Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Authors: Marta Skreta
Tara Akhound-Sadegh
Viktor Ohanesian
Roberto Bondesan
Alan Aspuru-Guzik
Arnaud Doucet
Rob Brekelmans
Alexander Tong
Kirill Neklyudov
Venue: ICML 2025 spotlightposter (2025)
Keywords: diffusion models
Sequential Monte Carlo
classifier-free guidance
product of experts
annealing
Primary area: deep_learning->generative_models_and_autoencoders
TL;DR: Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts

Abstract: While score-based generative models are the model of choice across diverse domains, there are limited tools available for controlling inference-time behavior in a principled manner, e.g. for composing multiple pretrained models. Existing classifier-free guidance methods use a simple heuristic to mix conditional and unconditional scores to approximately sample from conditional distributions. However, such methods do not approximate the intermediate distributions, necessitating additional `corrector' steps. In this work, we provide an efficient and principled method for sampling from a sequence of annealed, geometric-averaged, or product distributions derived from pretrained score-based models. We derive a weighted simulation scheme which we call Feynman-Kac Correctors (FKCs) based on the celebrated Feynman-Kac formula by carefully accounting for terms in the appropriate partial differential equations (PDEs). To simulate these PDEs, we propose Sequential Monte Carlo (SMC) resampling algorithms that leverage inference-time scaling to improve sampling quality. We empirically demonstrate the utility of our methods by proposing amortized sampling via inference-time temperature annealing, improving multi-objective molecule generation using pretrained models, and improving classifier-free guidance for text-to-image generation.

Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Marta Skreta" * 12 Tara Akhound-Sadegh *3 Viktor Ohanesian * Roberto Bondesan Alán Aspuru-Guzik 12
Arnaud Doucet Rob Brekelmans?
Alexander Tong 748 Kirill Neklyudov 1 1749
Abstract
While score-basee based generative models are the model of choice across diverse domains, there are limited tools available for controlling inference- time behavior in a principled manner, e.g. for composing multiple pretrained models. Exist- ing classifier-free guidance methods sim- ple heuristic mix conditional and uncondi- tional scores to approximately sample from con- ditional distributions. However, such methods do not approximate the intermediate distribu- tions, necessitating additional 'corrector' steps. In this work, we provide an efficient and prin- cipled method for sampling from sequence of product distri-
QxO0)
axxx
1-1.0
I-
C- 0.0
aite)
annealed, caaometric-averggee butions derived from pretrained score-based mod- els. We derive a weighted simulation scheme which we call FEYNMAN- KAC CORRECTORS (FKCs) based on the celebrated Feynman- Kac formula by carefully accounting for terms the appropriate partial differential equations (PDEs). To simulate these PDEs, propose Sequential Monte Carlo (SMC) resampling algorithms that leverage inference- scaling to improve sam- pling quality. We empirically demonstrate the util- ity our methods by proposing amortized sam- pling via inference-time temperature annealing. improving multi-objective molecule generation using pretrained models, and improving classifier- free guidance for text-to-image generation. Our code is available at https: //gggghtth..oo com/ ma marrasktt// ffccffk--- di ffusion.
Figure I. FEYNMAN-KAN CORRECTOR Inference for annealed (t.s(z) x d=10
(=(x)==10 and product (z(2) x gl (x) g (x) densities.
1. Introduction
Score-based generative models, also known diffusion models, have emerged as the model of choice across diverse generative tasks such as image generation, natural language, and protein simulation (Saharia et al., 2022; Sahoo et al., 2024; Abramson et al., 2024). These models leverage the ability to estimate scores of the sequence of noise-corrupted distributions and then use the learned scores reverse the corruption process enabling high-quality generation. Thus, diffusion models aim to produce new samples from the same distribution as the training data.
However, the classical paradigm of generative modeling as the problem of reproducing the training data distribution becomes less relevant for many applications including drug discovery and text-to-image generation. In practice, genera- tive models demonstrate the best performance when tailored to specific needs at inference time. For instance, linear combinations of scores allow for concept composition (Liu et al., 2022) for increasing image-prompt consistency as in classifier-free guidance (CFG) (Ho & Salimans, 2021). However, by modifying the scores, one loses control over the marginal distributions of the generated samples. Various approaches from the Monte Carlo sampling literature have been adapted 'correct' samples along trajectory to more
"qqual contribution, ''Euual scnior-authorship 'University of Toronto Veecoor Institute 'MeGill University "Mila- Quebee Al In- stitute Impeeial College London Goooge DeepMind 'Université de Montréal "Current AT affiliation: Duke University "Institut Courtois. Correspondence to: AT <ayt14@duke.com edu>, KN <k.nccludov @ gomi.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Feynman- Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
closely match the preseribed intermediate distributions. Assuming access to exact score, additional Langevin corrector steps with the desired invariant distribution can be applied with additional simulation steps as the only practical overhead (Song et al., 2021; Bradley & Nakkiran, 2024). However, these corrector schemes are only exact in the limit of infinite intermediate steps. Accept-reject or Sequential Monte Carlo techniques may be used when the score is parameterized through scalar energy function (Du et al., 2023; Phillips et al., 2024), although these parameterizations require extra computation during training and may sacrifice expressivity in practice (Salimans & Ho, 2021; Thornton et al., 2025). While methods for sampling from mixtures or equiprobable regions of diffusion models have been proposed (Skreta et al., 2025), general solutions to accurately sample from combinations or temperings of parabeeepaa diffusion models with limited computational overhead remain elusive.
In particular, during training, one gradually destroys samples from the data-distribution Paua(z)) by simulating the following noising SDE:
dz, = fr + t,,dW, Ir= (t)nn()), (I)
where [+(++) is usually some linear drift function f+(x)) = a,I;, O- defines the scale of noise through time, and dW, - is the standard Wiener process. The drift fr and the diffusion coefficient o, are chosen so the final density is close the standard normal distribution pr=1 RI /N Ta)-
The generation process then can be defined as the family of denoising SDEs in the opposite time direction (t 1 7),
dx. (-J+(ma) + oiV log di + a,dW, (2)
where p: = P1-- is the density of the marginals induced by the noising process in Eq. (I); hence, the process starts with zy (V 0, Ig). By training a model of the score functions V log (c))). one can generate new samples from Pdnta (2) using Eq. (2) (Song et al., 2021).
To address these challenges, we introduce FEYNMAN- -ACC CORRECTOR (FKCs), which enable efficient and principled sampling from sequence of annealed, geometric-avvraged, or product distributions derived from pretrained diffusion models. To develop KEYNMAN-KAC CORRECTORS and test their efficacy, we make the following contributions:
2.2. Feynman-Kac PDEs
While Eq. (2) describes procedure for simulating individual particles, we can also derive Partial Differential Equations (PDEs) which describe the time-evolution of the density of samples (z()) under this SDE. We begin by describing the relevant equations for the standard SDE case. (1) Continuity Equation, which describes how the density changes when the samples move in space according to flow or ODE with drift Uz
We propose a flexible recipe for constructing weighted stochastic differential equations (SDEs), which account for additional terms appearing when manipulating the distribution of generated samples.
As our primary examples, we derive the correction terms for multiple heuristic schemes commonly used to approx- imate annealed, product, or geometric averaged distribu- tions, including CFG (Sec. 3).
Opt - -V. (w))(t))))))
de, ve vx(
(3)
Øt
where po indicates the evolution only according flow. (2) Diffusion Equation, which describes the change of the density for the pure Brownian motion with coefficient Tt.
To simulate these weighted SDEs, we propose a family of Sequential Monte Carlo (SMC) resampling schemes, which 'correct' a batch of simulated samples to closely approximate the intermediate target distributions (Sec. 4). For the problem of sampling from an unnormalized den- sity, we demonstrate that FKC allows for sampling from a variety of temperatures without retraining (Sec. 5.2). Moreover, we demonstrate that a high-temperature learn- ing, low-temperature inference scheme can be more effi- cient than the notoriously difficult task of directly training a sampler at lower temperature.
Opeiif (ar)
Ot = of diff (x).
dr. = o,dW
Ap, 2
(4)
where pitf denotes evolution due to the diffusion term only.
The SDE in Eq. (2) can be viewed as the composition of flow and diffusion terms, where the corresponding Fokker- Planckk PDE describes the combined evolution
Optdo(x) = - (V.pt" pt (x))(())) + of Ap."())). Øt
(5)
For pretrained diffusion models we demonstrate that adding FKC terms enhances compositional generation of molecules with multiple properties (Sec. 5.3) and classifier-free guidance for image generation (Sec. 5.1).
2
However, our main focus in this work will be to study a third type of PDE, which will yield weighted SDEs that we eventually use to simulate a sequence of marginals other those the forward noising process Pi-r (Sec. 3).
2. Background
(3) Reweighting Equation, which describes the change of density when samples have time-dependent log-weights w,
2.1. Diffusion Models
Generative modeling via diffusion models can be for- mulated as the simulation of the Stochastic Differential Equation (SDE) corresponding to the reverse-time process.
2

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
which are updated based the positions of samples Tt+
The reweighting equation adds an extra dimension to the interplay between different simulation schemes.
dw-; = (e(xx))t 7 Op.
= ge (x())
Continuity - Reweighting We first recast the continuity equation in terms of reweighting, in which case the simula- tion changes the density solely by adjusting the weights of samples (without transport),
Ot
where g.(x) = ge (x) f (z)))pp (x)
where the last equation guarantees the conservation of the normalization constant, i.e. f dx (e(x)p) (x) = O.
@p.(x) Øt -V (.) = (p(t)) (V (V (V.n(()))(())) pi(e))
Feynman-Kac Formula We now focus on the combination of all three components to describe the Feynman-Kae Kac PDE,
dw-- vv(,)) (V)gg(())) (x (12)
Diffusion + Reweighting We further observe that diffu- sion terms may be captured in the weights using
(x) 01
(E + of (x)
@p((2) of (p)( Ot 2
of pi (o) (z)
+ (x))) (x),
IV [Vlog:((=))
2
where to sample from p (I)), one first has sample 24 via the following SDE
du-- (A (ogg + V (ogg)(())))))) dt
(13)
In particular, using Eqs, (12) and (13) we now have an approach for translating arbitrary flow Us or diffusion oz terms into the reweighting factors, assuming access to an exact score function V log Such manipulations will play key role in deriving our proposed methods in Sec. 3.
(8) and then reweight the obtained samples using wt. Thus, pl (1) reflects the density of weighted samples, which dif- fers from the density ptt (z)) obtained via the Fokker-Plan Peaker-Planck PDE in Eq. (5) due to the addition of reweighting terms. In practice,
dx+- = (xa) + o,dW, du,- = b ddt
3. Modifying Diffusion Inference using Feynman-Kac Correctors
account for this difference by sampling
K
exp(wX+ K E
~ Categorical
In this section, we propose new sampling tools for combin- ing or modifying diffusion models at inference time using the Feynman K PDEs in Sec. 2.2. To this end, consider several different pretrained diffusion models with marginals { following
k=1
and returning ZT as approximate sample from pr. We discuss more refined resampling techniques in Sec. 4. For es- timating the expectation of test functions é, we account for the weights by reweighting a collection of K particles, i.e.,
(i)
Ögf = -V.
o V log di)) + of Agg. (14a) dr, = o Vog (z ) + o,dW, (14b)
- (V
Ot
2
exp(wot) E, exp(w?f)
Epp [o()]
E
((*t).
(10)
+
which is the denoising SDE from Eq. (2). Note that of may arise from training on different datasets or correspond to conditional models with different conditioning. Through- out this work, we assume access to an exact score model (;;; 0*) = V log qf (z), in part to facilitate the conversion rules introduced in Sec. 2.3 and summarized in Table 1.
For justification of the validity of this weighting scheme for Keynman-Kac PDEs, see App. A. The expression in Eq. (10) corresponds to Self-Normalized Importance Sam- pling (SNIS) estimation, which converges to exact expecta- tion estimators when K- so (e.g. Naesseth et al. (2019)).
2.3. Flexibility of Simulation for Given Marginals
At inference time, would like sample from modified target distribution involving these given models. While other variants are possible, we focus on the following examples:
Given a PDE describing the time-evolution of a particular density (x). there may exist multiple simulation methods. For instance, it is well-known that the diffusion equation (4) can be simulated using an ODE (Song et al., 2021).
1
Annealed:
Pt, (x)
Z,(B) (2)
Diffusion - Continuity Through simple manipulations, we can rewrite the diffusion equation using a continuity equation and change the simulation scheme accordingly
Product:
Pt
prod (x)
Z,
1
ai (z)
(15)
1
Geometric Avg:
(x) 2((3) at
@pe(x)
07 Ax) (V. v log pu((t) 2
0t
A common heuristic for sampling from the distributions in the form of Eq. (15) is to simulate according to the score function of the target density. For example, in classifier-free guidance (Ho & Salimans, 2021) we use the score of the ge-
= dx: = 0,
(11)
3

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
ometric average V Vlggg. h (1 -3V 3 + V log qi to simulate the following SIDE
In practice, of (1/00) may represent an unconditional model (or a model with empty prompt) whereas of (x/c) is con- ditioned on a text prompt, class, or other random variables (Ho & Salimans, 2021). Alternatively, in autoguidance techniques, & may be an undertrained version of a stronger conditional or unconditional model o (Karras et al., 2024a).
dx; = (-ft(zz) + V log 800 (rr)) + o.dW, (16)
However, despite the similarity to Eq. (2), this heuristic does not sample from the prescribed marginals (including the final distribution), except in special cases. We proceed by using the Pt.s example to illustrate our approach.
For our purposes, we will view CFG as an attempt to sample from the geometric average distributions (x) x (f (x)0 Using the conversion rules in Table 1, we derive the reweighting terms which facilitate consistent sampling along the trajectory.
3.1. Outline of Our Approach
To remedy this, we inspect the PDE corresponding to pi.g- which can be written in terms of the evolution of of and of
0pi xx)
1 de
Proposition 3.1 (Classifier- Guidance + FKC). Con- sider two diffusion models q (z), o (x) defined via (14). The weighted SDE corresponding the geometric average of the marginals pt.)(x) x qf (=x===0² [x) is
(17)
Øt
Ot (B)
Expanding and using our expressions for the Fokker-Planee Planck equation of of in (14), we proceed to locate terms cor- responding to the simulation of an SDE with the drift (xx))) = - ff(aa) + ofV log pf,
dr, = ((1 - 3)V log a (I:) + VV log of (x:))dt [1 + o,dWe,
Collecting all remaining terms of PDE (17) into weights Te (xxx we obtain the following Feynman-Ko Kac PDE, which can be simulated using the weighted SDE in Eq. (8), along with the resampling schemes described in Sec. 4
(19)
of B(B - 1)] 1)/l log (x:) V log of o dw,- 2
(x:))
0pt.0 (V,
(V. Pi,R Ut + oi APt + Pr,g Gt.
In Prop. D.3, we provide a more general formulation of this proposition outlining a continuous family of weighted SDEs sampling from the geometric average 200 (() x q. (xx)==0 q (z)². As a further example, we combine CFG with a product of experts in Prop. D.4.
(18)
0t
Conversion Rules To facilitate the construction of Feynman- Kac PDEs corresponding to existing simulation schemes, in Table we present the conversion rules that de- scribe how the corresponding PDEs change for the annealed densities and the product of densities. We use these rules as building blocks when deriving our practical schemes.
3.3. Annealed Distribution
Next, we consider a single diffusion model with the learned score V log 9: (z). which we use to sample from the annealed or tempered density
Pt, (x) (x(x)
panneal
(20)
Computational Considerations Our recipe above can yield many different weighted PDEs for given sequence of target distributions. In practice, we would like our simu- lation scheme to closely approximate the intermediate tar- gets distributions to limit the need for correction. On the other hand, for computational efficiency, we hope to obtain weights which avoid expensive divergence (V, (x)) or Laplacian terms ( (V, V log (z+)). Remarkably, for linear drift functions f(( commonly used in diffusion models (Song et al., 2021), we find that simulating according to the common heuristic in Eq, (16) yields a Feynman-Kae PDE whose weights can be estimated with no additional overhead. We focus on these schemes in our examples.
For 3 > 1, this can be used to generate samples from modes or high-probability regions of given models (Karczewski et al., 2025), while in Sec. 5.2 we explore the use of annealed inference in learning diffusion samplers from Boltzmann densities. The annealed target can be shown to admit the following Feynman Kac weighted simulation scheme.
Proposition 3.2 (Annealed SDE + FKC). Consider a dif-
fusion model (x) defined via (14). Sampling from the annealed marginals pfan promeat (z) qx(x)² , > 0 can be performed by simulating the following weighted SDE
dæ, = (-fe(xx)) + too V log (e(xt))dt + Sa,dWe,
3.2. Classifier-Free Guidance (CFG)
[x(rr)) (V. + of
o B B log uo)))()))) )at, di, 2
dw, = (3 (3- -
CFG (Ho & Salimans, 2021) is a widely-used procedure that simulates an SDE combining the scores of conditional and unconditional models with a guidance weight B,
with the coefficients (for (3 + (1 (oa-3)3a333300
r = 3 + (1 - (1-3)a, Ç = + - (21)
v log (x))) (1 - 3) of (x 0)) + sV log of (2r (c)
4

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Feynman-Kat Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Original FK-PDE
Original wSDE
Annealed PDE
Annealed SDE dr,
FK Corrector dwy +=
Proof
g(.)
ve(m))dt
(V.PLar))
vx(2x]dt
(8 1)(V, sx)dtt
Prop. C.1
(V. Pr.
8[3 - 1)( (V log 12 3x)eet
Prop. C.2
of Aq
o,dW,
of Apt.s
o,dW,
-B(B 1) o
2 V log @e di
Prop. C.3
23 Apt.
dW;
(B 1) 2 A log gedt
Prop. C.4
9:
dwy= = gedt
Bg.Pe.
Agedt
Prop. C.5
time-dependent annealing: 5 + 8t
0d.
log qedt 2C
Prop. C.6
Original FK-PDE
Original wSDE
Product PDE
Product SDE dr,
FK Corrector dwy +=
(V.gu)2) o Ag
1/4 dt
Pv. + xx)) o Ap [o] + 9p
(ut + xf)ett
((V log logst uf) + (V of vf))
Prup. C.7
a,dW;
o,dW,
of (V log V log qi dt (g) + gi)dt
Prop. C.8
gt 9/
du.-
1.2 dt
Prop. C.9
Table I. Conversion rules for different terms of the original Feynman- Kac PDEs (FK-PDEs) and the corresponding weighted SDE (wSDE), For every term term corresponding to the original densities qi (first two columns), we present the terms corresponding to the annealed marginals (z) x (x))) (top part) and the terms corresponding the product of marginals pe (x)) xx qf (z))f(a) (z) (bottom part). Importandy, the correctors are additive in the weight space, e.g. when transforming the Fokker-Plann equation, we transform both the continuity & diffusion equation terms and sum the corresponding correctors. References to proofs provided in the right-most column.
See Prop. D.1 for proof, and note that linear drifts fe(x) will lead to constant divergence terms which cancel upon reweighting in (9) and (10). We detail two choices of a.
conditions simultaneously.
Again, a natural heuristic is to use the score of the target product density in the reverse- SDE (2),
Target Score Simulation For = O, we have 1j ,9 and { = I, which yields the target score SDE whose drift corresponds to the score of the annealed target,
V log p prod (z) = V log यो (zz) + of
(25)
In the following proposition, we further combine these rules with the annealing procedure to present the weighted SDE prod that samples from the marginals p,, (x) x (ql ( (z))
dr. (-fe(x)) + Bo log (e )dtt + o. (22)
Tempered Noise Simulation For a = 1/2, we have v) = (1+33/22 = 1/vB). We refer to this as an SDE with tempered noise, namely
Proposition 3.3 (Product of Experts + FKC). Con- sider two diffusion models qe (z), of (x) defined via (14). The weighted SDE corresponding to the product of the marginals rhe (x) x (x) (x))² with 3 > is
3 +1 of log (x:))) + VB
de, = (-J((a) + 2
dW. (23)
We focus on these two choices of a, but note that for differ- ent 3, we found that either target score or tempered-non noise simulation could perform better in practice (Sec. 5).
dr, = oi (V log of (xx) + V Vlog (z)) - (x, + Ço,dW,
(26)
3.4. Product of Experts (PoE)
dw, = 8(3- - 1) 2
V log (x:) + V log of (xt)
Intuitively, samples from the product of densities corre- spond the generations that have high likelihood values under both models. The product can also be interpreted as unanimous vote of experts, since sample is not accepted if one of the densities is zero. Formally, consider the density
+ +00 (V log of (2+), V log of ( + -288 () S (d:
(27)
with the coefficients (for (B + (1 -9)2a)/33
11 3 + B)a, <= /(3 + -B)2a)BBB (28)
p. (rod = qe (z)/z.
(24)
For conditional generative models, the product of densities can describe samples satisfying several condi- tions. For example, in image generation, we could use q(x| ""orre"))(""""" "a sandy beach") to generate images of "a horse on a sandy beach" (Du et al., 2023). In Sec. 5.3, we demonstrate that the PoE target can be used to improve molecule generations which satisfy multiple
See proof in Prop. D.2, Again, note that for linear drifts, the divergence term (V, fx)) is constant and can be ignored. Further, for / = 1, the first term in the weight evolution vanishes to leave only the inner product of score vectors. Similarly to Eqs. (22) and (23) for annealing, we have the target score SDE (a = 0, = 8. = 1) and the tempered noise SDE (a = 1/2, 1 = (8 + (
5

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Feynman- Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
More generally, we derive the weighted SDE that samples from Pt,a (2: II, g (x)),, i.e. the weighted product of marginal densities of (z) for arbitrary number of diffusion models (see Prop. D.5).
all particles are preserved in the case of uniform weights.
Jump Process Interpretation of Reweighting Finally, by reframing the reweighting equation in terms of a Markov jump process (Ethier & Kurtz (2009, Ch. 4.2)), a variety of further simulation algorithms for Feynman- Kac PDEs are possible (Del Moral (2013, Ch. 1.2.2, 5); Rousset & Stoltz (2006); Angeli (2020)).
3.5. Reward-tilted Target Density
Finally, our framework can easily incorporate a reward function r(z) defined on the state-space at inference time. Namely, we assume that the function (B)r(())) is normal- izable and consider the reward-tilted density pf y7 (x) x qu (z:) coxp(3r((z)) Despite its similarity to the product of densities, this case is different we do not assume exp()rr(z)) changes according to the diffusion process.
A Markov jump process is determined by a rate function >((z), which governs the frequency of jump events, and a Markov transition kernel Je(y(z), which is used to sample the next state when a jump occurs. The forward Kolmogorov equation for a jump process is given by
Oy:
Bypupp (x) Ôt
(/>wW))ttttD - (((
Proposition 3.4 (Reward-tte -ilted Target+ + FKC). Consider a
diffusion model qe (2) defined via (14). Sampling from the
where the two terms can intuitively be seen to measure the inflow and outflow of probability due to jumps.
reward-tilee tilted marginals pt (f (z) x qe(2) (z) exp(Ser(z)) is
performed by the following weighted SDE
Our goal is to find (z). Jı(y(z) such that p! matches the evolution of p." in Eq. (6) for a given choice of gt- In fact, there are many possible jump processes which satisfy this property (Del Moral (2013, Ch. 5) Angeli et al. (2019)) We present a particular choice here, with proof in App. B.2.
dx.+ = oVVVlog + 3: vr(x;))dl-
(29)
- f((r))dt + o,dW,
dw,- = 03: (xx))dl ( Jelae)) fe(x:))dl+
01
(30)
+ + 3 Vr(ar) o (ogg
Proposition 4.1. For a given gr. in Eq. (6), define the jump process rule and transition as
Ae(x)) (9((x) Ep. (9o1)
(31a)
See proof in Prop. D.6. Here, the weights increase when the vector field of the diffusson models aligns with the gradient of the reward function.
(9e(8) 19cl pi(v)
Je(y/x) J (gx(z) (e pc(2)dz
(31b)
4. Resampling Methods
where (u) = max(0, -u) and (u)+ = max(0, u).
Then,
In this section, we describe several options for utilizing the weights to improve sampling with a batch of K parti- cles. While the simplest technique would be to simulate the weighted SDE in Eq. (8) for K independent particles across the full time interval E [o, 1] and reweight using SNIS in (10), we expect these full-trajectory weights to have high variance in practice due to error accumulation.
Oy jump (x) 0pk (x) = p. (z) (g) -Ep. (el) (32)
0t Øt which matches Eq. (6).
In continuous time and the mean-field limit, this jump pro- cess formulation of reweighting corresponds to simulating
Sequential Monte Carlo Since our weights provide a proper weighting scheme for all intermediate distributions (Naesseth et al., 2019), App. A), we can leverage SMC techniques which reweight particles along our trajectories.
2''s Fe+dtt { {~ Je(Vl))))
w.p. 1 - (e( + o(dt) w.p. (x + oddt).
(33)
We expect this process to improve the sample population in efficient fashion, since jump events are triggered only in states where (ge (z) Ep. (9()) 2 0 = (x(x) S Ep, 19). and transitions are more likely to jump to states with high excess weight (9x()) Ep. [9.)) > 0.
In practice, we find that resampling only over an 'active interval' E [/minn m is useful for improving sample qual- ity and preserving diversity, and set weights zero outside of this interval. Within the active interval, we resample at each step based on the increment (n(h)) = g((x(()) dt. using systematic sampling proportional to exp [x()) (Douc & Cappé, 2005). For small discretizations dt, we might expect relatively low-variance weights. From this perspective, sys- tematic resampling is an attractive selection mechanism as
In practice, pf (z) K Ek= rate (x()) and transition Jz(2)x:) Instead of simulating Eq. (33) directly, one can also adopt an implementation based on birth-death eexponential clocks' (BDC, Del Moral (2013, Ch. 5.3-4). see App. B.3).
an empirical approximation 1 (x(())) to approximate the jump
6

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Feynman- Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Starting Disribution
Ground Truth Samples
Target Score
Target Soore
FKC
Tempered Noise
Tenpered Noise
FKC
29
20
28
28
-20
-20
28
-20
28
Figure Samples from Mixture of 40 Gaussians.
Table 2. Comparison of EDM2+FKC (O) with EDM2+CFG for image generation using EDM2. We sweep over noise level (r)
EDMB
and steps (N). For all metrics, we report CLIP and ImageReward
(IR) scores averaged over 10,000 images.
FKC
CLIP (1)
IR (1)
FKC
N
CLIP (1)
IR (*)
10
32
28.74
-0.25
40
16
28.67
-0.30
10
32
28.97
0.03
0
16
29.12
-0.01
-0.24
0
40
32
29.00
0.04
32
29.14
0.05
80
32
28.76
-0.44
40
64
28.81
-0.19
80
32
28.99
0.04
40 64
29.12
0.07
Figure 3. Samples from EDM2+CFG (top). EDM2+FKC (bottom).
FKC resampling against vanilla CFG. We find that adding
5. Empirical Study
FKC improves performance in both ImageReward and CLIP score, indicating both higher prompt adherence and aesthetically better images. While this comes at the cost of extra computation due K 1, we find that FKC demon- strates benefits even for K 2, with K 8 performing the best (Table A3). Qualitative results in Fig. 3 further support the finding that FKC can improve image quality.
In this section, we compare our Feynman- corrector (FKC) resampling schemes against their corresponding SDEs without resampling. We consider both target score and tempered noise SDEs. While we show results for BDC sampling in App. F.2 Table AI, we proceed with systematic resampling throughout the remainder of our experiments.
5.1. Image Generation with EDM2
In App. F.5, provide additional analysis using FKC with latent diffusion image models.
In this section, study the effect of FKC resampling for image generation in RGB pixel space, using CFG with EDM2-XS model trained on ImageNet-511 512 (Karras et al., 2024b). In particular, we test whether resampling to more closely match the intermediate geometric average distribu- tions translates to improvement two downstream image quality metries: CLIP Score (Radford al., 2021) and Im- ageReward (Xu al., 2024). CLIP Score measures the cosine similarity between the image and text prompt em- beddings; IeageReward assigns score that reflects human preferences (aesthetic quality and prompt adherence).
5.2. Samplers from the Boltzmann Density
As described in the Sec. 1, our FKC inference techniques suggest flexible schemes for learning diffusion samplers at given temperature and sampling according to a different temperature. Since we are given an energy function in these settings, are not restricted to learning with temperature for for base model qt- Thus, use (Tz, Ts) refer to the learning (gt) and sampling target (Pr,a) distributions, with B = TL/Ts in the notation of Sec. 3.3.
For fixed simulation scheme, we compare the effect of adding FKC resampling (O) versus the standard baseline without resampling
Mixture of 40 Gaussians with Ground Truth g.¹ To verify our tools in a tractable setting, we consider a highly multimodal distribution where we can calculate the optimal le and V log gt for (small) integer Tz. We show qualitative results in Fig. 2. We find that target score + FKC performs best, while tempered noise has a tendency drop modes. We also find that FKC outperforms SDE-only simulation in both tempered noise and target score settings. This is further supported by quantitative results in Table Al.
We report results across various simulation parameters, namely the number of sampling steps N and churn parameter r (which controls how the SDE integration scheme adds noise). For FKC, we additionally sweep over the batch size number of particles K, whereas K = corresponds to the no-resampling baseline (@). To calculate metrics on a single image for FKC, we resample from among K particles according to the weights since the last resampling step. Note that we often observe that final-step images from single batch with FKC are nearly identical visually due to weight degeneracy.
Sampling IJ-13 To demonstrate the utility of first learn- ing sampler at a high temperature then annealing to a lower temperature vs. directly learning at a lower temperature, we consider a Lennard-Jone base temperature Tr = 2. We train a Denoising Energy
(LJ) system of 13 particles at a
In Table 2, we compare the quantitative performance of our

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
-Target Score SIC
Table IJ-13 sampling task with various SDEs, with performance
Target Score
measured by mean + standard deviation over 3 seeds. The starting temperature is Tr= = 2. annealed to target temperatures Ts = 0.8 and Ts- = 1.5. The DEM samples are generated with model trained at those corresponding target temperatures.
Teaperren Noliew SDE PKC
12
Energy 19
10
0
Target Temp
SDF Type
FKC
11
Fnergy-99;
0.189 0.002
14.730 0.000
15:556 0.045
Target Score
0.048 1. 0 010
6.252
2.710
6.356 2.673
(8 15-2.0
0.8
Tempered Noese
0,308 =i 0.007
6.487
2: 0.056
5.501 4: 0.255
2
7.016 0.558
7.111 + 0.535
DEM
0.300 = 00001
9.794
0.801 0.101
2.0
1.3
1.5
1.4
1.2
1.0
8.8
1.5 (8 1.33)
Target Score
Temperol Novse
0.0666 # 0. 0.002
0.083 + 0.008
0.368 + 0.008
0.000)
0.766
2,154 0:048
5.340 + 0.064
5.396 0.083
#: 0.156
0.989 + 0.171
3,520 0.258
6,230 # 0.254
3:556 + 0.000
samples from the annealed target distribution and our methods different temperatures. Note the training temperature Ty, = 2.
Figure 4. 2- --ssssssssenn between energy distributions of MCMC
Target Temperature
DFM
0:258 + 0.005
4.471 0.100
5.211 + 0.017
-5.786
44
-9.35
-12 645
Matching (DEM) model (Akhound-Sadegh al., 2024) at this base temperature and perform temperature-anneale inference to lower temperatures. In Table 3 and A2 we com- pare the performance of a DEM model trained at a lower temperature against a DEM model trained at a higher temper- ature and annealed to the lower temperature using various SDEs. We evaluate methods using the 2- 2-Wasserstein metric between distance distributions, and the 1- and 2- 2-Wasserstein metrics between energy histograms reference (App. F.3). We find that tempered Foise+FKC performs best at higher temperatures. However, at lower temperatures, the target score SDE+FKC performs best. Both methods outperform DEM directly trained at the lower temperature for tempera- tures Ts E [2.0, 0.8] (Fig. 4). We find DEM qualitatively easier to learn at higher temperatures requiring much less tuning compared to lower temperatures (Fig. A1). This makes the train-then-anneal approach attractive in this set- ting. For extended results and discussion see App. F.
P41601
-7.600
1888
9.326
-19 535
P23021
Reference
TargetDiif
DualDiI
FKC [ours]
Figure 5. Molecules generated from our method (target score SDE with 8 = 2.0 and FKC resampling) and baselines in the binding pockets of two proteins: GRM5 (top row, UniProt ID P41594) and RRMI (bottom row, UniProt ID P23921). Docking scores for each molecule and target above each image; lower docking scores are better. Here, we display molecules with the best dacking scores that have QED > 0.4; more generations are in App. F.6. The binding pocket is shaded in light green.
showcase our best results in Table 4 and the full ablation in App. F.6. We evaluate the generated molecules on their docking scores to a protein pair, P1 and P2. We report the average of docking score products for each target, as well as the average maximum docking score for a pair. Lower dock- ing scores are better, and so lower maximum docking scores indicate the molecule is better binding to both targets. We compute the percentage of molecules that have better docking scores than known binders, as well as the number of valid and unique molecules generated, their diversity, their drug- likenssssesss (QED (Bickerton et al., 2012)), and their synthetic accessibility (SA (Ertl & Schuffenhauer, 2009)).
5.3. Multi- Marti-Target Structure-Based Drug Design
We apply FKC to the setting of seneture-based drug design (SBDD), where the goal is to design molecules (or ligands) using the three-dimensional structure of a biological tyrgel-yypically a protein-as ass a guide (Anderson, 2003). The ligands are then evaluated based on how well they fit into the protein's 's binding site. We focus on dual-target drug design, where molecule should interact with two proteins simultaneously. Dual-target drug design has become increasingly investigated for targeting complex disease pathways such as in various cancers and neurode- generation (Ramsay et al., 2018), as well as for diminishing drug resistance mechanisms (Yang et al., 2024).
We find that the target noise SDE at 3 > 0.5 generates molecules with better average docking scores for each of the target proteins compared with both basclines DualD- iff (Zhou et al., 2024) and TargetDiff (Guan et al., 2023). When we incorporate FKC, the average docking scores im- prove further. In Fig. A7, we observe a positive correlation between the FKC weights and docking scores. There is a slight sacrifice in terms of diversity and uniqueness when resampling with FKC, although this is a common trade-off for an increase in quality. Notably, our method achieves
We investigate the performance of PoE using both target score and tempered noise SDEs at various 8, with (g) and without (8) FKC. Ligand performance is determined by docking scores each protein target using AutoDock Vina (Eberbardt et al., 2021). We evaluate 100 protein pairs and average our results over tasks. We sampled 5 molecule sizes from the original training set from Guan et al. (2023): (15, 19, 23, 27, 35) generating 32 molecules per size. We
8

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Table Docking scores of generated ligands for 100 protein target pairs (P1, P2). We generate 32 ligands for 5 molecule lengths for each protein pair using the Target Score SDE. Lower docking scores are better. Values are reported as averages over all generated molecules in each run. "Better than ref." the percentage of ligands with better docking scores than known reference molecules for both targets (the
mean docking score for the reference molecules is - -7.915+2.441 We also report the diversity, validity & uniqueness, SA score, and
QED. Difg from Guan et al. (2023), DiualDiff from Zhou et al. (2024).
(P, (1)
max(P v, (4)
Py (4)
2a (4)
Better than ref. (1)
Div. (t)
Uniq. (?)
(L)
QED (t)
only'
59.355 190 139
669661 a.
-8 119
-7.213_ara
0.321 1o.am
0.886 oo.
1.918
0.5881o.0m
0.531 10:10
B
FKC
(Py P2)
max(Ps, (U)
Ps (D)
Pa (
Better than ref. (1)
Div. (t)
Uniq. (f)
SA (1)
QED (t)
64 554.55
- -.0090-2.06
-7 2.950022.11
-8.0 8.008-2.154
0.306
0.883-00012
0.94/200124
0.609
66.388 66.380ga5 747
6.966g3.20
8,085 saa
8.0
0.341 807
0.870ga
0.951 ono
0.506
0.587 205770..2
1.0 0
g51 15a
7.256_a.00
2. sas
2.2
018
0. 0.984_o 100
0.61110..0m owo
0.589 196
76, (136=83... asG
-7 -7.019-2.005
-8 657
-8. -..660-2.40
0. 1.13440.41
0.8
0,939 too
0.627=0.005
0.591
71. 180 730
-T -2.22122.007
-8 3.36622500 sa6
-..00122005
0 0.38320.500 s89
0.877=00015
0.961 LG
0.61120..06 2000
0. 0.59442... 194
2.0
77.271 1as 264
-7.720125mm
-8.682 12. 868
-8775__21w
0.450 L0.4as
0.8016_oots
1.8612
_0.112
0.582
0..59200.148 10.344
the lowest maximum docking score, meaning that generated ligands are able to better bind to both proteins (on aver- age across tasks). Our method also generates the highest fraction of molecules that are better than known binders (ref- erence molecules), which could motivate using our model in de novo drug design settings (the mean docking score of We visualize ligands for a sample target pair in Fig. 5 and Fig. A6.
Indeed, the celebrated Jarzynski equality (Jarzynski, 1997; Crooks, 1999) and its variants admit an elegant proof using the Feynman- -ac formula (Lelièvre et al. (2010, Ch. 4), Vaikuntanathan & Jarzynski (2008)).
rredictor-corrector simulation (Song et al., 2021) performs additional Langevin steps to promote matching the interme- diate marginals of Pt of a diffusion model. These schemes can be adapted for annealed or product targets, although Du et al. (2023) found best performance using Metropolis corrections. Bradley & Nakkiran (2024) interpret standard CFG SDE simulation (19) as a predictor-corrector where the corrector targets a different guidance or geometric mix- ture weight 3 = 1 (1 + 3). Our resampling correctors are instead tailored to the original guidance weight B.
reference molecules is - -7.91542.
In App. F.7, we further investigate the utility of PoE in generating molecule SMILES using a latent diffusion model, and show that FKC resampling improves generation for small molecules satisfying multiple functional properties.
6. Related Work
Sequential Monte Carlo methods have proven useful across wide range of tasks involving diffusion models, including for reward-guided generation (Uehara et al., 2024; 2025; Singhal et al., 2025; Kim et al., 2025; Chen et al., 2025a), conditional generation (Wu et al., 2024), or inverse problems (Dou & Song, 2024; Cardoso et al., 2024).
Finally, SMC methods have recently been extended to dis- crete diffusion models (Singhal et al., 2025; Li et al., 2024; Uehara et al., 2025; Lee al., 2025a), where the approach of Lee et al. (2025a) is analogous to FKC for discrete settings.
7. Conclusion
For compositional generation, Du et al. (2023) learn an energy-based score function and use the energy within MCMC procedures. Thornton et al. (2025) improve training of the energy-based score function by distilling an uncondi- tional score model, where the resulting energy can be used for SMC resampling from annealed or product densities.
In this work, we proposed FEYNMAN- KAC CORRECTORS, an array of tools allowing for fine control over the sample distributions of diffusion processes. These target distribu- tions may arise in compositional generative modeling (Du & Kaelbling, 2024), where we seek to combine specialist models capturing various chemical properties of molecules or different aspects of a complex prompt. Geometric av- eraging appears in widely-used CFG techniques while, via annealing, we demonstrate that an approach of first learning amortized sampler at higher temperature and then an- nealing using FKCs down to a lower temperature opens up new dimension for the construction of amortized samplers.
Within the context of diffusion samplers from Boltzmann densities, Phillips et al. (2024) consider SMC for energy- based score parameterizations. Chen et al. (2025b); Albergo & Vanden-Bijnden (2024) consider SMC resampling along trajectories with respect a prescribed geometric annealing path, where Albergo & Vanden-Eijnden (2024) is presented through the Feynman-Kat Kac perspective. The approaches in Vargas et al. (2024); Alhergo & Vanden-Fijnden (2024) cor- respond to the escorted Jarynski equality (Vaikuntanathan & Jarzynski, 2008; 2011), where additional transport terms are Icarned to more closely match the evolution of a given density path (Arbel et al., 2021: Chemseddine et al., 2025; Máté & Fleuret, 2023; Tian et al.. 2024; Fan et al., 2024; Maurais & Marzouk, 2024; Vargas et al., 2024).
Finally, our framework allows for the use of reward models (Prop. D.6) and for time-dependent annealing schedule 3,, (Prop. C.6), where the log-density terms needed for weights can be estimated using methods from Skreta et al. (2025).
9

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Feynman-Koe Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Impact Statement
Bickerton, G. R., Paolini, G. V., Besnard, J., Muresan, S., and Hopkins, A. L. Quantifying the chemical beauty of drugs. Nature Chemistry, 4(2):90-98, 2012.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
Bingham, E., Chen, J. P., Jankowiak, M., Obermeyer, F.,
Pradhan, N., Karaletsos, T., Singh, R., Szerlip, P. A., Horsfall, P., and Goodman, N. D. Pyro: Deep universal probabilistic programming. Mach. Learn. Res., 20: 28:1- 28:1-28:6, 2019.
Acknowledgments
This project was partially sponsored by Google through the Google & Mila projects program. The authors ac- knowledge funding from UNIQUE, CIFAR, NSERC, In- tel, and Samsung. The research was enabled in part by computational resources provided by the Digital Re- search Alliance of Canada (https:/ //as:///all aalianccacnnn ca), Mila (https: /t/mmmmaaa. quebec), the Acceleration utoronto. ca/), and NVIDIA. KN was supported by IVADO and Institut Courtois. MS thanks Ella Rajaonson for assistance with docking visualizations, as well as Austin Cheng and Cher-Tian Ser for providing feedback molecule genera- tion.
Bradley, A. and Nakkiran, P. Classifier-free guidance is predictor-corrector, arXiv preprint arXiv:2408.09000, 2024.
Cardoso, G. V., El Idrissi, Y. J., Le Corff, S., and Moulines, E. Monte Carlo guided diffusion for Bayesian linear in- verse problems. In International Conference on Learning Representations, 2024.
Consortium (https: /aaaaaccceeaaaacceeeeee
Chang, J. and Ye, J. C. Ldmol: Text-conditioned molecule diffusion model leveraging chemically informative latent space. arXiv preprint urXiv:2405.17829 2024.
Chemseddine, J., Wald, C., Duong, R., and Steidl, G. Neural sampling from Boltzmann densities: Fisher-Rao curves in the Wasserstein geometry. In International Conference on Learning Representations, 2025.
References
Abramson, J., Adler, J., Dunger, J., Evans, R., Green, T.,
Pritzel, A., Ronneberger, O., Willmore, L., Ballard, A. J., Bambrick, J., al. Accurate structure prediction of biomolecular interactions with alphafold 3. Nature, pp. 1-3, 2024.
Chen, H., Ren, Y., Min, M. R., Ying, L., and Izzo,
Solving inverse problems via diffusion-based priors: An approximation free ensemble sampling approach. arXiv preprint arXiv:2506.03397 2025a.
Akhound-Sadegh T., Rector-Brooks, J., Bose, J., Mittal, S., Lemos, P., Liu, C.-H., Sendera, M., Ravanbakhsh, S., Gidel, G., Bengio, Y., et al. Iterated denoising energy matching for sampling from Boltzmann densities. In International Conference Machine Learning, 2024.
Chen, J., Richter, L., Berner, J., Blessing, D., Neumann, G., and Anandkumar, A. Sequential controlled Langevin dif- fusions. In International Conference on Machine Learn- ing. 2025b.
Albergo, M. S. and Vanden-Fijnden, E. Nets: A non-equilibrium transport sampler. arXiv preprint arXiv:2410.022111, 2024.
Chizat, L., Peyré, G., Schmitzer, B., and Vialard, F.X. An interpolating distance between optimal transport and Risher-Rao metrics. Foundations of Computational Math- ematics, 18:144,,72 2018.
Anderson, A. C. The process of structure-based drug design. Chemistry & Biology, 1009):777-797,0000 2003.
Crooks, G. E. Excursions in Statistical Dynamics. Univer- sity of California, Berkeley, 1999.
Angeli, L. Interacting particle approximations of Feynman- Kac measures for continuous-time jump processes. PhD thesis, University of Warwick, 2020.
Davis, M. H. Piecewise-deetrmmiissi Markov processes: general class of non-diffusion stochastic models. Journal of the Royal Statistical Society: Series B (Methodologi- cal), 46(3):353- 1984.
Angeli, L., Grosskinsky, S., Johansen, A. M., and Pizzofer- rato, A. Rare event simulation for stochastic dynamics in continuous time. Journal of Statistical Physics, 176(5): 1185-1210, 2019.
De Bortoli, V., Hutchinson, M., Wirnsherger, P., and Doucet, A. Target score matching. arXiv preprint arXiv:2402.0886 2024.
Arbel, M., Matthews, A., and Doucet, A. Annealed flow transport Monte Carlo. In International Conference on Machine Learning, 2021.
Del Moral, P. Mean Field Simulation for Monte Carlo
Integration. Chapman and Hall, CRC press, 2013.
10

---

Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Dou, Z. and Song, Y. Diffusion posterior sampling for linear inverse problem solving: A filtering perspective. In International Conference on Learning Representations, 2024.
Markov processes. In International Conference on Learn- ing Representations, 2025.
Huang, K., Fu, T., Gao, W., Zhao, Y., Roohani, Y., Leskovec, J., Coley, C. W. Xiao, C., Sun, J., and Zitnik, M. Ther- apeutics data commons: Machine learning datasets and tasks for drug discovery and development. Proceed- ings of Neural Information Processing Systems, NeurlPS Datasets and Benchmarks, 2021.
Douc, R. and Cappé, O. Comparison of resampling schemes for particle filtering. In ISPA 2005. Proceedings of the 4th International Symposium Image and Signal Processing and Analysis, pp. 64-69, 2005.
Du, Y. and Kaelbling, L.. Compositional generative model- ing: A single model is not all you need. 2024.
Jarzynski, C. Equilibrium free-energy differences from nonequilibrium measurements: A master-equation ap- proach. Physical Review E, 56(5):5018, 1997.
Du, Y., Durkan, C., Strudel, R., Tenenbaum, B., Dieleman, S., Fergus, R., Sohl-Diicsttenn, J., Doucet, A., and Grath- wohl, W. S. Reduce, reuse, recycle: Compositional gen- eration with mergy-based diffusion models and MCMC. In International Conference on Machine Learning, 2023.
Karczewski, R., Heinonen, M., and Garg, V. Diffusion mod- els as cartoonists! the curious case of high density regions. International Conference on Learning Representations, 2025.
Eberhardt, J., Martin, D., Tillack, A. F., and Forli, S. Autodock vina 1.2. O: New docking methods, expanded force field, and python bindings. Journal of Chemical Information and Modeling, 01(8):39113888 2021.
Karras, T., Aittala, M., Kynkäänniemi, T. Lehtinen, J., Aila, T., and Laine, S. Guiding a diffusion model with a bad version of itself. In Neural Informaton Processing Systems, 2024a.
Ertl, P. and Schuffenhauer, A. Estimation of synthetic acces- sibility score of drug-like molecules based on molecular complexity and fragment contributions. Journal of Chem- informatics, 1:1-11, 2009.
Karras, T., Aittala, M., Lehtinen, J., Hellsten, J., Aila, T., and Laine, S. Analyzing and improving the training dynamics of diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 24174-24184, 2024b.
Ethier, S. N. and Kurtz, T. G. Markov Processes: Charac- terization and Convergence. John Wiley & Sons, 2009.
Kim. S., Kim, M., and Park, D. Test-time alignment of diffusion models without reward over-optimization. In International Conference on Learning Representations, 2025.
Fan, M., Zhou, R., Tian, C., and Qian, X. Path-guided particle-based sampling. In International Conference Machine Learning, 2024.
Köhler, J., Klein, L., and Noé, F. Equivariant flows: exact likelihood generative learning for symmetric densities. In International Conference on Machine Learning, 2020.
Gardiner, C. Stochastic Methods. Springer, 2009.
Ghosh, D., Hajishirzi, H., and Schmidt, L.. Geneval: An object-focused framework for evaluating text-to-image alignment. In Advances in Neural Information Processing Systems, 2023.
Kondratyev, S., Monsaingeon, L., and Vorotnikov, D. new optimal transport distance on the space of finite Radon measures. Adv. Differential Equations, 2016.
Guan, J., Qian, W. W. Peng. X., Su, Y., Peng. J., and Ma, 3d equivariant diffusion for target-aware molecule generation and affinity prediction. arXiv preprint arXiv:2303.0355 2023.
Lee, C. K. Jeha, P., Frellsen, J., Lio, P., Albergo. M. S., and Vargas, F. Debiasing guidance for discrete dif- fusion with sequential Monte Carlo. arXiv preprint 4rXiv:2502.06079, 2025a.
Ho, J. and Salimans, Classifier-free diffusion guidance. In NewrIPS 2021 Workshop on Deep Generative Models and Downstream Applications, 2021.
Lee, S., Kreis, K., Veccham, S. P., Liu, M., Reidenbach, D., Peng, Y., Paliwal, S., Nie, W., and Vahdat, A. Genmol: A drug discovery generalist with discrete diffusion. 2025b.
Hoffiman, M. D. and Gelman, A. The No-U-Turn sampler: Adaptively setting path lengths Hamiltonian Monte Carlo. Journal of Machine Learning Research, 2014.
Lelievre, T., Rousset, M., and Stoltz, G. Free Energy Com- putations: A Mathematical Perspective. World Scientific, 2010.
Holderrieth, P., Havasi, M., Yim, J., Shaul, N., Gat, I. Jaakkola, T., Karrer, B., Chen, R. T., and Lipman, Y. Generator matching: Generative modeling with arbitrary
Li, X., Zhao, Y., Wang. C., Scalia, G., Eraslan, G., Nair, S., Biancalani, T., Ji, S., Regev, A., Levine, S., al.

---

Peynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Derivative-free guidance in continuous and discrete dif- fusion models with soft aalue-based decoding. arXiv preprint arXiv:2408..082. 08252, 2024.
Richter, L. and Berner, J. Improved sampling via learned diffusions. In International Conference Learning Rep- resentations, 2024.
Liero, M., Mielke, A., and Savaré, G. Optimal entropy- transport problems and a new Hellinger-Kantrrviich dis- tance between positive measures. Inventiones Mathemati- cae, 211(3)969-1117, 2018.
Rogers, D. and Hahn, M. Extended-connectivee finger- prints. Journal of Chemical Information and Modeling, 50(5):742-75) -544, 2010.
Rousset, M.
On the control of an interacting particle esti- mation of Schrödinger ground states. SIAM Journal on Mathematical Analysis, (863):824-844, 2006.
Liu, N., Li, S., Du, Y., Torralba, A., and Tenenbaum, B. Compositional visual generation with composable dif- fusion models. In European Conference Computer Vision, pp. 423 439. Springer, 2022.
Rousset, M. and Stoltz, G. Equilibrium sampling from nonequilibrium dynamics. Journal of Statistical Physics, 12::1211- 2006.
Lu, Y., Lu, J., and Nolen, J. Accelerating Langevin sampling with birth-death. arXiv preprint arXiv: 1xX.vv/005.0900 /905009853.3 2019.
Saharia, C., Chan, W., Saxena, S., Li, L., Whang. J., Denton, E. L., Ghasemipour, K., Gontijo Lopes, R., Karagol Ayan, B., Salimans, T., et al. Photorealistic text-to-image dif- fusion models with deep language understanding. In Ad- vances in Neural Information Processing Systems, 2022.
Máté, B. and Fleuret, F. Learning interpolations between Boltzmann densities. Transactions on Machine Learning Research, 2023.
Maurais, A. and Marzouk, Y. Sampling in unit time with kernel Fisher-Rao flow. In Party-first International Con- ference on Machine Learning, 2024.
Sahoo, S. S., Arriola, M., Gokaslan, A., Marroquin, E. M., Rush, A. M., Schiff, Y., Chiu, J. T., and Kuleshov, Simple and effective masked diffusion language models. In Advances in Neural Information Processing Systems, 2024.
Midgley, L. I., Stimper, V., Simm, G. N., Schölkopf, B., and Hermández-Lobato, J. M. Flow annealed importance sam- pling bootstrap. International Conference on Learning Representations, 2023.
Salimans, T. and Ho, J. Should EBMs model the energy or the score? In Energy Based Models Workshop-ICLR 2021, 2021.
Naesseth, C. A., Lindsten, F., Schön, T. B., et al. Elements of sequential Monte Carlo. Foundations and Trends@ in Machine Learning, 1233)3007-392, 2019.
Singhal, R., Horvitz, Z., Teehan, R., Ren, M., Yu, Z., McK- eown, K., and Ranganath, R. A general framework for inference-time scaling and steering of diffusion models. arXiv preprint arXiv:2501 06848, 2025.
Neal, R. M. Annealed importance sampling. Statistics and Computing, 11:125-139, 2001.
OuYang, R., Qiang, B., and Hemnnndez-Lobato, J. M. BNEM: A Boltzmann sampler based on boot- strapped noised energy matching. arXiv preprint D0Xi:2440900978,, 2024.
Skreta, M., Atanackovic, L., Bose, A. J., Tong, A., and Nek- Iyudov, K. The superposition of diffusion models using the Itô density estimator. In International Conference on Learning Representations, 2025.
Phillips, A., Dau, H.-D., Hutchinson, M. J., De Bortoli, V., Deligiannidis, G., and Doucet, A. Particle denoising dif- fusion sampler. In International Conference on Machine Learning, 2024.
Song. Y., Sohl-Dickstee J., Kingma, P., Kumar, A., Er- mon, S., and Poole, B. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021.
Radford, A., Kim, W., Hallacy, C., Ramesh, A., Goh, G., Agarwal, S., Sastry, G., Askell, A., Mishkin, P., Clark, J., et al. Learning transferable visual models from natural language supervision. In International Conference Machine Learning, 2021.
Thornton, J., Béthune, L., ZHANG, R., Bradley, A., Nakki- ran, P., and Zhai, S. Controlled generation with distilled diffusion energy models and sequential Monte Carlo. In International Conference on Artificial Intelligence and Statistics, 2025.
Ramsay, R. R., Copovic-Nikolice M. R., Nikolic, K., Uliasii, E., and Bolognesi, M. L. A perspective on multi-target drug discovery and design for complex diseases. Clinical and Translational Medicine, 7:1-14, 2018.
Tian, Y., Panda, N., and Lin, Liouville flow importance sampler. International Conference on Machine Learning, 2024.
12

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Feynman-Kee Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Uehara, M., Zhao, Y., Biancalani, T., and Levine, S. Un- derstanding reinforcement learning-based fine-tuning of diffusion models: A tutorial and review. arXiv preprint arXiv:2407.13734, 2024.
Zhou, X., Guan, J., Zhang, Y., Peng. X., Wang, L., and Ma, J. Reprogramming pretrained target-specific diffusion models for dual-target drug design. In Neural Information Processing Systems, 2024.
Uchara, M., Zhao, Y., Wang, X., Regev, A., Levine, S., and Biancalani, T. Inference-ting time alignment in diffu- sion models with reward-guided generation: Tutorial and review. arXiv preprint vrXiv:2501.0665 2025.
Vaikuntanathan, S. and Jarzynski, C. Escorted free energy simulations: Improving convergence by reducing dissipa- tion. Physical Review Letters, 100(11))19000,, 2008.
Vaikuntanathan, S. and Jarzynski, C. Escorted free energy simulations. The Journal of Chemical Physics, 134(5), 2011.
Vargas, F., Grathwohl, W. S., and Doucet, A. Denoising dif- fusion samplers. In International Conference on Learning Representations, 2023.
Vargas, F., Padhy, S., Blessing, D., and Nusken, N. Trans- port meets variational inference: Controlled Monte Carlo diffusions. In International Conference Learning Rep- resentations, 2024.
Wang, H., Skreta, M., Ser, C.-T., Gao, W., Kong, L., Strieth- Kalthoff, F., Duan, C., Zhuang, Y., Yu, Y., Zhu, Y., al. Efficient evolutionary search over chemical space with large language models. In International Conference on Learning Representations, 2025.
Woo, D. and Ahn, S. Iterated energy-based flow matching for sampling from Boltzmann densitics. arXiv preprint arXiv:2408. 16249, 2024.
Wu, L., Trippe, B., Naesseth, C., Blei, D., and Cunning- ham, Practical and asymptotically exact conditional sampling in diffusion models. In Advances in Neural Information Processing Systems, 2024.
Xu, J., Liu, X., Wu, Y.. Tong, Y., Li, Q., Ding, M., Tang. J., and Dong, Y. Imagereward: Learning and evaluating hu- man preferences for text-to-image generation. Advances in Neural Information Processing Systems, 36. 2024.
Yang. Y., Mou, Y., Wan, L.-X., Zhu, S., Wang, G., Gao, H., and Liu, B. Rethinking therapeutic strategies of dual-target drugs: An update on pharmacological small- molecule compounds in cancer. Medicinal Research Re- views, 44(6):2600-2222 2623, 2024.
Zhang, Q. and Chen, Y. Path integral sampler: stochastic control approach for sampling. In International Confer- ence on Learning Representations, 2022.
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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
A. Expectation Estimation under Feynman-Kac PDEs
We proceed in two steps, first finding Kolmogorov backward equation corresponding to evolution under weighted Feynman-Kac SDE. We then use this identity to derive the expectation estimator in Eq. (10). Throughout, we consider the evolution of density Pt defined via the following Feynman- Kac PDE,
a गe P( -V, (V))(()))(()))) Apr(as) + pe(as) + (g((te) - ff st su(ar)paaas)ddd
of 2
(34)
Our proof follows similar derivations as in Lelièvre al. (2010, Prop 4.1, Ch. 4.1.4.3) (see also (Vaikuntanathan & Jarzynski, 2008; 2011) 1) and references therein), where the authors are interested in sampling from a sequence of unnormalized distributions dr specified via a time-varying energy or Hamiltonian. The proofs often rely on Langevin dynamics that leave Pt invariant. We adopt similar proof technique, but focus directly on simulation with arbitrary vt 9r derived via our methods Sec. 3.
Proposition A.1. For bounded test function + R. and pt satisfying Eg. (34), we have
(or()))6(2T))) = E Efeli (ece)dep(()))
(35)
ZT
where dr, (z dt a,dW, where ZT is normalization constant independent of x. Eq. (35) which suggests that the self-normalized importance sampling approximation in Eq. (10) is consistent as K + co.
To Po
Proof. The proof proceeds three steps, delineated with bold paragraph headers. We first derive the backward Kolmogorov equation for appropriate functions, then specify the evolution of the Feynman-Ka PDE for the unnormalized density, before combining these results to prove the result in Prop. A.I.
Backward PDE: For given test function (xx), consider defining the following function
Φ (x, t)
E el," (m(m)) ((TT) xy = =r] Φr T) d(x)
(36)
where expectations are taken under the evolution of the SDE dr, = (x(x))dt + o,dW, In particular, for t, we have
Φ (z. t) = E eli (u(z.)ds_J? g.(a.)ds ((xT) x2 = 2] = E ell go(xx)de D (Ir, T) 0z = x
(37)
We will leverage this identity to derive a PDE which 2 (T t) must satisfy. Note, to link @T E) and (the expected value of) Φ T), we should account for the weights edi 9.(z,)ds Thus, we apply Ito's product rule and Ito's lemma to capture how eli (e(xa)de (2r, 7) evolves with T.
a( ell (x (T) ((T,, = eli dx(rx)ds d r) + @ (r(xr,T)del aa(aa)ds + d(( T), eel aloa)day (38)
In the final term, eli (x(x)d is non-stochastic and, assuming it has finite variation, the term d elf doledd vanishes. We use Ito's lemma to expand (2, r) and simple differentiation for del: 9.(2.)ds
dl delli alle.dde (e = eli dalz.)de
( + (v, (2r), V r)) + o 2 A&T((T:7) dr
Or
+ eli (x o (2,, dW.) + f eel (e (g+ dr
(39)
= ele dx(r))ds (00T(17:7)) (Ve(I),, ( T)
0i A (T dt
0t
+ eli 9.(2.)ds 0e (D dW.) Integrating Eq. (40) t to T and taking expectations under the simulated process from initial point I; x. the
(40)
14

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
stochastic term vanishes and we obtain
E jele 9. (ala)dds (TTT, T) =x]
E ae(diio g-(xx)d= (1,, | Fx = x
(41)
( + (ur(2), ( + o² ( + @r( r)g (x) dr
E
eli sa(xa)ds
Finally, we simplify the first line Eq. (41). Considering the definition and endpoint condition in Eq. (36), we have
E oof golz.)de (pr. : T) 21 I] - E eel g2(2, ga(z,)do jdb Dr (x, t) at
(42)
E Efel aa(a, sala.)as ((T)) I - =1] T (x, 1) =
by definition in Eq. (36). Since eli a (w,)dd O, this implies that the integrand in the second line of Eq. (41) should be zero for any Thus, we obtain backward PDE which is often used directly in the statement of the Feynman-Kan Kac formula,
a T) + (1)(1), (or( of
A T) + (r (x) =
(43)
Evolution of Unnormalized Density In practice, we cannot exactly calculate f gr (x:))) which appears in the reweighting equation in Eq. (6) (or Eq. (46) below) to ensure normalization. Eventually, we will account for normalization using SNIS as in Eq. (10).
For now, consider the evolution of unnormalized density (x)) = Pc(x) for particular Ut, de, 9t. and some normalization constant Z, With foresight, we define
ar (((1)) - (V)i(())))(()))) of ( (e(2)))e((4))
a
(44)
which we justify by noting that only the reweighting term does not preserve normalization. In particular, let
/ (z))).
0₂ log Ze
(45)
which seems to be natural candidate from inspecting pt (x) f Pi (x)dz)) which implies a log (1) = 9x(x) f pf (1r)g (z)dr. Defining terms such that O, po (x) 0 log P." (z) - θ log Z, yields Eq. (45). We finally confirm that the definitions in Eq. (44) and Eq. (45) are consistent with the original Feynman Kac PDE.
general, reweighting-only evolution O.p," (z)
i
of 2 Ap,( + px(1)) (ailze) - -f se(es))((as)ddr
ne (x.)) = - p. +
(46)
Namely, since (x,),) = (r( 2, the definitions in Eq. (44)-(45) should satisfy
0
0
Or Pc(x)
(pr (T))
(47a)
01
2
7,-
at (((r)) + (c(x)) 0 logoo 1
(47b)
z at pi(m)) - -,(x))), 8 1o Zo
(47c)
of A (((()))((())) - / pu(xs)gu
=,, (
(47d)
Noting that Vz, 24 0, we can pull 7-1 inside differential operators to obtain
_- v.
(xx)
+ of
(x()
+ (e(xx)
De(ars) / pa(zz)oe(se)ddr
ve(
Z₁ (x(2x)
(47e)
Z₁
2
Z,
Z,
0² + Apr(xx) + pe(xs) (9)(zt)) - / ((((s)))e((ss)d
(V
(470)
as desired.
15

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Feynman- Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Expectation Estimation: Now, we use Eq. (43) write the total derivative of the following integral under the unnormalized density Pr (a:).
d
(f
= (oDr(er.t)
pe(x)ddx + f (0)(2))
dx
(48a)
di
01
01
Using Eq. (43) and Eq. (44), we have
[J( (o(())) ( - -² (
- Dr ((r())))))))))))))))))))))))(( pa(t)
(48b)
/ ( + of
A)x()) (((())(()))
Integrating by parts in the second line, we have
{(- (o,,))
o² 2
( - (x, (e
(48c)
/ (11(s))
ot 2
A (2. 1) Φ (x. D2) ir (x))dd
0
(48d)
Integrating on the interval t = 0 to = T, we obtain
/ Dr ()po = S." d
dt d
(f
dt 0
(49)
Thus, we can set these two quantities equal to each other. Using the identity (x(x) = Px and assuming we initialize simulation with normalized po(x)= = (x) with Zo = 1, we can finally use the definitions in Eq. (36) (namely ( 7) )(Tr)) to write
/ D
/ @T((T: (ppr(eT)deT
() (zo, (To
(50)
Zo / (Ejelū asle,)ds M(T)) zul) Po -z / ((rr)p((())drs
(51)
1 E Bleli galewdo (pp(er)(((T)) Zr
(52)
which is the desired identity. In practice, we could estimate Zr = 1 EK cto (dl
= R Ex
e
and
Blel g. (x.))ss ((zT)) ~ 1 R Ek en o(z)))) (K)), which yields Eq. (10).
Note that our choice of upper limit T' in 4T was arbitrary, suggesting that we could repurpose the same reasoning for estimating expectations intermediate from initialization at time O. This suggests that samples properly weighted for estimating expectations and normalization constants Z, for intermediate pt (Naesseth et al., 2019).
Similarly, changing the lower limit of integration from t 0 to intermediate t, the analogue of Eq. (51) suggests estimating expectations using Ey1Ep, [ t)] = Zr [o Given properly-weighted particle approximations of Pt, 2., can continue calculating the appropriate weights along the trajectory to estimate Zr or terminal expectations under pr. These arguments can be similarly adapted to justify SMC resampling at intermediate steps, as we do in practice (Sec. 4).
B. Feynman-KKe Processes
B.1. Markov Generators for Feynman-Kac Processes
In Sec. 2, we described the adjoint generators L; [p)), -,100 [pr), (7:1)) corresponding to flows with vector field Vt, diffusions with coefficient o, and reweighting with respect to Gt- In particular, the Kolmogorov forward equation Oe at (x) = Li (x)) corresponds to our PDEs presented in Eqs. (3), (5) and (6). In the lemma below, we recall the generators
which are adjoint to those in Sec. 2 and operate over smooth, bounded test functions with compact support, e.g. _ [0].
16

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Feynman-Kee Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Lemma B.1 (Adjoint Generators). Using the identity J (x) (i)))(()) dr = (x)d(()) (r(1) da
Flow:
cl [o](x) - (o(())))xx))
(r](x) - - PV, vx(x))
(o(xx) of
of
Diffusion:
AΦ(x)
2 pt(x)
(53)
2
(d))) (t) (r(())) - f (((x) pe(t) de
(p)()) pe(x>) (g(()) / gx(1)) pe(x)dr)
Reweighting:
Proof. The proofs for flows and diffusions follow using integration by parts, with proofs found in, for example Holderrieth et al. (2025, Sec. A.5). For the reweighting generator, have
/ "( (p.(<)ddr=
/ M()) pi(x)) (1(x)= / gi(v) p.(u)ddy) de
= pi(w) ((w)) (s)())) (l((x)
(ola))
f sl(w)) m((s)dy) dd
= f pe(x) (ol de
Note that the weights 9r often chosen in relation the unnormalized density of p. (Lelièvre et al. (2010, Sec. 4)), and our attention will be focused on the pair of generator actions C
C,(9) [pe], c( [o] for possibly time-dependent ø.
B.2. Jump Process Interpretation of Reweighting
One way to perform simulation of the reweighting equation will be to rewrite it in terms of a jump process. We first recall the definition of the Markov generator a jump process (Ethier & Kurtz (2009, 4.2), Del Moral (2013, 1.1), Holderrieth al. (2025, A.5.3)) and derive its adjoint generator.
Lemma B.2 (Jump Process Generators). Using the definition of the jump process generator and the identity (xx) J, [p))(x) dr = J (x) (x) dx. Letting We(z, y) = Je(l)))(ylz) for normalized Je(y)a)),
F. (w)) (+) = / (o(v) (()))) T. (pi](t)) = (/ Xe(e) ( (r()))((x)
Jump Process:
(54a)
(54b)
Proof. Through simple manipulations and changing the variables of integration, we obtain
J o(x) Ji (z)x dx / (x) px)) dx
/ (J (b()) - +(x)) (())) pr (x) dx
/ / (w)ds(t)) dyda / /ooo)det))))( dydæ
/ / (o))) dxdy
(la)s dyda
/ ((w) ((Mno)SAA))))O (()))(())))))))))))))))()) (slole))))) )dr
(x))())) (ʃ ( (()))
using the assumption that (z(yz) is normalized.
Reweighting -+ Jump Process Our goal is to derive jump process such that the adjoint generators are equivalent
(, [p)(xx) (,(3) (xx) for given reweighting generator with weights dr (Eq. (53)).
While Del Moral (2013); Angeli (2020) emphasize the freedom of choice in such generators, Sec. 4 of (Angeli al., 2019)
For example, see Rousset (2006); Rousset Stoltz. (2006) particular instantiation comhining separate birth and death processes.
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Feynman-Kee Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
argues for particular choice reduce the expected number of resampling events. To define this process, consider the following thresholding operations,
(u) = max(0,- uu
(u)* = max(0, u),
which satisfy: (u) - (()) u.
(55)
We can now define the Markov generator using
J((ylz) = (ge(g) Ep, (gel)
+ pc(()
We(2, y) = (xx)))(())))))
{((x)= = (9((x)) - Ep.19el)
(56)
['(x(2) pn [gxl) Pr (z)dz
Since jump events are triggered based on (x(e)) = (ar(x)) Ey. |y:l)= and are more likely transition events with high excess weight (ge(y) - p. [gr])* pe + et al., 2019).
(v), we expect this process to improve the sample population in efficient fashion (Angeli
Proposition B.3. For a given weightinggfunction and the adjoint generator E (+ the adjoint generator J.(W) derived using in Eg. (56) satisfies ( -,,0) [pe](x). More explicitly, we have
[pr](x)=
[p]](x) = Je (.( (pol(x)
(57)
pr(x)) (s((e) ʃ (n(x) m(w)de)== = (ʃ (alt)) Bpelool) I (x)(=) - Epi [sx]) (=(dd=
(ge(x)) By, (gel) pe
pe (w)ddy) p(()) Byn loil)
Proof. We start by expanding the definition of J. [p)](x) (pe)()))
Je (po((r) = / Ac X - (c()))e(())
(58a)
(s (ar(w) Ex, (eal) [[((z) - Ep. pe (2)d2 pe (((uddy) - pe(x)) ()(x)( (se(x) - Eyp, Ep.leal)
(g)(x) - Ep, (ue)) (e(x)
(58b)
(/ (Da(s) Epr pe pe(w)dy)
(
(r(olte) (ae(()) Epp loel) Pe(ar)
Ep. (9(l) pc(2)dz) ) (i(x) - Ep, Ep.loal)
(58c)
[(9v(t) Ep, lod) Pe(y)dy
Pc(x)) (9l(a) Ep, Epi pe(0)) (c((x) - -Ep Ep.
(58d)
(g((=) Ey, luo)) ( pe(2) Using Eq. (55), note that
[ (p.l pe(z)dz- / dpu(t) (9((z) Ep, (ol) = -- -((o(t)-pppoollll) (s)d)))
(59)
which implies [0x(2) - Ep, lon)) (el))+)((>)d== = 1 (9x - -Ep, (gel) pt (z)dz. We proceed in two cases, handling separately the trivial case where the denominator in Eq. (58d) is zero.
Case (Ae(z)) V E supp(p:)): Note that J (gx(2) - Ep, [9:]) (=()d2 0 if and only if ge(2) Epp [9e], Vz, since (u) 2 0. In this case, the generators become trivial and we can confirm
c; c: (m](s) = (x()) p(()))
(x(x)= - / ((()) p(())d) = (Fx)) Ep.( = 0
(60)
J. (m)(() = / 0. 0 pp))
and thus Eq. (57) holds, desired.
Case 2 (3xx E supp(p:) s.t. (x( > 0): Under the assumption, 3r E supp(ut) s.t. (ge(x) [9e]) > 0. This implies J (gx(z) Ep, [ge) pr(=)d= = (gx()) Ep. (x(2d > 0.
In this case, we can conclude using Eq. (59) that I dd) (a.()) Ep, Bs.
J (d) (g((=) (sri)
18

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Continuing from Eq. (58d)
F,( (pe((t) - [(ar(t] - Ep, (9x)) pr(uidy
([(((z) Bp. (ee)) (c(2)dz) pe(x)) (((x((((())) (((()) - Ex Egg ler)) pe(x)) pe(c) - -Bp, -Ep.bol))
(61a)
= =p((T) (()((z)) Ep.lsil) +
((((x) Epr(
(61b)
()(()) Ep. (9 (r)(xr)
(6lc)
(61d)
as desired. Note that, in the second to last line, we used the identity in Eq. (55) that (u)* - -(u) = u.
B.3. Simulation Schemes
In practice, we use an empirical mean over K particles with as an approximation the expectation Ep, [9t], with
(9((x(()) Ep.l (90(x(())
Ea(al()) ()((=(()) Ep. (oil) + = (a(aalt)) K
1 1L
Σ (a(0)) +
See Del Moral (2013, Sec. 5.4) for discussion.
Discretization of the Continuous- Time Jump Process To simulate jump process with generator [Ø]. we can consider the following infinitesimal sampling procedure (Gardiner (2009, Ch. 12); Davis (1984); Holderrieth et al. (2025)).
With rate (xr) go (x) Ppp (91)
the jumps new configuration,
Zt
with probability 1 1 dt Ae(xx)) + o(d))
(g)(2(()) 1/ E/ Bc )(()(()))
K
Ft-dt
yi+di ~ Categorical
with probability >e(xx)) + o(dt)
j=1 ((((z(0)) - : Σ ge (z(1)))
k=1
The new configuration is sampled according to an empirical approximation of (y(lz) using p. (v) K Ek 8y (z))) where the outer K factor cancels.
Note that the jump rate zero for particles with (x)) 2 Ep. [9.]. Resampling new particle proportional (g. (zx(1)) R Ej dr (x) thus promotes the replacement of low importance-weight samples with more promising samples.
Interacting Particle System Following Del Moral (2013, Sec 5.4), the process may also be simulated using 'exponential clocks'. In particular, we sample an exponential random variable with rate 1, (()) ~ exponential(1) as the time when the next jump event will occur (see Gardiner (2009, Ch. 12)). We record artificial time by accumulating the rate function
Ataais = -i=tio (xa dt for samples along our simulated diffusion. Upon exceeding the threshold time (k) 2 7()) we sample transition according the empirical approximaton of Je(y)z) in Eq. (62). We report results using this scheme in App. F.2 Table A1, but found it to underperform relative to systematic resampling in these initial experiments.
C. Proofs for Table
C.1. Annealing
Proposition C.1 (Annealed Continuity Equation). Consider the marginals generated by the continuity equation
0q((z) (x))
(62)
Øt
The marginals (x) x of (x)) satisfy the following PDE
θ
σe ((,( - -V (V.rr,(())) + Pt. Pr,s(() - (.)))
(63)
(x(x)= = - (B - (1
(64)
19

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Proof. We want find the partial derivative of the annealed density
(x)2) (t. de (e(x)3
a
σ. Pto((r) =?
(65)
By the straightforward calculations we have
①
ff
log Pt. 33 0t
log qt- - dæ pr, Ot
log 4:
(66)
ōt
B(V, vt) 3(V log qt, u1) - f dx Pr, 1 v2) - B( lo gz, u4)
(67)
- (V, v lo vz) + (1 - -3)( v2) - / do pr, [ BB vv) - (V log Pt.s, vz)]
(68)
(V, V logpe, v1) + (1-11111))) (1) 04)
dx PL, d
(1)))))))))))))))))))))))))) (V((V
(69)
Thus, have
0 Dt Pt,((z) -
+ [1-))) v)VV,,())) (1 v(())
(70)
which can be simulated as
dx, = dz(zz)dtt,
(71)
dw;- = - (3 - ()( (t.
(72)
Proposition C.2 (Scaled Annealed Continuity Equation). Consider the marginals generated by the continuity equation
(q.(2) -VV,
(73)
0L
The marginals (3) x of (2) satisfy the following PDE
P(.(()) - -VV.PPr + (t.)(()) E
9c (1)]
(74)
(x = (B - 1) log (x),
(75)
Proof.
We want to find the partial derivative of the annealed density
(x)))
(t,)(x) d (x (rs(() =?
(76)
By the straightforward calculations have
0
log qt - f dæ Pt, Ot
θ
log Pt.O = 3 Øt
log lo
(77)
0t
3( V log v²) - / dr pt, (-3((V ((V log qt, wt)l
-- =-B(V, vx)
(78)
V Pt, Pt, vi) - -f dx pr, [ m4) (V logg wt)
(v,)e))
(79)
[- pr.ol- V log Pt.p, Bui) - -(1 (V-)(Voogpr, - / dr Pr.el
(V,
3) loVpo,,,, 4)]
(80)
Thus, we have
- (p.)) te P(e(()) - (V). + Pr.
(81)
(x(2) - (1 (1-3)(} V log Pt. ut),
(82)
20

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
which can be simulated
dx: = Ber (2, )dt, dw; B(3 1)( V lo dtt
(83)
(84)
Proposition C.3 (Annealed Diffusion Equation). Consider the marginals generated by the diffusion equation
(4)()) oī (g)(
(85)
at
Then the marginals (x) x (x) satisfy the following PDE
σt Pt,s(2) (x) = o/
(x) + (L))(1)) [9((x) De (z))
(86)
(x(x) -333 1) - o2
| log (xx]]
(87)
2
Proof. We want to find the partial derivative of the annealed density
(x (t,)(1)) _d (x(x)
(t.(s) =?
(88)
By the straightforward calculations we have
log 9: / dx p1 0t
log P
B Ot
log qt
(89)
0t
o?
o
V log 4: 11 / dx Pr,e
o
A log at + B oo
B
A loggr at +
B
V log dell
(90)
2
2
2
2
of
A log Pr. +
0
V log
Pr.pt..l''' 122
- dr pt,ee
o²
A
Pt +
o7
V
log
(91)
2
28
2
28
of
A log P Pt.
+
o
V
log
1
1
of
V
log s
(92)
2
2
B
- f du Pt,
[-1
1
1
07
V log log
(93)
2
Thus, we have
of 01 Pr (x) Apr, + (x)) [g((x)) (x).
0
(94)
(((x)= 3(3 -1) 07
| (2)1
(95)
which can be simulated (for 3 > 0) as
dx, = o.dWe
(96)
dur - 3(B 1) of
V log qi (z.)
(97)
2
21

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Proposition C.4 (Scaled Annealed Diffusion Equation). Consider the marginals generated by the diffitsion equation
(q((=) oi Aq((x).
(98)
Öt
Then the marginals (a)) x ue (z) satisfy the following PDE
①
0²
pt))(() (r)
(pr,)()) + Pr. (m))
(99)
01
2/3
(x(x) = (3 1) of
A log ge (x).
(100)
2
Proof. We want find the partial derivative of the annealed density
(x)
(r. de (0(x)3
Pr.((( =?
(101)
By the straightforward calculations we have
log 9: / dx pt Ot
log Pt. = Ot
log qe
(102)
Öt
log qt + 1 o
V log 4:11 112 - / dx pr,s
of
B
A
A log ai + +
| log qell
(103)
2
2
2
2
A Pt, Pt.S +
V log Pt,,sl'''
2
- 1 dx Pt.e
o
A
Pog +
of
V log Pt.
(104)
2
23
2
28
of
oi
1
o²
Pr.3 +
V
log Pt, 11
1
A log Pt,8
(105)
2/
2,3
3
2
- f de pr,
[11 (1
1
of
A Pt. pe,s
(106)
Thus, we have
0t (t,)(()) = of
0
(1)) + (2))(() (x)]]
(107)
2,3
gx(z)= (B- 1) of
A Alog (x),
(108)
2
which can be simulated (for 3 > 0) as
dx; = σt V3
dW.
(109)
du, (8 - 1) of
A log ge dt.)
(110)
Proposition C.5 (Annealed Re-weighting). Consider the marginals generated by the re-weighting equation
0q((z) qe (x) (.
(111)
Øt
The marginals (x))( of.( (x) satisfy the following PDE
0t Pt,((x) = -Pt,s (3g. - Epr. Bg (x)]]
(112)
22

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Feynman-Kan Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Proof. We want find the partial derivative of the annealed density
(x))) (r. ed (e(x))
① a Pt,((()) =?
(113)
By the straightforward calculations we have
i
log - / dr pe, Q
log Pt,s
log
(114)
Öt
0t
Ot
(g((x)) (x((())((())) -f do PI, Pr (B(g(())) (x)()) (x))
(115)
= -gg(x)) - dæ pr,))
(116)
Thus, we have
0 δt ((,) = - Epe, (x)],
(117)
which can be simulated as
dx. 0, du- = (gx
(118)
(119)
Proposition C.6 (Time-dependent annealing). Consider the annealed marginals pt,z(x) (x)))) following some F
dx, = (x,)(x1) + or,3dW,, dw: (x.)(
(120)
(121)
Then, for the time-dependent schedule Br we have
dæ, V1,3, (xt) + or,,,
(122)
0
dw, = 9r.0 (xt) +
log de (z.),
(123)
Ot
sampling from Pt,s, (2) x 9.
Proof. First, let's note that for the annealed marginals (x))(z) (=())) with constant 3, we have
log qt- - / dr Pt,s
log = B Ot
log q
(124)
Ot
Øt
1
1 The Apr.o + (g1,g Ep.,.
V V.,
(125)
Pt,
Pr,
2
Thus, for the time-dependent Br, we have
i θ
08 log 4:
fdx PL,a.
θ log log q:
08, at
Pt,o, Pt,
log qe
Br at
(126)
Ot
at
Ōt
oien Ape.o. +
(on.0 +
03,
03,
= (V, Ut., +
91.0.
log
- -pe..r
9t.0. +
log
Pr.
Pt.S. 2
01
0t
(127)
From which we have the statement of the proposition.
23

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
C.2. Product
Proposition C.7 (Product of Continuity Equations). Consider marginals 9:" (x) generated by two different continuity equations
(a) (V de (z)),
Ouk((t) o
(128)
0t
ōt
The product of densities (x()) x (x) satisfies the following PDE
0 (c = - (( - (x))))
(129)
(x(x) = (V
(130)
Proof. For the continuity equations
a (2) = -- 1.2 (z))
Øt 9t
(131)
we want to find the partial derivative of the annealed density
af (x) pt(x) J af ())
a (())) =?
(132)
By the straightffrward calculations have
0
0 01
θ log oi - -f dr Pt 01
logpt Pr
log de +
log qi a
log के
(133)
31
Ø1
= V, of + x,?) - (V log ot ut)
(v log oi vf
(134)
/ da pel [ vt + nつ? - (V log ut m
V log o u?)l
(135)
(V, + ut) (V log pr. oo + ot (V ot rt) + (V VVlog of ut))
(136)
- f dr pel ( log ut xg)) log /7 =
(137)
Thus, we have
① σe pc(s) (11 (x))))((()),
(138)
ge(x) = (V lf (00))
(139)
which can be simulated
dr: = (z))
(140)
dw, lo (xx), (2)) (V Vlogg((()) uf da.)))
(141)
Proposition C.8 (Product of Diffusion Equations). Consider marginals 1.2 (x) generated by two different diffasiio equations
dql (x)
= of (x), 0t 2
0q o7 (z))
Øt 222
(142)
The product of densities (x) x q (xx) satisfies the following PDE
o² at (c()) (p)( ( p((()) (x)()))(())),
(143)
2
(x = - of(V log qf (z), V log (c))
(144)
24

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Proof. We want find the partial derivative of the annealed density
(z) d af (x)
a ot Pc()) =?
Pc(x)
(145)
By the straightforward calculations we have
①
2
2
/ dx Pt
log Pt
log 1 +
log ci
1 log qi
+
log li
2
(146)
01
01
0l
of
A log के
+
of
V log qi
2
+
of
log
q.
+
oi
V log qi
2
2_
(147)
2
2
2
2
- de Pt.
of
o
of
oi
V log u² ?
A
log
qi
V
log
Li
log vi +
(148)
2
2
2
2
of
03 V loggt g V Vlogg?) - f dx pr pil- (V log V log df)
log Pt +
V
pil
(149)
2
2
Thus, we have
①
ot (2) + (9)(() (p))))(((z)),
P(())
(150)
2 (0(x)= (Vlgg (x). Vlogg
(151)
which can be simulated as
dæ, o,dWe,
(152)
dw:- -op Volog of (Ix), V log di (xx)) dt.
(153)
Proposition C.9 (Product of Re-weightings). Consider marginals (x)) generated by two different diffusion equations
(x)
0qf(x)
(g)(z) - Egg (x) (x).
(gx(x)) (x)) 28 (z).
(154)
Ot
δt
The product of densities (x) x qº (xr) satisfies the following PDE
πe P( (.(x)) Ep, (=))
(155)
(x(x) (x) + 9)(x,,
(156)
Proof. We want to find the partial derivative of the annealed density
(x) Pc(x) d af (x)
a =?
p())))=
(157)
By the straightforward calculations we have
a Ot log qी ff dx Pt
0 log at +
2 log
log Pt log at
(158)
Øt
(v) ( + ()((x)) E ( (x))
(159)
- -f dx p. (
- Eg/
(oR(0)) -B, (x))
(160)
= (x) - (2(x) - f dr pa (o) - {(()]] -(())))
(161)
25

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Feynman-Kee Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Thus, have
a σe p(()) ((() (z)) (+)) 9c(z) = 9 (z) + (1(x),
(162)
(163)
which can be simulated
dr. 0, dw, o) (ax) + (21).
(164)
(165)
D. Proofs of Propositions
Proposition D.1 (Annealed SDE). Consider the SDE
dæ; (-f((a)) + of log (x()) dt + o,dW,,
(166)
then the samples from the annealed marginals (x)) & (x))) can be obtained via the following of SDEs
VV 0Z (1 B)2aa) dW.
log (og (o dr. (-fe(xx) + (3 (B (1- Vog
(167)
dw; (8-1)\V,((<x:)) 20,3(68 V 2og di
(168)
where the parameter satisfies 0 < (8 + 332a)/3.
Proof. For the following SDE
dr. (-fe(a)) + Vog (xx)) dt + o,dW,,
(169)
let's consider everything but the drift fc. Thus, we can write the following PDE
0q: (V. 2e [1 V log ao (og (1 b) oo Aq: b oo 2
Aq.
(170)
Ot
2
We apply Prop. C.2, Prop. C.I, Prop. C.4, Prop. C.3 (rules from Table 1) to the corresponding terms of the PDE above. Hence, the formulas for the weights are
(x(x) - a)0,3(8 - 1) - log q: (xx++
(171)
+ (B - 1) (1 b)o b)o? A log (x:) - 3(B -
bo? 2 2
V log (x)))
(172)
2
Let's cancel out the term with the Laplacians, hence, we have 2a 1 band
(x)) (1 a1-a (1-6)((0----000800-1)))--
log -
3 1) log a (x)]1²
(173)
The PDE for the density is
Opt,a
1-
of
(V, (- (3(1 a) + Vog
Apr,s + Pt,s (g. Ep..9t)
(174)
Øt
B
2
3 + (1
2a of
= =-(ViPLa(-ftt (8+ 3)a) log u))
Apt,o + Pt, - Ep,,
(175)
3
2
26

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Feynman-Ka Correctors in Diffusion: Annealing, Guidance, and Product of Experts
This corresponds the following family of SDEs (0 S (1 3)2a)
o7 (1 3)20)
dr (-f((z)) + + (1-))3a)) (og (x+)) +
dW.
(176)
3
dw:- (8-1)(V
2 0088 V (0)g dl.
1
(177)
Proposition D.2 (Product of Experts). Consider two PDEE corresponding to the following SDEs
dx = (-f((z)) + Vog ( (zt)) + o,dW
(178)
which marginals denote of (xz) and qī (z1). The following family of SDEs (for 0 S (1=+(1-322o) corresponds to the product of the marginals (2) x (gl (z)
of (1 (1--3)) B3)aa)
dx. (-fe(1)) + (1o V log of (xx)))
dW.,
(179)
8
dw,= = Bo V VVllooogg 9 (z1), V l + 3(B 1) ?
o v log (xx) + V log ( (23 - 1)(V, f((e)))
dt.
(180)
Proof. First, according to Table 1, we have the following PDE for the product density (x) x al ( is
Op,(x)
ot 2
(x)+
(181)
Ōt
+ -
(182)
where
(x(x) (Vlogg f(( o (Vlogu((4) + of (
(183)
of log (z), V (log
(184)
V VVV (x), v log (x)) - (fr 1o + V log (
(185)
Now, combining Prop. D.1 and Prop. C.5, for the annealed density pt. x (x)) we have
sp) Øt
(186)
8 3+ (1 3)2a of
+
Am, + pr (z)()) - (x))
(187)
8 2
(x(x) = 3o? (V log d: (z), V log log (z)) - (z)
(z), (z) ( (8 () 2f))) + 3(3 1) o7
(188)
v (z)
(189)
The last step is interpreting
V (r.)( as the weight term, i.e.
0p),((x)
(V, Pr. (x) (-f((x) + oz 02 (1-3) V (z))))+
(190)
Ø1
3 + (1 - 3)2a o3
(p), + (t, - (x))
(191)
3
2
(.(x) = 30 (v log of (I), v llg (x)) + 3(3 1) o3
V of (x) + V (x)
(192)
2
+ +23 - (V
(193)
27

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Feynman-Kee Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Thus, we get the following family of SDEs (for 0 < (3 + (1-3)2aBBB
+ log /7 (sx))) + of (1 3)2aa
dx. = (-f((z)) +
dW.,
(194)
B
Vof(Vlogge (z)), Vloga (xx)) B(3 1) o
dwy=
v log क: (xx) + V log (z,) (28 - 1)(V,, f((.)) dt.
(195)
Proposition D.3 (Classifier-free Guidance). Consider two PDEs corresponding to the following SDEs
dx. (-fe(x)) V loga 2 (xx))dl + o,dW
(196)
which marginals we denote as qf (xx) and (2t). The SDE corresponding to the geometric average of the marginals Pe.a(x) x
(B + 2a(1 - 8))/B 2 0, is
dr = (fe((r)) + o (1 1 a(1 3) (11 B) 12 (2t) + BV log (mx))) ) dt + 1a 1/1
2a(1 - 5)
dW.,
B
(197)
dw; - (z) v logg 200 () (21), Vlog Vlog
(198)
Proof. First, according Prop. D.I, we perform annealing p(x) x -=) and pi, x of (z)2, i.e.
0p. 0px) - (V, (V.P)) + 0i + Ba, (loggl))))
1 1- + 23a1 of
App
(199)
01
1- 3 2
+ + (2))))())(9(())))))))))))))))))) - Epi, 3 ,((x)),
(200)
(x(0) = B(V. (x))) + 307 (=)
(201)
and
0pī, (2)
( + of(8 (10000000))00000--0-00000//////0 B 8+ (1 + (1) (g((() F,g,, (x))
3)222 of
Apt
(
(202)
0t
3
2
(203)
(x(x) = (B- 1) V) + 7 1) log (zx))
(204)
Now, we would like to match the diffusion coefficient (to directly apply Prop. C.8 for diffusion equations in the product case, and avoid the evaluation of additional Laplacian terms in the weights).
3 2.aa
3 + (1 - 9)222 = B - 32 + 2,2 a1 = 3 - 132 + (1 - --9)2202
(205)
1-8 8
B
Bag (1 - -3))2z = a2 = 01 a(1 - 18)2
(006)
B2
28

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Now, according to Table 1, for the product density Pr,3 x ( we have
v (og + V log ))
V, (1))) -2f((z) -20-
(207)
1
B + (1 3)2a of
(x) + Pr. (x) - (()))
(208)
B
2
(x(x) = ((VV)((x)) + - 1og (x)
(209)
2
Eq. (201)
1
+ - 1V +
(210)
2
Eg. (204)
+ (1 - 3) (V log of (x), fx) + ( (1 B)a) (1-))) log (f)
(211)
+ + v log 0 (z). - (z) + o. (1 - 3)
(8+ (1- 3)a Vlgg
(212)
B
of B) log (2)), v log (
(213)
where the terms in the last three lines arise from the conversion rules from the product in Table Finally, we obtain
(x(x) ]
log log (x) V oog 4 a(1 3) 2 208 - 1) ( (IV - (V. ((1- Log (x) v log (z), fx))).
(Vlog (x), V Vlog (o))(())))-
(214)
8
(215)
Finally, we re-interpret (V. (z)) the weighting term, and get
0pt,)(z)
of ( 1 a(1 a(1-))
V. (.P(((
(f((x)
B)V + a log VVggell()+++ooolltoo (+))
(216)
Ōt
3
+ o. (1
2a(1 3)
1
B (p) pr.) Fp.,o
(217)
2
gc(x)= = 3 - 1) -ogdd)) 200
(218)
2
Thus, for (3 + 2a(1 - B))/B3 2 0, we have
dx.- (fe(wz)) + of (1+ a(1 - ) (11 (V) + +or/1 1
2a(1 - 3)
dW,, (219)
B
B
1
dw:- 2
08 - -1)/ log (1x) - df + (2:), Vlog
(220)
29

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Feynman-Kat Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Proposition D.4 (PoE + CFG). Consider two PDEs corresponding to the following SDEs
+ dr = (- (z) + a,dW,, (zt)) + a,dWe,
(221)
dr. = (-f((z)) + V (
(222)
with corresponding marginals (((z)), (21) and (z1). The SDE corresponding to the product of the marginals Pr,((x) x (xx))21100( (z) is
dx, (-fc(z,) + +f(1(a) (z a.dW,
(223)
du 2 1)
- (
(zx) || + V (xg) - V log (): (2+)))))
(224)
+
(z:))
(225)
where we denote v. (x)) = V gt + V log de (x).
Proof. Using Prop. D.3, we start from the SDEs simulating the product q: (z)) (x)² and (rr),ii. i.e.
dx; (- f((z)) + of (11
+ Io yf (xx))) dt o.dW.
(226)
(m))
dw; 2 20 /) (z)) - v log g! (z))
1
(227)
dr, - f((w)) + 31 (og + 3V log of (x:))) + t,WW
(228)
mm(
1
dw:- 2 20 (000331)//V/// (x:)) v of
(229)
Then we consider the product of these SDEs, i.e.
@pr (V.rr (z)))) o(())))) + of
(p),((x) (9 - (x)
(230)
Öt
2
1 (000(-11) (1000-10((00n))))-ooo00111111111 [og - vlog 1P)
(x)
(231)
((000-10000000000000000----00101 (z)
(232)
00/0 - (07500-1((V/ggttt))-eoooo (1)
Vogg.(<) - Vlog
(233)
+ (10))
(234)
Re-interpreting (V, Pt. (.)) we get
0pt,z(x) Øt
(V. Pr.s (x) f((() + +z(())))))))))) + of
(x)))
(235)
9((x) 1 10 [og ologg
| (z) - V oz)
(236)
{x) ( f()))
(237)
which corresponds to
dr. = (-fe(zt) + o + v (xx))) ao,,,
(238)
dur- 5073(3 (| Vlog (z:) (Vogga))) - log olog
[log
(239)
+ (o))()))) (z))) + (V.
(240)
30

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Proposition D.5 (Target Score Product SDE). Consider two PDEs corresponding to the following SDEs
dx = (-fi(z)) + of log &i Jdt + ordW,,
(241)
with corresponding marginals (xx). The SDE corresponding to the product of the marginals Pt,o(x) x II, ()()) can be simulated as follows
dæ =
- ((((z) + to²
Σ 3 v log (xx)) dt - o,,W,
(242)
2
-1 (V
- of
o, 2
dw =
Σ
B V log (1))
Σ
3
V
log (xx)) dt.
(243)
2
Proof. The target log-density is defined as
(x) = Σ Be log (x) log Zt.
(244)
The dime-derivative of the target log-density is
log log = Σee Ba ot
log (x) Ex(()) Σ B: Ot
log (x).
(245)
Öt
We focus on the first term, and write for all the marginals their corresponding PDEs
i log (x) = (v. f((z) + (ogg((()) (V (og -f(xx) -V + (ogg
(246)
Øt
+ oº
+ o²
(z)
Vogg (x)
(247)
2
2
= -f(x)) + of v log logg((z)) - (V (ogg(()) + of a log log (ix)
of
v log af (x)|
2
2
Σ 3 ot
0
log (x)x = - (V.
- - o - (VVoo)))))(oo())))..oooa-. (1)o
(248)
0,
of
+
A logp((z)
Σ 3 [o
(249)
2
2
(V, (c(x)
V. o
(1) + o
(250)
of
Alogpt(a)
V [=g
(251)
+ Σ B. - [v)((z))) + of
of 2 E> Vo (x))
[Voop:(<))
log
(252)
2
Writing down the PDE (z). we get
Öp((z) - (V.
+ (f logg)(())) + of
+ ()())) - (x(x)
(253)
Øt
2
(x(a)= =
Σ 8 -11 (v.(((z)
of 2
[logp(<)))
of Σ 3 (1)
(254)
2
which ends the proof.
31

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Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
In partieular, for the target (x())) ( 8 (1)
have
+ 8 V Vlog (o dt + ordW,
dx =
- [1()
+ o. (1-g)V
(255)
2
2
of
- V log (x) + マ/2
3
||
dup
(1-3)V [)Vlog((>))
v log of (z)
(256)
2
2
2
(1 V)
+ $
( ||
+ (logg((ze)1" dt.
(257)
2
Proposition D.6 (Reward-tilted SDE). Consider the following SDE
dr. vx(2)dt + a,dW,,
(258)
which samples from the marginals (x)). The samples from the marginals pr(x) x (x()) exp(Ber(x)) can be simulated according to the following SDE
dr. (v+(34)) avr( + o,dW,,
(259)
dw,= = ( ((r(())) of Vog (2,) -
oi 3,
(260)
+ (a- 31 o 2
Ar(zz) + 03: Øt
r dt.
(261)
For the reverse-time SDE with drift u, (z,)) = (z, V log qe (z,) corresponding the original diffusion generative model and a 3,07/ /2, we obtain the following weighted SDE
dx. (-Je(ax) + of Vog (x (I+) + Br of 2 vr(xx))dl + o.dW.,
(262)
[83 Ot r(x) (.Vr(z), of (og f((z)) ) dt
dw:
(263)
2
Proof. First, consider the density qe (x) that follows the PDE
(4)()) - (V, (x())()a(()))) + ot Am(()).
(264)
Ot
2
We want to find the PDE for the reward-tilted density
qe (x) exp(((((()))
(x(x) = f dx ( exp(3er(x))
(265)
Straightforwardly, we get
i log
② log qe (x) + 08, r(x) - / do pe(a))
08,
rl()))
log (x) +
ai r(x)
(266)
01.
at
Hl
For the first term, we have
i
log qe (x) = - (V, (V,e((z)) - (V log (x) ex())) + o7
0
2
A log qe (x)
+
2
v log gr (x)
(267)
2
=V(V) (Vlogpi (x), (x))
A Alogpt(x)
+
| log (x))
(268)
2
o² 8, Vr(zz)
3, of
+ (VVr(()) ((( of (z)
Ar(z).
32

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Thus, have
0p((s) = (V.p((r))s(z)) + of A p(())(((())) - B((c))((()) 2
(269)
Öt
9r (z) (. (rVr))), v l 2 (.(((()))) -
Br Br of Ar(x)+ 0t r(x).
7())
03
(270)
Furthermore, we can add the gradient of the reward as additional drift tenn (r(x),,e.
@p((x) (V. + +aVr(z))) + of 2 )
(p)())
(271)
Ot
(x))) = adr(x)= + a (Vogg)< vr(a)) - 31 of
03: r(z)+
Ar(x)+
+
(272)
2
0t
of 3. 31(r(()))
8, 3.Vr(z). (xx) of log (x
Taking ve(x)= = fi) + V log (x) and a= /,0 we have
(p((x)
(- Vlogu( + + of of
Vr(z)) + of Ap((2) + (1(((((((()) - (x)(
0L
of (x())= = Be Vr(x),
V Vlogm((o))
+ 03. r(x) + -((()) - oi
(VV(()))
(273)
2
Ot
03:
Vr(x)) of 2
r(x)
+
Vlogg(1)) f((x))
(274)
0t
This can be simulated
dx: (-fe(za) V + t of Vr(xx))dt o.dW
(275)
103 dw, Øt r(x) + (Bvr(s)), of Voggg(()) - f((0)) /
(276)
E. Additional Related Work
Amortized Sampling Recently, there has been renewed interested in learning amortized samplers, and particularly diffusion-based amortized samplers particularly towards molecular systems. Midgley et al. (2023) explored learning normalizing flow using an a-divergence trained with samples using annealed importance sampling (Neal, 2001). Zhang & Chen (2022); Vargas et al. (2023; 2024); Richter & Berner (2024); Akhound-Sadege et al. (2024); Albergo & Vanden-Fijnden (2024); De Bortoli et al. (2024) learn diffusion annealed bridges between distributions using various methods.
While we use DEM this work as achieves state of the art results for our LJ-13 setting, there are several works that build upon DEM using bootstrapping (OuYang et al., 2024) and learning the energy function instead of the score (Woo Ahn, 2024). We note that our FKC approach can be applied to any diffusion based sampler.
The reweighting portion of our Feynman- Kac weighted SDEs corresponds to a non-parametric Fisher-Rao gradient flow of a linear functional [pv]] = f 9t P,dx, whereas gradient flows in the Wasserstein Fisher-Rao metric (Kondratyev et al., 2016; Chizat al., 201 8; Liero et al., 2018) have form similar to our weighted PDEs (Lu al., 2019) for an appropriate ODE simulation term Vt Vgt.. In sampling applications, Chemseddine et al. (2025) study the problem of when a given tangent direction in the Fisher-Rar space can be simulated using transport via tangent direction in the Wasserstein space.
(Vaaserrteiin--Fisher-aa Gradient Flows
F. Additional Experimental Details and Results
F.1. Sampling Metrics
We use a number of metrics to asses the quality of generated samples. These metrics capture different aspects of the distribution. Before computing metries we filter out samples with energy > 100. This only affects non-resampled metrics
33

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Feynman-Ka Correctors in Diffusion: Annealing, Guidance, and Product of Experts
and prevents numerical. We find this filters no samples for DEM or with FKC, and filters less than 3% of samples with target score SDE tempered noise SDE sampling. We justify this it is easy to set these filters for generated samples of very poor quality.
Distance-W2 For the LJ-13 task compute the 2- 2-Wasserstein distance between pairwise distance histograms. For this metric we take all pairwise distances for all samples and flatten them into single distribution. For a sample of 10,000 points this leads to distributions of size 700,000 as there are 70 pairwise distances for a 13 particle system. This is useful metric as it is equivariant and measures the global fidelity of the generated samples. It however is not useful for assessing fine grained details of the generated samples. For that we turn to the Energy-W1,p distances.
Energy-Wi/2 The Energy-Wi and Energy-W2 measures the deviation in the energy value distribution of samples from the reference distribution and the generated distribution. We find this metric is useful to assess the overall fit of a model, although it cannot assess whether a sampler drops modes well. A model that has a reasonably small Energy Wasserstein distance may still have missed a mode of a similar energy value.
We note that for the LJ-13 task we exclude samples with energy > 100 for all methods and metrics. In practice this only affects Target Score and Tempered Noise SDEs without FKC and DEM trained at lower temperatures. This excludes roughly 2-3% of samples for those models, which helps these baselines.
Maximum Mean Discrepancy (MMD) We use a radial- basis function MMD with multiple scales to assess distribution fit. This measures how well the reference distribution matches the generated distribution locally.
Total Variation distance For low dimensional sampling problems, it is useful to consider the total variation distance between empirical distributions that are discretized into a grid. This measures fit in terms of density, ignoring the underlying metric, and is less sensitive global reweighting of modes.
I-Wasserstein and 2- Wasserstein distances (W1 (W2 On 40 GMM we also measure the 1- I-Wasserstein and 2- Wasserstein distances between the generated and reference distributions with respect to the Fuclidean metric. We note that while this possible to measure in the LJ-13 case, is not as useful as particles in the LJ-13 setting are SE(3) equivariant, and therefore the Euclidean distance is not suitable ground metric.
F.2 Mixture of 40 Gaussians
The mixture of 40 Gaussians setting is a 2D energy function with 40 randomly initialized modes with equal standard deviation. This serves as a useful experimental setting where we are able to calculate true densities and scores efficiently without modelling error.
F.2.1. ADDITIONAL RESULTS
We include quantitative results for the tractable GMM example in Sec. 5.2, where we start at temperature Tz 3 and anneal to target temperature Tss 1/3. We used geometric noise schedule with amin = 0.01 and a dmax 500. We sample 10k samples with 1000 integration steps, with dt = 0.001. We observe that Target Score sampling (a 0) from Eq. (22) with systematic resampling performs best in more metrics. We also use this example as an ablation study for the impact of the resampling scheme, where we find that systematic resampling appears to outperform the birth-death exponential clocks implementation of the jump process resampling. See Sec. 4 and App. B.2.
On ground truth de A subtle point to note is that qf To is not a mixture of ||| Gaussians, but rather |T|T& Gaussians for integer Tr., This means that we are restricted to small integer Tr. We use Tr. = 3 for all experiments in the 40 Gaussians setting. Note that we reserve 3 T for the ratio of learning and sampling/target temperatures.
F.3. LJ-13 Sampling Task
The Lennard-Jones Potential. The Lennard-Jones (LJ) potential is an intermolecular potential, modelling interactions of non-bondee con-bonding particles. This system is studied to evaluate the performance of various neural samplers. The energy for the system is based on the interatomic distance between the particles is given by:
6
12
EW3 (W)(
E Σ 2T
Fim dey
(277)
d.
if
34

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
MCMC Dataset
MCMC Dataset
Tempered Noise SDE
FKC Samples
8.07
Target Score SDE + FKC Samples
Nor Porma (lal11 lized Densif ty
0.87
DEM Samples
Normalized Density
DEM Samples
0.06
0.06
0.05
0.05
0.04
0.04
0.83
0.03
0.02
0.82
10
00,
0.01
0.00
0.00
-40
-30
-20
28 25
-80
-70
-50
-40
-38 -24
E(x)
E(x)
Figure AI. Comparison between the energy distribution of the MCMC dataset, samples generated using DEM model trained the target temperature, and samples generated using temperature annealing from model trained at starting distribution T = 2. Left: the target temperature is and Right: target temperature
MCMC Dataset
MCMC Dataset
No Normalized lized Jensity
Tempered Noise SDE FKC Samples
DEM Samples
No Normaizeed Density
1.75
DEM Samples
Target Score SDE FKC Samples
8.6
1.58
0.5
1.25
0.4
1.00
0.3
0.75
0.50
0.25
8.88
3
4
5
7
8
4
5
Interatomic Distances
Interatomic Distances
Figure A2, Comparison between the distribution of the interatomic distances of the particles in the MCMC dataset, samples generated using a DEM model trained at the target temperature, and samples generated using temperature annealing from model trained at starting distribution T = 2. Left: the target temperature is 1.5 and Right: the target temperature is 0.8.
35

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Feynman-Kan Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Table Mixture 40 Gaussians. Sampling from an annealed distribution with inverse temperature 5 runs with 10k samples.
Metries are calculated over
SDE Type
FKC
Energy-
MMD
Total Var
W1
W2
Target Score
0.943 + 0.026
0.020 + 0.001
0.487 + 0.007
11.304 + 0.296
15.671 + 0.269
Tempered Noise
1.032 + 0.012
0.058 + 0.001
0.638 + 0.002
16.051 + 0.123
19.627 + 0.101
Target Score
BDC
1.064 + 0.369
0.010 = 0.004
0.402 + 0.029
7.797 3.990
12.451 5.417
Tempered Noise
BDC
1.228 = 0.401
0.056 = 0.029
0.572 0.055
12.598 = 4.155
17.679 = 4.178
Target Score
systematic
1.098 + 0.418
0.007 + 0.005
0.372 + 0.020
6,256 + 3.960
11,265 + 5.629
Tempered Noise
systematic
0.926 + 0.248
0.027 + 0.011
0.512 + 0.017
9.974 + 1.229
14.045 + 1.308
Table A2. Additional results for LJ-13 at different target temperatures. The model trained at starting temperature T, = 2.0 and metrics are computed over runs. DEM run for one seed only as the standard-doviation over seeds is negligible.
Target Temp.
SDE Type
FKC
distance-W2,
Energy-Wi
Energy-W2
0.9 (B=2.2)
Target Score
0.215 0.001
13.886 0.040
14.893 0.012
0.042 0.009
6.218 = 0.896
6.259 + 0.873
Tempered Noise
0.1 0.016
5.633 + 0.090
7.682 + 0.585
0.042 0.004
4.384 = 0.135
4.530 = 0.167
DEM
0.168 +
14.516 +
14.606 +
1.0 (3=2.0)
Target Score
0.221 + 0.001
12.915 + 0.054
13.558 1 0.112
0.039 0.008
2.629 = 0.665
2.876 = 0.548
Tempered Noise
0.094 0.002
5.215 = 0.095
6.560 + 0.000
0.053 0.008
3.205 = 0.462
3.538 = 0.468
DEM
0.127
1.352 +
2.050 +
1.2 (B=1.67)
Target Score
0.234 = 0.004
10.414 = 0.036
10.910 + 0.110
0.026 + 0.001
2.831 + 0.155
2.915 + 0.074
Tempered Noise
0.098 + 0.002
4.258 = 0.069
5.564 = 0.095
0.076 0.006
1.017 + 0.494
1.300 + 0.433
DEM
0.143
9.669 +
9.736 +
where we denote the Euclidean distance between two particles and j by d, = constants. As in Köhler et al. (2020), we also add a harmonic potential to the energy so that LLJ-eystem = ELJ (x) + The harmonic potential is given by:
||æ, æzyl2 and Fmm and are physical
posc
(x)
Σ
|| [ - IcOM|
(278)
2
where ICOM refers the center of mass of the system. We I'm 1. 2.0 and 1.0.
Training details. All DEM models are trained for 166 epochs on 4 NVIDIA A100 80GB GPUs. For all models, the best checkpoint with the lowest energy-W2 is used for inference. The model is an EGNN with the same architecture as in Akhound-Sadege et al. (2024). Similar Akhound -Saoound-Sddghhhhhe al. (2024), we use geometric noise schedule for all experiments. We set mmin 0.01 and mmax 4.0. We clip the score maximum norm of 1000 (per particle). For sampling, we use 1000 integration steps with dt 0.001. For inference with FKC, assume Gaussian distribution at time tstatt 0.99 and start integration with the annealed SDE and resampling at that time. We found that this helps significantly to reduce the variance of the results over different runs. For visualizations in Figs. Al and A2, we selected the best run for all methods for consistency.
In line with previous work, we find the DEM scores are noisy at high times, based on the score of the energy. This can be seen from the score estimator in DEM, which depends on the average gradient direction from a normal distribution sampled around 2t. The variance of this estimate grows with both time and gradient of the energy. This makes DEM style objective significantly easier to train on smooth energies, as quantified by norm of the score of the energy.
36

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Target Score SDE
Target Score SDE FKC
Tempered No1se
Interatomic Distances W1
Target Score
Target Score SDE FKC
Tempered Noise SDE FKC
Tempered Nolse SDE
0.20
Tempered Nolse FKC
Energy W1
18
0.10
0.85
2.0
1.8
1.6
1.4
1.2
1.9
8.8
2.0
1.8
1.6
1.4
1.2
1.9
0.8
Target
Temperature
Target Temperature
Figure A3. Left: 1- 1-Wasserstein between energy distributions and Right: 2- 2-Wasserstein between distributions of interatomic distances of MCMC samples from the annealed distribution and generated samples.
Target
Tenpered
Target
Terpered FKC
0.82
0.82
8.80
0.08
0.80
8.80
-88
-60
-48
-88
-68
-48
-98
-68
-48
-89
-50
-48
8.00
0.04
8.84
8.02
0.02
0.62
0.00
0.99
0.89
0.80
-60
-40
-29
-68
-48
-20
-60
-49
-28
-68
-48
-20
0.86
0.96
0.05
8.84
0.94
8.04
0.96
0.84
8.62
0.82
8.62
8.00
0.80
0.69
0.00
-60
-40
-20
-60
-49
-29
-69
-48
-28
-68
-40
-29
8.06
8.66
= 1.2
0.96
0.06
0.94
0.84
0.02
8.82
0.02
0.96
8.66
0.06
8.84
0.94
8.84
0.02
0.62
8.02
-20
28
-20
Figure A4. Energy distributions samples generated with temperature annealing compared to the MCMC samples (in dark green), at different target temperatures. The starting temperature is T,
2.0.
37

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Feynman-Kat Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Table A3. Comparison of EDM2+FKC ( with EDM2+CFG for image generation using EDM2. With sweep over batchsize K. For
all metrics, we report CLIP and ImageReward (IR) scores averaged over 10,000 images.
FKC
7
BS
CLIP (1)
IR (1)
40
1
32
28.75
0.24
40
2
32
29.03
-0.07
40
4
32
29.11
0.02
40
8
32
29.14
0.05
40
16
32
29.04
0.05
40
32
32
29.
0.04
40
64
32
28.95
0.01
Sampling Reference distributions To generate reference distributions from the Lennard-Jones-13 potential we Pyro (Bingham et al., 2019) and Noru sampler (Hoffman & Gelman, 2014) with default arguments. We use 20k warmup steps and collect 20k samples from the 10 chains for each temperature.
Additional results In Table A2 we can see additional results extending Table 3 for intermediate temperatures. Here we can see that generally the same patterns hold with one exception-DEM on target temperature of 1.0 is better than FKC with 3 = 2.0 on Energy-Wi and Energy-W2 metrics. This means that DEM has on this temperature has better local fidelity but slightly worse global fidelity. This quite interesting as we know DEM was originally developed and therefore tuned with temperature of 1.0 this dataset. Our hypothesis that some hyperparameters are specifically tailored this setting. It is quite interesting then that FKC can still perform better than DEM on global metrics for this temperature.
In Fig. A3 we can see the Energy-Wi and Interatomic Distance W2 metries plotted against the target temperature using model trained at temperature 2.0. Here see that FKC performs well across all temperatures for our global metric of interatomic distances and across energy Wi distances, although at low target temperatures the Energy-Wi metric gets worse for all methods. We note that this is after excluding roughly 2-3% of samples with unacceptably bad energy from the Target Score SDE and Tempered Noise SDE Therefore even though the lines are close here, we still prefer the FKC samplers.
F.4. EDM2 image generation
Additional experimental details For all experiments with EDM2 we use the default classifier free guidance weight 1.4. For the y ablation we use batch size 32 for FKC and for the steps experiment we use batch size for FKC.
In Table A3 we experiment with FKC batch size. Theoretically larger batch sizes should be better it corresponds larger sets for the importance sampling step. We see that FKC gets higher scores with larger batch sizes, even with batch size 2 with performance plateauing after batch size Rs 8.
Additional Results
F.5. Latent vs. ambient space image models
We apply the FKC method for generating images with Stable Diffusion-XI. (SDXL.). latent diffusion model. We show performance of SDXL+FKC on the GenEval benchmark in Table A4, but we do not observe any significant increase from SDXL with vanilla CFG. While there are examples where FKC improves the semantic accuracy of SD-XL generations, (see Fig. A5), the gain not consistent when evaluated on 1000 prompts. There multiple potential reasons for this behaviour, e.g. sampling from the geometric average is not better than CFG for these metrics, or the dimensionality of the latent space too high, resulting in weights with too much variance. Investigating the effectiveness of these methods with latent diffusion models could be an interesting area of future study; in the present study, we find FKC to be consistently helpful when applied to models in the ambient space.
F.6. Multi-Target Structure-Based Drug Design
Additional experimental details for main experiments Following Zhou et al. (2024), we align the target pockets in 3D space and generate sample coordinates for each pocket using an SE(3)-equivariant graph neural network over 1000 integration steps. We then use our Product of Experts (PoE) scheme (Prop. 3.3) to guide ligand generation.
We note that the PoE weight computation in Eq. (27) necessitates equal sample dimensionality, otherwise resampling would be skewed to favor samples of higher dimensions. This requires the molecules within a batch to have the same number of atoms. This motivates our use of fixed size molecule bins for generation.
38

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Feynman-Kan Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Table A4. Image generation using SDXL with classifier-free guidance (CFG) the GenEval benchmark (Ghosh al., 2023). For all
metrics mean values reported. indicates values directly taken from leaderboard.
Model
B Overall
Single object
Two object
Counting
Colors
Position
Color attribution
GenEval original 553 prompts
CLIP retrieval*
0.35
0.89
0.22
0.37
0.62
0.03
0.00
SD-1.5*
0.43
0.97
0.38
0.35
0.76
0.04
0.06
*
0.55
0.98
0.74
0.39
0.85
0.15
0.23
SDXL (our run)
7.5
0.57
0.99
0.80
0.46
0.86
0.11
0.22
SDXL+FKC
5.5
0.58
0.99
0.77
0.49
0.87
0.10
0.22
SDXL+FKC
7.5
0.57
0.99
0.78
0,46
0.83
0.13
0.23
GenEval 1000 prompts
SDXL (our run)
7.5
0.58
0.99
0.79
0.45
0.88
0.11
0.21
SDXL+FKC
5.5
0.57
0.99
0.79
0.42
0.86
0.13
0.21
SDXL+FKC
7.5
0.57
0.99
0.80
0.45
0.83
0.13
0.22
"a photo of blue TV and grean to- othbrush
*e photo of red hendbag end gre- en conputer nouse"
"e photo of an or- ange bench and e black refragerator"
CPC (B 7.5)
CPG FKC (p 5.5)
Figure A5. Examples where SDXL+FKC (hottom) outperforms EDM2+CFG (top).
Our baseline is the target score SDE with 3 0.5, which is equivalent DualDiff from Zhou et al. (2024) and also corresponds to averaging of scores (Liu et al., 2022). We also generate molecules conditioned on single protein pocket using TifgetDiff from Guan et al. (2023), but dock the molecules to both targets in protein pair to understand the need for conditioning on two pockets simultaneously.
SDE Component Analysis In Table A5, we show the performance of varying the following SDE settings for dual-target drug design: SDE type, 3, and the presence/absece FKC resampling. Here, we report metries for generating molecules on single protein pocket pair (UniProt IDs P23786/P05023) as the validation set. We generate molecules batch 32 molecules for different molecule lengths, which were sampled from the original training distribution (Guan et al., 2023):
(15, 19, 23, 27, 35).
We study the impact of the following changing the following settings:
Inverse temperature (5) We find that as we increase 3 from 0.5 to 2.0 the product of the docking scores of the protein pair increases, though the delta increase is larger at smaller fis.
FKC. Next, we try turning FKC on at a fixed 3. find that performance improves at both 3 = 1.5 and 2.0, although the improvement at | 2.0 is larger. However, this comes a cost of diversity and the uniqueness of molecules generated.
Fmax Given that resampling is helpful in terms of improving the quality of the final molecules but decreases molecular diversity, we investigate setting Imxx for our best / settings, where we resample only when <= 1max- We find that
39

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Feynman-Ka Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Table A5. Performance of generated molecules with different SDE settings. We generate 32 molecules for 5 molecule sizes for one protein pair for each setting. Lower docking scores are better. Values are reported as averages over all generated molecules in each run. "Better than ref." is the percentage of ligands with better docking scores than known reference molecules for both targets (the mean docking score for the reference molecules is -8.255+1.8a0 We also report the diversity, validity & uniqueness, and quality, which refers to the
percentage of molecules that are valid, unique, have QED 2 0.6 and SA S 4.0 (Lee et al., 2025b). Bolded values are the best metrics
within each set of midlines. For B 1, target score and tempering noise match (Prop. 3.3).
FKC
SDB Type
(P, P2)
max(1, (c)
(2)
Better than ref. (7)
Div. (t)
Val. & Uniq. (f)
Qual. (t)
0.5
O
Target Score
67.057
-7-077 20087
8.377
0.251 20.
0.886
2.002
1.0
0
Target Score
73.386
7:520
3.840gasm
8 174
0.378220a11
0.884-000s
0.9022 ona
0.231 29.10
1.5
0
Target Scure
75 75.213.15
-8.085_n
-8 8,9800amms
-8 -,225_11m
0.402 anm
0,850_
0.988
0,250 100
2.0
0
Target Score
75.551 300.545
1.089 aoo
8. 1.906
8.309
0.391 20.321
0.881
0.994.0.012
0.288, -o:88,_.100 10.179
1.5
0
Target Score
15.210
8.085 ass
0.980
8.258
0.880
0.988...o0r
0.250 :0:000000.0 150
1.5
0
Target Score
75.798.32 a64
-7.438_2.00 2,007
-8 3.0
-8822 181
0.446 454
0.651_o.00
0.475
2.0
Target Scure
75.551 116
-8.089
-8. 3,960 18s1
-8.309 1:113
0,391
0.881
0,904
0.288 179
2.0
Target Soure
91.8455:2.422
8.977
9 9.984,1.53 533
8.978 434
0.67
0.617
0. 0.473
0. 0.044go.0ma
1.3
0.4
Target Scure
74 314488
-7 7o
-8 883
-8 3..322=1.m
0.372
0 0..02
G. 8.981
0 0,2622...22
0.5
TTrget Soore
80.421 100.007
0.3142ua52
2.007
0.867
D. 0.9044-0.022
0.288 pas8
1.5
0.6
Target Seure
83. 13.405519000
-8 8.530
9.516
-8 66666.1.00
0 0.185
0.830
0.925aa.ora
0. 0.244
O
0.7
Target Soore
83. 19.34
8,434_om2
9.420 2o
1 8.23_1.mm 1.997
50830 sa
0.790_0.00
0. 0.162210.00 205
1.5
1.0
Target Score
75. 798
7.438
8.582
8. 8.829
0.446
0.651
0.475ga
0.1000
0.4
Target Score
79.734.m7 sa
9,280gooss
2.487.1113 133
20.
0.876
0.988_..om
1088_...00 20.015
0.306 as4
2.0
0.5
Turget Score
61.919
_.500__000
-9. g
-8. 796 1.00
0.51
0.851 _.s1__.ee
0.938 10.000
0.244
0
0.6
Target Scure
87 87.953
-9. 1.0
0.569.0.40
0.818_..m:
0.212 201
2.0
0.7
Target Score
65 168
-8,574=1.07
-9.5 9.510=112z
-8. 222
0 2.508
0.780=0.044
O. a.800
0. 0.231 20.224
Target Scure
91.8 91.845.za.co 421
-8.97741.45 453
9,98441 ass
1.977211.1 4s4
0.67
2.617g000
0.475ga 102
0.044go.oo
0..
0
Target Score
077657221 965
-7 -7.6672000
-8 37750001
-7 189
0.261 so is
0.896g..00
0. 0.969_.. 1969_..oe
0, 0.244 101
Tempered Noise
19.6
7.83820.872
8.727
8.085 uss
0.887
2.044ga.00e
0.300.2 2122
0.5
0.6
TTrrgrr Score
77. 20.
8 8.24320544
9.1122001
8.337
T.
2.877
0.975
0.21223.108
0.5
0.6
Tempered Noise
78.501
8.323
9.127
8 8.496 100
0.496 anw
0.87920.014
0.951ga.0
0.250 16s
2.0
0
Target Score
75.551
8.069 wno
8. a01
-8 -.50091111
0.301
0, 0.881
0.994_..om 011
0.288 179
2,0
Tempered Noise
75.868 136154
-8.015 9.90
-8. -8,352 1.977
0,060 380
0.874_c.m
0,984,
0,262 1..
2.0
0
0.6
Targer Score
87 87,983 22 asa
8.700 201
1.
8.988 11.
0.569 io ane
0.81810.011
0.888 ...om
0.212 10000
2.0
0.6
Tempered Noise
19.696
8 2.2
-9. 9,104anm
-8.88111.41
0. 0.464220.22
0.796
0.838
0.331:00.22
2.0
0
0.7
Target Score
85.168 85.168:21.25
8.57441.007
-9.5 9.514=1m2
-.882711.42
0.593 484
0.780
a.806
0.231
0.7
Tempered Noise
81,909 186
-8.531
-9,809, 170
-8.515.n.a
0.589 40.427
0.796.o.ozz
0,850 ...10...131
0.262 10........3
setting /max to a value in 00. generates molecules that are higher in quality compared to always resampling or no
resampling for 3 = 1.5. For ( 2.0, the performance slightly decreases, but the diversity and uniqueness of the molecules is much higher at the end. Setting mma 0.6 gives good tradeoff in terms of generating molecules that perform well vs. maintaining diversity, and so we proceed with 3 2.0 and tmax 0.6 for the final experiments.
SDE Type Finally, we try using different types of SDEs. We find that at lower 8, the Tempered Noise SDE performs better with and without FKC. At higher 3, however, using the Tempered Noise SDE does not significantly change performance or decreases performance. Thus, for the main experiments, we proceed with the Target Score SDE.
Visualizing docked molecules In Fig. A6, we visualize the molecules with the highest docking scores to the protein pair GRM5/RRM1 (UniProt IDs P41594 and P23921, respectively) at each molecule size.
Correlation between FKC weights and target metric In Fig. A7, we plot the Spearman rank correlation of FKC weights and docking scores of the final generated molecules for two protein pair examples to understand whether sampling according to these weights will select for better molecules. We observe that for a protein pair where molecules improve with FKC resampling, there tends to be a positive correlation between the FKC weight and docking score (avg. p = 0.275 for an example protein pair), while this trend is less clear for a case where FKC resampling does not improve the molecule (avg. p 0.087 for example protein pair).
E.7. Molecule SMILES generation using latent diffusion models
We also investigate generating molecular SMILES strings conditioned on functional properties, which describe the desired function that the molecule should have. Molecules often need to possess multiple properties (e.g. bind to protein X and be non-toxic) (Wang et al., 2025). By controlling for these properties during the molecular generation process (as opposed to host-hoe filtering), we aim to increase the probability of discovering molecules that exhibit all desired characteristics, thereby improving the efficiency of hit identification.
40

---

Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
15 atoms
19 atoms
-7.389
8.169
P41594
QED 8545
-7.648
452
-6. -6.898
-7.22
8.819
-8. 241
88. 741
P23921
DED: 8.65
QED: B, Target Diff
OED: 0.61 DualDiff
QED: 0.68 FKC [ours]
QED: 8.21 TargetDiff
@ED: 8.88
QED: 0.68 FKC [ours]
DualDifF
23 attoms
27 atoms
444
99.
-9.788
P41594
QED
-7.648
-7 887
1313
-9. BBB
-7.467
-9.326
18 352
P23921
OED: 8.65
QED:
8.59 FKC [ours]
QED: TargetDiff
QED: DualDiff
QED: 0.48 FKC [ours]
Reference
Tiff
DualDiff
35 atoms
10.370
-12,
P41594
OED:
-7.648
444
598
18. 536
P23921
OED: B865
QED: 0.24 TirgetDiff
QED: 8.28 DualDiff
QED: 8.41 FKC [ours]
Reference
Figure A6. Molecules generated from our method (target score SDE with / = 2.0 and FKC resampling) and baselines in the binding pock- cts proteins: GRM5 (UniProt P41594) and RRMI (UniProt ID P23921) for all molccule sizes considered (f15, 19, 23, 27, 35) atoms). Docking scores for each molecule and target are above each image; lower docking scores are better. The QED of the molecule is above each model name. The binding pocket is shaded light green.

---

Feynman-Ka Correctors in Diffusion: Annealing, Guidance, and Product of Experts
0.00
0.08
8.48
8.40
Spearman s
8.20
8.20
-8. 0.28
-8.28
-B.
-0.40
-8.60-
-0.60-
8.08
8.20
8.48 0.60
0.88
1.08
0.80
0.28
8.
8.68
8.80 1.08
(a)
(b)
Figure A7. Spearman rank correlation cocfficient (p) of FKC weights computed at each timestep T and the docking score of the final molecule for (a) protein pair where the final generated molecule improves with FKC and (b) failure case where the molecule has worse docking score than the baseline. Note that during inference, we only resample for r E [0.0, 0.6].
Table A6. Multi-property molecule generation results (PoE). For set of two larget properties (P1 and P2), we take the set of top- 10 best performing molecules from batch-size of 512 as the molecules with the highest P PP₂ scores. We report averages of the top- 10 molecules from 5 runs and the top-1 molecule overall, We also report the diversity, validity & uniqueness, and quality of all molecules.
P2 22
SDE Type
8
FKC
F1 to)
P2 top-10 (1). (P1, P2) top-1 (t)
Div. (t)
Val. & Uniq. (1)
Qual. (t)
Target Score
0.5
0.21220.016
0.35640.046
(0.500, 0.580)
0.91040.000
0.713+0.027
0. 0.12720.015
JNK3
20.00
U. 00.6000
(0.590,
0.88120.002
0.81340.022
0.352+0.012
GSK33
Tempered Noise
1.5
0.342 012
0.502 0.004
(0.500, 0.720)
0.882_a.002
0.832 021
0.360 021
Target Score
0.5
0.090
0. 0.43440.0os 006
(0.150,0.472
0.915 001
0.671 ;0
0.228
JNK3
0.132:10.052
0.55020.006
(0.280, 0. 0.469
0.884_0.001
0.650
0. 0.258 020
DRD2
Tempered Noise
1,5
0.141 020
0.617 0.000
(0.360, 0.655)
0.884_a.00s
0.661
0. 0.252
Target Score
0.5
0.1 g0.04
0.528
(0.051, 0.908)
0.914+0.00z
0.709
0.203
GSX38
0.228=00.166
0. 0.649=o.oo 084
(0,550, 0.655)
0.884_a.002
0. 0.774:0.015
0.303
DRD2
Tempered Noise
1.5
0.26610 061
0..6810.006
(0.520, 0.796)
0.885
0.77410.017
0.307
Model We select LDMol (Chang & Ye, 2024) to generate molecules, which is a latent diffusion model conditioned on natural language descriptions of molecule properties; this gives flexibility generating molecules with a wide range of properties.
Experiment setup: TDC oracles We consider three proteins oracles from Therpeutic Data Commons (TDC) (Huang et al., 2021): JNK3, GSK3,, 8, DRD2, which predicts whether or not a molecule binds to a protein. Note that while this task similar in nature to the objective of SBDD, we are generating molecules conditioned on a functional text description instead of 3D protein pocket. However, we could also consider other functional property descriptions, such as molecular solubility, toxicity, etc.
Prompts To generate molecules that inhibit specific protein, we prompt the model with "This molecule inhibits (protein_name) following Wang et al. (2025).
Metrics In addition to reporting the nop-perfrnnnng molecules, we report the percent of molecules that are valid and unique, as well as their diversity (evaluated using Tanimoto distance on Morgan fingerprints (Rogers & Hahn, 2010)) and quality, which is the set of unique and valid molecules that also have quantitative estimate of drug-likeness (QED) 2 0.6 and synthetic accessibility (SA)= < 4.0. This metric was taken from Lee et al. (2025b).
Results: TDC oracles We aim to generate molecules that satisfy the function of binding each protein when taking all combinations of the protein pairs. In Table A6, show the best performance for each set of molecules and in Table A8 ablate different SDE components. We find that the tempered noise SDE at higher 3 generates molecules that have higher fitness for binding to each pair of proteins. When we incorporate FKC, the average performance of the molecules further increases. We also note that PoE+FKC tends to generate more molecules that are unique, valid and are higher drug- like quality, although their diversity decreases slightly, which is a common tradeoff. practice, we find that the FKC weights with the latent diffusion model have a large variance during molecule generation. This is problematic, as large number of
42

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Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Table A7. Docking scores generated molecules to P1=ATPIA] and P2=CPT2 We used the Tempered Noise SDE with 8 = 1.5 and
generated 32 molecules.
FKC
(Fi, P2) top-10
(P,, P2) (
Div. (f)
4.05 2.36
(-8.87, -8.13)
0.921
7.49 20.11 8..1
-8441 99.73)
0.895
Opp Mrs
FKC
x FKC
44g
P1: -8.412
F2: -9.727
-8.870
P2: +8.133
M. MMM
HWy
-3.235
HyM
P2: -9.717
P1: -7. 548
HN
H₂M
P2: -9.291
HD
GH
OH
04
DH
PuN
Mrb
HD.
dH
Figure A8. Molecules with best docking scores for binding to ATPAI (P1) and CPT2 (P2) from PoE with FKC (left) and without (right).
samples are thrown away. Furthermore, we noted that the score was not always well-conditioned. To ameliorate this, for all experiments using LDMol, we divided the weights by a set temperature term (T = 100) to reduce their variance before resampling, clipped the top 20% to account for any score instabilities, and did early- early-stopping (only resampled for 70% of
the timesteps).
Experiment setup: protein docking Finally, we consider a more challenging setting of protein-ligand docking, where we generate molecules using LDMol based on text-based prompts of binding to the proteins ATPIA1 (UniProt ID PO5023) and CPT2 (UniProt ID P23786), and then evaluate them using docking. The protein pockets were obtained from Zhou et al. (2024) and the final generated molecules were docked using AutoDock Vina (Eberhardt al., 2021).
Results: protein docking Table A7 shows the docking scores of molecules, and we find that incorporating FKC generates molecules with better scores. While ligands are typically generated using SBDD, we find it interesting that text-prompt generation is able produce molecules that have reasonably good docking scores; known binders to ATP1A1 and CPT2 have docking scores -8.16 and -9. 144 respectively (Zhou al., 2024). We visualize the top molecules in Fig. A8.
43

---

Feynman-Ka Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts
Table A8. Multi-propese propeity molecule generation results. For set of two target properties (Pı and Pz). take the of the top-10 best performing molecules as the molecules with the highest P1 *P2 scores. We report the average properties of the top- molecules over five and the top-1 molecule overall. We also report the diversity, validity & uniqueness, and quality of all generated molecules, where quality is the percent of molecules that are valid, unique, have a QED 0.6 and SA < 0.4. For / = 1, target score and tempering noise match (Prop. 3.3).
P1
SDE Type
FKC
t0p-10 (t)
Pa t0p-10 (f)
(P1, top-1 (1)
Div. (t)
Val. & Uniq. (t)
Qual. (f)
P2
Target Score Tempered Noise
0.21220.016
0.35620.046
(0.500, 0.580)
0.910 000
0.713
0.127
0.5
0. 0..25=0.028
0.385
(0.440, 0.690)
0.909
0. 00.03
0.134=00 008
0.280
0,420=a.018
(0.470,
898 002
0.811 +0.00
0.205=0.001
1.0
JNK3
0.342
0.442 051
(0.600, 0.650)
0.897 10.002
19.015
0.205 10 015
3
Target Score
0.336 10.001
0.484_0.000 10.002
(0.480, 0.780)
0.886
0.81610.013
0.33610.022
Target Score
0.351 0340
0.447=0.025
(0.590, 0.780)
0. 0.886 003
0.823=00024
0,356+0.08
1.5
Tempered Noise
0.341 1.0009
0.468
(0.590, 0.560)
0.88 10.002
0.81320.025
0.352
Tempered Noise
0.342:20.012
0.502g90.04
(0.500, 0.720)
0.88220.002
0.802..0.021
0.360=z0022
Target Score
0.5
20.01
0.434_a.005
(0.150, 0.472)
0.915 0.915=0.00m 0.01
0.67 0.671g0.022
0,228 20.011
Tempered Score
0.066.0.015
0.571
(0.110, 0.943)
0.914:1:002
0.678 0.678...0188
0.0888
0.236. g: 020
0.087..0022
0.624
(0.100,
O 9,0000 001
0.675
0.24110:010
JNK3
1.0
0.09440.024
0 0.635
(0.413, 0.550)
0.899
0.686
0.263
DRD2
Target Score
0.136.20046
0.582_0.067
(0.490, 0.640)
0.886 00s
0.639
0.241 #0.017
Target Score
0.102 10.001
0.62010.144
(0.320, 0.541)
0.885 10.06
0.659La.022 L0.022
0.274 10 028
1.5
Tempered Noise
10.002
0..00_0.006
(0.250, 0.469)
0.884
0..50
0.258 10 020
Tempered Noise
0.141 00.000
0.617=9.010
(0.360, 0.655)
0.884
0..61
0,252
Target Score
0.146 20.04
0.528=0.077
10.051,0.900
0 0.91 0 001
0.709 1..09
0.203 = 015
0.5
Tempered Score
0.162:10.022
0.543 063
(0.430, (0.430,0.965)
0. 0.914 001
0.697
0.198
0.202:10.22
0.620
(0.660, 0. 72)
O. 0.908 002
0.77320.021
0.238 021
1.0
GSX3,8
0.190=a0022
6666+o.op
(0.240, 0.986)
0. 0.907
0. 00.04
0.254= 010
DRD2
Target Score
0.240.0.030
0.63620.066
(0.350, 0.804)
0.894=0.omm 002
0.750
0.290
Target Score
0.222 L0.006
L..64_o.068
(0.630, 0.580)
0.891 0.891 10.000
0.740 L0.027
0,283 1o.883
1.5
Tempered Score
0.228 20.016
0.64940.08. 084
(0.550, 0.655)
0.88440.002
0.77440.015
0.303
Tempeered Score
0.266 10.01
0.638_o.005
(0.520, 0. 796)
0.885
0.77410.017
0.307 10
44


Reviews:
[{"review_id": "w45jBiGRLU", "reviewer": "Reviewer_3HLs", "summary": "Prior work has shown that composing multiple pre-trained diffusion models is not straightforward in the context of energy-based models. This paper investigates Feynman-Kac correctors based on the Feynman-Kac formula to sample from annealed, geometric-average, and product distributions. Compared to the Fokker-Planck equation, a reweighting function is introduced, allowing for the incorporation of Sequential Monte Carlo resampling schemes. Experimental results demonstrate the effectiveness of the proposed algorithm.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their time, feedback, and positive appraisal of our work. We are heartened to hear that the reveiwer feels that the paper is \"theoretically sound\" and found that the paper was \"generally well-structured\". We now address the questions and suggestions raised by the reviewer. \n\n> When comparing Eq.(5) and Eq.(7), an additional term is added. I am curious whether this arises from the evolving normalizing constant $Z_t$ in the context of sampling scenario? In contrast, does the vanilla FP equation in Eq. (5) describe the evolution of distributions while keeping $Z_t$ unchanged?\n\n  Both PDEs in eqs. (5) and (7) preserve the normalization of the density in the sense that $\\int dx ~ p_t(x) = 1$ for all $t$, although the evolution of density changes with the addition of weighting terms. \n  \n  Note that the normalization constants $Z_t$ defined in eq. (13) change in time to guarantee that the density on the left hand side is normalized, e.g. $\\int dx ~ p_t^{\\text{prod}}(x) = 1$. This is due to defining the density up to a constant, i.e. $p_t^{\\text{prod}}(x) \\propto q_t^1(x)q_t^2(x)$.   \n   Note that, while the integral term $\\int g_t(x) p_t(x) dx$ in eq. (6) ensures normalization for the density evolution PDE, our SMC resampling can proceed with access to only the unnormalized weights $g_t(x)$ (see eq. (9))\n  \n  \n> To obtain FK PDE in Eq.(7), is it simply the addition of the FP equation and the reweighting equation? I guess there may be something missing before the definition of the FK PDE.\n  \n  FK PDE in eq. (7) is a more general type of PDEs than the FP equation in eq. (5). Indeed, the main difference between these PDEs is the reweighting term (the last term of FK PDE, i.e. eq. 6) that allows for simulation of processes not possible with FP equation. For instance, FK PDE can change the weights of disconnected modes by reweighting the samples without transporting them, while FP PDE would have to transport samples from one mode to another (via the vector field or noise) to adjust the relative weights. We will emphasize this point in the final version of the paper.\n\nWe would like to thank the reviewer for their time and feedback. We hope our answers here allow the reviewer to continue to positively endorse our paper, and we would happily clarify any addition questions which may arise.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes, I checked some of proofs.", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "This paper investigates Feynman-Kac correctors based on the Feynman-Kac formula to sample from annealed, geometric-average, and product distributions.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths:\n\n- The paper is generally well-structured and theoretically sound.\n\n- The paper proposes Feynman-Kac correctors based on the FK formula and apply it to different scenarios, including anealed, geometric-average, and product distributions derived from pre-trained diffusion models.\n\nWeaknesses:\n\n- Please see the questions below for the authors.", "other_comments_or_suggestions": "No.", "questions_for_authors": "- When comparing Eq.(5) and Eq.(7), an additional term is added. I am curious whether this arises from the evolving normalizing constant $Z_{t}$ in the context of sampling scenario? In contrast, does the vanilla FP equation in Eq. (5) describe the evolution of distributions while keeping $Z_{t}$ unchanged?\n\n- To obtain FK PDE in Eq.(7), is it simply the addition of the FP equation and the reweighting equation? I guess there may be something missing before the definition of the FK PDE.", "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": "PKyER2Wn0x", "reviewer": "Reviewer_jpZY", "summary": "The paper derives a suite of new tools for modifying pretrained diffusion models at inference time, using particle resampling techniques. In particular, they use the Feynman-Kac formula to derive the evolution of weights for particles when simulating the diffusion reverse SDE (or different variants of it) such that resampling using the weights results in samples from 1) tempered versions of the original diffusion marginals at different diffusion times 2) products of the marginals of two diffusion models 3) geometric averages of the marginals of two diffusion models. These weighting terms are derived for a plethora of different SDEs, and the paper focuses experimentally on ones where the 1) score function is scaled with a scalar 2) both the score and SDE noise are scaled with specific scalars. The paper explores sequential Monte Carlo resampling methods, and jump process reweighting methods. The proposed methods are evaluated on \n- different sampling problems, where the ability to change the temperature at runtime is validated\n- multi-property molecule generation, where the authors use the method to generate molecules that simultaneously inhibit multiple proteins\n- image generation, where the use of the method as a classifier-free guidance replacement is investigated", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their detailed feedback! We are thrilled to hear that they find our paper to be a 'useful contribution to the literature', 'well written and organized' and a new tool for the diffusion sampling community. Below, we address the reviewer's questions.\n\n> What is the interval $[t_\\min, t_\\max]$ in the experiments used? Does it require significant tuning?\n\nThe choice of the interval $[t_\\min, t_\\max]$ depends on the hyperparameters and, especially, the noise schedule, but does not require significant tuning. In practice, we chose the interval based on the fraction of unique samples resampled at every iteration during the resampling step. We report the corresponding plot in Fig. 2 of [the PDF](https://anonymous.4open.science/r/FKC-D7F0/rebuttal.pdf) for the annealing experiments. Note that the scale of the weights is proportional to the noise $\\sigma_t$ (see Prop 3.2), which results in a low number of unique samples close to $t=1.0$ where the variance of the noise is the largest due to the Variance Exploding schedule. We have added corresponding plots and discussion to the manuscript.\n\n> Why is dividing the weights by some constant factor $T$ is necessary? I suppose it makes sense that the particle resampling may be especially high variance in high dimensions?\n\nIndeed, as the reviewer points out, the variance of the weights grows with the number of dimensions due to being proportional to the norm of the score. For the latent molecule generation (8000 dimensions), we had to divide the weights by $T=100$ and clip lowest and largest $20\\\\%$ of the weights in order to achieve reasonable variance of the weights. To verify that the need for such heuristics is caused by the dimensionality, we did a new empirical study for molecule generation in the coordinate space, where the dim. is 3 * number of atoms in the molecule  (<100). Here, we neither divided nor clipped the weights and only chose the interval $t_\\max$ based on a validation set. We present new results below and have added them to the manuscript.\n\n> Is early stopping the right phrase to use here?\n\nThank you for suggesting a better explanation! The closest analog to selecting $[t_\\min, t_\\max]$ is the reducing the time interval for the integration of diffusion models, i.e. instead of integrating $t\\in[0,1]$, one usually integrates $t\\in[ε,1]$, where ε is small constant. We have changed this explanation in the manuscript.\n\n> Practical details on image generation\n\nFor image generation, we used Stable Diffusion XL with a variance-preserving SDE and an Euler–Maruyama solver, running for 100 steps. All the computations are done with float16 precision, which is crucial for stable generation via SDXL. All images are generated in 1024x1024 resolution. In the FKC portion of our experiments, we did not apply weight rescaling; moreover, we resampled all 64 particles after each time step throughout the entire time interval.\n\n## New molecule experiments\nWe did a new set of molecule experiments using a model that directly predicts the 3D coordinates of the atoms in a molecule from [1] on a harder set of tasks of generating molecules that dock to a pair of proteins simultaneously, expanding on the promising results from Table 4 of the original submission.  For 14 proteins pairs, we generated 32 molecules at 5 different molecule sizes (160 molecules per pair) using FKC product and found that it improved nicely over two SOTA methods:\n\n---\n**Table: Docking scores of generated ligands for 14 protein target pairs (P₁, P₂).**  Lower docking scores are better. \n|||(P₁ * P₂) (↑)|max(P₁, P₂) (↓)|P₁ (↓)|P₂ (↓)|Div. (↑)|Val. & Uniq. (↑)|\n|-|-|-|-|-|-|-|-|\n|P₁ only [2] ||62.77±23.74|-7.30±1.90|-8.38±1.51|-7.44±1.93|**0.89±0.01**|**0.95±0.07**|\n|β|FKC|||||||\n|0.5|no [1]|64.35±21.54|-7.14±2.12|-7.90±1.99|-7.96±1.67|**0.89±0.01**|0.89±0.21|\n||yes|64.05±31.21|-6.86±3.26|-7.89±2.90|-7.92±2.42|0.88±0.02|**0.95±0.11**|\n|1.0|no|69.03±21.61|-7.54±1.74|-8.24±1.71|-8.30±1.53|**0.89±0.01**|0.90±0.19|\n||yes|69.83±32.70|-7.40±2.93|-8.51±1.82|-8.27±2.88|0.85±0.02|0.92±0.10|\n|2.0|no|68.12±18.56|-7.40±2.03|-8.21±1.66|-8.11±1.62|0.88±0.01|0.94±0.16|\n||yes|**75.54±23.26**|**-7.91±1.62**|**-8.60±1.62**|**-8.66±1.55**|0.81±0.05|0.88±0.09|\n\n---\n[1] Zhou et al. \"Reprogramming Pretrained Target-Specific Diffusion Models for Dual-Target Drug Design.\" NeurIPS (2024).\n[2] Guan et al. ICLR (2023).\n\n> Implementation details \n\nFor molecules, we chose the time interval based on a validation set of 1 protein pair at 5 molecule lengths (32x5 generated molecules). We kept $t_{\\text{min}}$ at 0 and sweep over values of $t_{\\text{max}}$ for $\\beta=2$, SDE type=\"Target score\", and FKC=\"on\". We show this and additional ablations over $\\beta$ and SDE type in [this PDF](https://anonymous.4open.science/r/FKC-D7F0/rebuttal.pdf). Setting $t_{\\text{max}}$ to 0.6 gives a good tradeoff in terms of generating molecules that perform well vs. maintaining diversity, and so we proceed with $t_{\\text{max}} = 0.6$.", "claims_and_evidence": "I think that the claims are supported by evidence.", "methods_and_evaluation_criteria": "I think that all of the datasets chosen make sense, each tests one clear ability of the collection of methods proposed. The evaluation criteria are sensible as well.", "theoretical_claims": "I did go through Proposition B.1 and B.2 and didn’t find issues. But there are many more propositions in the paper.", "experimental_designs_or_analyses": "I think that the experimental design, and their analysis made sense, but I didn’t check the their validity in a lot of detail.", "relation_to_broader_scientific_literature": "The paper presents methodological progress in the context of applying inference-time controls to pretrained diffusion models using particle resampling methods. The paper presents many new tools, and, e.g., the ability to temper the target distribution has not been considered in previous work, to the best of my knowledge. This could serve as a very useful building block for diffusion methods targeting at sampling from unnormalised densities. Perhaps the most new important contribution is the systematic mathematical exploration of different reverse SDEs and their impact on the resampling schemes, and the paper could serve as a helpful reference for future work to build on.", "essential_references_not_discussed": "I am not aware of essential references not discussed, although I am not also particularly familiar with the literature on using SMC to guide diffusion models.", "other_strengths_and_weaknesses": "I think that this is overall a very nice paper, and a useful contribution to the literature. The paper opens up new tools for diffusion guidance to the community, is well written and organised, and experiments are done on multiple examples, showing the \n\nA weakness of the paper is that some practical details regarding the experiments are a bit more complicated than the theory would imply, and some hyperparameters exist that are not discussed at length in the main paper. I highlight two parts to kick of discussion: \n\nFrom section 4:\n“We find resampling only over an ‘active interval’ t ∈ [tmin, tmax] useful for improving sample quality and preserving diversity, and set weights to zero outside of this interval” \n\n- What is this interval in the experiments used? If this requires significant tuning for different data sets, that seems to be a downside of the method. Regardless, this should be detailed in the paper. \n\n\nFrom the Appendix, regarding the compositional molecule generation experiments: \n\n“In practice, we find that the FKC weights have a large variance during molecule generation. This is problematic, as a large number of samples are thrown away. Furthermore, we noted that the score was not always well-conditioned. To ameliorate this, we divided the weights by a set temperature term (T = 100) to reduce their variance before resampling, clipped the top 20% to account for any score instabilities, and did early-stopping (only resampled for 70% of the timesteps).”\n\n- Do the authors have intuition on why is it necessary to divide the weights by a set term 100 in this example? Is this done for the other experiments as well? 100 seems like a large number to use here, and a significant difference from the method implied by the mathematics seems, on the face of it, to raise into question the relevance of the mathematical method definition. \n- That said, I suppose it makes sense that the particle resampling may be especially high variance in high dimensions? \n- Is early stopping the right phrase to use here, as usually in machine learning it refers to a regularisation method for neural network training? \n\nThe image generation results seem quite okay, but there are not a lot of practical details. Weight stabilization techniques used? Active interval choices for resampling? How many particles? I understand that this is not necessarily the highlight of the paper, but since the experiments are there, it would be useful to include the details as well.", "other_comments_or_suggestions": "-", "questions_for_authors": "See “Other strengths and weaknesses”", "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": "OAdyEoe4s9", "reviewer": "Reviewer_pXP5", "summary": "This paper points out modifying score function of the pretrained generative modelings, such as classifier-free guidance might cause to generate samples that are not from the same distribution as the training data, and the corrector schemes used to address this problem either requires infinite many steps requiring more computation resources or is detrimental to the sampling quality. This paper proposes Feynman-Kac PDEs, aiming to generate samples from the same training data distribution and improve sampling efficiency.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the constructive feedback. We are glad that the reviewer finds our paper to be 'well-structured and complete' and the experimental design to be 'reasonable'. We also would like to thank the reviewer for bringing up reference [1] (Du et al.), as this is an excellent reference for clarifying our contributions.\n\nBefore addressing the concerns raised, we would like to clarify a potential misunderstanding about the proposed method.\n1. The main objective of the proposed method is to modify the target density at the inference time (e.g. by sampling from the annealed distribution or product of densities) rather than 'sampling from the training data and improving sampling efficiency'.\n2. In complete agreement with [1], our work shows that simulating the reverse SDEs with modified scores **does not** sample from the target densities. This is exactly the goal of introducing the Feynman-Kac corrector scheme, which, as we prove, allows for consistent sampling from the target densities by re-weighting and re-sampling. We have added a self-contained proof of Eq. 9 in the updated Appendix to further emphasize this claim.\n\nNext, we would like to address the salient concerns raised by the reviewer individually:\n\n> the choice of $t_\\max, t_\\min$\n\nThe choice of the interval $[t_\\min, t_\\max]$ depends on the hyperparameters and, especially, the noise schedule, but does not require significant tuning. In practice, we choose the interval based on the fraction of unique samples resampled at every iteration. We report the corresponding plot in Fig. 1 of [the PDF](https://anonymous.4open.science/r/FKC-D7F0/rebuttal.pdf) for the annealing experiments. Note that the scale of the weights is proportional to the noise $\\sigma_t$ (see Prop 3.2), which results in a low number of unique samples close to $t=1.0$ due to the Variance Exploding schedule.\n\nFor molecule generation experiments in Table 3, we selected $t_\\max$ based on a validation task by going over the grid $[0.6,0.7,0.8,0.9,1.0]$; $t_\\min$ was always set to $0$.\n\nWe have also added a new set of molecule experiments for a harder set of tasks using a model that directly predicts the 3D coordinates of the atoms in a molecule from [2] (Zhou et al.) and docks molecules to a set of target protein pairs. Again, we use a validation task to sweep over $t_\\max$ values, which we report in Table 2 of [the PDF](https://anonymous.4open.science/r/FKC-D7F0/rebuttal.pdf). \n\n> Ablation study for the resampling methods\n\nWe perform the ablation study of the two considered resampling methods in Table 6 of the Appendix. We find that systematic resampling always performs better or comparably and opt to use it throughout the rest of the empirical study.\n\n> Computational cost of the resampling step\n\nThe additional computational cost of the resampling step is negligible compared to the reverse SDE simulation. Indeed, all the weights depend only on scores, which are already evaluated at the forward pass. We purposefully avoid the computation of the divergence operators when deriving the integration scheme (see Lines 176-189 L). The cost of the systematic resampling step is equivalent to generating one uniform random variable, which is negligible.\n\nOur proposed method is notably more convenient than [1], which requires changing the diffusion model parameterization and performing additional MCMC steps or Metropolis-Hastings tests.\n\n> $\\beta = 7.5$ for FKC on SDXL\n\nAs requested by the reviewer, we provide results for both values of $\\beta$ on the Geneval benchmark [3] (Dhruba Ghosh et al.). See the detailed image-generation study in response to Reviewer 65RJ.\n|Model|$\\beta$|Overall|Single object|Two object|Counting|Colors|Position|Color attribution|\n|-|-|-|-|-|-|-|-|-|\n|SDXL (Rerun) |7.5|**0.57**|0.99|0.80|0.46|0.86|0.11|0.22|\n|FKC|5.5|**0.58**|0.99| 0.77|0.49|0.87|0.10|0.22|\n|FKC|7.5|**0.57**|0.99|0.78|0.46|0.83|0.13|0.23|\n\n> Derivation of equation (6)\n\nEquation (6) can be understood by the separation of variables. Indeed,\n$$\\frac{\\partial_t p_t^w(x)}{p_t^w(x)}=\\partial_t\\log p_t^w(x)=\\bar{g}_t(x)\\implies p_t^w(x)=p_0^w(x)\\exp(\\int_0^t ds \\bar{g}_s (x)),$$\nwhere the exponent part corresponds to the update of the weights $dw_t=\\bar{g}_t(x)dt$. Note that in eq. (9), $w$ appears as log-weights for Self-Normalized Importance Sampling.\n\nNote that Eq. (6) is meant to *introduce* the reweighting evolution. In later sections, we *derive* the weights which preserve the particular target distribution evolution under simulation or transport by an SDE with given drift/diffusion.\n\n### Closing remarks\n\nAgain, we thank the reviewer for their questions, which gave us the opportunity to clarify and improve our work. We hope that our answers fully address all the important questions raised by the reviewer, and we are happy to consider any additional questions or further suggestions.\n\nWe kindly ask the reviewer to consider increasing their score if our responses address their concerns satisfactorily.", "claims_and_evidence": "For Section 3, this proposes composing a few diffusion models during inference time, especially the product and Geometric Avg examples,  but I am not sure if this is a proper conduct. As [1] proves and empirically shows that the reverse SDE induced by composed score function do not correspond to sampling from the composed model and the reverse diffusion sampling will generate incorrect samples from composed distributions. Instead, they propose to sample with Langevin dynamics.  Based on my understanding, Feynmann Kac PDEs have SDE component embedded, so I doubt simply performing sampling process according to what this paper introduces will give wrong samples too. I might make the wrong connection and correct me if I am wrong.\n\n[1] Du, Y., Durkan, C., Strudel, R., Tenenbaum, J.B., Dieleman, S., Fergus, R., Sohl-Dickstein, J., Doucet, A. and Grathwohl, W.S., 2023, July. Reduce, reuse, recycle: Compositional generation with energy-based diffusion models and mcmc. In International conference on machine learning (pp. 8489-8510). PMLR.", "methods_and_evaluation_criteria": "I think the experimental designs for performance evaluation are reasonable.", "theoretical_claims": "I did not spot issues of the proofs or claims except the one I mentioned in the ``Claims and Evidence'' section.", "experimental_designs_or_analyses": "This paper proposes doing SMC and Jump Process for resampling to improve the performance, but I don't find the details specifically for each experiment. For example, this paper mentions SMC is only done when $t \\in [t_{\\mathrm{min}}, t_{\\mathrm{max}}]$, but those $t$'s are not mentioned in the experiment. I also cannot find if both of resampling methods are really used during the inference time, as the tables or figures do not show for them. An ablation study is required if so.\n\nWith all those two methods plugged in for resampling, I am also wondering the latency and computational cost-.\n\nIn table 5, the experiment is missing when $\\beta=7.5$ and FKC presents. I cannot see why this is not done.", "relation_to_broader_scientific_literature": "1. This paper derives PDEs to describe the time-evolution of sample density under the standard SDEs. This makes the sampling process more flexible.\n 2. This papers propose a few composition approaches, such as annealed, product and geometric average distributions. Again, as I have mentioned, the correctness is doubted as the reverse SDEs wouldn't give the right distribution [1]. \n\n[1] Du, Y., Durkan, C., Strudel, R., Tenenbaum, J.B., Dieleman, S., Fergus, R., Sohl-Dickstein, J., Doucet, A. and Grathwohl, W.S., 2023, July. Reduce, reuse, recycle: Compositional generation with energy-based diffusion models and mcmc. In International conference on machine learning (pp. 8489-8510). PMLR.", "essential_references_not_discussed": "I think it is complete.", "other_strengths_and_weaknesses": "The paper is written well-structured and complete.\n\nAgain, the weaknesses would be I did not notice the computation cost is reported to support the true efficiency of the proposed approach. Otherwise, the resampling process should be very costly.", "other_comments_or_suggestions": "No.", "questions_for_authors": "1. For equation 6, I cannot really see how $$\\frac{\\partial p_t^w(x)}{\\partial t} = \\bar{g}_t(x) p_t^w(x)$$ is derived. I quickly skimmed through the Appendix but did not find it explained anywhere.", "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": "0UsYv22v8W", "reviewer": "Reviewer_65RJ", "summary": "This paper proposed that the Feynman-Kac Correctors enhance the sampling of several types of compositional distributions. The main idea is based on the Feynman-Kac formulation and the transformation of transport and diffusion to reweight operation.\n\n## update after rebuttal\nMy view is not changed, so I maintain my original score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the detailed feedback and constructive suggestions. We are glad that they find the proposed idea 'novel' and its derivation 'supported by valid theory'. We are also happy to hear that our method 'contributes to the understanding of Diffusion Models.' We provide the suggested comparisons and answer the reviewer's questions below.\n\n> Testing the metrics on large-scale datasets (at least 1k prompts)\n\nWe thank the reviewer for the constructive feedback which has led to some interesting findings and a deeper understanding of this work. We evaluate FKC on two large-scale tasks: on ImageNet-1K using EDM2 [4] and on the Geneval benchmark [1] using SDXL, which evaluates the algorithms on a predefined set of prompts and measures adherence to the prompts. These additional large-scale experiments have led to two new insights:\n1. **FKC improves image quality and prompt adherence in ambient space but not latent space.**\nPoint 1 is supported by the following two tables. We can see substantial improvement in the performance of FKC on an ambient diffusion model but not on a latent diffusion model.  \n- **EDM2** We first conduct a new set of image experiments using the EDM2 model [4], which directly generates outputs in pixel-space. We compare generations from the baseline method, which uses CFG, with generations guided by FKC on ImageNet-1K and evaluate 10k samples using two metrics: CLIP Score and ImageReward [5], which reflects how closely they align with human preferences.\n\n||steps|churn|Clip Score(↑)|ImageReward(↑)|\n|-|-|-|-|-|\n|CFG|32|40|28.75|-0.24|\n|FKC|32|40|29.00|0.04|\n|CFG|48|10|28.83|-0.18|\n|FKC|48|10|**29.02**|0.06|\n|CFG|64|80|28.67|-0.21|\n|FKC|64|80|28.88|**0.09**|\n|CFG|128|40|28.71|-0.19|\n|FKC|128|40|28.94|0.06|\n         \nWe find that incorporating FKC improves both scores and qualitatively generates better images; we include examples in [this PDF](https://anonymous.4open.science/r/FKC-D7F0/rebuttal.pdf). We use default settings of $\\beta=1.4$ for both models. \n- **SDXL Geneval** The table below consists of three parts: Geneval benchmarks, performance of FKC on the prompts from Geneval (553 prompts), and on 1k prompts (553 + newly generated). We also rerun SDXL with the same hyperparameters as our algorithm for a fair comparison.\n\n|Model|$\\beta$|Overall|Single object|Two object|Counting|Colors|Position|Color attribution|\n|-|-|-|-|-|-|-|-|-|\n|CLIP retrieval||**0.35**|0.89|0.22|0.37|0.62|0.03|0.00|\n|SD 1.5||**0.43**|0.97|0.38|0.35|0.76|0.04|0.06|\n|SDXL||**0.55**|0.98|0.74|0.39|0.85|0.15|0.23|\n|**Geneval Prompts**|\n|SDXL (Rerun)|7.5|**0.57**|0.99|0.80|0.46|0.86|0.11|0.22|\n|FKC|5.5|**0.58**|0.99|0.77|0.49|0.87|0.10|0.22|\n|FKC|7.5|**0.57**|0.99|0.78|0.46|0.83|0.13|0.23|\n|**1k Prompts**|\n|SDXL|7.5|**0.58**|0.99|0.79|0.45|0.88|0.11|0.21|\n|FKC|5.5|**0.57**|0.99|0.79|0.42|0.86|0.13|0.21|\n|FKC|7.5|**0.57**|0.99|0.80|0.45|0.83|0.13|0.22|\n\n2. **FKC improves performance in the ambient space across tasks whereas improvements in latent space diffusion models are limited.** This is supported by Table 2 and Figure 2 of [the PDF](https://anonymous.4open.science/r/FKC-D7F0/rebuttal.pdf). Here, we present new and stronger results for the molecule generation task in the coordinate space instead of the latent space.\n\n> Is it possible to add particle interactive force to the proposed FKC method to mitigate the sampling covariance?\n\nIndeed, one could combine the proposed method with the Stein Variational Gradient Descent, targeting the intermediate marginals (annealed density or the product of densities) similar to [2]. This would require adding a term to the weights, which, theoretically, should reduce the variance of the weights.\n\n> Сonnection to the Fisher-Rao gradient flow\n\nThe reweighting equation $\\partial_t p_t^w(x) = p_t^w(x) \\bar{g}_t(x)$ does correspond to the Fisher-Rao gradient flow of a linear functional $G[p_t] = \\int {g}_t(x) p_t(x) dx$ (after constraining $p_t^w$ to remain normalized). The Wasserstein Fisher-Rao gradient flow has a form similar to the Feynman-Kac PDE with $\\sigma_t=0$ (deterministic evolution). We have included a note about this in the Appendix of the updated revision (see e.g. Thm. 3.1 in [3]).\n\n### Closing remarks\n\nWe thank the reviewer for suggesting this comparison, which significantly improves our empirical study. We hope that our answers address all the important questions raised by the reviewer, and we are more than happy to consider any additional questions or further suggestions.\n\n[1] Ghosh, Dhruba et al. \"Geneval: An object-focused framework for evaluating text-to-image alignment.\"  \n[2] Corso, Gabriele et. al. \"Particle guidance: non-iid diverse sampling with diffusion models.\"  \n[3] Lu, Yulong, et al. \"Accelerating Langevin sampling with birth-death.\"  \n[4] Karras, Tero et al. \"Analyzing and Improving the Training Dynamics of Diffusion Models.\"  \n[5] Xu, Jiazheng et al. \"ImageReward: Learning and Evaluating Human Preferences for Text-to-Image Generation\"", "claims_and_evidence": "The method of enhanced sampling from compositional distributions via reweighting is novel and reasonable. The derivation of the method is supported by valid MCMC/PDE theory.", "methods_and_evaluation_criteria": "The evaluation on SD-XL is not that convincing, the metric should be tested on large-scale prompts (at least 1k prompts) from standard datasets like MS-COCO.", "theoretical_claims": "I checked the theoretical claims. They are solid.", "experimental_designs_or_analyses": "sec 5.1: sound.\n\nsec 5.2: I am not familiar with the molecule generation task.\n\nsec 5.3: \n1. The result is not convincing enough. More experimental results on different CFG coefficients are needed. The Pareto curve between CLIP and FID across different beta can indicate the superiority of the proposed method. \n2. Tests on large-scale prompts (at least 1k prompts) from standard datasets like MS-COCO are necessary. Tests on other models like PixArt, and SD2b are beneficial.\n3. It is not clear why better adherence to the geometric average CFG distribution would contribute to enhancing the image quality.", "relation_to_broader_scientific_literature": "The proposed method contributes to the understanding of DMs and the potential enhanced performance of various downstream DM tasks.", "essential_references_not_discussed": "no", "other_strengths_and_weaknesses": "no", "other_comments_or_suggestions": "no", "questions_for_authors": "1. Considering the relation of MCMC and ParVI, it is possible to add particle interactive force to the proposed FKC method to mitigate the sampling covariance?\n\n2. I noticed that the reweight update has a similar form to the Fisher-Rao gradient flow. There may be some underlying theoretical connection.", "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:

Meta-review: The paper introduces the concept of Feynman-Kac Correctors (FKCs), which are a powerful and versatile tool for inference time control in diffusion models, enabling combination of different models (product of experts), annealing and guidance (conditional sampling). The work is a refreshing new perspective on an important problem which has mostly seen incremental advances and almost no conceptual leaps the last years. Simultaneously the paper is practical and readily applies to practical problems as evidenced by the extensive experimental section, which was further expanded during the rebuttal-discussion phase. The reviewers unanimously agree that this paper should be accepted.
Decision: Accept (spotlight poster)