemical Society. 137 (7):2695-2703, 2015.
Joseph L Watson, David Juergens, Nathaniel R Bennett, Brian L Trippe, Jason Yim, Helen E Eisenach, Woody Ahern, Andrew Borst, Robert Ragotte, Lukas F Milles, et al. De novo design of protein structure and function with rfdiffusion. Nature, 62097976):1089-11 2023.
16

---

Published as a conference paper at ICLR 2026
Luhuan Wu, Brian Trippe, Christian Naesseth, David Blei, and John P Cunningham. Practical and asymptotically exact conditional sampling in diffusion models. Advances in Neural Information Processing Systems,
2023.
Jiazheng Xu, Xiao Liu, Yuchen Wu, Yuxuan Tong, Qinkai Li, Ming Ding, Jie Tang, and Yuxiao Dong. Imagereward: learning and evaluating human preferences for text-to-image generation. In Proceedings of the 37th International Conference on Neural Information Processing Systems, pp. 15903-15935, 2023.
Claudio Zeni, Robert Pinsler, Daniel Zügner, Andrew Fowler, Matthew Horton, Xiang Fu, Zilong Wang, Aliaksandra Shysheya, Jonathan Crabbé, Shoko Ueda, et al. A generative model for inorganic materials design. Nature, pp. 1-3, 2025.
Leo Zhang, Peter Potaptchik, Jiajun He, Yuanqi Du, Arnaud Doucet, Francisco Vargas, Hai-Dang Dau, and Saifuddin Syed. Accelerated parallel tempering via neural transports. arXiv preprint arXiv:2502. 10328, 2025.
Mingtian Zhang, Osear Key, Peter Hayes, David Barber, Brooks Paige, and Francois- Xavier Briol. Towards healing the blindness of score matching. arXiv preprint arXiv:2209007396, 2022.
Xiangxin Zhou, Jiaqi Guan, Yijia Zhang, Xingang Peng, Liang Wang, and Jianzhu Ma. Reprogram- ming pretrained target-specific diffusion models for dual-target drug design. Advances in Neural Information Processing Systems, 37:87255-87255 2024.
Robert W Zwanzig. Nigh-temperature equation of state by perturbation method. i. nonpolar gases. The Journal of Chemical Physics, 1420-1442 1954.
17

---

Published as a conference paper at ICLR 2026
APPENDIX
A Implementation Outlines on Inference-time Control
20
Limitations
20
C Supplementary Methods
20
C.1 RNE for Diffusion Density Estimation
21
C.2 RNE for Diffusion Density Estimation with Importance Sampling
21
C.3 Heuristic Choice of a and b, for Inference-time Control
22
C,4 "FKC S RNC"
23
C.5 Exact SMC weight for Imperfect diffusion model
23
D RNE for Discrete Diffusion
24
D.1 RNC for Discrete Diffusion
................................
25
D.2 Discretisation of R for CTMC
25
Connection Between RNE and Other Approaches
26
F Additional Experiments and Analysis
26
F.1
RNDE and Ablation on Reference Process
26
F2
Analysis on RNDE-IS....
27
E3
Inference-time Annealing: more analysis...
27
F4
Empirical Analysis on Imperfect Score and Discretisation Error
28
F5
Ablation on Reference Process for Better Denoising Kernels
G
Discretisation, Stability and Convergence Guarantees
29
G.I Intuition Instability without Reference
29
G.2 Continuous formulation vs Discrete Gaussian kernels for R
30
G.3 RNE with Reference Convergence Rate
32
G.3.1 Convergence Rate for RNE
32
G.3.2 Convergence Rate for SMC Weights in Discrete Time and Approximated Score
36
H Proofs
37
H.I Correctness of SMC Weights in Eq, (16)
37
H.2 IS Perspectives for RNE, Connections to Huang al. (2021)
................................
38
H.3 Equivalence to Itô density estimator
40
H.4
Connections to Keynman- Kac Corrector (Skreta et al., 2025)
40
H.4.1
Anneal FKC
41
H.4.2
Product FKC
42
H.4.3
CFG FKC
44
H.4.4
Reward-tilting FKC
45
H.5
Connecting RNE Energy Regularisation with FPE Regularisation
47
I Additional Experimental Details
48
18

---

Published as a conference paper at ICLR 2026
L1
Additional Details for Inference-time Annealing
48
1.2
Additional Details for Multi-target SBDD
49
I.3
Additional Details for Trajectory Stitching Experiments
50
L.4
Additional Details for CTMC-RNE
50
I.5
Additional Details for Training Energy-based Diffusion Models.
51
1.6
Background and Details for Free Energy Estimation with Thermodynamic Integration
52
1.6.1
Background on Free Energy
52
I.6.2
Experimental Details
53
J Related Works in Sampling from Unnormalised Densities
55
19

---

Published as a conference paper at ICLR 2026
A IMPLEMENTATION OUTLINES ON INFERENCE-TIME CONTROL
In this section, we illustrate the key methods that need to be implemented when a user wants to implement different heuristic choices for a, and b, (the different sampling and target process choices).
We provide the RNC anneal weights as an example and illustrate how the R factor computation in the rne method does not need to modified as we change from task to task.
class GuidedSDE (self):
rne (fwd, bwd, t, a, ...): Discretise [t, a] with N steps
Model forward legprob def fwd_mu (telf,
for in range (N)
in = fwd ( tn)
Model back logprob def nu (self xt, xtml,
on - ( (......, tn)
1nR 4- bn fn return 1nR
Imp Implement target logprob def fwd lo (self, xtmml
def rnc ann (sde, beta, t, ...):
fmu, bnu = fwd mu, sde, bwd.bwd mu
fb, ba sde. fwd b, ade.bwd bwd
InRmunu rne (fmu, bnu, t, S,...) InRab rne (fb, ba, 8, S,...) Inw -1nRab InRmunu bata return Inw
Implement sample logprob def bwd (self, xt,ml,
Fig 9: Psuedocode illustrating development pipeline for RNC (exemplified with annealing). The user only needs implement the sampling and target kernels; changing these does not change the of the downstream code nor require re-derivation, unlike FKC (Skreta et al., 2025). In practice, we also add the analytical reference to the implementation of rne described in Section 3. To do this, we simply add/substract the forward/backward kernel and the corresponding Gaussian marginals to 1nR, which also does not need to be rewritten from task to task.
B LIMITATIONS
Our proposed method still encounters the following limitations: (1) RNC suffers from the common limitations of SMC, including bias from self-normalised importance weight when the sample size small, and low diversity in produced samples. Also, when the target is significantly far from the pretrained diffusion, SMC will not perform well. (2) One variation of RNC relies the assumption that our pretrained diffusion model is perfect. While this also the assumption for other previous approaches (Skreta et al., 2025), it will lead to biased amnealing/compooition results.
C SUPPLEMENTARY METHODS
This section presents supplementary methods and technical details that are not covered in the main manuscript:
In Appendix C.I, describe in detail how apply RNE for density estimation, and outline its connection to previous methods.
Appendix C.2, we extend the density estimation framework using importance sampling.
We then turn to the use of RNE for inference-time control. Recall that in Section 2.1.3, we discussed two scenarios: one assumes perfect diffusion model, while the other relaxes this assumption at the cost of more restricted design choices. We now provide additional details on these cases:
In Appendix C.3, we assume a perfect model and list several heuristic choices for the sampling and target processes (i.e., at, in Eqs. (13) and (14)).
20

---

Published as a conference paper at ICLR 2026
In Appendix C.4, under the perfect-modd model assumption, we show how RNC recovers FKC as a special case.
In Appendix C.5, we consider imperfect diffusion models. We explain how the SMC weight in Proposition 2.2 is derived and why it is only applicable in the reward-tilting setting.
C.I RNE FOR DIFFUSION DENSITY ESTIMATION
As we discussed in main text, a direct application of Eq. (11) is for density estimation: when T' = 1,
p- is tractable, typically as a Gaussian distribution. This leads to the following conclusion:
RN Density Estimator (RNDE). Consider a pair of forward and backward SDEs in Eqs. (6)
and (7) with drift j and V/2 which are time-reversal of each other. Let Y be the solution to an
arbitrary process with the same diffusion coefficient, going either forward or backward. With R defined in Eq. (10), the SDE's s marginal density Pt is given by
Pt = Pa (Y
(27)
Given perfect diffusion model, the RHS of the estimator is tractable up to discretisation error: P1 is a Gaussian density, and Ry can be calculated by the noising and denoising kernel with Eq. (10).
Connection to previous works. The relation in Eq. (27) coincides with the
SDE (Karczewski et al., 2024, Theorem 1), and equivalently, in their concurrent work, Itô density estimator (Skreta et al., 2024, Theorem 1). More precisely, their density estimator states that, for perfectly pretrained diffusion model defined in Eqs. (1) and (2) (e.g., or 6: F4 = fa and in = ft of V log Pt). letting Y be the solution to any backward process with the same diffusion coefficient as the diffusion model, log p. (Y,) follows the following SDE:
cl (logp.(Y)) = (v (((0) + v (X)) (J((Y)
of v V pog))(())) dt + V (Y) .r. (28)
2
Despite the theoretical equivalence, Eq. (27) is more flexible and practically applicable. Eq. (28) is only computationally feasible for diffusion models where ft is linear and its divergence term V ft is constant. By contrast, Eq. (27) can be applied efficiently to any bridge model whose marginal on one side is tractable, including stochastic interpolants (Albergo et al., 2023), bridge-matching models Jarzynski, 2008).
(Shi et al., 2023; Peluchetti, 2023), and escorted AIS samplers ( (Vaikuntanathan
Moreover, Eq. (27) provides an alternative- more flexible flexible-persper perspective on why the Itô density estimator remains valid even when any backward process generates Y: for perfect diffusion, the Radon-Nikodym derivative between the noising forward and the denoising backward process is always one and Eq. (27) always holds. From this viewpoint, there is even no need for Y to satisfy a backward SDE-it can follow processes in any direction, and the estimator still applies. Therefore, this estimator not only can be applied to estimate the density on samples Y, obtained from a backward SDE trajectory 1). but it can also estimate the density on arbitrary values Y: = yt, such as samples on a hold-out test set, like the cases considered by Kingma et al. (2021). To achieve this, we can simulate an SDE, forward in time, from yr, and apply Eq. (27) on this forward trajectory.
We additionally highlight that Skreta et al. (2024, Appendix D) also wrote down the Gaussian- based discrete-tima time estimator to derive Eq. (28). However, differently, advocate directly using the form in Eq. (27)-not not only because it's more computationally accessible, but also because the RND perspective behind Eq. (27) naturally facilitates enhancements via reference processes and importance sampling, yielding a more stable and accurate estimator, as we will discuss in Section 3.
C.2 RNE FOR DIFFUSION DENSITY ESTIMATION WITH IMPORTANCE SAMPLING
In the above sections, we make use of the fact that the Nikoymm derivative between process and its time-reversal is identically one. This provides us with an intuitive algorithm for density estimation. In this section, we provide an alternative approach for density estimation, without relying on the concept of time-reversal. Instead, it leverages the importance sampling perspective:
Proposition C.I. let Pe (xx) be the marginal density of X, It satisfying the backwards SDE:
dx, (X,)dtt Gdw, X1 P1-
(29)
21

---

Published as a conference paper at ICLR 2026
Consider a forward process Y, with drift uu and define R Eq. (10), have
pe(aa)
= xx]]
(30)
where the expectation is taken over the forward process within the time horizon [t. 1] conditional on Y, = It. When discretised with = t < t2
< tN
n= Paln+1 (Ye, Meats) Pu+1 (Y (X,,) Yes = I21
Pra (Tt,
E
Pix (Yix)
(31)
A detailed proof can be found in Appendix H.2. However, to provide more intuition, we showcase its derivation from the standard variational inference perspective (Blei al., 2017; Kingma et al., 2013):
Variational Inference Macros
Puthwise Counterparts
marginalise: p(a) =
plar, =jdr.
(s) p(Y, Y,,,,. )dYnN
condition:
p(z) = plols)p(z)dz,
(t((x) p(Yr, No.. Vin JdYnN,
p(x(=) o(=(ax)
P(Y, Yi New o(Y,,, DY, a)
re-weight:
p(x)
Cyta
p(a)
Pt, (2) (e
Pew (Kag)
Y. =x
RNEE
RNN.
We also note that Proposition C.I generalises the RN Density estimator in Appendix C.1. As in the setting of perfect rime-reversal, a single Monte Carlo sample suffices to recover pt. Hence, Eq. (30) degrades to Eq. (27). Proposition C.1 also offers an intuitive derivation of the reynman-Kac density relation proposed by Huang et al. (2021), as stated in the following Corollary:
Corollary C.2. (Huang al., 2021) The relation Eq. (30) can be simplified to
(f:V us (zx))))
p+(x))= = P. pi (z())) exp
74 = Jtt
(32)
where For represents a forward process with drift v.a.
C.3 HEURISTIC CHOICE OF de AND by FOR INFERENCE- TIME CONTROL.
After discussing RNE for density estimation, we now return to inference-time control. In Section 2, 1.3, we highlighted that we have the freedom to choose any of the sampling and target processes. this section, we list some heuristics that can be considered. We note that these by means exhaustive: any suitable heuristics can be used without altering the core SMC algorithm.
Consider the diffusion model defined in Eqs. (1) and (2) its SI characterisation in Eqs. (3) and (4). To align our notation with the standard diffusion model literature, recall that
v - 2 / /2 log = ft + (e o22/2V log
(33)
11 V 2/2 log = = - + o / log
Then, we may choose
Anneal:
a, ft + X logp: b, = + X log
(34)
0, ft 2 o.
2 2 v lo 1/g. (35)
Reward:
v log Pt + 2i b, ft
2
CFG & Product:
0t = ft X
V log (1) + xo V log (2)
b, = ft + x V log (1) + X 6.2 V log
pi (2)
(36)
where the hyperparameter can be heuristically selected or tuned, and gr can be designed//earned to approximate the h-traanffrmm (Uchara et al., 2025; Domingo-Entin Enrich al., 2024; Denker al., 2024) or set heuristically (Chung et al., 2023; Wu et al., 2023; Song et al., 2023b; Singhal et al., 2025).
*We emphasise the different hetween F.. and the time reversal of Eq. (29). The former directly runs in forward with drift ii, inducing new path measure, while the latter defines the same path measure Eq. (29).
22

---

Published as a conference paper at ICLR 2026
C.4 "FKC E RNC"
We now show the choices which recover FKC (Skreta et al., 2025) as special cases: Proposition C.3 ("FKC C RNC"). RNC with the following ar be and is equivalent to FKC:
Anneal:
ar Jt non V log be
(mo _e V log Pr. et (or,
Product:
a₂ ft no? Vlogpp (1) V loggt (2)
(37)
b. ft ioo) B) v log p! + V Vlogp! (2))
4 == {0r:
CFG:
at Jt of (1 - 3) V logh; (1)
v log (2)
be St: 6t Ct.
where 1 = 3 + (1 e and (= (1 (1- 3/2c/ for C 00, 1/2] following the definition in FKC (Skreta et al., 2025, Propositions 3.1, 3.2, 3.3).
FKC derives its weights via the Feynman- Kac PDE and then designs the sampling process to cancel the costly divergence term. Therefore, FKC has very restrictive design choices. By contrast, our RNC features higher flexibility in selecting these processes, yet still incurs no extra computational overhead, allowing to heuristically select process pair that may reduce variance (Jarzynski, 1997; Neal, 2001), Moreover, FKC requires deriving the weight formula for each task (anneal, product, etc), while RNC provides a macro-style "plug-and-play recipe for computing importance weights.
C.5 EXACT SMC WEIGHT FOR IMPERFECT DIFFUSION MODEL
We now consider the SMC weight for the imperfect diffusion model we discussed in Proposition 2.2.
Before discussing the results, distinguish two sources of error: (1) the pretrained diffusion model will not perfectly reproduce the training data distribution due to imperfect score and discretisation errors; (2) for RNC, when calculating the SMC weight using the relation in Eq. (11), this equation does not exactly hold due to imperfect rimerreversal and discretisation. The first error is intrinsic to the diffusion model and beyond control; aim to address the latter here. More concretely, define as the Law of samples under the imperfect diffusion model with discretisation error, and aim is generate samples from a: defined in terms of this Pt, following Eq. (15).
We still consider the time horizon [T, r'] as an example. To account for the error arising from both model imperfection and discretisation,
will conduct our discussion in discrete time with N steps 4 < l2 < ... < IN = r'. Let P#|441 (X.,1a) and ph (X,,, (x,,1)) be the denoising kernel for the imperfect diffusion model and our chosen sampling kernel, respectively. The SMC weight in Proposition 2.2 is exact. We repeat the result here for easier reference:
exp(r- (X,(+((X,=7)) IN Po (Xe,IXe.. exp(r= (p(r--(Xx==r) (Xtv=r)
W [[Tr]]
(38)
IT Paintz (Xe,IXe..t)
We now answer the following questions: Why is this SMC weight exact, and why does this only apply to reward-titingg?
First, analogous to the concept of time-reversal, we denote the posterior density for the diffusion denoising kernel as P2.Nji (Xtz.x (X1). According to Bayes's rule, we have
N-1
Pe, P2:NNI (Xi (X,,) = PIN (Xix) 3 II Ph (Xt, (X
(39)
Note that we do not know the tractable form of P2:N|I- However, as we will immediately see, we can cancel this term with our chosen target process and hence eliminate the need calculate it. Concretely, similar to the target process in Eq. (14), here we can also choose an arbitrary target conditional density g tagett (x1.x (x,), ), and the SMC weight is defined
exp(r, (X),,r)) Pr. (X,,)
garget (XI.N (x,,)
[ X exp(rr((Xex-))) PEN (Xex)
(40)
1==1 Pet Polat1 (Xe,, (X
23

---

Published as a conference paper at ICLR 2026
By Eqs. (39) and (40), we have
exp(r>(X,,,))) I Pajnt1 (Xe, (Xtes)) clangee (1 (Xaz.N (x,))
"[r,]] (X(x=r) P2INI1 (Xtx X1,) I Pnja41 (Xe, Xx,1X
(41)
The term P2:NI (X (11) is intractable, and quTraa we can freely choose without affecting the correctness of SMC. Therefore, we set glagel P2:N|1. these two terms will cancel and all terms left in the importance weight will be tractable. In continuous time, qlaree = P2:N|1 will converge to the time-reversed denoising SDE of the imperfect diffusion model.
important to note that the same cancellation cannot applied the annealing or the product case. In fact, the derivation is correct until the step of Eq. (41). Taking the annealing case as an example, this step gives us the following SMC weight:
Th Pulne (Xt, (Xto.)))
q garget XeaN (xe.))
"(rr'' X
(P2:N11 (X ta. (xtaxxX))) (x1))
II P Piintza (Xe, (
(42)
1 (Xezx (Xe,) to cancel the intractable term. This is because P2:N11 is normalised. We may set q (Xt,) PENN (Xezi (X,,) 3/Z. But Z = J V (Xtz (x,, )dXra.y. which is NOT constant but function of Xt.-
However, we then cannot set glage (Xt2, (x,,) P2.N11 &
D RNE FOR DISCRETE DIFFUSION
As we discussed in Section 2.3, RNE can be seamlessly adapted to discrete diffusion with Continuous Time Markov Chains (CTMC) (Campbell et al., 2022; Lou al., 2023; Shi al., 2024). To do first define the R quantity for CTMC.
Definition D.1. (CTMC R) Given a CTMC Y in the time horizon [T, r'] and rate matrices Qz, Qi corresponding to CTMC's evolving in different time directions in the time interval [T, r'J. we define
R (Y(rr)) = exp (J. O2 (Y,, (Y,, ds
Q O,O,,,y, Q. (Y,, Y.,
) (43)
Σ log
A,Y #Y
where Σ.y, *Y, sums over all points where Y, switches ("jumps"" between states.
It is casy to construct the marginal density estimator:
Proposition D.2. RN Density Estimator (RNDE). Consider pair of forward and backward CTMCs which are
of each other and with rate matrices Q2, Q'-
Let Y be the solution to an arbitrary CTMC, going either forward or backward. Then the marginal density Pt of the CTMC with rate matrix Q is given by
(X.) = (Yi) exp (S.
(Y,,, - (Y,, ds
(Y,
Σ log
Q₂ (Y,, Y,-)
S,Y
#Y
(44)
Proof. From (Holderrieth al., 2025, Proposition 5.1.) via reciprocating the RND we have that:
d dFP p d
(Yi.)) (Yt)
(Y) R (V
(45)
Then since dF² d =1, rearranging gives
P.(Y.) (Y)) (M..)).
(46)
n
24

---

Published as a conference paper at ICLR 2026
Notice that in discrete diffusion, one integrate the Kolmogorov forward equation in backward time numerically
Oept = -Q'
(47)
and then index into the vector Pt. with Y, obtain p(Y)), however this has the computational cost of (Number of Steps x (Vocabulary Size)²). Instead, our estimator can be run online whilst generating samples a cost of (Number of Steps Vocabulary Size), making our RNDE likelihood computation for general CTMCs much more tractable. We highlight that our speed gain holds even in settings where the concrete score is time- independent, and we are able obtain a matrix exponential solution to the Kolmogorov equation (Ou et al., 2024a). This is because the closed-form solution does not scale for large vocabularies, as it requires diagonalising Q/-
Unlike the Kolmogorov equation solvers, our RNDE introduces bias in practice due to the time reversal being approximate. To mitigate this, we can similarly derive an RNDE-IS-based estimator, which should coincide with the marginal density relation in (Campbell al., 2024, Section C.I.I., Page 23) used derive the ELBO objective in discrete diffusions from a continuous time setting.
D.1 RNC FOR DISCRETE DIFFUSION
Now that we have defined R for CTMC, our RN Corrector can be readily applied, yielding similar estimators to Lee et al. (2025), but generalising to more tasks such as annealing and reward-tilting
As with RNC for SDEs, we have the same setup.
1. backward sampling process: Dipe -A, Pt
a forward target process: 0ch, = B, hr
3. intermediate target marginal densities: ds which are function of pt satisfying d.p. Q7 Pr.
To give concrete example let write the weight for product:
Ro (x (X (X(w)))
Rngt (X (X(r))) } (X (xrr))
"[rr]]
(50)
Where Q7(0) Q(())(())))))))))))))))))) are the rate matrices of a CTMC and its reversal and B is the rate matrix corre- sponding target process CTMC, and A is the rate matrix of sampling process/proposal.
D.2 DISCRETISATION OF R FOR CTMC
As in the continuous state case, we can approximate as a product of discrete-time kernels.
IT Q' Pn|n+1 (Yen (Yn4s)
Ro (1
(51)
N-1 P/-1// (Venes (Yo..)
this can be seen formalised in Holderrieth et al. (2025, Appendix A), and the discrete kernels can be approximated using Fulers method (Campbell et al., 2022)
Potijn (Yitoe Yi) &y + Qr (Y,, Y(tAs) At At+ ooAt) O' PaIut1 (Yr Y(+s1) = dy. + D'+se (YosSe. Xi) At + +(At)
(52)
(53)
25

---

Published as a conference paper at ICLR 2026
E CONNECTION BETWEEN RNE AND OTHER APPROACHESS
perfect teve-seversal
Ió Density Estunator: Deasity Augmented SDE.
RN Density Estimator
Feynman-Kat Kac density relation
+importance sampling
special case
Feynman- Corrector
RN Corrector
Reward
Twisted Diftusion Sampler/
ciltngg
Feynmmnn Kac Steering
CTMC, special case
Guidance debiasing sampler
RNE Energy Regularisation
contimuous
Fokker-Planck Regularisation
-tiee
Fig 10: Connection between RNE and other density estimation & inference- -imee control & energy regularisation approaches. Methods with a grey striped background generally require divergence computation/estimatton and divergence-free options are available only in specific cases.
One of the key contributions of RNE is that it provides a unifying perspective, connecting several previously proposed approaches within single framework. While we have highlighted these connections throughout the method description, in this section, we present a concise summary:
For inference-time control, RNE recovers the Feynman-Kat corrector (Skreta et al., 2025) special case in continuous-time the Twisted Diffusion Sampler (Wu et al., 2023) and Feynman Kac steering (Singhal et al., 2025) for reward tilting, as well as the debiasing method by Lee al. (2025) for guidance in CTMC.
For density estimation, RNE is equivalent to the density-augmentted SDE approach (Kar- czewski et al., 2024) and its concurrent work, Itô's density estimator (Skreta et al., 2024) in continuous-time. When coupled with importance sampling, RNE further recovers the Feynman- Kac density relation as well as the diffusion density estimators proposed by Huang et al. (2021) and Premkumar (2024).
For energy- based training, RNE recovers the Fokker-Plane regularisation (Plainer et al., 2025), with simpler interpretation and cheaper calculation.
We summarise these connections in Fig. 10.
F ADDITIONAL EXPERIMENTS AND ANALYSIS
F.1 RNDE AND ABLATION ON REFERENCE PROCESS
To assess the effectiveness of RN Density Estimator (RNDE) proposed in Appendices C.I and C.2, we choose 10-D Mixture-of Soatture-of-Gaussiii Gaussian target with 40 modes, which was initially used by Midgley et al. (2022) to evaluate the performance of Boltzmann generators. Since we can access the analytical marginal density at any diffusion time step f, it is ideally suited for comparing different density estimation methods. Fig. 11, we compare four estimators: RNDE with reference, RNDE without www.eeneeesaallllnnbaaaeeeeeeeee
RNDE, and Itô density estimator (Skreta et al., 2024). We use the variance-explie explooiig (VE) diffusion with the exact score function, and we follow the discretisation schedule of Karras et al. (2022). For the reference process, adopt the same VE process starting from a standard Gaussian. The vanilla RNDE underperforms the Itô estimator; however, incorporating the reference process leads to substantially better density estimates. Incorporating IS further improves
reference,
26

---

Published as a conference paper at ICLR 2026
RNDE W. Ref
RNDE wo Ref
RNDE-I -5 wo Ref (N=50)
Ito Density Estimator
10
10
10
10
MSE
stap-200
step 440
100
100
107
Fig 11: MSE of the log diffusion density log Pt, against diffusion time t. Different colour represents different number of discretisation steps. We compare different approaches: RNDE with reference, RNDE without reference, RNDE-IS with 50 samples, and RNE Itô density estimator (Skreta et al., 2024). We use a VE process following Karras al. (2022) where E [0, 10), Pt + [(, 10 and po Pdata- Hence, the error increases when / gets closer to O.
performance, albeit at the expense increased computational cost. A more detailed analysis of importance sampling is provided in the next section.
F.2 ANALYSIS ON RNDE-IS
RIDE w.. Ref
RNDE iS w Ref (V=10)
RNDE-I5 IS w Ref (N=50)
RNDE-15 IS w Ref (N- 100)
2
stup-100
stop-200
10-
10
1.5
10
10-7
10-1
10F
10
10-2
10-1
10F
100
10-1
10-2
10-4
10F
10a
RNDE we Ref
RNDE-I we (N=10)
10)
RNDE-15 we Ref 50-
RNDE-I we (N-100)
stop -tep-100
200
10
10
10
1.55
30
15-4
10th
10-1
125
10
15-4
10
10-1
105
10
10-1
10th
10t1
125
10
Fig 12: MSE of density estimation by RNDE and RNDE-IS across varying sample sizes, shown with and without a reference process. Different colour represents different number of discretisation steps.
In this section, we provide a more comprehensive analysis of the performance of RNDE-IS and influence of the reference process. In Fig. 12, we show the MSE of density estimation by RNDE and RNDE-IS across varying sample sizes, both with and without a reference process. we can see,
when sample size is small (or without IS), using reference can significantly boost the performance; when the sample size is large enough, the reference will negatively influence the performance.
This behaviour is expected: when the sample size is small, reference used to address the instability RNE. This instability eliminated when the sample size is large enough. On the other hand, this reference will bring in its own discretisation error. This error is negligible compared to its benefits when the sample size is small, while becoming significant when the sample size is large enough. However, we stress that the application RNDE and RNC. we usually rely on estimation using just one sample, and hence reference is always favourable there.
motivated in Section 3, the
F.3 INFERENCE-TIME A ANNEALIN:: MORE ANALYSIS
0.
2.
22225
-279
1200
(a) Energy TVD
(b) Distance TVD
(c) W2
(d) Variance weights
Fig 13: Inference-time annealing for LJ-13.
27

---

Published as a conference paper at ICLR 2026
6:35-
Sample
3.0
0.30
p.3-
Sampla Energy
Onergy
1.-
0.5
0.20
6:11
8:15
6.2
8:0
SToompho Canerited
b00
STopppoo Conernted
-1.0
-30.
47. 4550
-425
-40.0
-1.5
-30
0.0
0.3
3.0
-30.0
47. -45.0
-425
-40.0
-1.
-05
0.5
80
1.0
Enerdy
Evergy
(a) Energy and samples for co
1.0, Cb
0.6.
(b) Energy and samples for Co = 0.8, cb 0.2.
Fig 14: Visualisation of inference- timee annealing on 10D Mixture of Gaussian target.
In this section, we provide a more comprehensive analysis of RNC with different Ca and cb-
We first consider the Lennard- Joness (IJ) system with 13 particles. We first train diffusion model for temperature hhgh = 2.0 and anneal down to Tow = 1.0. Similar to ALDP, we sample using a batch size of 500, and repeat this 50 times to collect all data. We chose this system it only has one peaked mode, showing different properties compared to the ALDP in the main text.
In Fig. 13, show the TVD between the energy histogram, the interatomic distance histogram, the W2 distance and the variance of accumulated weights. As we can see, the variance of accumulated weights features the same pattern as ALDP in Fig. 2, showing that c. + o within = 0.2 typically achieves better effective sample size (ESS) compared to other choices. However, energy TVD exhibits the opposite trend: actually improves in regions where the weight variance large. This can be explained by the property of the LJ-potential. As the distribution has very peaked mode, we generally do not need high diversity in the samples-even though ESS is so low that all 500 draws in batch collapse to the same particle, the SMC weights will still single out the sample that best aligns with the mode. By contrast, when ESS is high. the algorithm may occasionally select a suboptimal configuration, potentially due the intrinsic SMC bias associated with a finite sample size, and can be amplified by discretisation error and model imperfections.
This can also be observed in toy Gaussian Mixture target. Specifically, we consider anneal 10D) GMM target from Tiihh = 1 to Tow = 1/3. We run RNC with the analytical score from the GMM target, using batch size of 500, and collect 100 batches in total. In Fig. 14, we visualise two settings (a) Ca = 1.0, a = -0.6, resulting in a low ESS; and (b) c. = 0.8, C. = 0.2, resulting in a higher ESS. In case (a), exhibits significant mode collapse, with the samples being closer to the centre of each mode. However, the energy histogram shows good alignment with the ground truth energy. By contrast, case (b) the samples almost cover all the modes, but a few particles diffuse slightly, and hence the energy histogram deviates more from the ground truth.
In summary, SMC intrinsically struggles with sample diversity. In terms of RNC, different choices of Ca. yield different ESS values. This ESS (weight's 's variance) pattern is highly consistent across different targets. In general, a larger ESS preferable to ensure greater diversity. However, for certain targets, lower ESS can also be advantageous. Although there is universally optimal choice, we argue that if preserving the entire distribution is the goal, one should aim for the highest possible ESS; conversely, if the objective is merely to select the single best sample, it may be beneficial to pick Ca, C that leads to a slightly smaller ESS.
F.4 EMPIRICAL ANALYSIS ON IMPERFECT SCORE AND DISCRETISATION ERROR
In Proposition 3.2, we discussed the theoretical guarantee of the SMC weight calculated by RNE when the discretisation error and score network error are present. In this section, we evaluate the influence of these two errors empirically, taking inference-time annealing on ALDP, for example. To analyse the influence of discretisation error, we choose to run RNC with 20, 100, 200, 400 discretisation steps in the diffusion generation process; to analyse the influence of imperfect score, we choose to early stop the diffusion network training stage after different numbers of iterations. report the heuristic choice by simply annealing the score without SMC. As can see from Figs. 15 and 16, the performance of RNC increases w.r.. the number of steps and the number of training iterations, as expected from Proposition 3.2. Also, while RNC performs worse
For comparison,
28

---

Published as a conference paper at ICLR 2026
when using small number of steps and when the score has a larger error, it still presents empirical performance gain compared to the heuristic choice.
Energy TVD
Distance TVD
Energy TVD
Distance TVD
1.0
0.15
0.75
0.04
TVo
0.10
0.50
0.05
0.25
100
400
19
400
20/00
ZRang
50.90
100
200 mod
Training iteratioo
Training iteration
score
-00 Anneal score (wo SMC)
RNC
Fig 15: Influence of discretisation error.
Fig 16: Influence of score error.
F.5 ABLATION ON REFERENCE PROCESS FOR BETTER DENOISING KERNELS
One line of work (Bao al., 2022: Ou al., 2024b) aims estimate the variance of the denoising kernel, resulting in denoising kernel that is more accurate than the one obtained from simple EM discretisation. A natural question is whether our proposed reference process still provides gains when used together with such improved denoising kernels. In this section, we investigate this empirically. Specifically, we evaluate the performance of RNE for density estimation (RNDE) on a 10D GMM with 40 modes, for which the marginal density is available in closed form, allowing us to directly assess the accuracy of our RNE estimator.
We visualise the results in Fig. 17. In Fig, 17(a), we show the results obtained using EM discretisation, while in Fig. 17(b), we estimate the variance Var[#t..,, æt, following Ou et al. (2024b, Theorem 1). We can see that using the estimated variance substantially improves the accuracy of the RNE estimator. Furthermore, even in this setting, incorporating our reference process still provides an additional boost, further reducing the error and yielding highly accurate estimates.
RNDE Ref
RNDE Ref
RNDE Ref
RNDE Ref
10
10
10
Slep -100
Stap -100
Stop =200
stop =200
m600
-000
10a
10:1
10a
10:
107
10a
(a) EM discretisation.
(b) With estimated variance.
Fig 17: MSE of the RNE-estimated log diffusion density log Pt. against diffusson time t.
G DISCRETISATION, STABILITY AND CONVERCENCE GUARANTEES
In this section, we provide additional details and discussion on the discretisation error and convergence guarantees for RNE with reference.
G.1 INTUITION FOR INSTABILITY WITHOUT REFERENCE
To understand the instability without reference process, let's consider the following example: given a diffusion model with a forward VE-SDE dX, gdW, and backward score SDE dX, (X, + &,W, c. the contribution to from the final denoising step (from At to 0) N(x, XA1 + (X, At)
exp g2 2 2
(0.
(54)
N( Xo, ct
If e, decreases quickly as with many noise schedulers (Nichol & Dhariwal, 2021; Karras et al., 2022) and if eo is small, the term / becomes large and unstable. At high level, this issue arises because, at each discretisation step, the variances of the forward and backward kernels are misaligned.
In fact, this misalignment also introduces accumulated error, discussed in Appendix G.2.
29

---

Published as a conference paper at ICLR 2026
G.2 CONTINUOUS FORMULATION VS DISCRETE GAUSSIAN KERNELS FOR R
In the main text, we introduced RNE in the form a limiting ratio between sequences of Gaussian kernels, described in Eq. (10). Another equivalent formulation, expresses RNE in continuous time in terms of stochastic integrals.
discussed in Eq. (12), directly
In practice, for finite number of steps N, these two formulations have very different behaviours. The Gaussian kernel formulation (without reference) typically suffers from higher accumulated error when the diffusion coefficient e, is not constant; while applying Euler-Maruyama Eq. (12) will have this issue. Specifically, we have the following conclusion:
Denote the result obtained by applying Fuler-Maruyama to Eq. (12) with steps as Rv, and the result obtained by Gaussian kernel formulation (without reference) as GN. then
2
-G
A N log RN - log GN R0 Σ 7
d dlog
E1
(55)
26
6D
where is the dimensionality.
RNDE continuous formula
RNDE Gaussian kemels (wo Ref) RNDE Gaussian kemels (w Ref)
step = 100
L2 crrar
step 200
step
10-2
10-2
10a
10-2
10a
Fig 18: RN Density estimator calculated by three different ways: Left: Fuler-Maruyama continuous path integral (i.e., Eq. (12)); Middle: Gaussian kernels (i.e., Eq. (10)); Right: Gaussian kernels with reference (i.e., Eq. (24)). Different colour represents different number of discretisation steps.
We empirically verify this Fig. 18. As we can see, directly using Gaussian kernels will result in significant error, echoing our earlier discussion. However, fortunately, by adding the reference process as described in Section 3, we successfully address this issue and achieve the best performance out of the different estimators and discretisations we consider.
In what follows, we will analyse and discuss this error in more detail:
We first consider Eq. (12):
log RM (Y(())))) = L:d 2 14- "..
L-¹ og 2 .u, dv,+
1 J."
2 of (11mil -||,)1/
(56)
et
For simplicity, we consider discretising R using an equidistant step size (At is constant). Discretise using Malder-Mruyamma and denote its discrete version as RN, we obtain
log RN
cg Vto+i (Yents Yr.) Σ
1
Me. (Yiuyi Yi.))
e
Σ
1
||
2 + Σ 11,,,11AMm 2 n
(57)
22 2
11
Note that since (1) is a standard Riemann integral without a stochastic integrator, we can discretise it by evaluating the integrand anywhem in the interval i.e. Σn - ||P::| || Aln, V [M,, (.)) This
can be done since the upper and lower Darboux integrals are equal to each other. Following Vargas al. (2023b) we choose Σn 2:,_. |l At, to be more consistent with how discretised the backwards integral and better align with the Gaussian density ratio below.
30

---

Published as a conference paper at ICLR 2026
Now, let's consider the discrete Gaussian estimator of the RND (which denote by GN):
log GN log II W (Yio.i Yt, Men (Yi, ) At, C, At!)
(58)
W : Menta (V At, Enai Atl
expanding (where d is the dimensionality):
Yint Yr. M4, (Y,, Atil?
[| Yt. + Vint (Yin- 1ttl?
Cla-1
+ d log
(59)
2
2
72
expanding again:
||Vent
Yi, 112
(Yint1 -Vi.) Me.
Pe., 11
Σ
(60)
2
26
+ Σ
|| Yi.113 11.
(Yents .in.) -Vntt
+
|| 1|2 12
+ log
Caat1
(61)
2
1n+1
2
Pnts
21
Ctn
n
we can then see that:
AN = log RN logGN GN
Σ
||
||
Ye, 112
d dlog
Cla+1
(62)
2Atc
2
Cin
(a)
(6)
From the convergence rates the total variation and the Fuler-Maruyama discretisation,,, we know that (up to an error in Vtt),
(Ytn4,i Yim..)
Cn+1 At,
(63)
where is the i-th dimension index, and the approximation can be understood as a.s. or in L2 and can be formalised as an upper bound, but for ease of presentation, we chose =. Substituting this back into (a), we see:
| - 11
de
At
dN
2 bn
(64)
2 2
2
k
and for (b), we have
Σ
||ares Yeal 112
d
2
2
2
2
_
Gn
2
(65)
72
11
combining the two
2
AN R Σ
d log
E1
(66)
2c
Indicating that the deviation between Eq. (12) and Eq. (10) can be large in practice when N is not large.
In the limit (Berner al., 2025, Lemma B.7), we do note that this vanishes as d
A ~5 To have more intuition on the magnitude of this error, we consider a VE-SDE with e, a' where a² var /var into Eq. (66).
é 2 6 de? d dog
d(lne)
In eo) - dog é] O,
(67)
to
26, a 2A)
dN
Varmaa I/N
Σ
N
(68)
2
tm
2t
2
2
varmin
n
and thus
d
d.N varmax
I/N
varmax
A N
log
(69)
varmin
varmin
we choose varmux 102 varmin = 0.0012 and deviations from log Ry when in high dimensions.
100, then AN == 0.87d leading to large
31

---

Published as a conference paper at ICLR 2026
G.3 RNE WITH REFERENCE CONVERGENCE RATE
G.3.1
CONVERGENCE RATE FOR RNE
We now show our reference-based RNE's convergence rate. Furthermore the analytic reference based estimator we use allows us proof such a rate without needing to bound the error of estimating forward integrals with backwards samples or vice versa, as the reference cancels the mismatch in integral directions.
Whilst proof for the non-asymptotic guarantees of the continuous-time RNE estimator without reference should be possible, it will require lot more technical effort, which we will leave for future discussion.
Let's consider the case where we use the diffusion model defined in Eqs. (1) and (2), where μ4, = fo is linear function, and v4 = μ4 o/V V log Pr. is the backward drift. We now choose reference process whose forward drift di = (4,. and the initial marginal to be Gaussian. In this case, we can analytically solve the marginal at any time step t and hence can access the analytical time-reversal of the reference process. Let's s denote the drift of this time-reversal as d/- = μ4 of log Rt.
Proposition G.1. Let us consider the case where μr corresponds to a linear forward drift, as is the case in diffusion models and half-sided interpolants. Then we chose an analytic reference by setting its drift as μr and my to be Gaussian, and denote its time-reversal drift as 1/2. Assuming Y has bounded TP moments, and calculating R with reference, it follows that:
log Ri, (Y(ssr)) - log R- (Yrrr1))l S O(VAt)
(70)
Where 11-11za denotes the L norm, Y is the Enler-Maruyama discretisation Y,, and log RE (M is our discretised RNE estimator
R (Yrrrr)) = Tr (8x) n= Pmlaa1 (Yen (etts I[ Patiin
(re)) (Ct
(71)
Ny. (Y+) II P Prim+z (Pe,, (Y,,,)) II Ph+zjm (1
Proof. In this setting, the reference has the same forward drift as our diffusion and hence will be cancelled, leaving only two backward processes: one for diffusion, one for the reference. Therefore:
Tr (Y,)
1
log RM (X.r))) log
+
J of (v (1)) dr,
vo)
2 J of (110112- 1 (72)
n.. (Yn))
Tr (Y+)
pr'
Ii - welldtt + {" (ur - we) .Ww,
log
(73)
प., (Y,r)
2 oī
First, let consider the EM discretisation of Yt,
Yr. = Yemei -enti (Krurs ) - Ototi (W - Win): flatt
(74)
Yr, Pem1 poa
Menli Main(Yann (Bum )ct
T jaw,,
(75)
Now we can embed this discretisation into continuous time Y = En 1 Yr,, which we can express in integral form as,
Ye = Yr Yr- Do (Y())sss L.¹ a,
(76)
N where DY,).) = En A Ma++ (Feme)) and o. Dn [1 We also define in the same way, now by Lemma G.2 we have that
TT,
log R. (Z/rrr)) = llog
Σ
(Vin- - -ea-n )YG))(W...)......))))))))))))))) - W(,))
T (87,)
Oents
n
1 Σ of
1| Nea11 (1 )tt,
(77)
2
32

---

Published as a conference paper at ICLR 2026
Then, via Remark G.3 we have
log Ri (1,,11) log Rt (Y(r) 122 lo ((.r))) - log R. (Y(r+)))
where via Jensens inequality (i.e.
la +
3e
2 - (9 + ||||| 2 9||c|13) we have that
log (Y(.r) log nt
(% (1,r1) - - 3( + B + C)2
(78)
where (for brevity in this step we have assumed a time homogenous d, however its easy to see how this extends the time dependent setting)
A
((D, (Y,) (v)))) (vs 1 -6.))) dw
(79)
B = KgE
(0, - -.,)(( -wV, v.) (Y,))
(80)
C EE log n. (Y,) log क | + ||| log म. (Yr)) log n. (Y+)
(81)
now applying Itô's isometry:
1
A ==E
((V, - (T.( - (v. - (Y.
2o
[1( - (Y.)) (V. - ...
(82)
20
then by the Lip property of the SDEs coeficients, the strong convergence of the EM scheme (Kloeden et al., 1992), and that 0; > 0 (i.c. we bound fh,//.)h,dd & max, + hhds)
ASL
EIJY, Yallds < LAC,
(83)
For B, let's label f = (V,- vs), (D.-Wa):
|| - ff lI(F - f)(f f113.
(84)
Then we can use Cauchy-Schwarz inequality:
EIʃ of II( f)(F + f)([sss] & Ki ( ElIICF - EEI((f
1/2 (f." f)1
1/2
E[II(F + dd
(85)
Via the Lipschitz property of f, we have that
[ E(II(T ]d S K₂ [- ElIYY, Y,11*]ds,
(86)
inti E[ Y, [jdss
K1
(87)
Remark 1.2 in Gyöngy & Rásonyi (2011) (LP convergence of EM scheme) implies that:
E|||Y, Yall'ds KAAP,
(88)
and thus
( ElIIC (110--1119)
1/2 ds < KaAt.
(89)
33

---

Published as a conference paper at ICLR 2026
Also, (f- (11 bounded from Novikov's condition (which we assume all through- out) Jensensen's incquality. Thus,
(90)
Given log coneavity for #y (as it the density of the analytic reference, which is Gaussian), we have that
log n (Yr) log क. (Yr) S V lo (Y, - -Y))
(91)
thus by Cauuhh-Schwand
- log nT, (Y+)1 - | log (Y+)|1/[[,, Y-1191.
(92)
Applying Cauchy-Schwarz inequality, we have
TX Y-11 E|l/ log ,, (Y, 11 (1 - 111141//2
(93)
then since log n, linear, it follows that E|| log n, (Y, 1|///// is bounded and thus
C < HAI
(94)
Here L, Ko. K1, K2, K3. K4, K' and H are constants.
Now we introduce a few auxiliary results we needed to derive our convergence rate. Lemma G.2. this setting, the discrete RNE estimator with reference,
N-1 P (Ye, (8en))) II Patiln ( (Yaa)
N-1
Tr (X,)
Rd (8
(95)
Pnia+1 (Ye, D ( II Patiln (Yrou (rm))
Simplifies to
1
log RNN (f(1s1))
log
+ Σ
o - (Vlnts (8s))) Pem)
TT. (Yr')
(v² (9tmm1)) - 122 (Pinz)))
(96)
2 of
Proof. First note that, as the reference has the same forward drift as the diffusion process, we have
11 Ph (Ye, (PrII)) TT, (L,,) ( Pripol (Pen (Pres))
((=)) I
R RM (8(1r11))
(97)
For brevity, we will use log R.,, by substituting in the Gaussian kernels we have that:
log N llgg (Ý=) Pimes V. Me, At11 TT, (Yrr)
|8m.. Y,, + Ve
(Z
At
2A1o?
2A1o?
(98)
Expanding squares and collecting terms yields
(Lr)
log R. log
Σ E (Minss Dener) -Yc.)
T.. (Yr!)
Σ
(v2 (Pemss) - 1/2 (Prne)))
(99)
2 n 0
Remark G.3. By embedding the discretised Ý into continuous time Y, will not change the value of R.N i.e. R (fpr.))) R (Vrr)))
34

---

Published as a conference paper at ICLR 2026
Proof. Let's consider the continous time embedding formula of the RNDE estimator:
(Y+)
log RN
RRNNNNYNN (Y/,,)1) log
Σ
(vs
(Y.)d.
(100)
TT.(Yy')
In
0
1 in+1 1
Σ
(v.
(Y.))
(101)
now by construction inside the interval [tns tntil the integrands are constant and thus
log RN (Pprrr)) = log
T Tty (Yri)
(Jent1 Ventz))(s,
/ In
dy
(102)
1n
2 L. ds,
1 full
Σ F
1+t1 (Y,) 2 1 (Y.))
(103)
and,
(N (Yprr)) = log (T(
log R
+ ΣΣ
(Vtnt1 1/ents (P3)) - -Yr.)
(104)
Ty (Yri)
02 lat1
ΣS
Σ 02 02 (Wtuti (V,) W (Y,))
(105)
where by construction (Yenti Penn) (Yiuti - -Yr,) thus have embedded such that
log RN PRRRRNN (Mpmry)) = log
Tr (8r)
+ +E
1
(Ut V/ (1.4))((s))))))))) (P Pr..)
Tep (Yy))
0 1
Σ
(uZ, (Prm.1)) - -bb (Y, ,) = log R2 RN (P(r.19))
(106)
2
0
Int1
Proposition G.4. (RNC weights non-asymptotic error) We bound the error between exact discrete-time SMC weights (using exact likelihoods) and our RNC estimator,
log w exact (M)..r)) - loo w
RNC (Y(r,r7)lle= SO(A)1))
(107)
where
2r (Ý+) N-1 ph
Pn+ 1/n (Pents (Pa))
exact (brry)=
(108)
9- (Y,, I Po|n+ (Ye,, Vemiz)
and RNC (V (1rrg) the weight estimated by RN (Mrr)) (assuming perfect score).
Proof. Given the Lip assumption log Pr have:
log (X-)// (X,)) log (X-)p, S PIJY, Yr
(109)
Now combining with Proposition 3.1 via triangle inequality we have that
log (Fr)/p- - log R.\ (f(rr)))=71
log (oggr(Y+)//oe (Y+)/p+ (Y+)) log R.N (1)m,) -11 + || pogppo/ (VgD-
log R", (Y (nrg) - log R (Prrr)llzz + PIDY. - -Y-1lıa,
P'VAL
(110)
( P. (..)))
Now moving onto the weights (for simplicity we assume (Y,)
where log g is Lipchitz. Note that this assumption covers all cases but the reward-tilting one. However, for the
(Y,)),))
35

---

Published as a conference paper at ICLR 2026
reward-tilting one, we can still get similar bound assuming chosen intermediate reward is bounded or Lipchitz.)
gr (Y=) 4, (Yr:)
log u exxct (Vrrrr) log w RNC (M Mrrilll
v log
gogg(R (% (Mpr))))))
122
pr (Y+) pr.((rr))
$ L log
log R (P(trr9))
I P' P'v
12
G.3.2 CONVERGENCE RATE FOR SMC WEIGHTS IN DISCRETE TIME AND APPROXIMATED SCORE
Proposition G.5. (Discrete time and approximate score bound) Following (Lee et al., 2023; Chen et al., 2022) we assume:
| log pr (
,f(r,)112> - S Escores
(111)
then
log w/ (8 (orr))lle S Rescore + P'vAt log w exact (Vrorg)
RNC
(112)
where
II N- Px l/n (Frntz
2r (Y,)
w exact (Y (1rr))
(113)
gr (Yr,)
II - Pm|n+ (Ye, (Bent))
RNC (f(ss1)) the discrete time weight estimated using RN (Mrror) with the learned score st. and wg
Proof. To prove this conclusion, separate the error contributed by discretisation and by imperfect score first, and then bound them separately:
log w RNC (f(+)1) - log w exact (Yrr))))22>
(114)
=o| Ry (115)1)) logp-( (&))/p(((r)))ll
(115)
<|| log Ry (1,zg)) - log RM (Mmr))))>= + log pr (Ft)/p-((r)) log R2' (B(r,)))
(116)
log R (((r)))) log R (frr)))le + A't
(117)
Where the last line follows from Proposition G.4. Following the time embedding of Remark G.3 and Lemma G.2 we have
log RN (8r,"1) = A + BN + A +
(118)
TT (Y,)
1
log
(U, (Y.)aw.
(119)
T (Y,i)
5,
4
(vs (Y,) (..).) (V. D. (Y.))s 2 f-
0g (1,, (P.) (120)
1
and
log R2 (M/rory)
= A + BB C
(121)
= log
Jr. (Y₂)
+
1
(
J.)
/े
- (P.)1
1 -
(122)
Tip (Yr))
0g
2
Therefore.
log RN (8(rr)) log V (Mrrr))l2z < Mn BN 11 +IIC2 C. + | 11,.. (123)
36

---

Published as a conference paper at ICLR 2026
Let's focus on the first term first:
1
[| BN
112
=
(0? (.)((Y JdW.
(124)
0x
1/2
E
1/0 Ds dss
(125)
Escore
(126)
For the second term, let us use the shorthands f D2
V and f 1z ~ then
11Cg CN 122
(11/211 11fa 2 dss
(127)
(5% +fa) 1 fs)ds
(128)
1/2
1/2
< E
112
f 11
I
11
foll
2
(129)
0
0
< F Eacore
(130)
Where E 1 ff + + f1llds
1/2
<
F is bounded from Novikov's condition (which we assume
all throughout) + Jensen's 's inequality.
Finally, the exact same arguments Equations 128-130 we that || s D' D'cooee- Therefore, in summary, we have log Rg (Mror)) - log RN (Yrrr))lea I Eéscore-
H PROOFS
H. CORRECTNESS OF SMC WEIGHTS IN EQ. (16)
Eq. (16) is correct SMC weight for marginal 9r. as for measurable function h on Xy: Ex I [h(X,)] Ex, [h(x,]] (131)
Ex Extrr Girer (A,,,,(
Running SMC with this weight in Eq. (16) has the following convergence guarantee:
Proposition H.1. Let (x(m), w()) (X (o. ) }-M- be the particles and their (unnormalised) weights
returned at the last iteration (denoise from To 0) of the sequential Monte Carlo algorithmm with M
particles described in Section 1.1.1. If the weighting function w(Ooro] is positive and lf F [w] (o Xxo] is bounded, then for bounded, me-measurable function h, have
|
M (m))
E
h(Xu)go(Xo)dXX
m=1 Eee wld 1(x(m))
S CCM-
(132)
E- (w/b [xo] is bounded, we have
(m))
lim
h(x (m)
as |
(133)
Σ w(i)
Proof. Eq. (132) is direct Corollary of (Mbalawata & Särkkä, 2016, Theorem 3.4). To apply Theorem 3.4 of Mbalawata & Särkkä (2016), we need to verify Assumption 3.1-3.3. Assumption 3.1 is satisfied as define the marginal qi to have bounded density. Assumption 3.2 is satisfied as our scheme ensures their Eq. (7) to be 0. Assumption 3.3 is satisfied according to our assumption. Similarly, Fq.. (133) is direct Corollary of (Mbalawata & Särkkä, 2016, Theorem 4.7).
37

---

Published as a conference paper at ICLR 2026
Discussion Assumptions: For convergence in L we require the following 2nd Order Novikov like assumptions to satisfied
E exp (f
|xx
<C
(134)
here a= represents all drift terms we use in our algorithm (at, be, th, Φ. M. u, etc.). For the a.s. convergence, we additionally require E[exp (2 Jr (x=(X)))ldd X C. Note that, for applying Girsanov's Theorem, already require the following assumption:
E exp 2 J. 1| (x.)(
X <C.
(135)
exp (5 ||u (x.)) xx]
We
note
that
assuming
or
4eep (1= 2||u* (x.))ltt) and hence Proposition H.1 requires stronger assumption than the standard Novikov required for Girsanov theorem.
[x < C' implies the first-order Novikov condition in Eq. (135),
H.2 IS PERSPECTIVES FOR RNE, CONNECTIONS TO HUANG ET AL. (2021)
Proposition C.1. let (x,, be the marginal density of X, = satisfying the backwards SDE:
dX, ve (Xx edW, X1 ~ P1-
(136)
Consider forward process Y, with drift u4, and define R. Eq. (10), have
pt (xx) E
(a)) mz].
(137)
where the expectation is taken over the forward process within the time horizon [t. conditional on Y = zt. When discretised with t = t < t2 < .... < <tN 1:
II n= P (Ye,, Mtmts) 1 n (Y,,, (
Pty (Ftt
E PtN (Yix)
Vn = Itg
(138)
Proof. Let's define the path measure Eq. (136) as F. We also denote the path measure of the forward process Y, with drift u,, starting from some g., as Q. Recall the definition of R in Eq. (10), can see that
dF (Vr.1))
(c(Y)) dd Pi(Y))
(139)
Hence we have
dF (Yi)
(Yz)
(V)((Y)) (1 Y= =x
pi
[.11])
Y, Wx
(140)
de
Pr (Vi)
dF
E
(Yaat)
Yi Ft
(141)
1//: dQ)
E (Y,)
dl (e.111 (Yaa)) Y, x
(142)
(((Y))/ (Y.) d
-p((a)) dF (e.c.
(143)
p((x)).
(144)
Here we use P (s.1]] to represent the path measure F0 conditional on values X, = 24 at t.
38

---

Published as a conference paper at ICLR 2026
Corollary C.2. The relation in Eq. (30) can be simplified to Pr(zt) = Euappr pi(Zi)exp exp (/. V uy (2x (Zx)ddt))
Ze
(145)
Proof. Using the conversion formula (Vargas et al., 2023b, Remark 3) with Eq. (12), we have that
log (Y,a))
/
1
Vv dYe
/
.Y.
1
(1
-112011 )dt
(146)
G
= /
1
dY; + /
1
ue dYe
[| 11
-11
11 +ez V )dt
u
2
(147)
which for dV, dtdt + &dW, can be re-expressed as:
log Ry (Y(ea))
/ (n ur) dw /
-dW
1 1 ue vo.11 + V vy )dt.
1
(148)
Ey
Notice that, by Girsanov theorem, we have / (v - .dW - tor (blll |u w113)dt' log d
dF.
thus
(r(2)) = Z px(Z)) un (Z))dd ( v.
5z )
Zo =
(149)
Corollary H.2.
The RNE-IS relation can also be simplified to
Pe(xx) = E (x)) (I' 2. (ve . -5' llur 10 1 + V mrccdl)) |r x1
(150)
(-f, 2.222 vv 111 V vy )dt' |1= YY=rd Ize
exp [Iogpi()) (h)
20c
(151)
which recovers the density estimator of (Premkumar, 2024, Eq. 3.5) and the Variational objective of (Huang et al., 2021, Eq 16). Additionally, we note
E V.V vrdt' Y, Fi=z1 - [
(-Yo = =x4) vel dt'
(152)
and we will recover the divergence-free density estimator of (Premkumar, 2024, Eq. 3.8).
Proof. Eq. (152) follows Eq. (148) with Jensen's 's inequality. For the divergence-free estimator:
Ep. U. V mode xz
S. E E [V vel
(153)
f. / pu = xx (y(ddyd
(154)
VP' = xx) -v
(155)
/ Wol xx) 2.) xz) ex (v)dddd
(156)
(-|Y; x1) vel de'
(157)
39

---

Published as a conference paper at ICLR 2026
Additionally, if access the unnormalised version of Pr and P1. taking the expectation ps, will obtain Jarzynski equality (Jarzynski, 1997) and Escorted Jarzynski equality (Vaikuntanathan & Jarzynski, 2008), which be used to estimate the free energy difference between two states with learned transports (He et al., 2025a).
H.3 EQUIVALENCE TO ITÔ DENSITY ESTIMATOR
The RNDE in Eq. (27) is equivalent to Itō density estimator in continuous time:
d (lgg,, (v A(Y) (Y) (fi()) of
logg)(Y))))dt Vlogp.(Y) aY. (158)
Proof.
With expression of R in Eq. (12) and the conversion rule (Vargas et al., 2023b), we have
Pog))(Y{) = log Pi (Y1) + log R (Y1,1))
(159)
= =logp1(Y1) Pe dYe { &
2/2
Pdt +
1 vy S(
aY.
_ c.p
1
f
1
ex'
Er
2
(160)
{"
= =logpi(Y))
The dr. + .dd
I l|
Eg
&
J. Pe dy. 1 2
1 Eg
L. V pt
| dt
(161)
E
For pretrained diffusion model considered in (Karczewski al., 2024; Skreta al., 2024), set = ot. 14 = ft. and V = ft ofV log Pt. Therefore,
(Yi (X1) [ v Vlogpr(Ye) -Yr).dY. dY.
+ +f Vlogpe(Ye) (FelTr) Voogpe dt' + (( ('v V. fe(Yo))tt' (162)
o?
H.4 CONNECTIONS TO KEYNMAN-KAC CORRECTOR (SKRETA AL., 2025)
As stated in Eq. (37), for some special cases, our proposed RNC is theoretically equivalent to FKC. In this section, we will prove these connections. The proof directly applies Eq. (12) and the conversion formula (Vargas et al., 2023b) to the importance weights in Eqs. (17) to (19).
Before showing the equivalence between FKC and RNC detail, we need clarify one important concept. In our importance weight calculation, as shown in Eq. (16), we have made few important details implicit for the sake of brevity. In particular, notice the following more explicit notation for the RND:
b. b.g
d (mrr)) =
(XX(s))))))) = (X
(X
(163)
do
d5ag. [r,T']
where ( is used to denote that the target process has as its initial distribution and moves forward time from J to T', e.g.
dY, = (, di +e Y, (e.g Or = -.)) I [T,T]]
(164)
and similarly Fa.g-t [r.] used to denote that the proposal process has as its initial distribution q. and moves backward in time from r' to r. is important to notice, when simulating the target process 0.-9. from to r'. it will not necessarily result in samples following gr.. In other word, assuming two adjacent steps [s. and [T, 7'}, then T bb.g- is not the same continuing [ [r,rt]
2tb.4.
40

---

Published as a conference paper at ICLR 2026
to r'. Note this clarification needs highlighting abuse /dQp//Q2xx....a to denote RNDs at different time intervals without providing a time index on Q (only on X). In fact, we use it to represent a sequence of path measures indexed by time opposed to the path measure of the same SDE simulated within different time horizons.
In what follows, will demonstrate which choices of d and b, recover the FKC weights from the RNC weights, thus reinterpreting these weights as the RND between two SDEs.
The proof follows the same principle: (1) we first express the importance weight of RNC using continuous formulation defined by Eq. (12), and (2) apply the conversion formula (Vargas et al., 2023b) to convert forward and backward It6's integrals in the same direction. (3) If there additional terms, such as a reward, we will apply Itô's Lemma to further simplify it.
H.4.1 ANNEAL FKC
Proposition H.3. Anneal-FKC. states that, for a perfect diffusion model (as defined in Eqs. (1)
and (2) Eqs. (3) and (4)), one implement the following backward sampling SDE:
dX, = (f((X)) noo lo Pt (X1)) dt + ÇardWe.
(165)
and the importance weight Sequential Monte Carlo satisfies the backward ODE:
d log wt - -33 (1.))..((0)) f((x))
of
(XVlogm())))l dt,
(166)
where
n + 3)a <= -1 B)2a/B, Va E [0, 1/2].
(167)
This is equivalent to our proposed RNC (Eq. (17)), when
a₂ = ft na? log Pt.
b, = ft - (or} - Bee)) V log & = Co2)
(168)
Proof. We consider the SI characterisation of the diffusion model, defined in Eqs. (3) and (4).
Therefore, we have
V Vlgg = fit & - o logp:
= V +
1 2
(169)
2
& + o
1 Vo 2
V log = =
v log
(170)
Therefore, with the continuous-time expression of R in Eq. (12) and the conversion formula (Vargas al., 2023b):
d log Ry, 1t Et
... V Hodt +
(n Me) pr)dt
(171)
2c
.xx V. (si
- o² v
of v log )dt
Vog
Vlogpt dt V Vlog: (St
2
2
(172)
Similarly, we have
at 1 b,
1
log Rg
dX, bidt +
(ar - bz) + bt)dt
(173)
20?
= log
=
dx. V (J+ (11l - Be v Vlogrt) di
-BVlogpt (f= f ?ou) _e?
log pt dt
(174)
41

---

Published as a conference paper at ICLR 2026
Then, according to the RNC weight given by Eq. (17), we have d log wi = 3d log R, - dog Rg
(175)
- B.c + B 4 oî
v logat
Sa (no) - - log dl
2
(1)
o
Be v log Pt
loggtt
ft f--
V
log pr fet (oa)
dt
(176)
(2)
To compare with FKC, set:
a= (or,
7= (1-+(1-Ba0 -3a, 5= -=
(177)
We first look at (I):
Bf + 3 9 o
v logp: ft + (ou) BEt) log
(178)
2
of v Flog Pt + ?oo? Bo7 VlogPt
-(B 1)f;
(179)
3(1 + (1 - B)2a/ / 0) logpt
-B3 1)/2
(1+ 3)a)o
V logg
(180)
=(3 1)ft
(181)
We then look at term (2):
of
BeL v logp:
fx
v log Pt fit (ou}
(182)
2
di
V log P: + (an) O(
V log
(183)
2
2
0
(3 + (1- (1-3)aajo
V log + ((8 + (1 3
V log
(184)
2
2
-(3 1) V lognt
(185)
2
Putting things together, we have
log wi = -(3 1) (- 80, logp.1 V fe dt
(186)
which coincides with the expression by FKC.
H.4.2 PRODUCT FKC
Proposition H.4. Product-FKC states that, for perfect diffusion models, one implement
the following backward sampling SDE:
dx, = (f((..) - noo" log (X.) - ?o? logp: ( dt + So,dW,
(187)
and the importance weight Sequential Monte Carlo satisfies the backward ODE:
d log w/ - (23- (((()) o. log ( (X.) V Vlog (2)
(t (X.)
+ of
3(3 1) logr (1) (X,) V loggi (2) (x)]1) dt, (188)
where
n= 8 (1 3)a, $= (1
Va E [0, 1/2].
(189)
42

---

Published as a conference paper at ICLR 2026
This is equivalent our proposed RNC (Eq. (19)), when
az Jt ?oo (Vlogri" (1) + V Vlogrt (2)
b, ft - (oo} - e) log (1) + V logp!") (2)
(190)
= L02-
Proof. Similar the anneal case, for i {1,2), we have
(8)
(8)
d log 12al
& Pi
dx, - V Pi dl
(5)
+
1
(v( - ((( + +x())d
(191)
26
Vlogpt (1) -d,-- V. (i &
-
v loggi (8) dt
2
o7 v log Dt )dl
Vlogpt (1) (fr
(3)
(192)
and
d log Rg = at - bi
dx,- V d, + 2c; (a- - b,) + d,)dt
(193)
= 3(V log Pt.
(1)
+ log (2)) dX,
(194)
- v. (fr ion, - ( (V logp: (1)
+ V log dt
- 3V logrt (1) + v logg(?)) (2)) ( (on) - 3e?
(V logp! (2) + V Vlogg(")) (2)) di
2
(195)
Then, according the RNC weight given by Eq. (19), we have
d llgguy Wi= = 3
Σ
log R - d log R
(196)
(=(1,2}
(20 --V.
a of (Vlogpt (1) Vloggl - ft + (n07 -+e)( logp! ( V logr!") 21) di
(1)
of v
Bei
VV logr!" (1)
le
logyt (1)
St + ior)
) (Vlogpt" (1) V ( di
2
(2)
BV logpt
(2)
2 fe
log Pi (2) - ft + (nr
Be,
(Vogp
(V loggt (1) + v logpt 22) dt
2
2
(3)
(197)
First, let's look term (1); same as Eq. (178), we obtain (1) = (2 - 1) We now turn to term (2):
ot v Poggh Pt (1) - fi+ + (ouz
log (1)
V Vlogp (2))
fe
(198)
2
+ ion) - 3e?
=
Vloggt
(1)
(V log + Vlogn( (2)
(199)
2
2
=(3 -
v Pr
(1)
+ oi v logr?? (2)
(200)
2
2
Similarly, for term (3), we have
o² V
log
(2)
Jt + ior
3e3
(V
logpt (1) + V logp:") (2)
(201)
2
2
=(3 - 1)
V loggr
Pt
(2)
+ 3 at v loo Vi
(1)
(202)
2
2
43

---

Published as a conference paper at ICLR 2026
Therefore, putting them together, we have
log wt 8 Σ d log Pell d log Rig
(203)
(1,22
= - (23 - 1)f
(204)
BV logp!" (1)
(3
o7 V loggt (1)
+3- o.
V dt
2
of
Pt +3 + ot
3V log (2)
(3
log
(2)
v log (1) dl
(205)
2
V (23 ftl
(206)
07 V log (1)
1) ot
A log (1)
(3
+ (B
v Vlogg!
(2)
dt
2
2
1) ot
3V loggt
(3 2
logpt (2)
(
v logy!"
(2)
dl
Bo V log (1) V log (2)
(207)
1) ot
- (23 1) 3(3
V loggt (1) + V logpp 2 |
2
Bo, logrt (1) V Vlogr
(208)
H.4.3 CFG FKC
Proposition H.5. CFG-FKC states that, for two perfect diffusion models, one can implement the following backward sampling SDE:
dX, (fx(x.) (1 (1- V)o log (1) (X.) - -oo log log ( (x.) dt + o,dW, (209)
and the importance weight Sequential Monte Carlo satisfies the backward ODE:
o7 (33 - 1) log (x)) - Vlogp, (2) (X.))
loo w. = This is equivalent to our proposed RNC (Eq. (19)), when
(210)
2
a: ft f - of ( (11 (1- log p; (1)
+ BV logp! (2)) b, ft, 4 = ot
(211)
Proof. We directly consider 6 = at.
Similar to the anneal and product case, for i E {1,2}, we have
(2)
(1)
d log Peyl)
1/
14
dxx - V. ①
1
(1( 14 (1)) + 14e (
(
(3) (1)
(212)
of
20
of v log loggi (1) )dl
Vlogp( (1)
.xx V Fidt v log (1) (f+
(213)
and
a;
by y
1 (a: be)(aa + b.)dt
d log 11
dX, V b,dt +
(214)
o
20
=(1 B) log pt !1: + v log ) áx, V. f.dt
-((1 - ((1- log (1) + 2V log 21)) (A
(11 (B- logp!" (2) + +v (ollgg;))) ()) di (215)
2
44

---

Published as a conference paper at ICLR 2026
Then, according the RNC weight given by Eq. (19), we have
d log Wt = (1 - 3 yd log Ry + log R d log Rig.
(216)
= -11 3) logpi (1)
(r-
oi V (1) -
of 3(133\looppp (1) + 3V logpi (2)) di
2
2
of
ft of ((1 3)V log p + BV log 2))
- logo (2) ft
v Pog (2)
2
2
(217)
= (1 - 3)3 oz 2
(Vlog(()) logg!()) (v (Vlopp())
v logg logp dt
(218)
H.4.4 GEWARD-TILTING FKC
FKC also derive the reward-tilting formulas. Due to the update of their arXiv, they have two versions. In this section, we discuss that both are special cases of RNC.
Version In the appendix of FKC (V15, Skreta al., 2025, Proposition D.5), the authors derived FKC for reward-tilting. However, their conclusion requires the reward model to be twice differentiable, and it necessitates computing the Laplacian of the reward model in order to form the importance weight. We note that this reward-tille -illting formulation can be derived as special case of our RNC framework which in contrast Laplacian free.
Proposition H.6. Reward-FRR FKC states that, for the following backward SDE
dX,- = ux(X))dl - o,dW,
(219)
and the importance weight for Sequential Monte Carlo satisfies the backward ODE:
log wt [B.v((x)) (ui(X.) of (X,) 2 (vvr(())) + B, of Ar(X)) 8t
09 r(x,) dt
(220)
This is equivalent to our proposed RNC (Eq. (18)), when
== ut b, (X,,) + o log (X,) vro) with the intermediate reward rt Ber.
6 = =
(221)
Proof. For the processes considered in Eq. (219), 14 14 and 14 ue + of V log Pt. Similar the
anneal, product and CFG cases:
1/4 Pe āX, V u.dt (vr me) uc)dt
d log R²
(222)
of
207
Voggpt dx, V (n V logp: Vlogp: (mt + of 2 Vlogpiddt (223)
and
a; bi of
dog RB
dx.-
bpdt +
20 (a: bx) + bt)dt
(224)
= (Vlogg + 3 Vr)
ax, - -V. + o. V + 3,0
(w
-(Vlogpt + vrr Wt + ot V pr: + 3, o² Vr dl
(225)
2
2
Additionally, applying Ito's Lemma to rt Bir, we have
o2 dr,(X,) (),Ber(X,) + B Ar(x.) dt + (VV) Tx,
(226)
hhttps://arsvv.or//bb////// 0281
45

---

Published as a conference paper at ICLR 2026
Therefore:
d log w = d(rr(X()) + dlog R2 - log Ri
(227)
(gBer(X))) + Bt o7 2 Ar(xi) dt + -X
(228)
+ Voggp dx. v vu + of Vlggo V PlogPr (ue + of
v log
(229)
2
- (Vloga + P.FF) dx, + V. (u+ + Voggt + Vr)d
(230)
+ (Vlogp: + 3eVr) (4 + of
Vlogp: + +. ot Vr dt
(231)
2
2
(bBor(x))) + Bc o7 2 Ar(x)) de
(232)
V -V o Vlogp: (u + of v log ps dt
(233)
2
+V (4 + o Voogp: + =oo
(234)
+ + o 2
of
o dt
+ +
14
logg: + + 2
Vr
(235)
= (x1) + 8 2
Ar(xi) de
(236)
+ Vog 31 ot Vr + Rrr (4++ (w++
o² v log + +Pt of Vr dt
(237)
2
2
2
ot (...er(X.))+ + Ar())) di + B.r.
11 + o,2 log + B: o2
Vr
di
2
2
(238)
In a recent work, Chen et al. (2025) proposed and empirically explored a formula for solving inverse problems without the need for the Laplacian. Shortly after, in the updated version of FKC (V25, Skreta et al., 2025, Proposition D.6), the authors included a similar result for reward- tilting. By carefully designing the sampling process, they can cancel the Laplacian of the reward model. As with Version we now show how this reward- tiiting formulation can be derived as a special case of RNC.
Version 2
Notably, comparing the two FKC variants highlights RNC's greater design flexibility: FKC requires special design to eliminate the Laplacian term, while RNC relies single, unified formula that does not require the Laplacian, and hence supports a wider range of sampling processes, including the heuristic choices proposed by Chung al. (2023); Song al. (2023b).
Proposition H.7. for a perfect diffusion model (as defined in Eqs. (1) and (2)), one can implement the following backward sampling SDE:
0² dx.- (f((X.) oiV log (X.,) Bi vr(Xi))dt + o,dW,
(239)
and the importance weight for Sequential Monte Carlo satisfies the backward ODE:
d log w. (vV((X)) (Fr(X.) of [oop))())) r(Xx)) dt
0s, ot
(240)
-
This is equivalent our proposed RNC (Eq. (18)), when
02 3
be fo + 3 o Vr,
0, = ft ofV logpt
Vr,
4 = oi
(241)
2
2
htttps://rssvv.rr//abb 9v821
46

---

Published as a conference paper at ICLR 2026
with the intermediate reward ri Ber.
Proof. Similar to the anneal, product and CFG case, for the diffusion model, we have
of v Po
d log R loggp: P: åx, V fidt Vlogp: (fe
(242)
2
and for the sampling & target processes:
d dlog Rg a, be
1
dx, -.-V --bidt+ + (a. be) + bj)
(243)
of
20?
=Vloog Vr) dx,- V v.(ft 3g of
Vr)dt
o 2
-(V + Vr) (fr (n-
v
log Pt dt
(244)
Again, applying Itô's Lemma to rr = Ber, we have
o2 dr((), = - ()Ber(Xx)) + 31 Ar(xi) dt + &VV(()) -Xx,
(245)
Therefore:
d log wi = d(r) + d log Ry dog Ri
(246)
(x1) + B: o7 Ar(X.) di + xx. 2
(247)
of
+ Vloger dx, V fidt - Vlogp: (f:
v log dt)
(248)
2
3 of
(Vlogn 3Vr) dx + V (J+
Vr)dt
(249)
2
+ (100g+++VVi) (n- (ni-
ot
v Vlogmt st dt
(250)
2
= (0(8((X)) + Be of
Ar(X)) dt
(251)
2
o/
V Sidi - Vlogp: (J+
v Vlogg.d
(252)
2
+V +V. B1 of
Vr)dt
(253)
of v VoggPt dt
+ (Vlogpt+ + 3.Vr)
ft
(254)
2
o2 v VoggPt
(x)) + Vr.
dt
(255)
H.5 CONNECTING RNE ENERGY REGULARISATION WITH FPE REGULARISATION
Our proposed RNE regularisation is connected to the Fokker- Planck Equation (FPE) regularisation (Plainer et al., 2025) in the limit. We assume the diffusion model's noising drift is ft and the denoising drift is ft ofV log Pt. To make the discussion easier, now swap the diffusion direction, that Po corresponds to the Gaussian side, and p1 corresponds to the data side. Therefore, the diffusion's noising and denoising processes are given by
dX,- = - (xt)) + o,W, N1~M
(256)
dx,- f((x))dt t Voggo (t t, Xo po
(257)
47

---

Published as a conference paper at ICLR 2026
We first recall the FPE in log-space:
of Vlog of
log (X,, V (Xi)
Alogpt =0
(258)
2
We then look RNE:
log (ogptts(Xx+a)) - (ogp:(X,) ++ 1
H+ fx(X,) dx,
V log dx,
$% A fx))) ax,
(+
r+
logp. fall
1flll2ds (259)
20
20²
Using the conversion rule (Vargas et al., 2023b), have:
++
log log::+s((X+aa) (X) =
0 logps dx,+ +
fx(X))ds
(260)
++t
V log
follads
||. 2
(261)
20
20
When At + O, we have
1 dlog Px(, oz of logp: dx, V fx())dtt 10 log log 2
V
log Pr fidt
20
(262)
Due to Ité's Lemma, we have
of
llogp:(X) = 0e (o +
A Alogpidt + V (Xg);(X)) .X,
(263)
2
We can hence write the RNE relation as
of A po Vlag -XX
a lo
V logn, dx, V fa + 207 lo lo V Po = 0 (264)
ot
which gives us the same expression as the FPE relation.
I ADDITIONAL EXPERIMENTAL DETAILS
1.1 ADDITIONAL DETAILS FOR INFERENCE-TIME ANNEALING
Network and Diffusion Hyperparameters. For ALDP and LJ-13, we use the EGNN (Hoogeboom al., 2022) with 4 layers and 64 hidden units. Following Karras et al. (2022), parametrise the network as "denoiser" to output the mean value given noisy samples. We also rescale the input by Cin and add skip connections following Karras al. (2022). For GMM, calculate the analytical score instead of training diffusion models.
We choose VE-SDE: dX, / where t E [0.001, 10). We discretise the time horizon according to Karras et al. (2022) with N 200 steps, i.e.
in = (o +
n (t1fa N
n 1, NN
(265)
Dataset. Alanine Dipeptide (ALDP) a target with 22 atoms, each of which has dimensions. The target is defined in implicit solvent, with the AMBER f196 classical force field. Following He et al. (2025a), we gather samples from a 5-mccrosecond simulation under 300K with Generalised Born implicit solvent implemented in openmmtools Chodera et al. (2025). The Langevin middle integrator implemented by Eastman et al. (2023) with a friction of !/picosecond and a step size of fentoseconds used to harvest total of 250,000 samples.
48

---

Published as a conference paper at ICLR 2026
Lennard-Jones (LJ)-13 is system with 13 particles, with Lennard-Jones potential between all pairs of particles and j. Concretely, the entire potential of the system is defined as:
N
U= Σ U X,I)> iF
+
Σ
X
x
(266)
n=1
N'
where
(
12
Un(r) de
(267)
Here
N En=1
Xn - Ew= is a harmonic oscillator.
FKC Details. Skreta et al. (2025) discussed two choices for annealing: target score simulation, where one rescales the score by the temperature, and tempered noise, where one rescales the diffusion coefficient. In our experiment for ALDP,, we report the former, we found the latter achieves significantly worse performance with severe mode collapsing. This is in line with their observation for GMM (see Figure 2 in Skreta et al. (2025)).
RNE Details. For all experiments, we use an analytical reference with the VE process where To is a standard Gaussian.
Resample Details. For ALDP and LJ-13, we run SMC with a batch size of 500, and collect samples by repeating 50 batches. For GMM, to have clearer visualisation, accumulate the weight along the generation process, and calculate Effective sample size (ESS). ESS is smaller than 75%, we will perform resampling and reset the weight of all particles to O.
collect 100 batches. We
Computational Resources. All experiments are run on a single NVIDIA H100 GPU.
I.2 ADDITIONAL DETAILS FOR MULTI-TARGET SBDD
Introduction. Structure-based drug design (SBDD) (Blundell, 1996) is main paradigm in drug seecovery-given a protein target (i.c. pocket), we aim to design (small-monl molellll ligands that bind to it. Recently, multi-target SBDD has attracted increasing attention to design ligands that bind to more than one target (Bolognesi & Cavalli, 2016). This problem can be formulated as sampling from the product of multiple diffusion models because of the inaccessibility ligands that bind to multiple targets (Skreta et al., 2025). We take the pre-trained diffusion model from Guan et al. (2023), which trained conditioning on each different protein pocket and generates ligands conditioned on a single target. We consider the dual target scenario with 20 pairs of protein targets, randomly sampled from the setting by Zhou et al. (2024). We validate the performance mainly based on the docking score calculated by Autodock Vina, with additional basic statistics and physicochemical properties (Eberhardt et al., 2021).
Dataset. We randomly sampled 20 pairs of protein targets from the dataset provided in (Zhou et al., 2024) with indices: (356, 233), (186, 341), (36, 333), (84, 41), (406, 169), (255, 39), (423, 45), (277, 262), (21, 334), (36, 121), (378, 143), (274, 307), (16, 143), (36, 345), (421, 420), (264, 26), (230, 70). (350, 137), (324, 423), (110, 39).
Statistics. The diversity is calculated as the pairwise distance between any two generated ligands (I - Tanimoto similarity of their Morgan fingerprints). The quality is evaluated by the percentage of ligands that have QED 0.6 and normalized SA score 2 0.67.
Experiment Details. For each target, we sample 32 ligands with size 23 following (Skreta et al., 2025). We take pre-trained diffusion models conditioned each protein target (Guan et al., 2023).
RNE Details. For all experiments, we use an analytical reference with VP process at stationarity, following Vargas et al. (2023a).
(1) (2) 3
Product Details. We consider the product of two diffusion models, i.e., do x Po Po In our experiments, select
shows the best performance according Skreta et al. (2025).
Resampling Details. For each protein target, we run SMC with a batch size of 32, Following Skreta al. (2025), the resampling is performed when & [0.4T, T].
49

---

Published as a conference paper at ICLR 2026
Computational Resources. All experiments are run single NVIDIA H100 GPU.
1.3 ADDITIONAL DETAILS FOR TRAJECTORY STITCHING EXPERIMENTS
Network and Diffusion Hyperparameters. We use an MLP of layers and 512 hidden units. We use the VE (EDM) schedule with preconditioning (Cin; Cout; Cakip) following Karras et al. (2022). Following Luo al. (2025), when training the network, we normalise the data to [-1, 1J.
Dataset. We use the dataset poi dataset of short trajectories length 64.
(Park et al., 2024) This is
Choice of Reward Function Our reward function need to impose (1) first trajectory starts from the initial position and (2) the last trajectory ends at the target, and (3) consecutive trajectories are connected. Here, we follow He et al. (2025c) to define our reward function. For easy reference, we describe the design choice below. We will first define the reward for the clean space (t = 0), followed by the reward Tt when t > O. For each of the 3 constraints, we impose combination of L distance and distance. For now, we use X to represent the i-th point the j-th short trajectory. We 1 represent the last element, following the index convention of Python. We use O and P to represent the initial and target points.
Reward for initial point: r = - oo (Ar [Ar=||Xx00 - -011 + X - - 0111)
(268)
Reward for target point: r" - -xp(A2 -1,1111
- 1
+ | 1,
1|--1 -PH)
(269)
Reward for neighboring trajectories:
(270)
AE A -x
-
(271)
where we set do Ap 100 J. AN =100, d12 and dr. 10. and the final reward is:
r=r + p
(272)
For the intermediate reward rs, we define it following: re(x x?, xX') B,
(273)
We use X to represent the sample at diffusion time step in the j-th short trajectory. E X?] is calculated by Tweedie's 's formula. 3r is a smooth function between 31 0 and Bo = 1. We set Br. |3 & (8)//r 3//03 and p 10. When sampling, we discrete the diffusion process
into N 600 steps and apply resample at every step.
RNE Details. We choose be to the standard noising process, and at be reward-guided process. More precisely, we add an extra term Vr, to the original score. We apply SMC with a batch size of 10,000. The SMC weight is calculated easily by RNC:
exp(r, (Lx , x! D)
II R (X
[ (XX
(1) t
X
(L)
W|rrr1| exp(r=([X))
(274)
Computational Resources. All experiments are run on single NVIDIA RTX 4090 GPU.
1.4 ADDITIONAL DETAILS FOR CTMC-RNE
Network and Difflusion Setups. We use the MaskGIT model (Chang et al., 2022) as our network. This model reproduced PyTorch by Besnier et al. (2025) and we use their pretrained model weight. This is model with two components, a VQ-GAN and latent model to predict masked token value conditional on the unmasked region. This model on its own not a diffusion model. However, similar to Ren et al. (2025). we can turn MaskGIT into masked discrete diffusion by introducing a stochastic masking schedule. More precisely, following Shi al. (2024), we introduce a stochastic masking schedule a,, Following the notation of Shi et al. (2024) where we use em to represent one-hot vector where element at the mask index is 1, the forward (masking) process is defined by
p(æ, (x.)) Cat
It
I (1
0tt
1e
(275)
a+
a2
50

---

Published as a conference paper at ICLR 2026
and the backward (unmasking) process is defined as
O2 at Em (MT) cm) )
p(rg(()) - Cal Ti
2it
(276)
1 Ot2g
where (as)GIT(z:) predicts the probability of tokens given input 21-
Choice of Reward Function. We define the reward function r by ImageReward (Xu et al., 2023).
takes text prompt and image, outputting score reflecting the alignment between the image and the prompt. We amplify the IR value by 10 to amplify the reward strength. Note that IR is defined over images, while our masked diffusion is in the latent discrete space. Therefore, every time we evaluate the reward, we need to first reconstruct the image by the decoder of the VQ-GAN. We define the intermediate reward (3,,) r(og), where &n is obtained by directly input x into MaskGIT, and taking the token with the largest probability.
RNE Details. When generation, we use CFG strength of 2. We choose b, be standard unmasking process (with CFG), and a, to be the standard masking process. We discretisation the unmasking process with 128 steps. We apply a batch size of 32, and resample at every time step except the first and the last one to aviod numerical issue. We apply both CFG debiasing and reward-tillue -iwardttltting together, and hence the SMC weight is given by the combination of Eqs. (18) and (19):
exp(rr(X+)) (xp(r=((X+)))
(RRl) (X(rr1))
( (X
"(r,z') X
(277)
1.5 ADDITIONAL DETAILS FOR TRAINING ENERGY-BASED DIFFUSION MODELS
To obtain X2 and X1+ for the objective Eq. (22), we simply add noise to the training data. Plugging in the definition of R, we obtain the regularisation term:
R1 = F a sgogo p" (xtl
log ( ax(z))
+ log Pt+ Ae (1++ log p: (278)
We can also use the reference process as introduced in Section 3, leading to the following objective:
R2= = Fr, - a sg
- (or
logg (xe+sel)))) + log (EL+Ar1)))) + log ne (x:) log (ogTTA((EEta)))
IogPt+s -ogggt log))
(279)
The entire training loss is
L -= Ezi
120 Eny aile, [eDSM + R1
(280)
where choose the strength AR 103 and At 10-4 and found these hyperparameters generalise well across difference targets.
For both GMM and ALDP, we use a VE process following Karras et al. (2022) (dX, = 22dW, where E [0.001, 10)) and take the inner product between the input and network output to obtain scalar value as the (negative) energy, i.c., log Pr (xx) R ge (æ,, t) = N (, t) Ir. where NN is the neural network, Gn is the rescaling factor used by Karras al. (2022). For GMM 100D, found it is beneficial to add another scalar network: Pe(ee) = go (x,, t) NC f) It N N22 (Ci t) where N N,, outputs scalar. We use a standard MLP for GMM and an EGNN for ALDP. We scale up the RNE regularisation by AR as it is generally close to O, which results in the following loss function:
L = E EE Exr, Ino Ez scla [esM + X RR].
fDsM
V Log 2) V (Iz. 1)
(281)
where we choose Ax = 10 and At 10 and we sample t by logt ~ (N(11.211.2 following Karras et al. (2022). We found this set of hyperparameters works well for both 2D GMM and ALDP.
51

---

Published as a conference paper at ICLR 2026
Details on Dual SM Baseline. Dual SM (Guth al., 2025) proposed Time Score Matching regularisation term, adding to the DSM objective, to learn a better energy-based based diffusion model:
TimesM || (ze, t) 0, log p(z t)ll
(282)
We use the same VE process following Karras et al. (2022), and hence log p( no. 1) (a (*) We also reweight the Time SM and DSM term following (Guth et al., 2025), leading to the following loss:
12 [ (DSM + &TimesM
1 dp
f = F, 2e
(283)
where d is the dimensionality. Note that the exact form of the objective is different from that used by Guth et al. (2025) because they assumed log p(r (20.) t) A (zzl200.) However, we follow their principle to ensure unitless of both Time SM and DSM terms.
1.6 BACKGROUND AND DETAILS FOR FREE ENERGY ESTIMATION WITH THERROODYNMMI INTEGRATION
1.6.1 BACKGROUND ON FREE ENERGY
We first provide a brief introduction the background on free energy, following the discussion of He et al. (2025a). For more comprehensive treatment, we refer the reader to He et al. (2025a). Precisely, the free energy is expressed as:
2 / exp(=U(<))dz
F = log Z,
(284)
where Ω C Rd, U is the energy function, assumed to be such that Z < cc. In many cases, rather than calculating F directly, one may be interested in the free energy difference between systems (or states) S. and S, with energies Ua and U. This is important for biological conformational changes, ligand-macromoleuul binding, or chemical reaction mechanisms (Wang et al., 2015):
AF Fb F. - log / Za)
(285)
Zwanzig (1954) reformulated the problem as importance sampling, where one system serves as the proposal and the free energy difference is estimated via Monte Carlo sampling. This is known as the free energy perturbation (FEP) method:
AF - log( Zz/Za) log Ea [exp(U. Ub)].
(286)
where we use E, to denote the expectation with respect to the equilibrium distribution pa (drr 2-1e T--e-UUalee -Ux()) of system S.-
On the other hand, the Thermodynamic Integration (TT) approach introduces sequence of distribu- tions that connects the two marginal distributions and estimates free energy difference as follows:
OF
AF
=
dt
(287)
J0
01
1 87 0t dt
(288)
Z,
f exp( -U:)
0U, )d 8t
dl
(289)
J exp(- -U,)d
TOU:
Em
di
(290)
In our experiment, we will aim to estimate the free-energy difference using the TI formula with a learned energy path U., similar neural TI (Máté et al., 2025).
52

---

Published as a conference paper at ICLR 2026
1.6.2 EXPERIMENTAL DETAILS
System Details. We estimate the solvation free energy for alanine dipeptide following He al. (2025a). Concretely, we consider the free energy difference between ALDP in the vacuum en- vironment and with implicit solvent, defined with AMBER f196 classical force field. We train our model with samples used by He et al. (2025a). The author gathered the training set from microsecond simulation under 300K with Generalized Born implicit solvent implemented in openmntoc) 1s (Chodera et al., 2025). The Langevin middle integrator implemented in Eastman al. (2023) with a friction of 1/picosecond and step size of 2 femtoseconds was used to harvest total of 250,000 samples.
Below are settings for S, and Ss:
Sa: ALDP in the vacuum environment;
Ss: ALDP in implicit solvent.
Similar to He et al. (2025a), when training the network, we rescale each target scale by 20, i.c., we define the energy U (m) Note that this will only change the scale of input and the score, with no influence on the free energy difference as long as we apply the same scaling to both targets.
Stochastic Interpolant Training Details. We train stochastic interpolant model bridging between S. and Sg The model has two networks, a vector field and an energy network:
Given pairs of samples (I mb) from systems S. and S., we first define an interpolant:
1 = aq + 3,0b + 7sE, E NN Id)
(291)
where on 1, 0; Bn 0, 1; and Ta 71 0 ensure proper boundary conditions:
4==0 = Ta and /==1 = Tb. AlAbbrgo et al. (2023), the vector field and energy path is defined as
ve (x) = |1 =
VÜr(x) = " [[ =
(292)
where the dot denotes the time derivative and El-14, xx] denotes expectation over the law of I, conditional on I, = x. Using the L formulation of the conditional expectation, we can write objective functions for the function U2 and VU, defined in Eq. (292); if we parametrize these functions
neural networks v," (I)) and Uz()., (z)) depending both t and x. this leads to the losses:
(v) 1) [delu (4: 1d (0) = E E 1 Ex..... [md VU; (T,) - ----1=7 [
(293)
(294)
where Xa and n) are weighting functions to balance optimisation across different times. In practice, we follow He et al. (2025a) to set X = 1 and 17. %
Additionally, also target score matching (TSM, De Bortoli et al., 2024) to enhance the energy learning following (Máté et al., 2025; He et al., 2025a):
TSM0 (0) = EL,i((0...)) C
(I.) - (v
(295)
CTSM. (0) = E mn. m [V.) -U,
(296)
At optimality, the following two SDEs become time reversals of each other (Vo, 2 0):
dx, = 0 VU + of (X,)dt + oe aB, Xo ~ Pa:
(297)
dx, (X + iy (X.)dt + or TB X1 ~ sbb
(298)
where pa and μb are the distribution defined via energy Uo and Ub.
We train the model with a batch size of 20. Also, to improve results and to accelerate convergence, we apply mini-bated Optimal Transport Tong et al. (2024) when sampling the pair (Ia, 27). We also follow He et al. (2025a) to use a different batch size for OT (500) and for training the network (20). We train both methods for 40,000 iterations.
We parametrise the energy network with inner product in the same way as Appendix L.5, i.c., U (x,,t) () = NN 1)- Ft. Note that in Máté al. (2025), the author add preconditioning
53

---

Published as a conference paper at ICLR 2026
ensure (-,0) U.(.) and (., U.()). However, this will require calling the target energy during training and will be less efficient (He et al., 2025b). Also, this requires designing a smoother parameter for t E (0, 1), which is easier for the LJ system considered by Máté et al. (2025) but non-trivial for ALDP. Therefore, we drop this preconditioning and fully parametrise the energy with neural network.
For the baseline without RNE regularisation, we only train the energy network using Lu + - TTM. I, For the results with RNE regularisation, we train both the vector field and the energy network, with additional RNE regularisation: Lu CTSM. 0 LTS.. + >x R, where
TSM.
R = E
((LA)))))))+++++ (299)
+ + (X)). We found the hyperpa- where μ VU, + e (X.), and = VO (X,)
rameters used in the diffusion setting generalise well here: At = 10-1 and Aæ 10 Different from the diffusion setting, we have the freedom to choose any o, 2 for the forward and backwards pair of SDE. We hence choose d; v0.2, Vt. We did not find that this choice had a significant influence on the results. Also, o, is a constant for any time step t, the instability issue discussed in Section does not happen in this setting, and hence we do not use the analytical reference here.
Estimation Details. After training the network, we estimate the free-energy difference using the TI formula. This estimation is identical for both the baseline and the RNE regularisation. One caveat, however, is that when training without preconditioning, the boundary conditions not satisfied. This not only makes the TI estimation inaccurate but also can introduce a constant shift at the boundaries. In other words, the free energies of U, and Ug differ, and the same holds for Ug and U..
To account for this mismatch, we estimate the free energy difference between Ua and Ug using FEP formulation as described in Eq. (286) with samples from Maa exp(-U.). Similarly, estimate the free energy difference between Un and UP using FEP with samples from jlb xx exp( exp(-Ux). Ub).
We then estimate the free energy difference between Uß and Uf using the TI formulation in Eq. (290). The final free energy difference between U, and U, will be the summation these three estimates:
A AFU., = Auu..u A Fugguy + AFug.
(300)
Note that Eq. (290) requires the sample from Pt. exp(-Uf)- However, we only have samples from Paa and 14a Therefore, we take the assumption that I, ~ Pt. where 7, defined as Eq. (291). similar Máté al. (2025). When the energy path is learned poorly, this assumption breaks down severely, resulting in inaccurate free-energy estimates. Conversely, a well-learned energy network improves the accuracy of the estimation. Hence, this serves good metric for assessing whether our proposed RNE regularisation provides improvements in learning more accurate energy network.
We estimate AFU.. and AFFe UUs using 5,000 samples each. For AFug.0g .uf. we use 5,000 samples with 1,000 steps, uniformly discretising the interval (0, 1). We repeat baseline and RNE regularisation 3 times, and report the mean and standard deviation in Tab. 5. The reference value is taken from He et al. (2025a), which was obtained with MBAR (Shirts & Chodera, 2008).
We also provide visualisation comparing Ua with the learned Ug and Ub with the learned U. in Fig. 19. In summary, the TI estimates reported Tab. 5 show that RNE improves the energy along the path, while Fig. 19 shows RNE improves the energy two ends.
54

---

Published as a conference paper at ICLR 2026
State Sa: 0.61
State Spi R 0.63
State Sai R²: 0.86
State Sui R²: 0.87
-20
-20
Energy
GT Energy
GT Energy
Energy
-2
-700
-680
-720
-700
-280
-260
-280 Learned Energy
Learned Energy
Learned Energy
Learned Energy
(a) Without RNE.
(b) With RNE.
Fig 19: Comparing Ua and U3 with the learned Ug and UP without with RNE regularisation.
RELATED WORKS IN SAMPLING FROM UNNORMALISED DENSITIES
is important to highlight that in the task of sampling from unnormalised densities, we have seen recent uptake in methods that exploit the RND between SDEs for Sequential Monte Carlo (Chen et al., 2024; Albergo & Vanden Eijndenn, 2024; Tan et al., 2025). Among these, Chen et al. (2024) is most closely aligned with methodology, it also the RND between forward and backward SDEs. However, their approach is not directly applicable to generative models, as it relies on access a geometric interpolation
the intermediate densities, which was manually designed in their between the prior and the target. In our case, these intermediate densities are not tractable, and this is precisely where our RNE framework comes in and provides a principled solution.
55


Reviews:
[{"review_id": "bxQ43LAGoj", "reviewer": "Reviewer_WESr", "summary": "The paper introduces the Radon–Nikodym Estimator (RNE). The authors make the observation that in diffusion, the forward and backward processes induce the same probability measure over the paths, which translates to their Radon–Nikodym derivative or, equivalently, their ratio, being equal to one. This allows them to calculate marginal densities using just the individual transition kernels, usually learned with a neural network, and cheap to compute. Using this estimator, the authors present different applications in inference-time control and regularization for training energy-based diffusion models.", "questions": "- Does the RNE compute a different weighting term than the Feynman–Kac formula? When are the two equivalent?\n\n- How does the number of steps affect the Radon-Nikodym estimator? If the sampling is performed with 10 or 20 steps, then the backwards and forwards kernels will be significantly different. If I understand this correctly, the estimator relies on the two kernels being 'similar enough'.\n\n- Are the importance weights for RNE \"better\" than the weights computed with FKC? Is there any intuition of why one should be better than the other? Do better importance weights work with fewer particles during sampling?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The proposed estimator provides a clean and elegant way to translate transition kernel probabilities to marginal densities. The idea of exploiting the intrinsic properties of the diffusion process to derive the estimator is novel and does not require any additional simulations, having potentially better scaling properties than previous approaches.\n\n- From my understanding, the advantage of the RNE over the Feynman-Kac Corrector (FKC) is that, compared to FKC, RNE provides an easier way to compute the importance weights for inference-time control. For FKC, the weights are accumulated over the backwards trajectory, which may be problematic, especially when introducing the necessary resampling in Sequential Monte Carlo.\n\n- Since there is no simulation required, the RNE can be computed for two neighboring timesteps using the forward and backward kernels, allowing the authors to utilize it in training an energy-based diffusion model. This extends the applicability of the proposed estimator to more applications than inference-time control.", "weaknesses": "- The main comparison regarding inference-time control seems to be with the Feynman-Kac Corrector. Although it is established that using density-ratio estimation scales badly with the number of dimensions, the FKC paper does showcase some results on high-dimensional data (images). The proposed RNE is only applied to low-dimensional settings.", "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": "eah5s8BxDm", "reviewer": "Reviewer_hh4L", "summary": "This paper introduces the Radon–Nikodym Estimator (RNE) to enhance inference-time control and energy-based training for diffusion models. RNE expresses marginal density ratios along the diffusion trajectory as products of ratios between forward and reverse transition kernels across time. The central idea is that a diffusion process and its time-reversal induce the same path measure, yielding a Radon–Nikodym derivative of one; this identity links marginal densities to transition kernels. For inference-time control, RNE supplies importance weights within Sequential Monte Carlo (SMC), providing a plug-and-play recipe that unifies several prior approaches. For energy-based diffusion, RNE serves as a lightweight regularizer that enforces consistency between model-implied marginals and transition kernels during training.", "questions": "Questions:\nOne confusing issues are in Section 2.1.1. To ensure the reverse-process marginal matches q_t, and if one interprets the importance weight as a rejection rate, why do the authors use self-normalized importance resampling (SMC) rather than enforcing exact q_t via rejection at each step under a suitable design of a_t and b_t (guarantee the rejection rate)? Under self-normalized SMC, with a finite number of particles, the empirical distribution of X_τ may deviate from q_τ, and this error can accumulate over time. How should M be chosen to control this error in practice?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Strengths:\n1. Generality and unification. The framework applies broadly across settings by leveraging Bayes’ rule and time-reversal: it covers diverse inference-time control tasks (e.g., annealing, reward tilting, model composition) and extends beyond SDEs to continuous-time Markov chains (CTMCs), demonstrating genuine plug-and-play utility.\n2. Comprehensive empirical evaluation. The method is validated on inference-time annealing (ALDP, LJ-13), model product for multi-target SBDD, scaling with particle count, and energy-based diffusion training (2D and 100D GMM, ALDP). The breadth and relevance of experiments substantiate the approach’s practicality.", "weaknesses": "Weaknesses:\n1. Limited algorithmic novelty for control design. While RNE offers a unifying lens over existing methods, the new algorithmic insights for inference-time control are modest. The experimental instantiations primarily focus on drift reweighting, leaving broader design spaces underexplored.\n2. Insufficient non-asymptotic guidance. Much of the theory is asymptotic, abstracting away time discretization and finite-particle effects. Clear prescriptions for step-size selection, resampling schedules, and particle budgets (M) are limited, leaving a potential gap between theory and practice in terms of variance, bias, and weight stability.\n3. Convergence guarantees are implicit. The convergence narrative relies on standard Feynman–Kac SMC theory and RNE discretization analysis, but the paper does not present a single, unified theorem asserting that, as M→∞ and Δt→0, the particle system converges to q_t and ultimately q_0 under the proposed RNE-corrected scheme. Stating assumptions and conclusions explicitly in the main text would improve readability and reduce ambiguity about the limiting operator and target-matching guarantees.", "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": "OIJKK5KUAy", "reviewer": "Reviewer_TNy1", "summary": "The paper introduces the Radon-Nikodym Estimator (RNE), a framework designed to unify various diffusion models by connecting marginal densities with transition kernels through the density ratio between time-reversal processes.  The paper also presents RNE as generalizing existing methods like FKC, TDS, and Itô density estimators, offering increased flexibility and computational efficiency.   Empirical results validate RNE across inference control, energy training, and density estimation on molecular dynamics, drug design, and Gaussian mixtures.", "questions": "1. Does the performance gain from RNC's flexibility (tuning $c_a, c_b$) consistently outweigh the tuning effort compared to methods like FKC? How can these parameters be chosen efficiently?\n\n2. How does the proposed method handle the numerical instability issue arising from the schedules in diffusion models as $\\sigma_t$ approaches 0? \n\n3.  How sensitive is RNC's performance to score model inaccuracies in practice?  Are there theoretical principles governing this sensitivity or ensuring robustness, apart from the specific case noted for reward-tilting (Proposition 2.2)?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The RNE framework offers a conceptually appealing and potentially unifying perspective on several recent techniques for controlling and analyzing diffusion models.  \n\n2.  The proposed RNC method appears to offer increased flexibility in designing the sampling ($a_t$) and target ($b_t$) processes compared to methods like FKC, which often have more constrained designs to avoid divergence terms.   \n\n \n3.  The identified potential numerical instability and the introduced reference process are validated by empirical results.", "weaknesses": "1. The introduced reference process, while stabilizing the RND estimator's variance, appears to increase the computational cost.\n\n2.  How reference process approach relates conceptually to methods that aim to directly minimize the conditional variance $Var(x_{t_i}|x_{t_{i+1}})$ or reconstruction error during the reverse process itself, rather than stabilizing the estimator of a ratio?  \n \n\n2. While RNC offers flexibility via parameters like $(c_a, c_b)$, the practical advantage over established methods like FKC seems conditional on tuning these parameters.  \n\n3. The paper proposes extending RNE to Continuous Time Markov Chains for discrete diffusion, but this extension is discussed briefly in the appendix (Appendix D) and lacks empirical validation in the experiments section.", "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": "Rk3dlltqCh", "reviewer": "Reviewer_zcEE", "summary": "The paper proposes a novel **Radon–Nikodym Estimator (RNE)** framework that unifies marginal density estimation, inference-time control, and energy-based training in diffusion models. By leveraging time-reversal theory, RNE connects marginal densities and transition kernels without explicitly solving Fokker–Planck equations. Building on this, the **Radon–Nikodym Corrector (RNC)** enables plug-and-play inference-time control—including annealing, reward-tilting, and model composition—via sequential Monte Carlo weighting. Additionally, RNE serves as a principled regularizer for energy-based diffusion training and generalizes beyond standard diffusion models to stochastic interpolants, bridge models, and continuous-time Markov chains.", "questions": "### A little typo\n\nI believe there is an issue with the use of Girsanov's theorem at line **1763**. The formulation of the Radon-Nikodym derivative seems to be incomplete, as it is missing the standard **exponential term**.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The paper presents a theoretically elegant and practically flexible framework for diffusion-based sampling. By introducing the Radon–Nikodym Estimator (RNE) and Corrector (RNC), it unifies marginal densities, transition kernels, and various correction techniques under a single probabilistic formalism. This approach enables plug-and-play inference-time control, bridges diffusion and energy-based modeling, generalizes beyond Gaussian diffusions, and demonstrates clear empirical improvements on tasks such as annealing, ligand design, and free-energy estimation, all while maintaining mathematical rigor and conceptual coherence.", "weaknesses": "### About the bound of the importance weight\n\nOne potential weakness of the proposed method is that the guidance weight w(t) used in the conditional or composite sampling scheme is not explicitly constrained. When w(t) becomes too large or varies significantly over time, it can cause several issues:\n\n1. **High variance in importance weights** – In the sequential importance sampling / SMC interpretation, unbounded w(t) directly amplifies the RN derivative, which may lead to weight degeneracy, where only a few particles dominate the distribution. This reduces effective sample size and lowers approximation quality.\n2. **Numerical instability or reducing diversity** – Large w(t) can excessively scale the conditional drift/score term, potentially causing gradient explosion or unstable particle trajectories in continuous-time SDE sampling. And excessive weighting favors certain modes strongly, leading to collapse of particle diversity and biased approximation of the target marginal qt.\n\n### About CFG\n\nIn your formulation, **CFG** is considered as\n$$\nd \\tilde X_t=\\left(f_t(X_t)-(1-\\beta )\\sigma^2\\nabla\\log p_t^{(1)}-\\beta \\sigma^2_t\\nabla\\log p_t^{(2)}\\right)dt+\\sigma d\\bar W_t,\n$$\nhowever, in some practical or generalized settings, the CFG formulation takes the form:\n$$\nd \\tilde X_t=\\left(f_t(X_t)-\\sigma^2\\nabla\\log p_t^{(1)}-\\gamma \\sigma^2_t\\nabla\\log p_t^{(2)}\\right)dt+\\sigma d\\bar W_t,\n$$\ni.e., \n$$\n\\alpha \\neq (1-\\beta).\n$$\nUnder this more general setting, certain derivations or propositions in the appendix (e.g., **Proposition H.4**) may not hold exactly.\n Could the authors clarify:\n\n- whether their framework can accommodate the case \n  $$\n  \\alpha+\\beta\\neq 1;\n  $$\n\n- and if not, what theoretical or empirical assumptions justify enforcing \n  $$\n  \\alpha+\\beta=1\n  $$\n  in CFG-like processes?\n\n### Reading difficulty / dense derivations\n\nSome sections, especially the appendix and derivations of the CFG and RNE formulas, are mathematically dense. Readers may find it challenging to follow all steps without carefully tracing the equations. Adding more intuitive explanations or diagrams could improve accessibility.", "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)